Download Notes from Craigfest - University of Melbourne

Notes from Craigfest
TriThang Tran
Acknowledgements
These are (incomplete) notes from a homotopy theory seminar series in the first half of 2012 at the University
of Melbourne. They were organised by Craig Westerland. Different speakers presented talks each week. The
speakers were Drew Heard, Craig Westerland, Narthana Epa, TriThang Tran, Jeffrey Bailles, Yi Huang,
Tarje Bargheer and Robert Sayer.
Contents
1 Fibrations and cofibrations
1.1 Fibrations . . . . . . . . . . . . . .
1.2 Pull-back of a fibration . . . . . . .
1.3 Fibre bundles . . . . . . . . . . . .
1.4 Cofibrations . . . . . . . . . . . . .
1.5 Pushout of a cofibration . . . . . .
1.6 Thinking of (X, A) as a cofibration
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4
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2 Cofibre sequences and higher homotopy groups.
2.1 Fibre sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Higher homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Cohomology theories
10
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Simplicial objects
12
4.1 Nerve functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Operads and loop spaces
5.1 Statements of recognition principle . .
5.2 Operads . . . . . . . . . . . . . . . . .
5.3 Monads and C-algebras . . . . . . . .
5.4 Construction a monad from an operad
5.5 Approximation Theorem . . . . . . . .
5.5.1 Idea of proof . . . . . . . . . .
5.6 The bar construction . . . . . . . . . .
5.7 Proof of recognition principle . . . . .
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6 Operads in action
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14
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16
17
19
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21
22
1
7 Group Completion Theorem
7.1 Two-sided bar construction . . .
7.2 Homology fibrations . . . . . . .
7.3 The telescope . . . . . . . . . . .
7.4 The theorem . . . . . . . . . . .
7.5 Step 1: M∞ ×M EM → BM is a
7.6 Step 2: hof ib 'w.e ΩBM . . . .
7.7 Step 3: H∗ M∞ = H∗ M [π0−1 ] . .
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homology fibration
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24
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8 Cobordism
27
8.1 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Moduli Spaces and Mapping Class Groups
33
9.1 Mapping Class Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9.2 How do we get characteristic classes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.3 Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
10 Harer Stability
10.1 General nonsense on homological stability
10.2 The maps . . . . . . . . . . . . . . . . . .
10.3 The Arc Complexes . . . . . . . . . . . .
10.4 Four properties of O . . . . . . . . . . . .
10.5 Spectral sequence argument . . . . . . . .
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36
36
37
38
38
40
11 Madsen-Weiss Theorem (Wonderful Copenhagen)
11.1 The topology on Ψk (Rn ) . . . . . . . . . . . . . . . .
11.2 Defining the maps . . . . . . . . . . . . . . . . . . .
11.3 αn is a homology equivalence . . . . . . . . . . . . .
11.4 Another way to look at classifying spaces . . . . . .
11.5 The monoid . . . . . . . . . . . . . . . . . . . . . . .
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41
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12 Algebraic K-Theory: Introduction and examples
46
12.1 K-theory of a symmetric monoidal category . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
13 K-theory of general linear groups over a finite
13.1 Topological K-theory . . . . . . . . . . . . . . .
13.2 Adams operations . . . . . . . . . . . . . . . .
13.3 Fixed points and homotopy fixed points . . . .
13.4 Quillen Plus construction . . . . . . . . . . . .
13.5 More about the plus construction . . . . . . . .
field
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49
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14 Brauer Groups
56
14.1 Some computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
14.2 Some Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
15 Waldhausen K-theory
60
15.1 Exact categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
15.2 Waldhausen Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
16 Higher K-theory
63
16.1 S• -construction (wS• ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
16.2 How to think of S2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
16.2.1 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2
17 K-theory spectrum
65
17.1 examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
17.2 Waldhausen’s Splitting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3
1
Fibrations and cofibrations
References: [1, 2, 7]
1.1
Fibrations
Definition 1.1. Let E, B be topological spaces. A map p : E → B has the homotopy lifting property (HLP)
with respect to Y if there exists a h̃ that makes the following diagram commute:
f
Y
i0
h̃
Y ×I
/E
<
p
h
/B
where i0 : Y ,→ Y × I is given by y 7→ (y, 0).
Definition 1.2. A (Hurewicz) fibration is a map p : E → B that has HLP for all spaces Y .
Remarks:
1. We have defined a Hurewicz fibration. There is also the concept of a Serre fibration which has ‘all
spaces Y ’ replaced with ‘all I n ’.
2. If b0 ∈ B, then we call p−1 (b0 ) the fibre.
3. The fibres need not be be homeomorphic, but they are in fact homotopic. (See eg [2])
p
4. We can denote a fibration by F ,→ E −
→ B. F is the fibre, E is the total space and B is the base space.
Examples:
1. Covering spaces p : X̃ → X are obviously fibrations. In fact, they satisfy the unique homotopy lifting
property.
2. E = B × F , p : E → B is the trivial fibration.
3. S 1 ,→ S 3 → S 2 is the Hopf fibration.
1.2
Pull-back of a fibration
Given a fibration p : E → B and a map g : A → B, define
A ×g E := {(a, e) ∈ A × E|g(a) = p(e)} ⊂ A × E.
Then the following diagram commutes:
p
A ×g E 0
/E
p
π
A
g
/B
NOTE: A ×g E is the pullback of the data given by g and p. In particular it satisfies the following universal
property: If π 0 : W → A and p00 : W → E are two other maps that commute with g and p, then they factor
4
through A ×g E. ie, there exists a map that makes the following commute:
W
p00
π
0
#
A ×g E
Y
/% E
p0
p
π
/B
g
Lemma 1.3. π : A ×g E → A is a fibration.
Proof. We want to show that, if Y is a topological space, and h : Y × I → A is a homotopy, then we can
find a lift h̃ : Y × I → A ×g E. We can use the universal property of a pullback to give the existence of a
lift. Pictorially:
/5 E
/ A ×g E
Y
:
p
/A
Y ×I
1.3
g
/B
Fibre bundles
Definition 1.4. A fibre bundle E over B with fibre F consists of a surjection p : E → B with the property:
If x ∈ B, then there exists an open set U containing x and a homeomorphism ϕ : U × F → ϕ−1 (U ) such
that the following diagram commutes:
∼
= /
ϕ−1 (U )
U ×F
ϕ
p
y
U
proj
Theorem 1.5. For paracompact base space B, a fibre bundle is a fibration.
Proof. See [7]. Here is a sketch.
Let p : E → B be a continuous map such that B is paracompact. Let {Uα }α be an open cover of B such
that pα : p−1 (Uα ) → Uα is a fibration for all α. This can be done by local triviality of fibre bundles. Then
we can use paracompactness to glue everything together to get that p : E → B is a fibration.
Remarks:
• The converse is false. Fibre bundles have homeomorphic fibres but fibrations only need to have homotopy equivalent ones.
• Propisition 4.4.8 in [1] proves the result for Serre fibrations without the paracompactness condition.
Definition 1.6. If X and Y are topological spaces, then X Y is the set of all continuous functions Y → X.
We can endow X Y with the compact-open topology to turn X Y into a topological space. The compactopen topology on X Y is the topology having sub-basis being all subset (K, U ) where Let
(K, U ) := {f ∈ X Y |K is compact , U is open , f (K) ⊂ U }.
Exercise 1.7. Show that X Y together with the topology having sub-basis being all the subsets of the form
(K, U ) is a topological space.
5
Definition 1.8. Let (Y, yo ) be a based space. The path space Py0 Y is the space of all paths in Y , starting
at y0 . That is,
Py0 Y := M ap((Y, y0 ), (I, 0).)
We somtimes drop the subscript y0 and just write P Y .
Let p : P Y → Y be given by p(α) = α(1) be evaluation at the end of that path.
Theorem 1.9.
1. The map p : P Y → Y is a fibration. The fibre is ΩY .
2. P Y is contractible.
Proof. See [2].
Remark: The fact that P Y is contractible allows us to use the Serre spectral sequence to compute the
cohomology of ΩY if no know the cohomology of Y .
1.4
Cofibrations
Definition 1.10. Let A and X be topological spaces. A map i : A → X satisfies the homotopy extension
property (HEP) with respect to Y if there exists a h̃ that makes the following diagram commute:
h
A
i
/ YI
>
h̃
X
f
/Y
p0
where p0 (ξ) = ξ(0).
Definition 1.11. A map i : A → X is a cofibration if it satisfies the homotopy extension property for all
spaces Y .
Remark: We often think of A as a subset of X with i the inclusion map. Then i : A → X is a cofbiration
if it satisfies: given a Y and a homotopy h : A × I → Y , then h can be extended to a homotopy on all of X.
1.5
Pushout of a cofibration
Given a map g : A → B and a map i : A → X we can form the pushout B ∪g X := (B t X)/(i(a) ∼ g(a)).
Pushouts are dual to pull-backs and satisfy a similar universal property. Here is the associated diagram
showing the universal property (uou can guess how to write it down explicitly):
A
g
B
i
/X
g0
/ B ∪g X
#) W
Lemma 1.12. The pushout of a cofibration is a cofibration.
Proof. Dualise the proof for fibration.
6
1.6
Thinking of (X, A) as a cofibration
Often we consider pairs (X, A) where A is a subspace of X. Then the inclusion i : A ,→ X is a cofibration if
it satisfies: if f : X → Y is a map of topological spaces and h : A × I → X is a homotopy, then there exists
a map making the following diagram commute:
f ∪h
X × {0} ∪ A × I
8/ Y
i
X ×I
Definition 1.13. A pair (X, A) is called a neighbourhood deformation retract (NDR) pair if there exists a
u : X → I and a h : X × I → X such that
1. A = u−1 (0);
2. h(−, 0) = idX ;
3. if t ∈ I and a ∈ A, then h(a, t) = a;
4. if x ∈ X and u(x) < 1, then h(x, 1) ∈ A.
Moreover, (X, A) is a DR pair if it satisfies: if x ∈ X then u(x) < 1. We say that A is a deformation retract
of X.
Remark: We can think of these conditions meaning that there is a neighbourhood of A that deformation
retracts to A. In particular, all CW-pairs (X, A) where X is a CW-complex and A is a subcomplex are NDR
pairs.
Theorem 1.14. Let A be a closed subspace of X. The following are equivalent.
1. (X, A) is a NDR pair.
2. (X × I, X × {0} ∪ A × I) is a DR pair.
3. X × {0} ∪ A × I is a retract of X × I.
4. The inclusion i : A → X is a cofibration.
Proof. Exercise.
2
Cofibre sequences and higher homotopy groups.
Definition 2.1 (Mapping cylinder). Let f : A → X. The mapping cylinder Mf is
Mf := A × I t X/((a, 1) ∼ f (a)).
Remarks:
1. Mf is the pushout of f : A → X and A ,→ A × I.
2. Mf deformation retracts onto X by sliding each point (a, t) along {a} × I ⊂ Mf to f (a) ∈ X.
Definition 2.2 (Mapping cone). Let f : A → X. The mapping cone Cf is
Cf := A × I t X/((a, 0) ∼ f (a), (a, 0) ∼ (a0 , 0)) = Mf /(A × 0).
Remark: The mapping cone is the pushout Cf = X tf CA, where CA is the cone of idA : A → A.
7
Definition 2.3 (Smash product). Let X and Y be topological spaces. Define
X ∧ Y :=
X ×Y
.
X ∨Y
If X and Y are based spaces, then we have
X ∧Y =
X ×Y
.
X × {y0 } ∪ Y × {x0 }
Remarks:
1. If you are familiar with the category of R-modules, then you may recall that there is an adjunction
from between the functors − ⊗ B and Hom(B, −). This means that if A and C are R-modules, then
there is an ismorphism Hom(A ⊗ B, C) ' Hom(A, Hom(B, C)).
2. By analogy, (so we aren’t being too precise here), there is an adjunction between −∧A and M aps∗ (A, −),
giving isomorphisms M aps∗ (X ∧ A, Y ) ' M aps∗ (X, M aps∗ (A, Y )). The anology is to think of smash
product like tensor product. If we want to make this more precise, we need to show that the category
of modules with tensor product is symmetric monoidal and so is the category of pointed topological
spaces with smash product (!! - can clarify in future).
Definition 2.4 (Reduced suspension). Let X be a topological space. The reduces suspension of X is
ΣX :=
X ×I
.
((X × {0, 1}) ∪ ({x0 } × I))
Theorem 2.5. X ∧ S 1 = ΣX, where S 1 = I/∂I.
Proof. Exercise.
Theorem 2.6. The suspension functor is adjoint to the loop space functor. That is [ΣX, Y ] ' [X, ΩY ],
where [A, B] are the maps from A to B up to homotopy.
Proof. We have M aps∗ (X ∧ A, Y ) ' M aps∗ (X, M aps∗ (A, Y )) from the above comment. Taking A = S 1
and using Theorem 2.5 we have
M aps∗ (ΣX, Y ) ' M aps∗ (X, M aps∗ (S 1 , Y ))
M aps∗ (ΣX, Y ) ' M aps∗ (X, ΩY )
And taking π0 of both sides gives the result.
Theorem 2.7. Let f : X → Y and Cf be the cone of f . Let i : Y ,→ Cf be the inclusion of Y into Cf given
by y 7→ y. Let Ci be the cone of the inclusion map.
Ci is homotopy equivalent to ΣX, and we get a long exact sequence
f
i
X−
→Y →
− Cf → ΣX → ΣY → . . .
called the cofibre sequence.
Proof. There is an inclusion Cf → Ci . Thus we have a sequence X → Y → Cf → Ci . Once we have shown
that Ci is homotopy equivalent to ΣX, we are done since we have
f
i
X−
→Y →
− Cf → Ci ' ΣX → ΣY → . . .
[I STILL WANT TO FLESH THIS OUT A BIT MORE FOR MYSELF]
Theorem 2.8. If X and Y be pointed, path connected, CW complexes and Z is any pointed space, then the
cofibre sequence induces a long exact sequence:
f∗
i∗
[X, Z] ←− [Y, Z] ←− [Cf , Z] ← [ΣX, Z] ← [ΣY, Z] ← . . .
This is a LES of groups to the right of [Cf , Z] and is abelian for [Σ2 , −] or pointed sets for the first three
sets.
8
2.1
Fibre sequences
For a based map f : X → Y , define the homotopy fibre to be
Ff := X ×f P Y = {(x, λ)|(f (x) = λ(1)} ⊂ X × P Y.
Equivalently, it is the pullback of the data f : X → Y and p1 : P Y → Y , where p1 (λ) = λ(1).
/ PY
Ff × g E
p1
π
A
/B
f
Note that π : Ff → X is a fibration since P Y → Y is a fibration. Let m : ΩY → Ff be the inclusion
m(x) = (∗, x). We have the fibre sequence
Ωπ
Ωf
m
π
f
ΩFf −−→ ΩX −−→ ΩY −→ Ff −
→X−
→ Y.
Theorem 2.9. For any based space Z, the induced sequence
. . . → [Z, ΩFf ] → [Z, ΩX] → [Z, ΩY ] → [Z, Ff ] → [Z, X] → [Z, Y ].
is a long exact sequence. This is called the Puppe sequence. The last three terms are pointed sets, while after
that the terms are groups.
Proof. No proof in notes... you do it!
2.2
Higher homotopy groups
We can define the higher homotopy groups similar to how we define π1 .
Definition 2.10. πn (X, x0 ) = [(S n , s0 ), (X, x0 )], where s0 = (1, 0, . . . , 0).
Usually we write πn (X) for any path-connected space X since in this case, πn (X) is independent of choice
of basepoint. Observe that
πn (ΩX) = [S n , ΩX]
= [ΣS n , X]
= [S n+1 , X]
= πn+1 (X).
Theorem 2.11. For a fibration F → E → B, we have a long exact sequence in homotopy given by
. . . → πn (F ) → πn (E) → πn (B) → πn−1 (F ) → . . . → π0 (F ) → π0 (E) → π0 (B)
Proof. Putting Z = S 0 in the Puppe sequence gives
. . . → [S 0 , Ωn F ] → [S 0 , Ωn E] → [S 0 , Ωn B] → . . . .
Then using the above observation gives the result:
. . . → [S n , F ] → [S n , E] → [S n , B] → . . .
Remarks:
9
1. Covering spaces p : ˜(X) → X are fibrations with discrete fibres. Therefore pγ : πn (˜(X) → πn (X) is an
isomorphism for n ≥ 1 and injective for n = 1.
2. The Hopf fibration S 1 ,→ S 3 → S 2 can be used to show that πn (S 3 ) = πn (S 2 ) for n ≥ 3.
Theorem 2.12. If n ≥ 2, then πn (X) is abelian.
Proof. INSERT PICTURE PROOF ( THE ONE YOU SEE ALL THE TIME WITH g and f IN A BOX
MOVING IN A CIRCLE )
3
Cohomology theories
3.1
Definitions
Definition 3.1. A reduced cohomology theory Ẽ ∗ is a collection of contravariant functors
Ẽ q : hT op∗ → Ab,
for q ∈ Z≥0 satisfying
1. If i : A → X is a cofibration, then there exists an exact sequence
Ẽ 2 (X/A) → Ẽ q (X) → Ẽ q (A)
i
induced by X/A ← X ←
− A.
∼
=
2. There exists a natural isomorphism Σ−1 : Ẽ q+1 (ΣA) −
→ Ẽ q (A).
ji
3. If {Xi }i∈I is a set of based spaces, then the inclusions Xi −→ ∨l∈I Xl induces an isomorphism
π(ji )∗ : πi∈I Ẽ q (Xi ) ∼
= Ẽ q (∨l∈I Xl ).
4. If f : X → Y is a weak equivalence (ie. an isomorphism in πq for all q) then f ∗ : Ẽ ∗ X → Ẽ ∗ Y is an
isomorphism.
Example: H ∗ (−; A), singular cohomology with coefficients in A.
Definition 3.2. A contravariant functor F : C → Set is representable if there exists and object XF in C
and a natural isomorphism
η : F (Y ) → HomC (Y, XF ).
Example: Let C = k-Vect, F : C → C given by V 7→ V ∗ = Hom(V, k). This is the dual of a vector space
we see in first year linear algebra. To make it match with our definition of a representable functor, we can
compose F with the forgetful functor to SET .
Definition 3.3. A covariant functor F : C → Set is corepresentable if there exists an object ZF in C and a
natural isomorphism
η : F (Y ) = HomC (ZF , Y ).
Note: In the cases we consider, the morphisms of C will be maps up to homotopy. In this case, a functor
F is corepresentable if there is a natural transformation
η : F (Y ) = HomC (Zf , Y ) = [ZF , Y ]
Example: C = hT op∗ , F = πn (−) = [S n , −], so Zπn = S n .
Theorem 3.4 (Brown). If F satisfies the wedge and M.V axiom, then it is representable.
10
The MV axiom is: If X = U ∪ V and αu ∈ F (U ), αv ∈ F (V ) and αU |U ∩V = αV |U ∩V , then there exists
α ∈ F (X) such that αU = α|U and α|V = αV .
NEED A REFERENCE.
Theorem 3.5 (Yoneda Lemma). If F and G are representable functors and η : F → G is a natural
transformation, then there exists fn : XF → XG such that the following diagram commutes.
η
F (Y )
∼
=
/ G(Y )
∼
=
HomC (Y, XF )
/ HomC (Y, XG ).
NEED A REFERENCE.
3.2
Spectrum
Definition 3.6. A prespectrum T is a collection (Ti )i∈Z of based spaces with the homotopy type of a CW
complex and maps
fi : ΣTi → Ti+1 .
T is a spectrum if the adjoint maps f˜i : Ti → ΩTi+1 are homeomorphisms.
T is a Ω-spectrum if the adjoint maps are homotopy equivalences.
CAUTION: In this section, we are using Σ to mean the suspension (not the reduced suspension).
If T is a Ω-spectrum, we can define
T̃ q (Y ) = [X, Tq ] = HomhT op∗ (X, Tq ).
We can also define
T̃q (X) = lim πn+q (X ∧ Tq ).
n→∞
∗
Theorem 3.7. T̃ is a cohomology theory. Moreover, every cohomology theory Ẽ ∗ is built this way from a
Ω-spectrum E.
Proof. Two parts:
(=⇒)
• Homotopy invariance: automatic from [ , ].
∼ [X, Ω2 Tq+2 ] and the last of these is clearly an abelian group.
• Abelian group: T̃ q (X) = [X, Tq ] =
i
• Axiom 1: Given A →
− X → Xi ' X/A, we have [A, Tq ] ← [X, Tq ] ← [X/A, Tq ] which is what we
want.
• Axiom 2: T̃ q (X) = [X, Tq ] = [X, ΩTq+1 ] = [ΣX, Tq+1 ] = T̃ q+1 (ΣX).
• Axiom 3: [∨i∈I Xi , Tq ] = πi [Xi , Tq ].
• Axiom 4: Whitehead theorem.
(⇐=) Suppose we are give E ∗ . The wedge axiom and the LES together with Brown’s Theorem give Ẽ Q (X) =
[X, Eq ]. Here we use the fact the we can derive the MV axiom from LES.
∼ Ẽ q (X) Gives [X, Eq ] = [ΣX, Eq+1 ] = [X, ΩEq+1 ].
Suspension: Ẽ q+1 (ΣX) =
Lastly, Yoneda implies that Eq → ΩEq+1 gives f∗ : πn Eq ∼
= πn ΩEq+1 , taking X = S n .
11
4
Simplicial objects
Definition 4.1. Let C be a category. A simplicial object in C consists of (Xn )n≥0 where Xn ∈ C together
with morphisms
∂in : Xn → Xn−1 ,
and
sni : Xn → Xn+1 ,
for 0 ≤ i ≤ n such that
1. ∂i ∂j = ∂j+1 ∂i , if i < j

