Download 23 Introduction to homotopy theory

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Introduction to homotopy theory
The objects of homotopy theory are the categories of spaces S and spectra Sp. In this
overview I want to explain certain features and constructions with these categories which
will become relevant in the seminar. One needs 1-categories or model categories in order
to capture theses objects on a technical level. In this overview I want to explain how
one can work with spaces and spectra properly by simply obeying some rules of correct
language.
As a preparation I recall the notion of limits and colimits. For categories C and I (the
latter assumed to be small) let C I be the functor category. There is a constant diagram
functor
c : C ! CI .
The limit (colimit) is a (in general partially defined) right (left) adjoint of c:
c : C ⌧ C I : limI ,
colimI : C I ⌧ C : c .
We will illuminate this abstract definition by giving examples.
Example 23.1. A typical example of a limit is a pull-back or fibre product:
X ⇥Z Y
/
✏
X .
✏
/Z
Y
A typical example of a colimit is a push-out
Z
X
/
✏
Y
/
.
✏
X tZ Y
The quotient by an action a of a group G on X is the colimit of the diagram
a
G⇥X
,2
pr2
X
/
X/G .
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We start with a discussion of spaces S. One can characterize S as the presentable 1category (this essentially means that it admits all limits and colimits) generated by a
point. There are many models for S.
Example 23.2. One can start from topological spaces Top. A morphism f in Top is
called a weak equivalence if ⇡0 (f ) is a bijection and ⇡n (f ) : ⇡n (X, x) ! ⇡n (Y, y) are
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isomorphisms for all n 1 and x 2 X. If we invert weak equivalences (technically this
happens inside 1-categories), then we get a model for the category of spaces
1
Top ! Top[W
]'S .
2
op
Example 23.3. One can define a model for S starting from simplicial sets sSet := Set .
Recall that op is the category whose objects are the posets [n] = {0, . . . , n} and whose
op
morphisms are order preserving maps. For a category C we call C
the category of
simplicial objects in C.
In order to explain the notion of a weak equivalence in sSet we need the geometric
realization. The standard simplices n ⇢ Rn+1 provide a functor
! Top ,
:
[n] 7!
n
.
Considering sets as topological spaces, a simplicial set X 2 sSet provides a diagram of
topological spaces
F
F
m
:[m]![n]
⇥ X[n]
id⇥
✏
[m]2
m
⇤ ⇥id
⇤
/
F
⇥ X[m]
n
[n]2
/
⇥ X[n] ,
(4)
✏
kXk
whose colimit is the called the geometric realization of X. A more condensed, but equivalent, definition uses the notion of coend
Z op
kXk :=
⇥X .
By definition, a morphism f in sSet is a weak equivalence, if kf k is one in Top. We get
the desired model of spaces
sSet ! sSet[W
1
]'S .
Note that by construction k k descends to a functor k
which happens to be an equivalence.
k : sSet[W
1
] ! Top[W
1
]
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In the following I list some of the basic properties of the category of spaces S. The model
independent notion of isomorphism is equivalence.
1. Topological spaces or simplical sets represent spaces.
2. There exists a one-point space ⇤, i.e. a final object.
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3. A space X has a functorially defined set of connected components ⇡0 (X).
4. For each map x : ⇤ ! X we have functorially associated homotopy groups ⇡n (X, x),
n 1.
5. A morphism f : X ! Y between spaces is an equivalence if and only if ⇡n (f ) is an
isomorphism for all n 0 and base points x 2 X (for n 1).
6. For two spaces X, Y 2 S we have a functorially associated mapping space Map(X, Y ) 2
S. Two morphisms f0 , f1 : X ! Y are equivalent if they represent the same point
in ⇡0 (Map(X, Y )).
7. In spaces we can form limits and colimits.
Our language and pictures are adapted to Top.
Example 23.4. The fibre f
be written as a limit
1
(y) of a morphism f : X ! Y in Top at a point y 2 Y can
Fiby (f )
/
⇤ .
y
✏
f
X
/
✏
Y
In general, limits in Top do not preserve weak equivalences. In general, the fibre defined
as above for a map of topological spaces does not represent the fibre of f considered as a
morphism in S. For example, we have a weak equivalence of diagrams
'
⇤
!