 sj−1 ∂i i < j
id
i = j, j + 1
2. deli sj =

sj+1 ∂i i > j
3. si sj = sj+1 si if i ≤ j + 1.
The ∂i are called boundary maps and the si are called degeneracy maps.
Definition 4.2. Let C be a category. The simplicial category sC is the category with
• Objects = simplicial objects in C;
• HomsC (X, Y ) = {(fn )n≥0 |fn ∈ HomC (Xn , Yn ), ∂i fn = fn−1 ∂i , si fn = fn+1 si }.
Another way we can think about the simplicial category is as follows.
Let [n] be the category whose objects are the integers 0, . . . , n, with morphisms given by i → j if i ≤ j.
We can draw this category as as
0 → 1 → . . . → n.
Let ∆ be the category with
• Objects = [n];
• Hom([n], [m]) = order preserving maps from [n] → [m].
Then an object of sC is a contravariant functor ∆ → C. In particular, Xn = X([n]).
Lemma 4.3. Every morphism in ∆ can be factored as f = ∂ . . . ∂s . . . s where ∂ i : [n − 1] → [n] is given by
j
j<i
j 7→
j + 1 j ≥ i,
and si : [n + 1] → [n] is given by
j 7→
j
j≤i
j − 1 j > i,
To get the boundary and degeneracy maps for sC in this picture, we have ∂i = X(∂ i ) and si = X(si ).
The morphisms of sC are natural transformations and the relations they satisfy correspond to commuting
squares.
Homework: Complete the story. Remember you must also show that given an object in sC as given in
the first definition, we get contravariant functors ∆ → C.
12
4.1
Nerve functor
Let Cat be the category of small categories.
Definition 4.4. The nerve (in C) is a functor
N : Cat → sC
D 7→ F unc(−, D)|∆ : ∆ → C
In particular, N D : ∆ → C is given by N D([n]) = F unc(∆, D)([n]) = {T (0) → . . . → T (n)}.
Let G be a group. Consider the category Ĝ where Ob(Ĝ) = ∗ and Hom(∗, ∗) = G and composition is
h ◦ g = gh.
gn
g2
g1
We have N Ĝ([n]) = {∗ −→ ∗ −→→ . . . −→ ∗|gi ∈ G} = Gn .
Bar notation: we can instead write N Ĝ([n]) = {[g1 | . . . |gn ]|gi ∈ G}. Using this notation, the ∂i and si
are easy to describe.
We have ∂i : Gn → Gn−1 is given by
[g1 | . . . |gn ] 7→ [g1 | . . . |gi−1 |gi gi+1 |gi+2 | . . . |gn ], for 0 < i ≤ n.
For ∂0 we just get rid of g1 . ie
∂0 [g1 | . . . |gn ] = [g2 | . . . |gn ].
Similarly, si : Gn → Gn+1 is given by
[g1 | . . . |gn ] 7→ [g1 | . . . |gi−1 |1|gi | . . . |gn ], for 0 ≤ i ≤ n.
This is called the nerve of G.
[COMMENT: IF WE WANT THIS TO BE A FUNCTOR, WE SHOULD SAY WHAT IT DOES TO
MORPHISMS]
4.2
Geometric realization
Definition 4.5. Let X be a simplical space. The geometric realization of X, denoted |X|, is the space
G
|X| := (Xn × ∆n )/ ∼,
where ∆n = {(t0 , . . . , tn ) ∈ Rn+1 |ti ≥ 0,
P
ti = 1} and ∼ is the equivalence relation generated by
• if x ∈ Xn , u ∈ ∆n−1 , v ∈ ∆n+1 , then (x, δ i u) ∼ (∂i x, u),
• if x ∈ Xn , u ∈ ∆n−1 , v ∈ ∆n+1 , then (x, σ i u) ∼ (si x, u),
where
δ i (t0 , . . . , tn ) = (t0 , . . . , ti−1 , 0, ti+1 , . . . tn−1 ); and
σ i (t0 , . . . , tn+1 ) = (t0 , . . . , ti−1 , . . . , ti + ti+1 , ti+2 , . . . , tn+1 ).
Definition 4.6. Let G be a group. The classifying space of G is BG := |N Ĝ|
p
With some work [REFERENCE SOMETHING], we can get a fibre bundle G ,→ EG −
→ BG, with
EG weakly contractible. But this means EG is simply connected, so EG is the universal cover of BG, so
π1 (BG) = G.
If we use the long exact sequence of homotopy groups, we can also show that πi (G) ∼
= πi−1 (G) If G is a
discrete group, then πi (G) = G if i = 0 and is 0 is i 6= 0.
If G is abelian, we can iterate this process to get the spaces K(G, n).
13
5
Operads and loop spaces
References: [3]
NOTATION: In this section, we use Σ’s for the symmetric group and S’s for suspension.
5.1
Statements of recognition principle
Motivation: Let X be a topological space. How can we tell if there exists an n ∈ Z≥0 and a space Y such
that X = Ωn Y ?
This question is essentially answered by the following theorem:
Theorem 5.1 (Recognition principle). There exist Σ-free operads Cn for 1 ≤ n ≤ ∞ such that every n-fold
loop space is a Cn -space and every connected Cn -space has the weak homotopy type of an n-fold loop space.
There is some history about the development of the recognition principle in the preface to [3] that may
be of interest.
In the next section, we define what an operad C and a C-space is.
5.2
Operads
Let U be the category of topological spaces (compactly generated, Hausdorff) with continuous maps, and
let T be the same category, but with based spaces and based maps.
Definition 5.2. An operad C is a collection of spaces C(j) ∈ U for j ≥ 0, with C(0) = ∗ with maps
γ : C(K) × C(j1 ) × . . . × C(jk ) → C(j1 + . . . jk ).
such that:
1. An associativity formula holds:
2. There is and identity 1 ∈ C(1) such that γ(1; d) = d and γ(c; 1, . . . , 1) = c.
3. There is a right action of the symmetric group Σj on C(j)
[BE MORE EXPLICIT LATER, MAYBE WITH A DIAGRAM]
Definition 5.3. A morphism of operads ψ : C → C 0 is a sequence of Σj equivariant maps ψj : C(j) → C 0 (j)
such that ψ(1) = 1 and the following diagram commutes:
C(K) × C(j1 ) × . . . × C(jk )
γ
ψk ×ψj1 ×...ψjk
C 0 (K) × C 0 (j1 ) × . . . × C 0 (jk )
/ C(j)
ψj
γ
0
/ C 0 (j),
where j = j1 + . . . + js
Example 5.4 (The endomorphism operad). Let X ∈ T The endomorphism operad EX of X is the operad
given by
• (E)x (j) = { based maps from X j → X}.
• γ(f, g1 , . . . gk ) = f (g1 × . . . gk )
• 1 ∈ EX (1) = id : X → X.
• The Σj action permutes the coordinates of X j .
14
Example 5.5 (Little n-disks operad). The little n-disks operad is the operad with
Cn (j) = {j -tuples of ‘nice’ embeddings of an n-disk into a single n-disk such that their images are disjoint}.
Exercise: How is γ defined? What do we mean by nice?
In some sense, the endomorphism operad is the mother of all operads. The associativity, unit and
symmetry axioms for an operad are defined precisely to mimic what happens in the case of the endomorphism
operad. If we are given a j-tuple of elements in X, then elements of EX (j) can act on it by plugging in the
tuple to the j functions. For a general operad, we can define actions by going through EX
Definition 5.6. A C-space is a pair (X, θ) where X ∈ T , θ : C → EX is a morphism of operads.
As an aside, we can define morphisms of C-spaces and get a category of C-spaces. We don’t use this
language much in this section so this can be ignored.
Definition 5.7. A morphism of C-spaces is a map f : (X, θ) → (X 0 , θ0 ) such that f is a based map
f : X → X 0 and f ◦ θj (c) = θj0 (c) ◦ f j where c ∈ C(j).
The category of C-spaces is denoted by C[T ].
Now, we can think of C spaces in another way. Given a morphism of θ : C → EX , we can consider the
maps θj : C(j) → EX (j) = {X j → X}. Noting the adjunction
∼
=
Hom(C(j), EX (j)) −
→ Hom(C(j) × X j , X),
we can specify a C-space by maps θj : C(j) × X j → X satisfying certain conditions. We state this more
precisely as a lemma.
In particular, given θj : C(j) → EX (j), we get a map C(j) × X j → X by
(x1 , . . . , xj ) 7→ θj (c)(x1 , . . . , xj ), for c ∈ C(j).
We now state the above discussion a bit more precisely in the form of a lemma.
Lemma 5.8. An action θ : C → EX determines and is determined by maps θj : C(j) × X j → X, for j ≥ 0
such that
1. θ0 : ∗ → X.
2. The following commutes
C(K) × C(j1 ) × . . . × C(jk ) × X j
γ
/ C(j) × X j
θj
$
:X
shuffle
θk
C(K) × C 0 (j1 ) × X j1 × . . . × C(jk ) × X jk
γ0
/ C(k) × X k .
3. If x ∈ X then θ1 (1; x) = x.
4. If c ∈ C(j), σ ∈ Σj , y ∈ X j , then θj (cσ; y) = θj (c; σy).
[INSERT FLOWER EXAMPLE TO DESCRIBE THE DIAGRAM]
We can now prove one half of the recognition principle. That is, we can show that ever Ωn Y is a Cn space, where Cn is the little n-disks operad. In particular, we are claiming that the operads in the recognition
principle are the little disks operad. To show that Ωn Y is a Cn -space, we need to define the action.
15
[INSERT PICTURE].
Given a map f : S n → Y , we can think of it as a map Dn → Y that sends the boundary of Dn to the
basepoint in Y . Then θj : Cn j × (Ωn Y )j → Y is given by
fi (y) , y ∈ ei (Dn ),
θj (e1 , . . . , ej ; f1 , . . . , fj ) =
∗
, y ∈ D \ {e1 (Dn ), . . . ej (Dn )},
where (e1 , . . . , ej ) ∈ Cn (j), fi : (Dn , ∂Dn ) → (Y, ∗) is in Ωn Y . This can be more easily understood if we
draw a map of the domain of the new function: [INSERT PICTURE]
In [3], May uses the little n-cubes operad instead of the little n-disks operad. Both of these work fine
and in fact, the proof of the recognition principle allows for operads that are homotopy equivalent.
5.3
Monads and C-algebras
Operads are closely related to monads. In fact, it turns out that an operad C determines a monad C and a
C space becomes an algebra over the monad C (a C-algebra).
We are particularly interested in this because of the following.
Theorem 5.9 (Approximation theorem). For the operads Cn of the recognition principle, there is a natural
map of Cn -spaces
αn : Cn → Ωn S n X
for 1 ≤ n ≤ ∞ and αn is a weak homotopy equivalence.
In summary, Cn X ' Ωn S n X, which is pretty cool, because we are interested in loop spaces. In particular,
while Ωn S n X is a priori hard to understand, the monads Cn will be quite concrete, and yield to computations
more easily. We will see some applications later on.
We begin with some definitnions.
Definition 5.10. A monad in a category T is a triple (C, µ, η), where C : T → T is a contravariant functor,
and µ : C 2 ⇒ C, η : 1 ⇒ C are natural transformations such that the following diagrams commute.
CηX
CX
o
/ C 2 X η(CX)
CX
µ(X)
=
# {
CX
C 3X
Cµ(X)
.
µ(CX)
/ C 2X
µ(X)
C 2X
=
µ(CX)
/ CX
The first diagram tells us that η acts as the unit (in particular we can think of it as saying Cη = ηC = C),
while the second diagram says that µ is associative (think µ(C 2 )C = Cµ(C 2 )). Thus, we can think of monads
as monoids in the functor category.
Definition 5.11. A morhpism ψ : (C, µ, η) → (C 0 , µ, η 0 ) of monads is a natural transformation of functors
ψ : C ⇒ C 0 such that the following commute for all X ∈ T .
X
η0
η
CX
}
ψ
16
"
/ C 0X
CCX
ψ2
/ C 0C 0X
µ
CX
ψ
µ0
/ C 0X
Definition 5.12. An algebra (X, ξ) over a monad (C, µ, η) is an object X ∈ T together with a map ξ :
CX → X in T such the fo following commute.
X
/ CX
η
ξ
=
CCX
! X.
µ
Cξ
CX
/ CX
ξ
ξ
/X
For brevity, we will often just call these objects C-algebras. So why the ‘algebra’ ? If we instead pretend
that C were a group G, and ξ : G × X → X was such that it satisfied the above diagrams, then this would
mean precisely that X was a space with a G action. Thus we can think of a C-algebra as a space X with
some notion of an action of C on it.
Definition 5.13. A morphism f : (X, ξ) → (X 0 , ξ 0 ) of C-algebras is a map f : X → X 0 in T such that the
follow diagram commutes.
CX
Cf
ξ0
ξ
X
/ CX 0
f
/ X0
Given an operad C we can construct a monad C. Then for the little disks operads Cn , we get associated
monads Cn . Recalling that monads are functors (in our case from T → T ), we can apply Cn on a space X.
When X is connected, the approximation theorem is the statement that
Cn X ' Ωn S n X,
where ' here means weak homotopy equivalence. In the next section, we discuss how to construct these
monads.
5.4
Construction a monad from an operad
Let C be an operad. Define maps σi : C(j) → C(j − 1), 0 ≤ i < j by σi (c) := γ(c; si ), where si =
(1, . . . , 1, ∗, 1, . . . , 1).
Example: In EX , σi f )(y) = f (si y) where si : X j−1 → X j given by
si (x1 , . . . , xj−1 ) = (x1 . . . , xi , ∗, xi+1 , . . . , xj−1 ).
Thus, we si puts an element of X j−1 into X j in the simplest way possible. In particular, σi turns the
i h part of c ∈ C(j) to the basepoint ∗.
t
17
[INSERT A PICTURE]
Now construct C as follows:
For X ∈ T ,
CX =
G
C(j) × X j / ∼
j≥0
where ∼ is the equivalence relation given by
1. (σi c, y) ∼ (c, si y) for c ∈ C(j), 0 ≤ i < j, y ∈ X j−1
2. (cσ, y) ∼ (c, σy) for c ∈ C(j), σ ∈ Σk , y ∈ X j
F
Note: Sometimes, if we want to be explicit about condition 2, then we will write CX = j≥0 C(j) ×Σk
X j / ∼, using ∼ only for relation 1.
We can think of the first condition as a kind of degeneracy condition. For example, in the endomorphism
operad, we would regard (f1 , f2 , f3 ) × (x1 , ∗, x3 ) ∼ (f1 , f3 ) × (x1 , x3 ). The second condition says that the
quotient respects the Σj action on C(j). For example (f1 , f2 , f3 ) × (x1 , x2 , x3 ) ∼ (f2 , f1 , f3 ) × (x2 , x1 , x3 ).
Exercise: What does CX look like for the little disks operad?
For C to be a monad, we need F
to check that CX ∈ T . We can topologise
CX as follows.
Fk
k
Let Fk CX be the image of j=0 C(j) × X j in CX. Note that j=0 C(j) × X j is a topological space, and
we can give Fk CX the quotient topology. Observe that Fk−1 CX is a closed subspace of Fk CX and so we
can give CX the topology of the union of Fk CX. Then F0 CX is a single point which will be our base point
for CX.
Check C is functorial: If c ∈ C(j), y ∈ X j , let [c, y] denote the image of (c, y) in CX. For a map
f : X → X 0 in T , define Cf : CX → CX 0 by
Cf [x; y] = [c; f j (y)].
We should then check that C(f ◦ g) = Cf ◦ Cg. [EXERCISE]
We also need to define µ and η for our monad C.
Define µ : C 2 X → XC by
µ[c, [d1 , y1 ], . . . [dk , yk ]] = [γ(c; d1 , . . . , dk ), y1 , . . . , yk ]
for c ∈ C(k), ds ∈ C(j), ys ∈ X js .
Define η : X → CX by
η(x) = [1, x]
for x ∈ X. We can check that µ and η satisfy the diagrams in the definition of (C, µ, η). In particular, the
definition of an operad shows that µ is well defined and gives the associativity square. Moreoever, we can
use the unit formulas for operads give the unit square for η.
Example 5.14. If C(j) = ∗, then CX = (tk ∗ ×Σk X k )/ ∼. Elements of CX can be represented as tuples of
elements of x. The multiplication is just concatenation, and the Σk action makes things commutative. The
equivalence relation says (x1 , ∗x3 ) = (x1 , x3 ). In particular, CX is the free commutative monoid on X with
unit ∗.
Example 5.15. If C(j) = Σk , then CX = (tk Σk ×Σk X k )/ ∼ is the free monoid on X.
18
5.5
Approximation Theorem
In this section, Cn will refer to the operads of the recognition principle. In particular, we should think of Cn
as the little n-disks operad. We will use Cn to denote the monad associated to Cn by the construction in the
previous section.
Theorem 5.16 (Approximation Theorem). If X is connected, then there is a natural map αn : Cn X →
Ωn S n X, which is a weak homotopy equivalence. In fact, Ωn S n is a monad, and αn : Cn → Ωn S n is a
morphism of monads.
For a general pair L, R of adjoint functors, we can get a monad. Observing that S n and Ωn are adjoint
functors this is how we get the monad Ωn S n . In briefly describe this process below.
Let L : C → D and R : D → C be adjoint functors. That is, for X ∈ C and Y ∈ D, HomD (LX, Y ) ∼
=
HomC (X, RY ). Then M = RL is a monad. We can explain how this works as follows. Consider
Hom(LX, LX) ∼
= Hom(X, RLX)
which we obtain from replacing Y with LX. There is and identity element id ∈ Hom(LX, LX), which we
can push through to Hom(X, RLX to get a natural transformation
η : I → RL.
η is the called the unit of the adjunction.
Similarly, consider
Hom(LRY, Y ) ∼
= Hom(RY, RY )
which we obtain from replacing X with RY . We can similarly get the co-unit of the adjunction by taking
id ∈ Hom(RY, RY ) and pushing it to the left to get a natural transformation
: LR → I.
We can define the natural transformation µ : RLRL → RL by
RLRL 7→ R(RL)L.
Then we can check that (RL, µ, η) is a monad.
5.5.1
Idea of proof
We did not prove the recognition principle in the seminar. The full proof can be found in gils. The crux of
the argument is to construct a quasi-fibration
Cn X → En (T X, X) → Cn−1 (SX)
such that the following diagram commutes.
Cn X
αn
Ωn S n X
/ En (T X, X)
αn−1
/ P Ωn−1 S n X
/ Cn−1 (SX)
αn−1
/ Ωn−1 S n−1 (SX)
The maps αn are the morphisms of monads Cn → Ωn S n we discussed (but did not construct) earlier. The
bottom row is the fibration disccused in 1.8. Lastly, although we do not describe it here, the En is an
attempt to construct a ‘path’ version of the little disks algebra. Given this, and the fact the when n = 1,
α0 : SX → SX is an isomorphism, we can apply the long exact sequence of homotopy groups together with
the 5-lemma to get the approximation theorem.
Exercise: Fill in the details.
19
5.6
The bar construction
Recall the recognition principle (Theorem 5.1). We have already described how one side of the theorem
works. In this section, we aim to make headway into showing the other direction. In particular, we want to
show the following.
Every connected Cn -space has the weak homotopy type of an n-fold loop space.
We will do this via the following string of equivalences.
X ' B(Ωn S n , C, X) ∼
= Ωn B(S n , C, X)
In particular, we will construct the space B(S n , C, X) which we can think of as a de-looping of X. The way
we construct the B is similar to the discussion earlier on the construction of classifying spaces of groups,
and hence the use of the name ‘bar construction’.
We will build the space B similar to the nerve construction used to build classifying spaces in section 4
Recall that an object X ∈ sT is a sequence of objects Xq ∈ T , q ≥ 0, together with maps
∂i : Xq → Xq−1
and
si : Xq → Xq+1
in T , for 0 ≤ i ≤ q. These maps were called boundary and degeneracy maps respectively and they satisfied
the relations given below.
1. ∂i ∂j = ∂j+1 ∂i , if i < j

 sj−1 ∂i i < j
id
i = j, j + 1
2. deli sj =

sj+1 ∂i i > j
3. si sj = sj+1 si if i ≤ j + 1.
We also defined morphisms f : X → Y in sT to be a sequence fq : Xq → Yq of maps in mcT such that
∂i fq = fq−1 ∂i and si fq = fq+1 si .
Definition 5.17. A homotopy h : f → g in sT between maps f, g : X → Y consists of maps
hi : Xq → Yq+1
for 0 ≤ i ≤ q such that
1. ∂0 h0 = fq and ∂q+1 hq = gq

 hj−1 ∂i i < j
∂j hj−1 i = j > 0
2. ∂i hj =

hj ∂i−1 i > j + 1
hj−1 ∂i i ≤ j
3. si hj =
hj sj−1 i > j
NOTE: The important thing here is that we have notions of homotopy equivalence, deformation retracts
etc. When T = T op, these things translate to the usual notion of homotopy after we take geometric
realizations.
Chapter 9 of [3] goes through key properties of the simplical category which we use in the construction
of B. In this section, we will outline the main arguments.
20
Definition 5.18. Let (C, µ, η) be a monad in T . A C-functor (F, λ) in a category V is a functor F : T → V
together with a natural transformation λ : F C → F such that the following commute.
F
Fη
/ FC
λ
=
! F.
Fµ
F CC
/ FC
λ
λC
FC
λ
/C
Compare this with the definition of a C-algebra (X, ξ). In particual, one can think of a C-functor as
something with a right action of C.
[PUT SOME EXAMPLES HERE]
We can now define the categorical 2-sided bar construction.
Let B(T , V) be the category whose objects are triples
((F, λ), (C, µ, η), (X, ξ))
where (F, λ) is a C-functor, (C, µ, η) is a monad in T and (X, ξ) is a C-algebra. The morphisms are triples
of morphisms (π, ψ, f ) of C-functors, monads and C-algebras respectively.
Then we can define a functor
B∗ : B(T , V) → sV
by Bq (F, C, X) = F cq X. There are q + 1 action maps with which to form a map F X q X → F C q−1 X.
• The right action of C on F induced by λ is ∂0 .
• The q − 1 multiplication of C induced by µ are the ∂i for 0 < i < q.
• The left action of C on X induced by ξ is ∂q
The si correspond to the q + 1 insertions of the unit of C to get a map F C q X → F C q+1 X.
The morphisms B∗ (π, ψ, f ) are given by Bq (π, ψ, f ) = πψ q f : F C q X → F 0 (C 0 )q X 0 .
With these definitions, one can check that B∗ is well defined by checking the conditions required to be
in sV.
5.7
Proof of recognition principle
We didn’t actually complete the proof in the seminar. I might type this up slowly in my spare time. Note:
S and Σ have switched again!
Here is an outline. We want to show the following.
X ' B(Ωn S n , C, X) ∼
= Ωn B(S n , C, X)
We claim that Ωn S n is a C-functor, so B(Ωn S n , C, X) makes sense. Lemma 9.7 of [3] gives the second
equivalence. It states the following.
Lemma 5.19. Let G : V → V 0 be a functor. Then GF : T T → V 0 is a C-functor and for a C-algebra X,
B∗ (GF, C, X) = G∗ B∗ (F, C, X).
In particular, B(Ωn S n , C, X) ∼
= Ωn B(S n , C, X). The first inequality is analogous to ΩBG ∼
= G for a
group G.
21
6
Operads in action
In this section, we illustrate some of the cool things you can do with the approximation theorem and the
recognition principle. Instead of using the little n-disks operad, we will use the little n-cubes operad. We
note that this does not change our two theorems - in their proofs, we can replace the operads that are
equivalent.
Definition 6.1 (Llittle n-cubes operad). The little n-disks operad is the operad with
Cn (j) = {j -tuples of ‘nice’ embeddings of an n-cube into a single n-disk such that their images are disjoint}.
Exercise: How is γ defined? What do we mean by nice?
Claim: the map Cn (k) → P Confk (I n ) that maps a little cubes to its lower left hand corner is a homotopy
equivalence.
Definition 6.2. P Confk (X) = {(x1 , . . . , xk )|xi 6= xj if i 6= j} ⊂ X k
P Confk (X) is called the pure configuration space of X. Notice that the symmetric group Sk acts on
P Confk (X) by permuting coordinates. In particular
σ.(x1 , . . . , xk ) = (xσ(1) , . . . , xσ(k) ).
The configuration space of X is Confk X := P Confk (X)/Sk . We can think of Confk X as subsets of k
distinct elements of X. To get an idea of what these look like, we study P confk (I n ). We start with the case
n = 1.
◦
◦
◦
Proposition 6.3. P Confk (I) ' Sk × ∆k . As a set, it is k! copies of ∆k .
◦
Proof. A k-tuple of distinct elements in I is a set {x1 , . . . xk } ⊂ I and a permutation σ such that
0 < xσ(1) < . . . < xσ(k) < 1.
◦
In particular, {(x1 , . . . , xk ) ⊂ I k |0 < x1 . . . < xk < 1} ∼
= ∆k .
Now consider the case n = 2. That is, we now look at P confk (I 2 ). Observe that C2 (k) ' P Confk (I 2 ).
In particular, the map in the forward direction is to send each square to its left hand corner. The homotopy
inverse put a small square with bottom left hand corner being the points in P Confk (I 2 ), chosen small enough
so they do not intersect. Further note that pi1 (P confk (I 2 )) ∼
= P βk , the pure braid group.
[INSERT PICTURE OF BRAIDS IN A BOX]
We now want to give an example of the approximation theorem in action. Recall that the approximation
theorem says the following. If X is a connected based space, then
Cn X ' Ωn Σn X,
where ' here denotes weak homotopy equivalence.
Recall,
G
Cn X =
Cn (k) ×Sk X k / ∼ .
k≥0
22
In particular, if n = 1, then
G
C1 X =
C1 (k) ×Sk X k / ∼
k≥0
G
'
(Sk × ∆k ) ×Sk X k / ∼
k≥0
G
=
∆k × X k / ∼
k≥0
G
'
Xk/ ∼
k≥0
=: M X
Definition 6.4. M X is the free topological monoid on X with unit ∗.
Observe that the name is justified, since an arbitrary element of M X is just a tuple of elements of X,
(x1 , . . . , xk ) for some k, say. The multiplication in M X (when we look back at the original definition) simply
becomes concatentation. ie (x1 , . . . , xk ).(y1 , . . . , yl ) = (x1 , . . . , xk , y1 , . . . yk ). The equivalence relation gives
us the unit ∗, satisfying (x1 , . . . , ∗, . . . , xk ) = (x1 , . . . , ˆ∗, . . . , xk ), where ˆ∗ denotes removing ∗ from the tuple.
The upshot of the recognition principle is that we now know
ΩΣX ' M X.
So what was a priori hand to understand (ΩΣX) is actually something quite concrete. Actually, it is also
true that the equivalence also preserves the multiplication, which we have not shown, but is easy to see if
we draw a picture.
[INSERT PICTURE]
Corollary 6.5. If the ring of coefficients is such that H̃∗ X is flat over R, then H∗ (ΩΣX) ∼
= T (H̃∗ X) as
rings, where T (H̃∗ X) is the tensor algebra over H̃∗ X.
Proof. We will inductively show:
H∗ (
n
G
X k / ∼) =
k=0
L
n
M
H̃∗ (X)⊗l .
k=0
L
Base: H∗ (M X) = H∗ (X) = H0 (X) H̃∗ X = R H̃∗ X
i
Inductive step: Consider the cofibration Mn X →
− Mn+1 X → Xi . The cofibre C(i) = Mn+1 X/Mn X =
X ×(n+1) / ∼, where (x1 , . . . , xi , ∗, xi+2 , . . . , xn ) ∼ ∗.
Therefore C(i) = X ∧(n+1) . Moreover, by the flatness assumption, H̃∗ (X ∧n ) = H̃∗ (X)⊗n .
Thus we have the long exact sequence
. . . → H∗ M )nX → H∗ Mn+1 X → H∗ Ci → . . .
Checking that this splits gives the result. [FILL IN THE DETAILS]
Corollary 6.6. For n ≥ 2, H∗ (ΩS n ) = R[x], where dimX = n − 1.
Proof. ΩS n = ΩΣS n−1 . H∗ (ΩS n ) = T (H̃∗ (S n−1 )) = R[ generator in dim n − 1]
Recall that the recognition principle states that a connected space X is a Cn -algebra (we used the
terminolgy Cn -space in the previous section) if and only if X ' Ωn Y for some Y , where ' is weak equivalence.
One
that gives us an idea of what is going on is the following.
F (non)-example
n
0
k≥0 P Confk (I ) is a Cn -algebra. It is a Cn space in the usual way. It is Cn (S ) and it wants to be
n n
n n 0
0
Ω S = Ω Σ S , but fails to be because S is not connected. [QUESTION: How is this failing?]
23
Here is an example.
F
F
Let M be a closed smooth manifold. Let Xi = k≥0 Emb(M tl , I n ) and X 0 = k≥0 Emb(M tk , I n )/Dif f (M tk ).
This is an algebra over Cn X. For example, in the case where M = S 1 , we can define θ in the usual way.
[Draw picture].
This leads us into considering the group completion theorem.
Theorem 6.7. If X is a C1 -algebra, then there exists a space BX (it is the same B as earlier!) such that
H∗ (X)[π0−1 ] ∼
= H∗ (ΩBX)
Note that previously we had X = Ω[BX] if X were connected. The group completion theorem tells us
part the story when X is no longer connected.
7
Group Completion Theorem
References: [4]
7.1
Two-sided bar construction
We have already seen many incarnations of the bar construction. They are all essentially the same thing.
Nevertheless, we will now describe the bar construction for a topological monoid.
Let M be a topological monoid. Let X be a right M -space. Let Y be a left M -space. For p ∈ Z≥0 , define
Bp (X, M, Y ) = X × M p × Y , with maps

i=0
 (xm1 , m2 , . . . , mp , y)
(x, m1 , . . . , mi mi+1 , . . . , mp , y) 0 < i < p
∂ i (x, m1 , . . . , mp , y) =