⇤ ,
1
⇤
0
/
✏
⇤
[0, 1]
/
✏
⇤
but the map of limits is ; ! ⇤, surely not a weak equivalence.
One needs a modification of the construction of the fibre in order to obtain a model inside
Top for the fibre taken in S. In this situation one speaks of homotopy fibres. A model
for the homotopy fibre is the limit of
Fiby (f )h
/
⇤ .
y
✏
X ⇥f,Y,ev0 Y I
ev1
/
✏
Y
In general, if the morphism X ! Y in Top is a so-called fibration, then the fibre taken
in Top represents the fibre in S. Note that the map ev1 is the replacement of f by a
fibration.
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For a map f : X ! Y of spaces we can control the homotopy groups of the fibre by the
long exact sequence
⇡n (Fiby (f ), ⇤) ! ⇡n (X, x) ! ⇡n (Y, y) ! ⇡n 1 (Fiby (f ), ⇤) .
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Example 23.5. The cofibre of a map f : X ! Y is defined as the colimit of
X
/
⇤
.
f
✏
Y
/
✏
Cofib(f )
A good model for the homotopy cofibre is given by the cone of f
X
/
⇤
.
(0,idX )
✏
[0, 1] ⇥ X t{1}⇥X,f Y
✏
/ Cofib(f )h
Here the left vertical map is the replacement of f by a cofibration. It is difficult to control
the homotopy groups of the homotopy cofibre. The appropriate invariant is homology. 2
Example 23.6. The functor
colim
op
:C
op
!C ,
X 7! |X|
is called realization. If applied to simplicial spaces one again observes that it does not
preserve weak equivalences and therefore does not model the realization in S.
A weak equivalence preserving model for the realization is the geometric realization given
by the formula (4) and will be denoted by |X|h .
The homotopy groups of |X|h can be approached via the Bousfield-Kan spectral sequence
whose first page is
1
Es,t
:= ⇡s (X[t]) .
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Example 23.7. The nerve N (C) of a category is the simplicial set given by
N (C)[n] := Cat([n], C) .
We have a functor N : Cat ! sSet. A categorical equivalence induces a weak equivalence
between nerves. This provides a natural factorization
N : Cat[W
1
] ! sSet[W
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1
]'S .
One can define the notion of a category is a diagrammatic way. Interpreting these diagrams in topological spaces we obtain the notion of a topological category with space of
objects Ob(C) and space of morphisms Mor(C). In this case the nerve naturally refines
to a simplicial space,
N (C)[0] := Ob(C) ,
N (C)[2] := Mor(C) ⇥s,Ob(C),r Mor(C) , . . . .
N (C)[1] := Mor(C) ,
From now one we consider N (C) as a simplicial space, i.e. as an object in S
classifying space of a topological category is the space defined by
op
. The
BC := |N (C)| 2 S .
A functor between topological categories C ! D is called a weak equivalence if N (C) !
N (D) is an equivalence. It induces an equivalence of classifying spaces BC ! BD.
In the seminar we will study the topological bordism category of n 1-dimensional manifolds with a tangental ✓-structure C✓ . The main goal is to identify the classifying space
BC✓ .
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Example 23.8. Let G be a topological group. It gives rise to a topological category G with object ⇤ and morphism space G. The classifying space of this category
is the usual classifying space BG of G. A homomorphism between topological groups
which is a weak equivalence induces a weak equivalence between topologically enriched
categories and therefore a weak equivalence of classifying spaces. For example, the
usual inclusion O(n) ! GL(n, R) is a weak equivalence, hence we get an equivalence
BO(n) ' BGL(n, R).
Let G act on a space X. Taking the quotient X/G in Top does not preserve weak
equivalences. A good model for the homotopy quotient is
X/h G := EG ⇥G X ,
where EG is the geometric realization of the simplicial G-space [n] 7! G[n] with the usual
simplicial structure. So the topological space EG ⇥G X is a good model for the quotient
X/G taken in S.