(x, m1 , . . . , mp y)
i=p
for 0 ≤ i ≤ p. Intuitively, we think of ∂ i as removing the it h comma (start counting at 0).
Then B• (X, M, Y ) is a (semi-)simplicial space.
Define Bp M := Bp (∗, M, ∗) and Ep M := Bp (∗, M, M ). Then we can define the classifying space of M to
be the geometric realization BM := |B• M |. Moreover EM := |E• M |.
In particular,
∞
G
BM =
M n × ∆n /∼
n=0
where ∼ is the equivalence relation generated by
1. (m0 , . . . , mi−1 , 1, mi+1 , . . . , mn , t0 , . . . , tn ) ∼ (m0 , . . . , mi−1 , mi+1 , t0 , . . . , ti + ti+1 , . . . , tn ),
for i = 0, . . . n − 1.
2. (m0 , . . . , mn , 0, t1 , . . . , tn ) ∼ (m1 , . . . , mn , t1 , . . . , ti + ti+1 , . . . , tn )
3. (m0 , . . . , mn , 0, t0 , . . . , ti−1 , 0, ti+1 , . . . , tn ) ∼ (m1 , . . . , mi−1 mi+1 , . . . , mn , t0 , . . . , ti−1 , ti+1 , . . . , tn )
4. (m0 , . . . , mn , t0 , . . . , tn−1 , 0) ∼ (m0 , . . . , mn−1 , t1 , . . . , tn−1 )
These equivalence relations precisely reflect the simplicial structure of B. In particular, we can summarise
the last three relations as (∂ i (x), y) ∼ (x, di (y)), where di : ∆n → ∆n−1 is the inclusion of the ith face. If
we had defined degeneracy maps and and had a simplicial space, then the first condition would reflect the
degeneracy maps, (σ i (x), y) ∼ (x, si (y)).
Note: EM/M ∼
= BM .
If M were a group, we have a fibration X → EM ×M X → BM . However, we M is a monoid, we do not
always get X as the fibre back.
24
7.2
Homology fibrations
Homology fibrations are a weakening of the notion of a fibration. That is, if we consider a map p : E → B.
It can be one of the following.
1. A fibre bundle.
2. Fibrations.
3. Quasi-fibrations.
4. Homology-fibrations.
If p : E → B satisfied is one of the things on the list, then it is also everything below it. A homology
fibration is something at the very bottom of the list, which we describe below. We first give the definition
of a quasi-fibration.
If b ∈ B, the define the homotopy fibre of F at b to be Fp,b := E×B Pb B, where Pb B = M AP∗ ((I, 0), (B, b)).
The following diagram commutes.
F
; p,b
ϕb
p−1 (b)
/E
{b}
/B
p
In particular, if we have γ : I → B such that γ(0) = b and γ(1) = b, then ϕb (e) = (γ, e).
Definition 7.1. A map p : E → B is a quasi-fibration if it satisfies the following condition: if b ∈ B, then
there exists ϕb : p−1 (b) → Fp,b which is a weak homotopy equivalence.
Definition 7.2. A map p : E → B is a homology-fibration if ϕb is a homology equivalence.
Theorem 7.3. Le M be a monoid, X a topological space with a M action, x 7→ mx be an isomorphism in
H∗ for every m ∈ M . Then
EM ×M X → BM
is a homology fibration
Theorem 7.4. If X is a weakly contractible space with a free action of M , then there exists an M -equivariant
∼
homotopy equivalence X → EM . So X/M −
→ EM/M = BM
Example 7.5. Let M = Z≥0 . Then R is a contractible space with a free action of M . Thus R is a model
for EM . Then BM = EM/Z≥0 = S 1 .
Here, when we mod out by M , we need to turn the monoid relations x ∼ mx into an equivalence relation.
(Essentially, the smallest equivalence relation that turns ∼ into an equivalence relation). After doing this,
we see the fibres are Z instead of Z≥0 as one might expect.
7.3
The telescope
Definition 7.6. Let M be a topological monoid. Assume π0 M = Z≥0 . Let m ∈ M such that m ∈ 1, that is,
m is in the component 1 of π0 M . Define
M∞ = T elm M =
∞
G
Mn × I/ ∼,
n=1
where the ∼ is given by (x, 1)n ∼ (mx, 0)n+1 and Mn = M .
Note that this is a homotopy colimit. [insert joke about microscope + picture]
Proposition 7.7. If p : EM → BM , then H∗ (Fb ) ∼
= H∗ (T elM ), where Fb is the homotopy fibre.
25
7.4
The theorem
Theorem 7.8 (Group Completion Theorem [4]). Let M be a topological monoid and BM be its classifying
space. Let π = π0 M . If π is in the centre Z(H∗ M ) of M , then
H∗ (M )[π −1 ] ∼
= H∗ (ΩBM ).
We do this in 3 steps which are as follows.
1. Show that M∞ ×M EM → BM is a homology fibration.
2. Show that hof ib(p) ∼
=w.e ΩBM .
3. Show that H∗ M∞ = H∗ M [π0−1 ].
Given these three, the first implies that H∗ (hof ib(p)) ∼
= H∗ M∞ . The second then give H∗ M ∼
=w.e H∗ (ΩBM ).
Lastly the third give H∗ M [π0−1 ] ∼
H
(ΩBM
)
which
is
the
theorem.
= ∗
The rest of this section will be to describe the three steps.
7.5
Step 1: M∞ ×M EM → BM is a homology fibration
We use another definition of homology fibration that is stronger than our earlier definition (ie. it implies the
other definition) and will show that M∞ ×M EM → BM is a homology fibration in this sense, and so is a
homology fibration in the old sense (see [4] for details).
Definition 7.9. A map p : E → B is a homology fibration if whenever b ∈ B, then there exists arbitrarily
small contractible neighbourhoods U of b such that the inclusion p−1 (b0 ) ,→ p−1 (U ) is a homology equivalence
for b0 ∈ U .
Proposition 7.10. Let
E1 o
p2
p0
p1
B1 o
/ E2
E0
f1
B0
f2
/ B2
be a commutative diagram of spaces such that each pi is a homology fibration satisfy: if b ∈ B0 , then
'
p−1
→ p−1
0 (b) −
i (fi (b)) is a homology equivalence. Then the induced map of double cylinders
p : cyl(E1 ← E0 → E2 ) → Cyl(B1 ← B0 → B2 )
is a homology fibration.
Proof. Let b ∈ Cyl(B1 ← B0 → B2 ). Then there exists arbitrarily small neighbourhoods U in the form of
mapping cylinders
V0 → Vi
and p−1 (U ) is the mapping cylinder of P0−1 (V0 ) → p−1
i (Vi )
Proposition 7.11. Let p : E → B be a map of simplicial spaces satisfying the following condition: if
k ∈ Z≥0 , the Ek → Bk is a homology fibration.
Assume further that we have a simplicial map θ : [k] → [l] and if b ∈ Bl then p−1 (b) → p−1 (θ∗ b) is a
homology equivalence.
Then the map of realistions |E| → |B| is a homology fibration.
Proof. Observe that |E| and |B| are made up of skeletons. In particular |B|k is Cyl(|B|( k−1) ← ∂∆k ×Bk →
∆k × Bk ). Similarly |E|k is a cylinder. We can then apply the previous proposition.
26
Note that M∞ ×M EM and BM are the realisations of simplicial spaces E and B with
Ek = M∞ × Bk and Bk = M K
Now applying the proposition gives the result. Note that we have not yet checked the condition on the
fibres that is required to make the proposition work. We leave this for another time (see McDuff, Segal).
7.6
Step 2: hof ib 'w.e ΩBM
We want to show that M∞ ×M EM ' ∗.
m
m
m
Consider M∞ = T el(M −→ M −→ M −→ . . .). Then
id×m
id×m
M∞ × EM = T el(EM × M −−−−→ EM × M −−−−→ . . .)
and
M∞ ×M EM = T el(EM ×M M → EM ×M M → . . .) = T el(EM → EM → . . .) ' ∗
Here, the action of M on M∞ is defined as follows. If n ∈ MF, then n acts on Mk × I by (q, t).n = (qn, t).
We need M to act on the right since the telescope is given by Mk × I/ ∼, where (q, 1)k ∼ (mq, 0)k+1 . In
particular, we need (q, 1)k .n ∼ (mq, 0)k+1 .n which works only if M is acting on the right.
Given that M∞ ×M EM ' ∗, we get a diagram of fibre sequences
/ EM ×M M∞
Fp
'w.e
p
'
Fq
/∗
/ BM
=
q
/ B2
Note that Fq = {(e, f : I → BM )|f (0) = ∗, f (1) = q(e)} = {(∗, f : I → BM )|f (0) = ∗, f (1) = q(∗) =
∗} = ΩBM .
Therefore, hof ib(p) = Fp 'w.e Fq = ΩBM .
7.7
Step 3: H∗ M∞ = H∗ M [π0−1 ]
Observe that
[m]
[m]
H∗ M∞ = lim(H∗ M −−→ H∗ M −−→ . . .)
→
where [m] : H∗ M → H∗ M is multiplication by [m] ∈ π0 M = H0 M . If we use the Mayer-Vietoris sequence,
draw a picture and use some magic (see [4] for details), we see that the limit is H∗ M [π0 ]−1 which gives the
desired result.
[insert picture of M-V bits]
8
Cobordism
Things we might be interested in:
• cobordism of manifolds
• cobordism of oriented manifolds
• cobordism of spin manifolds
• cobordism of framed manifolds
• cobordism of stably almost complex manifolds
27
Recall from differential geometry.
Theorem 8.1 (Whitney Embedding). If M n is a smooth n-manifold, then there exists an embedding
i : M n ,→ RN +n
and N may be taken to be n.
Since we can think of a n-manifold as sitting inside RN +n , we can consider its normal bundle.
Definition 8.2. The normal bundle of i is ν → M , where
νx := di(Tx M )⊥ ,
where di : Tx M → Ti(x) RN +n ∼
= RN +n is the derivative.
Observe that ν = ν(i) is a N -plane bundle.
Definition 8.3. The Grassmannian of N planes in RN +n is
GrN (RN +n := {V ⊂ RN +n )|dimV = N }
Definition 8.4. The Stiefel manifold of orthogonal N frames in RN +n is
VN (RN +n ) := {(v1 , . . . , vN )|{vi } is orthonormal }
In particular, GrN (RN +n ) = VN (RN +n )/O(N )
Definition 8.5. The tautological bundle on GrN (RN +n ) is γN → GrN (RN +m ), where
γN := {(V, v)|V ∈ GrN (RN +n ), v ∈ V }
and the bundle map is given by (V, v) 7→ V .
Definition 8.6. The classifying map is
fν : M → GrN (RN +n ),
where fν (x) = νx .
Note that fν∗ γN ∼
= ν, where fν γN = {(x, (V, v))|fν (x) = V }.
Proposition 8.7. πi (VN (RN +n )) = 0, for 0 ≤ i < n.
Proof. Use induction and the fibre bundle
p
FN −1 (RN +n−1 ) → VN (RN +n ) −
→ S N +n−1 ,
where p((v1 , . . . , vN )) = vN . Complete the proof as an exercise.
So we have
VN (RN +n ) 
/O(N )

GrN (RN +n ) / VN (RN +n+1 
/ ...
/O(N )
/ GrN (RN +n+1 
induced from RN +n < RN +n+1 < . . .
28
/ ...
Taking limn→∞ , and using the proposition, we have
VN (R∞ )
'∗
/O(N )
GrN (R∞ )
' BO(N )
Since at each stage, we have a principle O(N ) bundle.
Assume that there exists a family of groups G(N ) and representations ρn : G(N ) → O(N ) and commutative diagrams
/ G(N + 1)
G(N )
ρN
O(N )
ρN +1
/ O(N + 1)
iN
M 0
where iN =
.
0 1
Here are some examples that we should have in mind.
1. G(N ) = O(N ), ρn = id gives rise to bordism.
2. G(N ) = SO(N ), ρn = (SO(N ) ,→ O(N ) gives rise to oriented bordism.
3. G(2N ) = U (N ), gives rise to complex bordism.
4. G(N ) = 0, gives rise to framed bordism.
Z/2
5. G(N ) = Spin(N ), ρn = (Spin(N ) −−→ SO(N ) ,→ O(N ) gives rise to spin bordism.
6. G(N ) = String(N )), gives rise to string bordism.
Definition 8.8. ν admits a stable G(N )-structure if there exists a lifting:
2 BG(N )
f˜ν
M
fν
/ GN (RN +m )
BρN
/ GrN (R∞ )
k
BO(N )
Definition 8.9. The nth stable G-bordism group is
(M G)n := {M n compact without ∂ s.t there exists i : M ,→ RN +n along with a choice of lift f˜ν of k◦fν }/ ∼
where (M, f˜ν ∼ (N, g̃ν ) if there exists l : W n+1 ,→ RN
0
+n+1
with G(N 0 ) structure on ν(l) such that
1. ∂W n+1 = M t N
2. h̃ν |M tN = i0N . . . iN f˜ν t iN 0 . . . iN g̃ν . Here, h˜ν is the lift of hν : W → BO(N 0 ) to BG(N 0 ) obtained
from the G(N 0 ) structure on V (l).
Example 8.10.
1. If G(N ) = O(N ), then the second assumption is vacuous. In this case, M On =
{ compact n-manifolds }/ bordism .
2. If G(N ) = SO(N ), M SOn = { compact oriented n-manifolds }/ oriented bordism
29
Claim: M is oriented ⇐⇒ T M is oriented ⇐⇒ ν is oriented for all i : N ,→ RN +n .
[insert picture of craig demonstrating orientation with arms and legs]
Definition 8.11. Let ξ → X be a vector bundle with a metric on ξ (ie. a fibre wise dot product).
The disk bundle on ξ is
D(ξ) := {v ∈ ξ : kvk ≤ 1}.
The sphere bundle is
S(ξ) := {v ∈ ξ : kvk = 1}.
The Thom Space is
X ξ := D(ξ)/S(ξ).
Claim: If X is paracompact, we can always define a metric on ξ. Moreover, any two ξ give homeomorphic
Thom spaces.
Proposition 8.12.
X ξ⊕1 = ΣX ξ ,
where 1 = X × R.
For examples,
T hom(1) =
X × [−1, 1]
= ΣX.
X × {−1, 1}
Proposition 8.13. Given a continuous map f : Y → X, there is an induced map f ξ : Y f ∗ξ → X ξ .
Exercise.
We want to study
Bi∗N (γN +1 ) = γN ⊕ 1
/ γN +1
BO(N )
/ BO(N + 1)
BiN
GrN (R∞ )
/ GrN +1 (R∞ )
f
Here, f maps V ⊂ R∞ to V ⊕ R ⊂ R∞ ⊕ R = R∞1 .
So we get maps
∗
ΣBO(N )γN = BO(N )γN ⊕1 = BO(N )BiN (γN +1 ) → BO(N + 1)γN +1
Moreover, we have a map ΣBG(N )
Bρ∗
γ
N N
→ BG(N + 1)
Bρ ∗
γ
N +1 N +1
from the commutative diagram
/ BG(N + 1)
BG(N )
BρN
BO(N )
/ BO(N + 1)
Definition 8.14. Let M G be the spectrum associated to the prespectrum BG(N )
M Gk = lim ΩN BG(N + k)
Bρ∗
γ
N N
, ie
Bρ∗
γ
N +k N +k
n→∞
1 As
a somewhat related but off topic note, the reader may want to look up the phrase Eilenberg swindle.
30
Theorem 8.15 (Thom).
(M G)n ∼
= πn M G
What has been computed:
1. G = O- Thom (not too hard);
2. G = U - Novikove/Milnor;
3. G = SO- Stong (messy, nice away from p = 2);
4. G = {1}- G(n) = {1}, BG(n) = ∗, BG(n)γn = Dn /S n−1 = S n , so M Gk = limn→∞ Ωn S n+k (which is
hard).
Moral: n-dimensional framed bordism = πns (S 0 ) = limk→∞ πn+k (S k ), the stable homotopy groups of
spheres is hard to compute.
8.1
Sketch of proof
We briefly summarise the main idea of the previous section and give a sketch of Theorem 8.16, which is a
restatement of the theorem earlier.
ρn
Setup: If G(n) −→ O(n) is a sequence of groups and group homomorphisms with commuting diagrams
/ G(n + 1)
G(n)
ρn+1
ρn
O(n)
/ O(n + 1),
then we define the G-bordism groups to be the bordism of manifolds whose stable normal bundle admits
the structure of a G-bundle. ie. there exists a morphism X → BG(n) that makes the following commute.
BG(n)
<
Bρ(n)
X
fν
/ BO(n)
Theorem 8.16. The nth such bordism group := (M G)n ∼
= πn (M G), where M G is the spectrum associated
to the prespectrum {BG(n)γn }.
We will use the following two facts.
1. Let Yi be a sequence of spaces such that Yi ⊂ Yi+1 ⊂ . . . and let X be compact. If f : X → ∪∞
i=1 Yi ,
then im(f ) ⊂ Yj for some j. Note that here, we endow Ui Yi with the topology of the union. [exercise:
prove this using the usual compactness type arguments.]
2. (Smooth approximation theorem). If M, N are smooth manifolds and f : M → N is continuous, then
there exists f 0 ' f such that f 0 : M → N is smooth. ie. C ∞ (M ) is dense in C 0 (M ).
00
3. Let Z ⊂ N be a smooth submanifold. There exists f : M → N which is smooth and transverse to Z.
00
ie if x ∈ (f )−1 (Z), then
00
Tf 00 (x) Z + dfx Tx M = Tf 00 (x) N.
00
00
Then Sard’s Theorem implies that (f )−1 (Z) is a submanifold of M and codimM ((f )−1 (Z)) =
codimN (Z). [See Guillemin and Pollack for a reference]
31
Sketch of Theorem 8.16. We want to construct maps T : πn M G → (M G)n and P : (M G)n → πn M G such
that T P = id and P T = id.
We construct T as follows. Let α ∈ πn M G = πn limk→∞ Ωk BGγkk . Then we can use the above fact to
find an α̃ and a k such that the following commutes.
/ lim Ωk BGγk
k
k→∞
O
α
Sn
α̃
$ ?
Ωk BGγkk
Then via the loops suspension adjunction, we get and
α : S n+k → BGγkk .
For simplicity, assume G = O. In particular, we have
α : S n+k → BO(k)γk .
Moreover, BO(k) = limk→∞ Grk (Rk+p ). So again, using fact 1, we can find an α̂ and a p such that the
following commutes.
α
/ BO(k)γk
S n+k
O
α̂
& ?
GRk (Rk+p )γk
Observe that S n+k = Rn+k ∪ {∞} and GRk (Rk+p )γk = γk ∪ {∞} and since we are dealing with based
maps, we have α̂(∞) = ∞. In particular, we can think of α̂ as a map
α̂ : Rn+k → γk ,
◦
and Grk (Rn+k ) sits inside γk as the zero section. By fact 3, there is a α ' α̂ which is smooth and
transverse to Grk (Rn+k ). Define
◦
T (α) := α−1 (Grk (Rn+k ).
Note in defining T , we made several choice:
1. Representative of homotopy class of α.
2. k
3. p
◦
4. Homotopical replacement α for α̂.
Homework: Think about why T is well defined. Moreover why is T (α) ∈ (M G)n ?
We now construct the inverse map P : (M G)n → πn M G. Let X be an n-dimensional manifold. Then
by the Whitney embedding theorem, there is a k such that there is an embedding i : X ,→ Rn+k . Then ν(i)
is a G(k)-bundle.
Recall that the classifying map is fν : X → BO(k). We have f ∗ γk = ν(i), and so the following diagram
commutes.
32
; γk
BG(k)
<
ν
X
f
fν
/ BO(k)
We now use the the tubular neighbourhood theorem from differential topology:
Theorem 8.17. Let X ⊂ Rn+k be a compact n-manifold. Then X has an open neighbourhood homeomorphic
to ν.
If n = 1, then the theorem corresponds exactly to drawing a tube around our X. Let U be the tubular
neighbourhood as in the theorem, ie. ϕ : U → ν is a homeomorphism.
We construct a map P (X) : S n+k → γk ∪ {∞} (again ∞ 7→ ∞) by
∞
y∈
/ U,
P (X)(y) =
F ϕ(y) y ∈ U.
Observe that the map is continuous since if we are near the end of the tubular neighbourhood, then ϕ
maps U to larger and larger values.
Again, we have made some choices in defining P , but one can check that this does not make a difference.
We can observe that T P (X) = X since T P (X) = P (X)−1 BG(k) = X. Craig intimidates us into thinking
that the rest of the proof works. That is, P T = idπn M G .
Note again that in constructing P we made the following choices.
1. A representative X ∈ (M G)n .
2. A choice of embedding X ,→ Rn+k and choice of k.
3. Diffrent radii of tubular neighbourhoods.
Different choices will give homotopic maps.
9
Moduli Spaces and Mapping Class Groups
9.1
Mapping Class Group
Denote by Sg,r the topological surface of genus g with r labelled boundary components.
Definition 9.1. The mapping class group of Sg,r is
Γg,r :=
Homeo+ (Sg,r , ∂)
,
Homeo+
0 (Sg,r , ∂)
where Homeo+ (Sg,r ) is the group of orientation preserving homoemorphisms of Sg,r that fix the boundary,
and Home+
0 (Sg,r , ∂) the homeomorphisms that are also isotopic to the identity.
We also define a related object,
Γrg :=
Homeo+ (Sg,r )
,
Homeo+
0 (Sg,r )
33
which consists also of homeomorphisms that do not fix the boundary. While we will not talk about this
other object too much, we note that there is a short exact sequence
1 → Zr → Γg,r → Γrg → 1.
There are several reasons to care about mapping class groups.
1. Homeo+ (S, ∂) is cool.
2. Diff+ (S, ∂) is cool. Moreover it turns out that
Diff+ (Sg,r , ∂)
.
Γg,r ∼
=
Diff+
0 (Sg,r , ∂)
3. Characteristic classes are invariants of a F -bundle with respect to F -bundle isomorphism. In particular,
we are interested in Sg -bundles.
Fact: There is a bijection
{Sg → E → X}/iso ←→ M ap(X, BDiff+ (Sg ))/homotopy equivalence
between isomorphism classes of Sg -bundles over X and homotopy maps from X to BDiff+ (Sg ).
This is analogous to the situation we saw in the previous sections where rank n vector bundles over X
were classified by homotopy maps from X to BUn .
9.2
How do we get characteristic classes?
Characteristic classes of surface bundles are given by H ∗ (BDiff+ (Sg )). To do this, we take a cohomology
class [α] ∈ H i (BDiff+ (Sg )). Then given an Sg -bundle
π:E→X
we can go via our bijection to get a homotopy map
[f ] : X → BDiff+ (Sg ).
By applying the induced map on cohomology to [α], we get a cohomology class f ∗ ([α]) ∈ H i (X). In
particular, f ∗ [α] is an invariant of Sg -bundle isomorphisms. WINNAR.
Thus, if we are interested in characteristic classes of surface bundles, we want to study BDiff+ (Sg ).
A theorem of Earl and Eels tells us that, for g ≥ 2, Diff+ (Sg ) 'h.e Γg . In particular, Diff+
0 (Sg ) ' {∗}.
Therefore, rather than studying the rather complicated infinite dimensional space BDiff+ (Sg ), we can look
at the much nicer BΓg .
To get BΓg , we want to find EΓg which is contractible and has a free Γg action. Then BΓg = EΓg /Γg .
This leads us to the study of moduli spaces.
9.3
Moduli Spaces
Key Idea: A moduli space is a space that parametrizes a class of objects we care about.
Example 9.2. Here are some examples that may be familiar.
1. RP 1 = S 1 /x ' −x parametrizes lines through the origin in R2 .
2. RP n parametrizes lines through the origin in Rn+1 .
3. Grn (Rn+m ) parametrizes n-dimensional vector subspaces of Rn+m .
34
In our construction, both our models for EΓg and BΓg will be moduli spaces.
◦
Our model for EΓg will be the moduli space of complete hyperbolic metrics on Sg,r up to isotopy. It is
called Teichmüller space, and can be written as


◦


f : Sg,r → S,




f
is
a
homeo,
r
Tg := (S, f ) |
/∼

S is a complete hyperbolic surface with labelled cusps and; 




f preserves labelling
where (S1 , f1 ) ∼ (S2 , f2 ) ⇐⇒ f2 ◦ f1−1 : S1 → S2 is isotopic to an isometry.
Example 9.3. T03 ∼
= {∗}.
Fact: Specifying the length (li ∈ R≥0 , for i = 1, 2, 3) of the boundary components of S0,3 fixes a unique
hyperbolic metric on it.
Question: What about T04 ?
◦
Fix a curve α on S0,4 . Then given a complete hyperbolic metric on S0,4 , there is a unique geodesic in
the same homotopy class of α. If we cut along α, do a small twist and stick the surfaces back together along
α, we get a different metric on S0,4 .
[INSERT A PICTURE]
The idea then is that choosing a length in R>0 and a twisiting angle in R specifies a metric on S0,4 . Thus
we have a heuristic argument that
T04 ∼
= R>0 × R.
More generally
3g−3+r
Tg4 ∼
× R3g−3+r .
= R>0
To see this, one uses similar arguments to the S0,4 case. The 3g − 3 + r comes from the fact that there are
3g − 3 + r non-intersecting essential closed curves on Sg,r .
Observe that Tgr is contractible. Moreover, Γrg acts on Tgr by
[ϕ].[S, f ] = [S, f ◦ ϕ].
Here, ϕ : Sg,r → Sg,r , so f ◦ ϕ : Sg,r → S and [S, f ◦ ϕ] ∈ Tgr . Unfortunately, the actions is not free.
Recall that a group action G × X → X is free if g.x = x only when g = id. To see an example
of the mapping class group not acting freely, then imagine a surface with a symmetry, say it has a π
rotation symmetry, call it ι. Then ι is a homeomorphism of S, so ι ∈ Γrg . But because of the symmetry,
[ι].[S, f ] = [S, f ] for [S, f ] ∈ Tgr . Surfaces with this type of symmetry are called degenerate surfaces.
Thus Γrg does not act freely on Tgr . However it is not far off since in most non-degenerate cases the action
the action is free. Thus
BΓrg ≈ Tgr /Γrg ,
where ≈ means almost equal to.
Definition 9.4. The moduli space of complete hyperbolic surfaces of genus g with r labelled cusps is
Mrg := Tgr /Γrg .
Example 9.5. M30 = {∗}, since T03 = {∗}.
Example 9.6. M40 ' S 2 − {3 points.}.
35
While it is not true that BΓg equals Mg because of the non-free action, the two are still very closely
related. In particular, it is true that their rational cohomologies agree. That is H ∗ (BΓg , Q) ' H ∗ (Mg , Q).
Thus, we are interested in H ∗ (Mg , Q). The Mumford conjecture states that H ∗ (M∞ , Q) ∼
= Q[k1 , k2 , . . .],
where deg(ki ) = 2i. A solution to the Mumford Conjectures comes in the form the the Madsen-Weiss
Theorem.
Philosophically, the idea is that the homology of Γg,r in a stable range is the homology of an infinite loop
space. In later lectures, we will look in more detail at the Madsen-Weiss theorem. For now we give a big
outline of the proof. There are three main steps.
1. Diff+ (Sg,r ) has contractible components for g ≥ 2.
2. Harer stability: Hi (Γg,r ) is independent of g and r for i g.
3. Group completion theorem.
We have already encountered step 1. Step 3 will allow us to show that the classifying space of compactly
supported diffeomorphisms of an infinite genera surface will be homology equivalent to some infinite loop
space. Step 1 relates the former with BΓ∞ , which we can hope to understand because of Harer stability.
10
Harer Stability
Recall the key players from the previous section: Sg,r , Γg , Tgr and Mg . We also had maps
BDiff(Sg ) → BΓg → Mg .
The first map is a homotopy equivalence, while the second is a rational homology equivalence. We also gave
a rough statement of Harer stability which said the following.
Theorem 10.1 (Harer Stability). Hi (Γg,r ) := Hi (BΓg,r ) is independent of g and r for i g
In this section, we will describe the proof of Harer stability. We will be following closely [8].
10.1
General nonsense on homological stability
Let G1 ⊂ G2 ⊂ . . . ⊂ Gn ⊂ . . . be a sequence of groups and group inclusions. Let X1• ⊂ X2• ⊂ . . . ⊂ Xn• ⊂ . . .
be a sequence of simplicial complexes with a simplicial Gn action such that
1. The Xn• are highly connected. ie. πk (Xn• ) = 0 for some k ≤ f (n), where f is some increasing function
of n.
2. The stabilizer of a p-simplex is StabXn• (σp ) ∼
= Gn−p−1 .
3. Gn acts transitively on Xnp (or at least as transitively on the vertices).
Then one might hope for: H∗ (Gn ) is independent of n for n ∗. Here are some examples.
1. The symmetric group Sn . There are inclusions Sn ⊂ Sn−1 . Sn acts on the standard (p − 1)-simplex
∆p .
2. Braid groups.
3. Unordered configuration spaces.
36
10.2
The maps
[THERE WERE PICTURES TO MAKE THE MAPS CLEARER]
What are the inclusion maps (also called stabilization maps)? Let
α : Sg,r+1 ,→ Sg+1,r
be the map the obtained from gluing two boundaries of a pair of pants to two boundary components. Then
this induces a map
αg : Γg,r+1 ,→ Γg+1,r
obtained by sending a map f to f ∪ id. Similarly from the map
β : Sg,r ,→ Sg,r+1
obtained from gluing one boundary component of a pair of pants, we get the map
Γg,r+1 ,→ Γg,r+1 .
Lastly, let
δ : Sg,r+1 ,→ Sg,r
be the map obtained by gluing a disk to a boundary component and let
δg : Γg,r+1 ,→ Γg,r
be the corresponding map of mapping class groups. Note that δ is the left inverse of β.
Let H∗ (αg ), H∗ (βg ) and H∗ (γg ) be the induced maps on homology. We can now state the theorem more
precisely as it appears in [8].
Theorem 10.2 (Harer Stability 1). Let g ≥ 0, r ≥ 1.
1.
H∗ (αg ) : H∗ (Γg,r+1 ) → H∗ (Γg+1,r )
is surjective for ∗ ≤
2
3g
+
1
3
and is an isomorphism for ∗ ≤ 23 g − 23 .
2.
H∗ (βg ) : H∗ (Γg,r ) → H∗ (Γg,r+1 )
is always injective and is an isomorphism for ∗ ≤ 23 g.
For the statement for closed surfaces, consider δg : Γg,1 → Γg,0 .
Theorem 10.3 (Harer Stability 2).
H∗ (δg ) : H∗ (Γg,1 ) → H∗ (Γg,0 )
is surjective for ∗ ≤ 32 g + 1 and is an isomorphism for ∗ ≤ 23 g.
In particular, if we combine the second theorem with the first theorem, then we get homological stability
for closed surfaces.
37
10.3
The Arc Complexes
Let S be a connected, oriented surface with boundary and let b0 , b1 ∈ ∂S.
An arc on S is an embedded path in S such that
1. the path intersects ∂S only at the endpoints; and
2. the intersection is transverse.
A collection {a0 , . . . , an } of arcs is non-separating if S − {a0 , . . . , an } is connected.
Definition 10.4. O(S, b0 , b1 ) is the simplicial complex with
vertices : isotopy classes of non-separating arcs with boundary {b0 , b1 }.
simplices : A p-simplex of O(S, b0 , b1 ) is a collection of p + 1 distinct isotopy classes of arcs ha0 , . . . , ap i such
that
1. There exists representative a0 , . . . , ap with disjoint interiors,
2. {a0 , . . . , ap } are non-separating, and
3. The anticlockwise ordering of a0 , . . . , ap at b0 agree with the clockwise ordering at b1 .
[Insert Picture]
We will denote by O1 (S, b0 , b1 ) to be the arc complex O(S, b0 , b1 ) where b0 and b1 are on the same
boundary component. Similarly, O2 (S, b0 , b1 ) will be the arc complex where b0 and b1 are on different
boundary components.
Note that Γ acts on O(S, b0 , b1 ) by [f ]ha0 , . . . , ap i = hf (a0 ), . . . , f (ap )i. There are four properties of
O(S, b0 , b1 ) that we will use in the proof of 10.2. Essentially, they will relate to the idea of want a “transitive
action, the stabilizer condition and high connectivity” which we described in the general nonsense section.
We will also describe relationships between the O1 and O2 . These are the four ingredients the Wahl refers
to in [8, Section 2].
10.4
Four properties of O
Proposition 10.5.
1. Γg,r acts transitively on the p-simplices of O(Sg,r , b0 , b1 ) for each p.
2. There exist isomorphisms
s
1
StO1 (σp ) −→
Γg−p−1,r+p+1
s
2
StO2 (σp ) −→
Γg−p.r+p−1
where σp is a p-simplex of Oi (Sg,r , b0 , b1 ) and StOi (σp ) is the stabilizer of a p-simplex in Oi under the
action of Γg,r .
∼ Sg ,r . In
Proof. (Sketch) Let σp = ha0 , . . . , ap i. Consider S − σp . Since σp is non-separating, S − σp =
σ σ
particular, we can show that gσ and rσ depend only on p (by an argument of fattening up arcs and removing
them from our original surface). So for any other p-simplex τp , S − τp ∼
= Sgτ ,rτ . By the classification of
surfaces, there is an isomorphism f : Sgσ ,rσ ∼
= Sgτ ,rτ . Then looking at the lift of f to our original surface,
we see that f is a homeomorphism of S that sends σ to τ , hence the action is transitive.
For the second part, we similar consider the cut surface and need to argue that all homeomorphisms that
fix the boundary of the cut surface are stabilizers of σp .
Proposition 10.6. Let α : O2 (Sg,r+1 ) → O1 (Sg+1,r ) be the map that connects the two components of S
that contains b0 , b1 via a strip.
[PICTURE]
38
Let β : O1 (Sg,r ) → O2 (Sg,r+1 ) be the map that separates the component of S that contains b0 , b1 via a
strip.
[PICTURE]
Given a p-simplex σp in O2 (Sg,r+1 ), we have the following diagram which commutes.
Γg,r+1 o
? _ StO2 (S
g,r+1 )
αg
(σp )
s2
∼
=
/ Γg−p,r+p
βg−p
α
Γg+1,r o
? _ StO1 (S
g,r )
(α(σp ))
s1
∼
=
/ Γg−p,r+p+1
(σp )
s1
∼
=
/ Γg−p−1,r+p+1
Similarly, given a p-simplex in O2 (Sg,r ) we have
? _ StO1 (S
Γg,r o
g,r )
βg
αg−p−1
α
Γg,r+1 o
? _ StO2 (S
g,r+1 )
(β(σp ))
s2
∼
=
/ Γg−p,r+p
Proof. See [8, Proposition 2.3]
The intuition though is that the right square is given by the previous proposition. The inclusions to the
left are just including a stabilizer of a group into the whole group. The square then reads, αg on Γ induces
βg on the stabilizers (and vice-versa for the second diagram).
Proposition 10.7. Let Sα and Sβ denote the surface S union a strip glued via α and β respectively. The
maps
α : Γ(S) → Γ(Sα )
β : Γ(S) → Γ(Sβ )
are injective. Moreover, for any vertex σ0 of Oi , there are conjugations cα and cβ such that the following
diagrams commute.
StO2 (σ
 _ 0)
w
Γ(S)

cα
/ StO1 (α(σ0 ))
_
StO1 (σ
 _ 0)
/ Γ(Sα )
w
Γ(S)