The algebraic tool to calculate the homotopy groups of X/G is again a spectral sequence
derived from the Bousfield-Kan spectral sequence with first term
1 ⇠
Es,t
· · ⇥ G} ⇥X) .
= ⇡s (G
| ⇥ ·{z
t⇥
In topological spaces we can consider:
1. a point in a space
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2
2. a homeomorphism
3. a locally trivial fibre bundle
4. a (model of a) homotopy quotient
In spaces we can talk about:
1. a connected component
2. an equivalence
3. some space is equivalent to the fibre of a morphism
4. a quotient
Example 23.9. There are similar notions of pointed spaces S⇤ related to Top⇤ and sSet⇤ .
The suspension ⌃X of an object X in a pointed category is defined as a colimit
X
✏
⇤
/
/
⇤
.
✏
⌃X
Note again, that the suspension in Top⇤ does not model the suspension in S⇤ . To represent the latter in Top⇤ we must use homotopy push-outs. For example, in Top⇤ we have
⌃S 0 ⇠
2
= ⇤, while in S⇤ we get ⌃S 0 ' S 1 .
We consider the suspension in C as an endofunctor
⌃:C!C
provided it is defined on all objects. The category of spectra Sp can be characterized as
the universal presentable 1-category generated by one object on which the suspension
acts as an equivalence. More concrete constructions (in the world of 1-categories) are as
Sp ' S⇤ [⌃ 1 ] ,
or as the colimit of the diagram
⇣
⌘
⌃
⌃
Sp ' colim S⇤ ! S⇤ ! . . . .
These descriptions yield the main features of the category of spectra:
1. There is an adjunction
⌃1 : S⇤ ⌧ Sp : ⌦1 .
Note that ⌃1 X is just the image of X under S⇤ ! S⇤ [⌃ 1 ] ' Sp. We call ⌃1 X
the suspension spectrum of X, and ⌦1 (E) the infinite loop space of the spectrum
E.
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2. A spectrum E has well-defined homotopy groups
⇡n (E) := ⇡2 (⌦1 ⌃2
n
E) ,
n 2 Z ,k  n .
We can replace 2 by any other k 2. A morphism between spectra is an equivalence
if and only if it induces a morphism in homotopy groups.
3. A sequence of pointed spaces and maps
(Xn )n
0
,
⌃Xn ! Xn+1
defines a spectrum E by
E := colim ⌃0 ⌃1 X0 ! ⌃ 1 ⌃1 X1 ! ⌃ 2 ⌃1 X2 ! . . .
.
From this we get the formula for the homotopy groups
⇡n (E) ⇠
= colimk ⇡n+k (Xk ) .
We can also start with such a sequence of topological spaces and maps using the
maps ⌃h Xn ! ⌃X ! Xn+1 .
4. A sequence of morphisms in a category is called a fibre sequence if at every segment
··· ! A ! B ! C ! ...
the morphism A ! B is represents the fibre of B ! C. Similarly we define the
notion of a cofibre sequence. In spectra the notions of fibre and cofibre sequences
coincide.
A map between spectra X ! Y can functorially be extended to a fibre sequence
· · · ! Z ! X ! Y ! ⌃Z ! . . . .
5. The category of pointed spaces and spectra have symmetric monoidal structures
denoted by ^. The functor ⌃1 is a symmetric monoidal functor, i.e.
⌃1 (X ^ Y ) ' ⌃1 X ^ ⌃1 Y .
For two spectra X, Y we have a mapping spectrum map(X, Y ) such that
⌦1 map(X, Y ) ' Map(X, Y ) .
We have the rule
map(X ^ Y, Z) ' map(X, map(Y, Z)) .
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(5)
6. The (reduced) homology of a pointed space X with values in a spectrum E is defined
by
H⇤ (X, E) := ⇡⇤ (⌃1 X ^ E) .
The functor ⌃1 (as a left adjoint) preserves cofibres. Given a map X ! Y of spaces
we get a cofibre sequence
⌃ 1 ⌃1 Cofib(f ) ! ⌃1 X ! ⌃1 Y ! ⌃1 Cofib(f ) .