/ StO2 (β(σ0 ))
_
cβ
/ Γ(Sβ )
Proof. For a proof, see [8, Proposition 2.4].
In terms of utility of the proposition, the injectivity statements as allowing us to consider the relative homology groups H∗ (StO1 (α(σ0 )), StO2 (σ0 )) and H∗ (StO2 (β(σ0 )), StO1 (σ0 )). Then if we want to map these to
H∗ (Γ(Sα ), Γ(S)) and H∗ (Γ(Sβ ), Γ(S)) respectively, this map will be trivial, since the commutating diagrams
tell us the map goes through Γ(S).
Proposition 10.8. Oi (Sg,r .b0 , b1 ) is g − 2 connected.
This is the high connectivity statement that we would like to hold for our arc complexes. The proof is
technical and is the contents of section 4 of [8]. We will not say anything about it here.
39
10.5
Spectral sequence argument
We will use what follows as a black box.
Setup: Let G and H be groups. Let X and Y be simplicial complexes with G acting on X and H acting
on Y (simplicially). Let φ : G → H be a homomorphism and let f : X → Y be an φ-equivariant map. If G
and H act transitively on the p-simplices for each p, X is g − 2 connected and Y is g − 1 connected, then
there exists a spectral sequence with
1
Ep,q
= Hq (StY (σp ), StX (σp )) =⇒ 0
for p + q ≤ g − 1.
We will now set ourselves up to prove Theorem 10.2.
Recall the maps αg : Γg+1,r + → Γg,r+1 and βg : Γg,r → Γg,r+1 . We can then form the relative homology
groups
H∗ (αg ) := H∗ (Γg+1,r , Γg,r+1 )
H∗ (βg ) := H∗ (Γg,r+1 , Γg,r )
We can restate Theorem 10.2 as follows.
Theorem 10.9. Using the above notation, we have
1. Hi (αg ) = 0 for i ≤
2g+1
3 ;
2. Hi (βg ) = 0 for i ≤
2g
3 .
and
Proof. We will do double induction on g and r by proving the following propositions.
(1g ) : Hi (αg ) = 0 for i ≤
2g+1
3 .
(2g ) : Hi (βg ) = 0 for i ≤
2g
3 .
If g = 0 for (1g ) or g = 0, 1 for (2g ), then both statements are talking about H0 . But the spaces we are
talking about are classifying spaces of groups, and so are connected. Thus, stability holds for these cases.
Inductive steps: We will show the following:
1. For g ≥ 1, (2≤g ) =⇒ (1g ).
2. For g ≥ 2, (1<g ) =⇒ (2g ).
Step 1. Using the above spectral sequence setup, take G = Γg,r+1 , H = Γg+1,r , X = O2 (Sg , r + 1), Y =
O (Sg+1,r ). Also, the maps φ : G → H and f : X → Y , both are induced by α : Sg,r+1 → Sg+1,r . Proposition
10.5 gives us the transitive conditions, while Proposition 10.8 gives us the connectivity conditions.
Thus there is a spectral sequence with
1
1
Ep,q
= Hq (StY (σp ), StX (σp )) =⇒ 0
for p + q ≤ g − 1.
Now, when p = −1, we can interpret σ−1 as an empty simplex, thus the stabilizers are the whole groups.
So
1
E−1,q
= Hq (Γg+1,r , Γg,r+1 ) = Hq (αg ).
When p ≥ 0, we can use Proposition 10.5 to recognise the stabilizers as mapping class groups of cut
surfaces to get
1
Ep,q
= Hq (Γg−p,r+1 , Γg−p,r ) = Hq (βg−p ).
In particular, we are now in a position to use induction on Hq (βg−p ).
1
We want to show E−1,q
= Hq (αg ) = 0 for q ≤ 2g+1
to complete step 1. We will do this by showing three
3
claims:
40
∞
1. E−1,q
= 0 for q ≤
2g+1
3 .
1
1
2. Ep,q
= 0 (or equivalently Ep,q−p
= 0) for q ≤
2(g−p)
.
3
1
1
3. d1 : E0,q
→ E−1,q
is the 0 map.
Given these three claims, the result then follows since
r
• Step 1 tells us that E−1,q
eventually dies in our desired range.
r
• Step 2 and 3 tell us that there can be no differentials into E−1,q
in our desired range, so nothing can
kill it.
1
• Thus, it had to be dead to begin with. ie. E−1,q
= 0 in our desired range.
Proof of claim:
∞
∞
1. Since Ep,q
= 0 for p + q ≤ g − 1, we have E−1,q
= 0 for q ≤ g and for g ≥ 1, (2g + 1)/3 ≤ g.
1
2. By induction, Ep,q
= Hq (βg−p ) = 0 for q ≤ 2(g − p)/3.
1
1
3. Lastly, d1 : E0,q
→ E−1,q
is a map Hq (StO1 (α(σ0 )), StO2 (σ0 )) → Hq (Γg+1,r , Γg,r+1 ). By Proposition
10.7, this map is 0.
To do step 2, we need to show that for g ≥ 2, (1<g ) implies (2g ). To do this, do a similar argument with
G = Γg,r , H = Γg,r+1 , X = O1 (Sg,r ), Y = O2 (Sg,r+1 ) and φ, f induced by β : Sg,r → Sg,r+1 .
11
Madsen-Weiss Theorem (Wonderful Copenhagen)
Recall the Mumford conjecture.
Theorem 11.1.
H ∗ (Mg , Q) ∼
=stably Q[x1 , . . . , xn , . . .],
where |xi | = 2i.
Stably means that the isomorphism holds for ∗ ≤ f (g) for some increasing f . In this case f is linear. This
was answered by the Madsen-Weiss theorem, but is still an active area of research. Some ongoing questions
are: What happens unstably? What happens in higher dimensions?
Our strategy will be to look for a string of isomorphisms that look like the following:
∼
H ∗ (Mg ; Q) ∼
= H ∗ (BΓg ; Q) ∼
= H ∗ (BDiff(Sg ); Q) ∼
=stably H ∗ (Ω∞
0 M T SO(2); Q) = Q[x1 , . . . , xn ].
We have already seen many of these isomorphisms in the previous sections. In this section, we focus on
the stable isomorphism, which is essentially the contents of the Madsen-Weiss theorem.
We are after the following zig-zag of equivalences:
G
Ω∞ M T SO(2) → Ω∞ Ψ ←
BDiff(Sg ),
g≥0
which will specialise to
∞
Ω∞
0 M T SO(2) →w.e Ω[Sg ] Ψ ← BDiff(Sg ),
where the right map is a homology isomorphism for ∗ ≤ 32 g. We will now try to describe what Ω∞ ψ is.
We will use the convention that a subset W ⊂ Rn is topologically closed if W is a closed subset. In
particular, we do not want to confuse this notion with a “closed” manifold.
Definition 11.2. Let
Ψk (Rn ) = {(W, w) ⊂ Rn | W is a topologically closed k-manifold and w is an orientation}.
41
11.1
The topology on Ψk (Rn )
While we did not say what that topology on Ψk (Rn ) was explicity, we noted that a neighbourhood basis
around ∅ ∈ Ψk (Rn ) is given by
U ({K}) = {W ∈ Ψk (Rn ) | W ∩ K = ∅}.
In particular, this is meant to capture the notion of being “close to infinity”. This topology is meant to feel
a bit strange. For example, if {K} is a family of compact subspaces of Rn , the map
f : R → Ψk (Rn )
given by f (t) = {1/t} × R2 for t 6= 0 and f (t) = ∅ when t = 0 is a continuous map, given the topology on
Ψk (Rn .
Definition 11.3. Let
Bn := {W ∈ Ψk (Rn ) | W ⊂ (0, 1)n }.
In particular, if X is a k-manifold and f : X → Bn , then the graph of f ,
Γf ⊂ X × (0, 1)n
has the projection Γf → X as a fibre bundle with fibre a compact surface of a fixed diffeomorphism type.
On the other hand, if f : E → X is a fibre bundle with fibre a surface of a fixed diffeomorphism type,
then E is a k + 2 manifold and we can use the Whitney embedding theorem to have an embedding
j : E → X × (0, 1)n
for n > 2(k + 2) = 2k + 4.
Thus we can define f : X → Bn by
f (x) = j(EX ) = {x} × i(FX ).
Homotopies of maps X → Bn correspond to isomorphism classes of surface bundles.
Proposition 11.4. Let X be a k-dimensional manifold. [X, Bn ] is isomorphic to isomorphism classes of
surface bundles over X whenever n > 2k + 4.
If we let n → ∞, we have the following.
Proposition 11.5. B∞ classifies surface bundles over X.
Not that we have a homeomorphism
Bn =
G
Emb(Sg , (0, 1)n )/Diff(Sg ).
g≥0
If we let n → ∞, the Whitney embedding theorem tells us that Emb(Sg , (0, 1)∞ ) ∼
=w.e . The idea is that
πn (Emb(Sg , (0, 1)∞ ) ∼
= π0 (Emb(Sg × S k , (0, 1)∞ )).
F
So Bn = BDiff(Sg ) and B∞ = g≥0 BDiff(Sg ).
42
11.2
Defining the maps
Let ∅ be the basepoint of Bn . Let S n = Rn ∪ {∞}. Define
α : Bn → Ωn Ψ2 (Rn )
by
α(W )(v) =
W +v
∅
, if v ∈ Rn
, if v = ∞.
This gives a map
B∞ → colimn→∞ Ωn Ψ2 (Rn ) =: Ω∞ Ψ.
Madsen-Weiss says that H ∗ (α) is an isomorphism for α restricted to a specific diffeormorphism type Sg
when ∗ ≤ 32 g.
For the grassmannian Gr2 (Rn ), we have the canonical bundle
γn → Gr2 (Rn ).
We also have
γn⊥ → Gr2 (Rn ),
where γn⊥ consists of (V, w), where V ∈ Gr2 (Rn ) is a 2-plane and w ∈ V ⊥ .
γn⊥ has Thom Spaces T h(γn⊥ ). We can define a map q : T h(γn⊥ ) → Ψ2 (Rn ) by
V + w , if (v, w) = ∞
q(V, w) =
∅
, if (V, w) = ∞
Taking the Ω∞ functor, we get
Q : Ω∞ M T SO(2) → Ω∞ Ψ,
where
Ω∞ M T SO(2) := colimn→∞ Ω∞ T h(γn⊥ )
and
Ω∞ Ψ := colimn→∞ Ωn Ψ2 (Rn ).
Q is the first of the maps the we wanted to defined at the start of the section.
Proposition 11.6. Q is a weak equivalence.
11.3
αn is a homology equivalence
Last time, we looked at the maps
αn : Bn → Ωn Ψ(Rn )
where Bn was the space of 2-manifolds in (0, 1)n and Ψ(Rn ) was the space of toplogically closed 2-manifolds
in Rn . (k = 2 if we look back at our original definition). The map αn was given by the formula
W + v , if v ∈ Rn
α(W )(v) =
∅
, if v = ∞,
where S n = Rn ∪ {∞}.
We would like to show that αn is a homology equivalence. Our strategy will be to decompose αn by
looking at one direction at a time.
43
Definition 11.7. Define
Ψ(n, k) := {W ∈ Ψ(Rn ) | W ⊂ Rk × (0, 1)n−k }.
Define α̃i : Ψ(n, i) → ΩΨ(n, i + 1), where
, if t ∈ R
, if v = ∞,
W + tei+1
∅
α̃(W )(t) =
where S 1 = R ∪ {∞}.
We can decompose αn into the following sequence.
α̃
Ωn−1 α̃n−1
Ωα̃
1
0
Ω2 Ψ(n, 2) → . . . −−−−−−−→ Ωn Ψ(n, n) = Ωn Ψ(Rn )
ΩΨ(n, 1) −−−→
Bn = Ψ(n, 0) −→
We want to show
1. α̃1 , . . . , α̃n are weak equivalences. This almost uses the group completion theorem.
2. α̃0 is a homology equivalence in a range. This uses the group completion theorem and Harer stability.
Firstly, we would like to describe another model for the classifying space of a monoid.
11.4
Another way to look at classifying spaces
For M a topological monoid, we can think of BM as pairs (A, f ) where A ⊂ R is a finite subset, and
f : A → M.
INSERT PICTURE of ai in a line mapping to f (ai )
We allow the ai and aj to collide, giving rise to a multiplication f (ai ).f (aj ) ∈ BM . [I don’t quite understand
this interpretation]
We will also be using the following theorem.
Theorem 11.8. For M a topological monoid, the map Define
β : M → ΩBM,
by
β(m)(t) =
({t}, t 7→ m) , if t ∈ R
∅
, if t = ∞.
Then β is a weak equivalence if and only if M has a homotopy unit, and π0 (M ) is a group.
11.5
The monoid
For i ≥ 1, we now want to define a monoid M that fits into the diagram
MO
/ ΩBM
β
'w.e
Ψ(n, i)
α̃i
'w.e
/ ΩΨ(n, i + 1),
since if we can show that β is a weak equivalence using the previous theorem, this we give that α̃i is a weak
equivalence, which is what we wanted to show.
Here is the monoid M .
44
Definition 11.9.
M := {(t, W ) ∈ (0, ∞) × Ψ(n, k + 1) | W ⊂ Rk × (0, t) × (0, 1)n−k−1 }.
Define the multiplication in M to be
(t, W ).(t0 , W 0 ) = (t + t0 , W ∪ (W 0 + tek+1 ).
The way to think about M is that M is a space of subsets with one component that can extend off
arbitrarily far (up to t). This, multiplication is to put the two subspaces next to each other, where the
+tek+1 term is to make the two subspaces disjoint.
There is a map Ψ(n, i) → M defined by M 7→ (1, M ) which is a homeomorphism onto its image. There
is also a map BM → Ψ(n, k + 1) defined by putting the labels on points in R next to each other on the
(k + 1)st coordinate. Looping this map gives a map from ΩBM → ΩΨ(n, k + 1). The also claim (without
proof - is there a quick reason why???) that this map is a weak equivalence.
We now want to show that M satisfies the monoid conditions so that β : M → ΩBM is a weak equivalence.
In particular we need the following theorem.
Theorem 11.10. M is grouplike. That is M has a homotopy unit and π0 (M ) is a group.
Proof. The homotopy unit of M is ∅. We want to show that mm0 and m0 m are in the same path component
as ∅. (I think we mean, for m ∈ M , there exists m0 ∈ M such that mm0 and m0 m are in the same path
component.)
[INSERT PICTURE PROOF - m0 is a copy of m but upside down. Then do some pushing of the pictures]
What’s left to show is that α0 is a homology equivalence. We cannot do the previous trick since things
in the P si(n, 0) do not stretch to infinity.
Definition 11.11. Let Lt = [0, t] × [0, 1] ⊂ R2 . Let Mn0 consist of pairs (t, W ) where t ∈ (0, ∞) and
W ⊂ Lt × (−1, 1)n−2 such that W agrees with Lt × {0} near ∂Lt × Rn−2 . We can define multiplication on
Mn0 by
(t, W ).(t0 , W 0 ) = (t + t0 , W ∪ {W 0 + te1 }).
0
classifies surface bundles with one boundary component.
Recall that B∞ classifies surface bundles. M∞
So
G
0
M∞
'
BDiff(Σg,1 , ∂).
g>0
As before, we want to construct a diagram
β
Mn0
'w.e
Ψ(n, 0)
/ ΩBMn0
'w.e
α̃0
/ ΩΨ(n, 1).
Due to time, we did not say what the maps were.
We now want to study the map β. Since Mn0 is not group like. eg. π0 (Mn0 ) = N when n is sufficiently
large (≥ 5).
But we can apply the group completion theorem using
m
0
M −−→
M,
45
where m0 is multiplying by a surface of genus 1. In the group completion theorem, we consider the telescope
of
m
0
0
M∞
−−→→
...
0
In our case, M∞
=
F
g>0
BDiff(Σg,1 , ∂). By the group completion theorem, we have
0
H∗ (T elM∞0 ) = H∗ (M∞
)[m−1
0 ] = H∗ (Z × BDiff∞ ),
L
0
where BDiff∞ is the limit BDiff(Σg,1 ) → BDiff(Σg+1 , 1) → . . . and H∗ (M∞
) = g≥0 H∗ (BDiff(Σg,1 )).
For g → ∞, we have isomorphisms. But Harer stability gives that this is an isomorphism for ∗ ≤ 23 g.
12
Algebraic K-Theory: Introduction and examples
Let R be a ring. An R-module M is projective if it satisfies the following universal lifting property: Given
a diagram of R-modules and R-module homomorphisms as follows,
M
f
A
f
~
g
/B
/0
there exists an f making the diagram commute. Equivalently, M is a summand of a free module (exercise).
Let
P (R) := {finitely generate projective R-modules}/iso.
Note that P (R) is a commutative monoid. The unit is 0 and addition is ⊕.
Definition 12.1. Let M be a commutative monoide. The group completion or Grothendieck group of M
is
Groth(M ) := M × M/ ∼
where (a, b) ∼ (a + c, b + c) for all a, bc ∈ M .
Definition 12.2. The K-theory of a ring is
K(R) = K0 (R) := Groth(P (R)).
Here are some examples of why one might care about K-theory. The idea is that we can study R via
studying its category of R-modules.
1. Let R be a field. An R-module is a vector space which is free and thus is projective. So
P (R) = { finite dim R-vector spaces / ∼
=.
As a monoid under ⊕, P (R) ∼
= Z≥0 by looking at the dimension. We thus have
K(R) = Z.
2. Let R be a Dedekind domain. ie. every ideal of R admits a unique factorisation into a product of prime
ideals. There is a theorem by Steinitz (?) that states the following.
Theorem 12.3. Every finitely generated projective R-module is isomorphic to Rn ⊕ I where I is a
fractional ideal. ie. I is an R-submodule of the field of fractions of R such that there exists α ∈ R such
that αI ⊂ R.
46
Thus
P (R) ∼
= Z≥0 ⊕ {fraction ideals}
and
KR() = Z ⊕ Cl(R),
where Cl(R) is the ideal class group, or sometimes called the Picard group P ic(R). Fact: Cl(R) =
0 ⇐⇒ R admits a unique factorisation of elements.
3. If G is a discrete group, then consider K0 (Z[G]), where Z[G] is the group ring. There is a map
Z = K0 (Z) → K0 (Z[G])
induced by the map Z → Z[G].
Aside: If R → S is a ring homomorphism, the map R-mod → S-mod that send M 7→ M ⊗R S defines
a map from P (R) → P (S). This then induces a map K(R) → K(S), which is what we are using above.
Definition 12.4. The Whitehead group of G is
W h0 (G) := K0 (Z[G])/Z.
Definition 12.5. A topological space X is finitely dominated if there exists a finite CW complex K
and a map f : K → X which admits right homotopy inverse. ie. there exists a h : X → K such that
f h ' idx .
Theorem 12.6 (Wall). Assume X is finitely dominated. Then there exists an invariant w(X) ∈
W h0 (π1 X) which is 0 ⇐⇒ X is homotopy equivalent to a finite CW complex.
While example 1 was rather simple, examples 2 and 3 are two good reasons as to why we might care
about K-theory. As another wishy washy example, orders and ranks of K(R) should also be related to
special values of L-functions. For example, if R is the ring of integers of a number field, then the Dedekind
zeta functions is
X
1
.
ζR (S) =
nm(I)S
I⊂R
Then the analytic class number formula relates the residues of ζ with the size of the Cl(R).
12.1
K-theory of a symmetric monoidal category
Let C be a small symmetric monoidal category. ie. there is a map ⊗ : C × C → C satisfying . . .
Let BC be the classifying space of C. B is a functor and B(C × D) ∼
= B(C) × B(D). Thus we get a map
B(⊗) : BC × BC → BC
One might be worried that this might not be a strictly unital, associative multiplication, but it turns out
to be E∞ [WHATEVER THAT MEANS]
Define
K(C) = ΩB(BC),
where the second B is the classifying space of BC with respect to the monoidal structure defined above.
Definition 12.7. The algebraic K-theory groups of C are
Ki (C) := πi (K(C)).
Here are some example.
47
1. Let C be the category with
Ob(C) = finitely generated projective R-modules
M or(C) = isomorphisms of R-modules
Then
K0 (C) = π0 (ΩB(BC))
= π0 (BC)[π0−1 ]
via group completion theorem
= Groth(P (R)
Which is the K-theory of a ring.
2. Let C be the category with
Ob(C) = vector bundles over X
M or(C) = isomorphisms of vector bundles over X
Then similarly to before,
K0 (C) = Groth(Ob(C)/ ∼
=)
= Groth(V ect(X))
=: K(X)
which is topological K-theory.
As a slightly more in depth example, Let X be a point. Then in this case, C = {C-vector spaces, isos} =
{0, C, C2 . . .}. The morphisms are Cn → Cn given by an element of GLn (C). Thus
BC = ∗ t BGL1 C t BGL2 C t . . .
So
ΩB(BC) = BGL∞ C × Z
Here again, get the last line from the group completion theorem since
H∗ (ΩB(BC)) = H∗ (BC)[π0−1 ]
= Z[Z] ⊗ lim H∗ (BGLn C)
→
= Z[Z] ⊗ H∗ (BGL∞ C)
= H∗ (Z × BGL∞ C).
Lastly, we can get a homotopy equivalence using the Whitehead theorem.
3. Let C be the category with
Ob(C) = closed, smooth, oriented topological surfaces
M or(C) = diffeomorphisms
We have
C∼
= {Σg |g = 0, 1 . . .}.
So
BC =
∞
G
BDiff(Σg )
g=0
and
Ki (C) = πi ΩB(BC) = πi (Ω∞ M T SO(2)).
48
4. Let K be a field. A central simple algebra (CSA) over K is an associative, finite dimensional K-algebra
A which is simple, with centre K. Eg. if K = R, then R, H and Mn (R) are CSAs, but C is not a CSA.
Let C be the category with
Ob(C) = CSAs over K
M or(C) = isomorphisms
This is symmetric monoidal since A ⊗K B is central simple if A and B are. The Brauer group of K is
Br(K) := K0 C = π0 ΩB(BC).
13
K-theory of general linear groups over a finite field
This section describes Quillen’s paper [5]. We have also used Daniela Egas Santander’s master’s project [6]
as a guide.
Here is the setup: Let q = pd for some prime p. We will show that for j ≥ 0,
K2j−1 (Fq ) = Z/q 2j − 1
K2j (Fq ) = 0
Combining this with our previous calculation of K0 (F) = Z for any field F, this means we have a complete
description of the K-theory of finite fields.
In order to do this, we will define K-theory in terms of Quillen’s plus construction
Ki (R) := πi (BGL(R)+ ).
We will then define a space F ψ q , whose homotopy groups we can compute, and show that
h.e
F ψ q −−→ BGL(Fq )+ .
13.1
Topological K-theory
1. K 0 (X) = K(X) = [X, BU × Z].
2. K 1 (X) = [X, U ].
3. K̃(X) = [X, BU ] for X compact and connected.
[WHAT COMMENTS SHOULD I ADD?]
13.2
Adams operations
Proposition 13.1. Let B be a compact, Hausdorff space. Then for all l ≥ 0, there exists ring homomorphisms ψ k : K(B) → K(B) (called Adam’s operations) with the following properties.
1. If f : B → B 0 induces f ∗ : K(B 0 ) → K(B) then ψ k f ∗ = f ∗ ψ k : K(B 0 ) → K(B);
2. If L a line bundle, then ψ k (L) = Lk ;
3. ψ kl = ψ k ◦ ψ l
Proof. Omitted.
49
Example 13.2. K̃(S 2 ) ∼
= Z is generated by H − 1, with (H − 1)2 = 0, where H is the canonical line bundle
1
over CP and 1 is the trivial bundle over S 2 . Then ψ k : K̃(S 2 ) → K̃(S 2 ) is a map Z → Z. Then
ψ k (H − 1) = H k − 1 = (H − 1 + 1)k − 1 = k(H − 1),
using the fact that (H − 1)2 = 0. Thus, ψ k is multiplication by k.
More generally, we can use induction and the fact (didn’t prove this) that there is an isomorphism
K̃(S 2 ) ⊗ K̃(S 2n−2 ) → K̃(S 2n ), we have
ψ k (αβ) = ψ k (α).ψ k (β) = (k.α)(k n−1 .β) = k n (αβ).
Thus ψ k : K̃(S 2n ) → K̃(S 2n ) is multiplication by k n .
For a more detailed account of this example, see Hatcher’s Vector Bundles and K-theory book in progress.
Now consider ψ k : K̃(B) → K̃(B). We have K̃(B) = [B, BU ]. By the Yoneda Lemma, ψ q : BU → BU .
13.3
Fixed points and homotopy fixed points
We fixed define fixed points in terms of a pullback diagram. We do this so that when we define homotopy
fixed points, they will look analogous.
Let ∆ : X → X × X be given by (x 7→ (x, x)). And let φ : X → X be any self map.
Definition 13.3. The fixed points of φ is the pullback X φ of the following diagram.
Xφ
/X
X
/ X × X.
∆
(id,φ)
In particular, X φ = {(x, y) | ∆(x) = (y, φ(y))}, which agrees with our intuition that a fixed point is a
point such that φ(x) = x
˜ : X I → X × X be given by γ 7→ (γ(0), γ(1)). That is, ∆
˜ takes a path to its
Now let I = [0, 1] and let ∆
endpoints. Again, let ϕ : X → X be any self map.
Definition 13.4. The homotopy fixed points of ϕ is the pullback X hϕ of the following diagram.
X hϕ
/ XI
X
/ X × X.
˜
∆
(id,ϕ)
˜ : X I → X × X is a fibration, and the fibre is ΩX.
Note that ∆
Definition 13.5. We say that (X, x) has additive structure if it has maps
1. (addition) + : X × X → X, that is associative and commutative up to homotopy.
2. (additive inverse) : X → X, which is an additive inverse up to homotopy.
(1,)
+
The map d : X × X −−−→ X × X −
→ X is called subtraction
Lemma 13.6. If X has an additive structure, then X hϕ is the homotopy fibre of 1 − ϕ
50
Proof. We have the following commutative diagram.
X hϕ
/ XI
X
/ X ×X
λ
/ PX
η
(1,ϕ)
d
/X
The map η : P X → X is the map in the path space loop space fibration, that sends a path to its endpoint.
The map λ : X I → P X is given by
λ([t 7→ w(t)]) = [t 7→ d(w(t), w(0)]
which changes the starting point of the path to the origin (up to homotopy at least). Thus X hϕ is the
homotopy fibre of 1 − q = d(1, ϕ) since it is the pullback of
X hϕ
/ PX
X
/ X.
1−ϕ
We now list some properties of a space with additive structure, but without proof.
Lemma 13.7. If X has an additive structure, then so does X hϕ .
Lemma 13.8. Let X be a space with additive structure and ϕ : X → X be a self map. Let Y be a space such
that Y → ΩX is nullhomotopic. Let [Y, X]ϕ be the fixed points of [X, Y ] under composition with ϕ. Then
[Y, X hϕ ] = [Y, X]ϕ .
Recall that the Adams operations were maps ψ q : BU → BU . Let F ψ q be the homotopy fixed point of
ψ . That is, it is the pullback
/ BU
F ψq
q
BU
˜
∆
(id,ϕ)
/ BU × BU.
Theorem 13.9. For j ≥ 0, the homotopy groups of F ψ q are
π2j−1 (F ψ q ) = Z/(q 2j − 1)
π2j (F ψ q ) = 0
Proof. Observe that F ψ q is the homotopy fibre of 1 − ψ q . The long exact sequence in homotopy then gives
(1−ψ k )∗
. . . → πj (BU ) −−−−−→ πj (BU ) → πj−1 (F ψ q ) → πj−1 (BU ) → . . .
Z , if j is even
j
j
But πj (BU ) = [S , BU ] = K̃(S ) =
0 , if j is odd
51
From our example, we know that ψ q acts on K̃(S 2 ) by multiplication by q and on K̃(S j ) by multiplication
by q j . Thus the long exact sequence breaks off into short exact sequences of the form
×(q 2j −1)
0 → Z −−−−−−→ Z → π2j−1 (F ψ q ) → 0
×(q 2j )
0 → π2j (F ψ q ) → Z −−−−→ Z → 0
from which we get the result of the theorem.
Lemma 13.10. Let R(Fq G) be the representation ring. There is a map R(Fq G) → [BG, F ψ q ].
Proof. For M a representation of G, there is a map
EG ×G M → EG ×G {pt} = BG
ie. EG ×G M is a complex vector bundle over BG. Thus, given a complex representation of G, we get an
element of K(BG).
Consider the following commutative diagram.
/ K̃ 0 (BG)
R(CG)
ψq
ψq
/ K̃ 0 (BG).
R(CG)
Here, the left map ψ q : R(CG) → R(CG) is an Adams operations defined on representation rings, which
are analogous to the Adams operations we have been looking at.
q
k
Assuming this, then a map R(CG) → K̃ 0 (BG) = [BG, BU ] gives a map R(CG)ψ → [BG, BU ]ψ .
The Atiyah-Segal completion theorem states that
0 ' K 1 (BG) = [BG, U ] ' [BG, ΩBU ].
By Lemma 13.8, we have that
q
q
[BG, BU ]ψ ' [BG, BU hψ ].
q
Then the Brauer lift [a fact we’ve taken without explaining], gives R(Fq G) ' R(CG)ψ . Altogether we
have maps
q
q
R(Fq G) ' R(CG)ψ → [BG, BU ]ψ → [BG, F ψ q ].
If we consider modular representations of GLn+1 (Fq ), this gives us a map
BGLn+1 (Fq ) → F ψ q .
Composing with BGLn → BGLn+1 and taking a limit gives a map
θ : BGL(Fq ) → F ψ q
that is well defined up to homotopy.
Theorem 13.11. θ induces a homology isomorphism.
Proof. Omitted. (This is a big theorem)
Note that θ is not a homotopy equivalence. In particular, both spaces have different fundamental groups.
52
13.4
Quillen Plus construction
The idea of the Quillen Plus construction is to define a new space that changes the fundamental group, but
not homology.
Theorem 13.12. Let X be a CW-complex and H be a normal subgroup of π1 X. ie. [H, H] = H. Then
there is a space X + with an inclusion
i : X → X+
satisfying the following properties.
1. X + is obtained from X by attaching 2 and 3 cells.
2. i∗ : π1 (C) → π1 (X + ) is surjective with kernel H.
3. Let f : (X, x0 ) → (Z, z0 ) be a map of connected spaces with ker(f∗ ) = H. Then there is a map
f 0 : (X + , x0 ) → (Z, z0 ) such that the following diagram commutes.
XO +
f0
i
X
!
f
/Z
4. H( X) = Hn (X + ) for all n.
Proof. Omitted.
Theorem 13.13. BGL+ and F ψ q are homotopy equivalent.
Proof. Omitted.
13.5
More about the plus construction
This section aims to clear up the relationship between the plus construction and group completion. I think
at some point, this section would be better if it were amalgamated into the previous section.
The overall moral of this story is that their is a way to constuct the plus construction - and just like
with singular homology, we really don’t care. More important are the properties this space has, which is the
main focus in the following few pages.
Let X be a connected topological space. Let G ≤ π1 X be a normal, perfect2 subgroup. The plus
construction X + (with respect to G) is a space which is obtained from X by attaching 2-cells to kill G ≤ πX ,
and 3-cells so that the inclusion
H ,→ X +
is a homology isomorphism.
Note that we can do the first bit (attach 2-cells) since π1 X is normal. In particular,
H1 (X) = (π1 X)ab =
π1 X
[π1 X, π1 X]
Since we kill off G in X + , this means
H1 (X + ) = (π1 X + )ab = (π1 X/G)ab =
π1 X/G
.
[π1 X/G, π1 X/G]
Since G = [G : G] ≤ π1 X, π1 X, we see that
H1 (X) ∼
= H1 (X + ).
Perfection can be used to show that X ,→ X + is a homology isomorphism.
2G
= [G, G]
53
Proposition 13.14. If R is a ring and s ∈ R, then
s
s
s
R[s−1 ] ∼
→R−
→R−
→ . . .)
= lim(R −
→
Proof. The map from the direct system to R[s−1 ] is as in the diagram below.
/R
R
r7→r
/R
r7→ rs
" |
R[s−1 ]
/ ...
r7→ sr2
Now given any A and maps fn that make the following diagram commute,
/R
R
f0
f1

A
/R
/ ...
f2
we can define f : R[s−1 → A by f (r/sk ) = fk (r) which will ensure that R[s−1 ] satisfies the universal
property of direct limits.
F
Now let M be a topological monoid with π0 M = N. Our main example is M = k BGLk (R), from
which we get K-theory defined as Ki R = πi ΩBM .
Remark: π0 (ΩBM ) ∼
= Z.
Question: Why is the group completion of π0 M = π0 (ΩBM )?
Denote by Ωk BM the component of ΩBM corresponding to k ∈ Z. For example, Ω0 BM are the null
homotopic loops in BM . There exists a natural map
i : M → ΩBM
given by i(m) = loop in BM indexed by m3 . This induces a map
I∗ : π0 M → π0 ΩBM = π1 BM.
Observe that the left hand side is a monoid, while the right hand side is a group. We study H0 (M ) = Z[π0 ].
We have
Z[π0 ΩBM ] = H0 (ΩBM ) = H0 (M )[π0−1 ]
= Z[π0 M ][π0−1 ]
= Z[Groth(π0 M )]
So π0 ΩBM = Groth(π0 M ). Thus for M as above, π0 ΩBM = Z.
F
Now we have H∗ (ΩBM ) = H∗ (M )[π0−1 ]. Pick s ∈ M1 , where M = k∈N Mk so that [s] is a generator of
π0 M . Then
H∗ (ΩBM ) = H∗ (M )[π0−1 ]
= H∗ (M )[s−1 ]
s
s
= lim(H∗ M −
→ H∗ M −
→ . . .)
→
3 recall
that the 1-skeleton of BM is a wedge of loops indexed by M
54
We also have
s
s
s
H∗ (Ω0 BM ) ∼
→ H∗ M2 −
→ H∗ M 3 −
→ . . .)
= lim(H∗ M1 −
→
∼
= H∗ M∞
where M∞ = lim→ Mn (the telescope).
For our example, we have Ω0 B(tk BGLk (R)) ∼
=H∗ BGL∞ (R).
Lemma 13.15. For all spaces S, every component of ΩX is homotopy equivalent.
Proof. We show that π0 (ΩX) ' πi X. Pick g̃ ∈ Ωg X. consider the map
Ω 0 X → Ωg X
given by f 7→ f∗ g̃.
It has a homotopy inverse given by h 7→ h∗ [g̃]−1
Theorem 13.16. If π0 M = N and [π1 M∞ , π1 M∞ ] is perfect, then
+
ΩBM ' Z × M∞
.
Proof. We have the following diagram.
M1
s
i
Ω1 BM
/ M2
i
∗s
/ M3
s
s
/ ...
i
/ Ω2 BM
∗s
/ Ω3 BM
∗s
/
This gives a map i : M → ΩBM . Thus there is a map
I : lim Mn → lim Ωn BM 'lemma Ω0 BM.
By the previous calculation, I is a homology isomorphism. By the universal property of the plus construction, I factors as
+
/ M∞
M∞
I
{
Ω0 BM
I˜
So we have
+
π1 M∞
=
π1 M∞
[π1 M∞ , π1 M∞ ]
= (π1 M∞ )ab
= H1 M∞
∼
= H1 Ω0 BM
= π1 (Ω0 BM )ab
= π1 Ω0 BM
since π1 ΩBM = π1 ΩBM = π2 BM is abelian.
So I˜ is an isomorphism in H∗ and π1 , so is a weak equivalence by the Whitehead theorem.
Example: Ki (R) = πi [ΩB(tk BGLk (R))] = πi [(Z × BGL∞ (R)+ )]
55
14
Brauer Groups
Let k be a field, and C be the category of finite dimensional central simple algebras (CSA’s) over k, where
the morphisms are isomorphisms. This is a symmetric monoidal category since the tensor product of a CSA
is a CSA. Previously, we defined the Brauer group of k as
Br(k) := K0 (C)
This section will go into more detail on Brauer groups.
Definition 14.1. A central simple algebra over k is a simple k-algebra A such that the centre, Z(A) = k.
A simple algebra is an algebra with no non-trivial two sided ideals. Examples of simple algebras are
fields and division algebras.
Example 14.2 (CSA’s). Examples of central simple algebras are
1. H = R[i, j, k]/(i2 = j 2 = k 2 = −1, ij = −ji = k) is a central simple algebra over R;
2. Mn (k) is a central simple algebra over k; and
3. If D is a division algebra over k, then D/Z(D) is a central simple algebra over k.
Theorem 14.3 (Artin-Wedderburn). The following are equivalent:
1. A is a finite dimensional central simple algebra over k
2. There is a division algebra D over k such that A ∼
= Mn (D) ∼
= D ⊗k Mn (k)
Sketch of proof. We use Schur’s Lemma which states
Lemma 14.4 (Schur’s Lemma). If M is a simple A-module, then EndA M is a division ring.
4
We also use a result due to Riefel.
Lemma 14.5. If A is a finite dimensional algebra over k, and L < A is a non-zero left ideal, then the
following holds: If D = EndA L, then A ∼
= EndD L.
We now proceed to prove the Artin-Wedderburn theorem.
If A is finite dimensional over k, then A is Artinian. This means there exists a minimal non-trivial left
ideal L < A. Thus L is simple. By Schur’s lemma, D := EndA L is a division ring. By Rieffel, A ∼
= EndD L,
and the latter is the matrix ring Mn (D).
14.1
Some computations
Proposition 14.6. For k an algebraically closed field, Br(k) = 0.
Proof. By Artin-Wedderburn, if A is a central simple algebra over k, then A ∼
= Mn (D), for some (finite
dimensional) division algebra D. We will show that D = k, so that A ∼
= Mn (k), which means the class of A
is zero in Br(k).
Let D be a finite dimensional division algebra over k. Pick d ∈ D \ {0}. Then for some n, the set
{1, d, d2 , . . . dn } is becomes linearly dependent. Thus there exists a polynomial f ∈ k[X] \ 0 such that
f (d) = 0. Thus, there are a, b ∈ k such that ad + b = 0 and so d ∈ k. Thus D ⊂ k. To see that k ⊂ D, note
that 1 ∈ D and so k = k-span(1) ⊂ D.
Proposition 14.7. If A is a central simple algebra over k and B is a simple algebra over k, then
1. The center Z(A ⊗k B) = Z(B); and
4A
quick sketch: If f : M → M then ker(f ) is a submodule of M , which means that ker(f ) = 0
56
2. A ⊗k B is simple; and
Proof.
1. Let z ∈ Z(A⊗k B). It is enough to show that z ∈ 1⊗k B, since this will mean the map b 7→ 1⊗k b
is a bijection between the two centres.
Let {bi } be a basis for B as a k-module. Then {1 ⊗k bi } is a basis for A ⊗k B as an A module. Thus
we can write z uniquely as
z = a1 ⊗ b1 + · · · + an ⊗ bn
Now,
aa1 ⊗k b1 + . . . + aan ⊗k bn = (a ⊗k 1)z
= z(a ⊗k 1) = a1 a ⊗k b1 + . . . an a ⊗k bn
So, if a ∈ A, then aai = ai a. Thus ai ∈ Z(A) = k. Thus
z = 1 ⊗ a1 b1 + . . . 1 ⊗ an bn = 1 ⊗k b
2. Let I be a non-trivial 2-sided ideal of A ⊗k B. We want to show that I = A ⊗k B. Since clearly
I ⊂ A ⊗k B, we need to show that A ⊗k B ⊂ I.
We want to find z ∈ I such that
z = 1 ⊗k b ∈ I.
Then (1 ⊗k B)(1 ⊗k b)(1 ⊗k B) = 1 ⊗k BbB = 1 ⊗k B ⊂ I. So (A ⊗ 1)(1 ⊗ B) ⊂ I and we would be
done.
To find the element z, pick a basis {bi } as before. Pick 0 6= ζ ∈ I, so that
ζ = a1 ⊗k b1 + . . . + an ⊗k bn
with n minimal. Since ζ 6= 0 we know that a1 6= 0. We have Aa1 A = A. Thus there are elements
a0 , a00 ∈ A so that
a0 a1 a00 = 1.
Then, let
z = (a0 ⊗k 1)ζ(a00 ⊗k 1) = 1 ⊗k b1 + a0 a2 a00 ⊗k b2 + . . . + a0 an a00 ⊗k bn
000
= 1 ⊗k b1 + a000
2 ⊗k b2 + . . . + an ⊗k bn
is in I.
Now for a ∈ A, consider
000
000
000
az − za = (aa000
2 − a2 a) ⊗k b2 + . . . + (aan − an a) ⊗k bn = 0
000
by minimality of n. Thus aa000
i = ai a, so ai ∈ Z(A) = k. Thus
z =1⊗b
for some b ∈ B, which is what we want.
Corollary 14.8.
over k.
1. If A and B are central simple algebras over k, then A⊗k B is a central simple algebra
2. If K is a field extension of k, then K ⊗k A is a central simple algebra over K. Moreover, we get a map
CSA0 s/k → CSA/K by A 7→ K ⊗k A.
57
Definition 14.9. Let (C, ⊗) be a symmetric monoidal category. A full subcategory T of C is cofinal if it
satisfies the following conditions.
1. (T, ⊗) is a symmetric monoidal category; and
2. If A ∈ Ob(C), then there exists B ∈ C such that A ⊗ B ∈ T .
Example 14.10. For R a ring, we defined K0 (R) := Groth(P (R)), where P (R) is the symmetric monoidal
category of finite dimensional projective modules over R, with isomorphisms. Recall that P is projective if
and only if there exists a module Q such that P ⊕ Q is a free module. Thus the free modules are cofinal in
the category P (R).
Proposition 14.11. The matrix algebras {Mn (k)|n ∈ Z≥0 } are cofinal in the category of central simple
algebras over k, with isomorphisms.
Proof. We check the two conditions.
1. Mn (k) ⊗ Mm (k) ∼
= (Mn (Mn (k)) ∼
= Mnm (k). Essentially, it is the idea that an matrix of matrices is
just a big matrix if we forget the parentheses inside our big matrix.
2. If R is a ring, define the ring Rop to be the set R, with addition a+Rop b = a+R b, and a×Rop b = b×R a.
Claim: Let R and S be rings, with inclusions k ,→ R and k ,→ S turning them into k-algebras.
Whenever we have the following commutative diagram of k-algebras such that the images of R and
the images of S commute, then the dotted arrow exists, and is unique.
;R
-
kq
R ⊗k S
#
∃!
#
/
;A
S
In our case, Let A be a central simple algebra over k. Let
R = {La ∈ Endk A | La (x) = ax}
and let
S = {Tb ∈ Endk A | Tb (x) = xb}.
Observe that R ∼
= A and S ∼
=A
op
and both embed into Endk (A). We have
La (Tb (x)) = axb = Tb (La (x)).
Thus La Tb = Tb La , so the images of R and S commute so we have
s
< R
-
kq
R ⊗k S
∃!
&
/ Endk (A)
8
" +
S
So there is a unique map
A ⊗k Aop ∼
= R ⊗k S → Endk (A) ∼
= Mn (k)
58
In particular, we have done the necessary work to show that
Br(k) = {finite dim CSA’s over k /isomorphisms}/(A ⊗ Mn (k) = A)
is a group.
We now list some Brauer groups that have been calculated.
Example 14.12 (Wedderburn). Every finite division algebra is a field. Hence Br(Fq ) = 0.
[Is this because of Artin-Wedderburn?]
Theorem 14.13 (Frobenius). Two calculations.
1. Br(R) = {R, H} ∼
= Z/2Z; and
2. H ⊗R H ∼
= Mn (R).
Theorem 14.14. If K is a local field, then Br(k) = Q/Z.
An example of a local field is Qp .
14.2
Some Galois theory
We saw earlier that if A is a central simple algebra over k, and K is a field extension of k, then A ⊗k K is a
central simple algebra over k. This induces a map from
Br(k) → Br(K)
Definition 14.15. The relative Brauer group is
Br(K/k) := ker(Br(k) → Br(K))
Definition 14.16. K is a splitting field of A if A ⊗k K ∼
= Mn (K). We also say that K splits A.
Theorem 14.17 (Noether). Every central simple algebra over k has a finite Galois splitting field.
S
Corollary 14.18. Br(k) = K⊂K s Br(K/k), where k s is the separable closure of k.
There is a result that relates the Brauer group to Galois cohomology. It says the following.
Br(K) ∼
= H 1 (Gks /k , P GL∞ (k s ))
∼
=Hilbert90 H 2 (Gks /k , (k s )× )
As a result of this, we have the following. If G is a finite group and G acts on A, which is abelian, then
H n (G, A) are torsion, so Br(k) is torsion. If K is a local field,
Br(K) = H ( Gks /k , (k s )× )
∼
= H 2 (Gkum /k , (k um )× ),
where um is the largest unramified separable extension.
59
15
15.1
Waldhausen K-theory
Exact categories
Definition 15.1 (Exact category). An exact category is a pair (C, E), where C is an additive category and
E is a family of sequences in C of the form
j
i
0→B→
− C−
→ D → 0,
(†)
satisfying the following condition: there is an embedding of C as a full subcategory of an abelian category A
so that
1. E is the class of all sequences (†) in C which are exact in A;
2. C is closed under extensions in A in the sense that if (†) is an exact sequence in A, with B, D ∈ C,
then C is isomorphic to an object in C.
Definition 15.2. C is closed under kernels of surjections in A, if it satisfies the following property: If
f : B → C in C is a surjection in A, then ker(f ) ∈ C.
The condition of being closed under kernels of surjections is sometimes taken as part of the definition of
exact category.
Definition 15.3 (K0 of an exact category). Let C be a small exact category. K0 (C) is the abelian group
with one generator [C] for each object C ∈ C, and relations [C] = [B] + [D] for every short exact sequence
0→B→C→D→0
in C.
Example 15.4. The category P(R) of finitely generated projective R-modules is exact since every exact
sequence of projective modules splits. We have K0 P(R) = K0 (R).
Here are some simple properties of K0 (A).
∼
=
1. [0] = 0 since 0 → A −
→ A → 0 → 0 is a short exact sequence.
2. If A ∼
= A0 then [A] = [A0 ].
3. [A0 ⊕ A00 ] = [A0 ] + [A00 ]
Any additive category is a symmetric monoidal category under ⊕. In general, by the above K0 (C) is a
quotient of the group K0⊕ (C) which we defined in previous sections, but they need not be equal.
Theorem 15.5 (Localization theorem). Let A be a small abelian category, and B a Serre subcategory of A.
Then the following sequence is exact:
loc
K0 (B) → K0 (A) −−→ L0 (A/B) → 0
Proof. Omitted. We didn’t talk about this in the lecture.
Theorem 15.6 (Resolution theorem). Let P ⊂ C ⊂ A be an inclusion of additive categories with A abelian.
Assume the following.
1. Every object C of C has finite P dimension; and
2. C is closed under kernels of surjections in A.
Then the inclusion P ⊂ C induces and isomorphism K0 (P) ∼
= K0 (C)
60
Proof. We claim the map ψ : K0 (P) → K0 (C) given by [P ] 7→ [P ] is an isomorphism.
To show surjectivity, let [C] ∈ K0 (C). Let P• → C be a finite P-resolution. The exact sequence
0 → Pn → . . . → P0 → C → 0
P
P
has χ = 0, where χ[C• ] = (−1)i [Ci ] (the sum is in K0 (C)). In particular, [C] = (−1)i [p] = χ(P• ), so
K0 (P ) surjects onto K0 (C).
The map is injective, since the map χ : K0 (C) → K0 (P) defined by χ(C) = χ(P• ) is a left inverse for
ψ.
Example 15.7. Let R be a ring. Let H(R) be the category of all R-modules M with a finite projective
resolution of finitely generated projective modules, and let Hn (R) be the subcategory in which the resolutions
have length ≤ n. It can be shown that these Hn (R) and H(R) satisfy the conditions of the resolution theorem.
Thus we have
K0 (R) ∼
= K0 H(R) ∼
= K0 Hn (R)
for n ≥ 1. In particular, H1 (R) = P(R). This is one reason why it is ok to define the K-theory of a ring
just in terms of its finitely generated projective modules.
15.2
Waldhausen Categories
Inuitively, a Waldhausen category is a category where the notions of cofibrations and weak equivalences
make sense. More precisely, we have the following definitions, from Weibel’s K-book.
Definition 15.8. Let C be a category equipped with a subcategory co = co(C) of morphisms in a category C,
called “cofibrations” (indicated by ). The pair (C, co) is called a category with cofibrations if the following
axioms are satisfied:
W0 Every isomorphism in C is a cofibration;
W1 There is a distinguished zero object ‘00 in C, and the unique map 0 A in C is a cofibration for every
A in C;
W2 If A B is a cofibration, and A → C is any morphism in C, then the pushout B ∪A C of these two
maps exists in C, and moreover the map C B ∪A C is a cofibration.
Definition 15.9. A Waldhausen category is a category C with cofibrations, together with a family w(C)
∼
of morphisms in C called “weak equivalences” (abbreviated ‘w.e’, −
→). Every isomorphism in C is to be a
weak equivalence, and weak equivalences are to be closed under composition. Thus w(C) can be regarded as
a subcategory of C. In addition, the following “Glueing axiom” must be satisfied:
W3 Glueing for weak equivalences. For every commutative diagram of the form
Co
A /
∼
C0 o
∼
A0 /
/B
∼
/ B0
the induced map
B ∪A C → B 0 ∪A0 C 0
is also a weak equivalence.
A Waldhausen category is a triple (C, co, w), but we often just write C. Usually, for a given category C,
the cofibrations are given and there is not much choice in what they are. However, there is often a choice
on the type of weak equivalences w are. Thus, we sometimes write wC instead of C to emphasise the w we
are using.
61
Definition 15.10. Let C be a Waldhausen category. K0 (C) is the abelian group with one generator [C] for
each object C ∈ C, with relations
∼
1. If C −
→ C, then [C] = [C 0 ]; and
2. If B C C/B is a cofibration sequence, then [C] = [B] + [C/B].
Example 15.11. An exact category is a Waldhausen category, with cofibrations being the admissible monomorphisms and weak equivalences being isomorphisms. By definition, the Waldhausen definition of K0 agrees
with the exact category definition of K0 in this case.
Example 15.12. This is an example of a Waldhausen category that is not additive. Let R(∗) be the
category whose objects are CW-complexes with countably many cells and morphisms are cellular maps. This
is a Waldhausen categorie. The cofibrations are cellular inclusions, and weak equivalences are weak homotopy
equivalences.
Instead of direct sum, since this is not an additive category, the coproduct is the wedge product A ∨ B,
which is the disjoint union of A and B with basepoints identified. The Eilenberg Swindle shows that K0 R = 0.
This is because C ∞ = C ∨ C ∞ , hence the cofibration sequence
C C ∨ C∞ C
shows that [C] = 0.
Example 15.13. Let Rf (∗) ⊂ R(∗) be the full subcategory of finite based CW complexes. This is a Waldhausen category with the same cofibrations and weak equivalences. In this case, we have
K0 Rf ∼
= Z.
Proof. We have
S n−1 Dn S n ,
so
[S n−1 ] + [S n ] = [Dn ] = 0.
Hence, [S n ] = (−1)n [S 0 ]. If C is obtained from B by attaching an n-cell, then C/B ∼
= S n and [C] = [B]+[S n ].
0
Hence K0 Rf is generated by [S ].
Moreover, there is a surjective homomorphism χ : K0 Rf → Z, given by
X
χ(C) =
(−1)i dimH̃ i (X, Q).
Hence [S 0 ] 7→ 1 is an isomorphism.
Theorem 15.14 (Cofinality). Let B be a Waldhausen subcategory of C closed under extensions. If B is
cofinal in C, then K0 (B) is a subgroup of K0 (C)5 .
Proof. The proof is the same`as for exact
` categories. If we consider B and C as symmetric monoidal categories
`
with product , we have K0 (B) ⊂ K0 (C). Given a cofibration sequence
C0 C1 C2
` 0
`
` `
in C, chooes
and
in C so that B0 = C0 C0 and B2 = C2 C20 are in B. Setting B1 = C1 C00 C20 ,
we have a cofibration sequence
B0 B1 B2
C00
C20
`
in C. Since B is closed under extensions in C, B1 ∈ B. Therefore, in K0 (C), we have
[C1 ] − [C0 ] − [C2 ] = [B1 ] − [B0 ] − [B2 ].
`
Thus the kernel of K0
injective.
5 Here,
`
→ K0 (C) is the kernel of K0 (B) → K0 (B), which means K0 (B) → K0 (C) is
cofinal means: for all C ∈ C, there is a C 0 ∈ C so that C
62
`
C 0 ∈ B.
`
`
Note: The proof shows that K0 (C)/K0 (B) ∼
= K0 (B)/K0 (C). Moreover, every element of K0 (C) has the
form [C] − [B] for some B ∈ B and C ∈ C.
Theorem 15.15 (Approximation). Let F : A → B be an exact functor between two Waldhausen categories.
Suppose also that F satisfies the following conditions:
1. A morphism f in A is a weak equivalence if and only if F (f ) is a weak equivalence in B.
2. Given any map b : F (A) → B in B, there is a cofibration a : A A0 in A and a weak equivalence
∼
b0 : F (A0 ) −
→ B in B so that b = b0 ◦ F (a).
3. If b is a weak equivalence, we may choose a to be a weak equivalence in A.
Then F induces an isomorphism K0 (A) ∼
= K0 (B).
Proof.
Example 15.16. Let Rhf (∗) be the Waldhausen subcategory of R(∗) of all based CW-complexes homotopic
to a finite CW-complex. The approximation theorem applies to the inclusion Rf (∗) into Rhf (∗)6 . Hence
K0 Rhf (∗) ∼
= K0 Rf (∗) ∼
=Z
16
16.1
Higher K-theory
S• -construction (wS• )
Let C be a category with cofibrations. We will define a simplicial category S• C.
S0 C is the zero category. S1 C is the category C but whose objects are though of as cofibrations 0 A.
More generally Sn C is the category whose objects are sequences of n cofibrations in C:
A• : 0 = A0 A1 . . . An
together with a choice of sub quotients Aij = Aj /Ai that are compatible in the sense that some diagram
commutes.
A morphism A• → B• in Sn C is a natural transformation of sequences.
0
A0 /
/ A1 /
/ ... /
/ An
0
B0 /
/ B1 /
/ ... /
/ Bn
Note: If C is a Waldhausen category, then so too is Sn C
B
A weak equivalence in Sn X is a map A• → B• such that each Ai −
→i are weak equivalences in C.
A map A• → B• is a cofibration if for every 0 ≤ i < j < k ≤ n, the map of cofibrations sequences
Aij /
/ Aik
/ / Ajk
Bij /
/ Bik
/ / Bjk
is a cofibration in S2 C.
6 By
the Whitehead theorem
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16.2
How to think of S2 C
The objects of S2 C can be thought of as cofibration sequences A1 A2 A12 in C.
A cofibration is a commutative diagram
A1 /
u1
B1 /
/ / Ajk
/ A2
u2
/ B2
u3
/ / Bjk
such that the maps A1 → B1 , A12 → B12 and B1 ∪A1 A2 → B2 are cofibrations. These conditions ensure
that the Waldhausen category axioms are satisfied, turning the Sn C’s into Waldhausen categories.
Proposition 16.1. S• C is a simplicial (Waldhausen) category.
Proof. The simplicial boundary maps are ∂i : Sn C → Sn−1 C by forgetting the ith term. ∂0 is slightly
different, in that we quotient the sequence by the first term so that our sequence starts at 0.
The simplicial degeneracy maps si : Sn C → Sn+1 C are defined by repeating the ith term. We can then
check that the simplicial axioms are satisfied.
Thus the Sn C fit together to form a simplicial Waldhausen category S• C. The subcategories wSn C where
all morphisms are weak equivalences fit together to form a simplicial category wS• C.
16.2.1
Geometric realization
We can do the classifying space construction to wSn C for each n to form a simplicial space BwS• C. We
write |wS• C| for the geometric realisation of BwS• C. Since S0 C = 0 is trivial, the space |wS• C is connected.
Note that secretly, we have a bi-simplicial complex - in one simplicial direction, the maps are the cofibrations, while in the other simplicial direction the maps are the homotopy equivalences of sequences. |wS• C|
is just the geometric realisation of this.
Definition 16.2. Let C be a small Waldhausen category. The algebraic K-theory space of C is
K(C) := Ω|wS• C|
The K-groups of C are defined to be the homotopy groups
Ki (C) := πi K(C) = πi+1 |wS• C|
Proposition 16.3. If C is a Waldhausen category then π1 |wS• C| ∼
= K0 (C), where K0 (C) is K0 of a Waldhausen category we defined last time.
Proof. If X• is any simplicial space and |X0 | is a point, then |X• | is connected and π1 |X• | is the free group
on π0 (X1 ) modulo the relation
∂1 (x) = ∂2 (x)∂0 (x)
for every x ∈ π0 (X2 ). In our case, we have the following:
1. X• = BwS• C;
2. π0 |BwS1 C| = { objects in C }/ ∼;
3. π0 |BwS2 C| = { cofibration sequences in C}/ ∼;
4. The maps ∂i : S2 C → S1 C map the cofibration sequence A B B/A to B/A, B and A respectively.
We can then match things up.
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17
K-theory spectrum
We can iterate the S• construction to form the iterates
wS•n C = wS• . . . S• C.
One can show moreover that there are natural homotopy equivalences
|wS• C| ' Ω|wS• S• C.
So we have a chain of homotopy equivalences
|wC| → Ω|wS• C| → Ω2 |wS• S• C| → . . .
Note that the first map is not a homotopy equivalence [check this]. As suspension is adjoint to the loops
functor, we have maps
Σ|wS•n C| → |wS•n+1 C|.
If we let En = |wsn• , we can show that E∗ is a Ω-spectrum. This spectrum is usually denoted by KC, and is
called the K-theory spectrum of C. Moreover, the coproduct on C induces a product on KC, so KC is a ring
spectrum.
17.1
examples
Example 17.1 (Exact categories). Let A be an exact category. Recall that it is a Waldhausen category where
the cofibrations are the admissible monica and the weak equivalences are isomorphisms. If i(C) denotes the
category of isomorphisms of C, then
K(A) = Ω|iS• A|.
Waldhausen proved that there a homotopy equivalence between |iS• A| and BQA, where BQA is the K-theory
space of an exact category gotten via the Q-construction.
Example 17.2 (Algebraic K theory of a point). Let Rf (∗) be the Waldhausen category of based CW
complexes. We write A(∗) for K(Rf (∗)) = Ω|hS• Rf |. From last time we have
A0 (∗) = K0 Rf (∗) = Z.
Example 17.3 (K-theory of spaces). Let X be a CW complex. Let R(X) be the category whose objects
are CW complexes Y obtained from X by attaching cells, and having X as a retract. The morphisms are
the cellular maps. R(X) is a Waldhausen category; the cofibrations are the cellular inclusions and the weak
equivalences are weak homotopy equivalences.
Let Rf (X) be the subcategory of R(X) where Y is obtained from X by attaching only finitely many cells.
We write
A(X) = K(Rf (X)) = Ω|hS• Rf (X)|.
A(X) is called the algebraic K-theory space of X.
Exercise: Show that A0 (X) = Z.
17.2
Waldhausen’s Splitting Theorem
One of the motivating reasons for studying Waldhausen categories was to prove a result known as Waldhausen’s splitting theorem. We state it here out of interest, but do not attempt to prove it. A proof can be
found in [REFERENCE]. Waldhausen’s splitting theorem roughly says the following.
Theorem 17.4. For any finite CW complex X,
A(X) ' W h(X) × Q(X+ ),
Where W h(X) is the Whitehead space of X (which can also be thought of as the moduli space of h-cobordisms
of X) and Q(X+ ) = Ω∞ Σ∞ (X+ ) is the stable homotopy space of X.
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References
[1] Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002.
[2] J May. A concise course in algebraic topology. University of Chicago Press, Chicago, 1999.
[3] J. P. May. The geometry of iterated loop spaces. Springer-Verlag, Berlin, 1972. Lectures Notes in
Mathematics, Vol. 271.
[4] D. McDuff and G. Segal. Homology fibrations and the “group-completion” theorem. Invent. Math.,
31(3):279–284, 1975/76.
[5] D. Quillen. On the cohomology and k-theory of the general linear groups over a finite field. Annals of
Mathematics, 2nd Ser, (96):552–586, 1972.
[6] Daniela Egas Santander. Algebraic k-theory of a finite field. 2010.
[7] Edwin Spanier. Algebraic topology. Springer-Verlag, New York, 1989.
[8] Nathalie Wahl. Homological stability for mapping class groups of surfaces. to appear in the Handbook of
Moduli, last version Dec 2011.
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