(6)
If we take the product with E (which again yields a (co)fibre sequence as a consequence of (5)), then the associated long exact sequence is the long exact homology
sequence
· · · ! H⇤ 1 (Cofib(f ); E) ! H⇤ (X; E) ! H⇤ (Y ; E) ! H⇤ (Cofib(f ); E) ! . . . .
Note that if we want to interpret this in pointed topological spaces Top⇤ , then we
must use homotopy cofibres e.g. represented by the cone of f .
7. The functor ⌦1 preserves fibre sequences (as a right adjoint) and hence we get a
long exact sequence in homotopy groups
· · · ! ⇡n (Z) ! ⇡n (X) ! ⇡n (Y ) ! ⇡n 1 (Z) ! . . . .
The (reduced) cohomology of a space X with coefficients in a spectrum E is defined
by
H ⇤ (X; E) := ⇡ ⇤ (map(⌃1 X, E)) .
If we insert (6) into Map( , E) we get a fibre sequence in spectra which yields the
long exact sequence in cohomology.
· · · ! H ⇤+1 (Cofib(f ); E) ! H ⇤ (Y ; E) ! H ⇤ (X; E) ! H ⇤ (Cofib(f ); E) ! . . . .
Example 23.10. Using the symmetric monoidal structure of S given by the cartesian
product we can define the notion of a commutative monoid. A monoid X in spaces is a
group if ⇡0 (X) is a group. Let CGrp(S) ✓ CMon(S) be the categories of commutative
groups and monoids in spaces. Note that commutative mopnoids in Top represent a very
restrictive class of commutative commutative in spaces. In order to model the general
case one needs the notion of E1 -spaces.
Since Sp is stable, the forgetful functor CGrp(Sp) ! Sp is an equivalence. As a rightadjoint of a symmetric monodical functor the infinite loop space functor ⌦ is lax symmetric
monodidal. It refines to a functor
⌦1 : Sp ' CGrp(Sp) ! CGrp(S)
which after restriction to the subcategory of connective spectra (i.e. those spectra with
trivial homotopy groups in negative degree) induces an equivalence
⌦1 : Sp
0 '
! CGrp(S) .
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Models for the inverse (denoted by sp) which turn a commutative group in S into a
connective spectrum are usually called 1-loop space machines.
Let A be a commutative topological abelian group, i.e. an object in CGrp(Top). Then
HA := sp(A) is called the Eilenberg-MacLane spectrum of A. We have
⇢
A n=0
⇡n (HA) ⇠
.
=
0 n 6= 0
The ordinary homology and cohomology of a space X is given in this language by
H⇤ (X; Z) ⇠
= H⇤ (X+ ; HZ) ,
H ⇤ (X; Z) ⇠
= H ⇤ (X+ ; HZ) ,
(X+ is obtained from X by attaching a disjoint base point).
For a spectrum E and a space X there is a Atiyah-Hirzebruch spectral sequence con2 ⇠
verging to H⇤ (X+ , E) with natural E 2 -term Ep,q
= Hp (X+ ; ⇡q (E)). In this way ordinary
cohomology can be considered as a first approximation to the cohomology of a space with
coefficients in a general spectrum.
Assume that X is a CW -complex. Then the associated E1 -term of the Atiyah-Hirzebruch
spectral sequence is the cellular chain complex
1 ⇠
Ep,q
= Cp (X+ ) ⌦ ⇡q (E) .
2
Example 23.11. We consider the topological symmetric monoidal (with respect to )
category of complex vector spaces VectC as a commutative monoid CMon(CatTop [W 1 ]).
Since the nerve and the realization preserve products we get BVectC 2 CMon(S).
The inlcusion of monoids into groups fits into an adjunction
GrCompl : CMon(S) ⌧ CGrp(S) : incl .
The connective topological K-theory spectrum is defined by
ku := sp(GrCompl(BVectC )) .
This is actually a ring space with ring structure induced by the tensor product of vector
spaces. Let b 2 ⇡2 (ku) ⇠
= Z be a generator. Then we get the periodic K-theory spectrum
as
KU := ku[b 1 ] .
It gives rise to K-theory H ⇤ (X+ ; KU) and K-homology H⇤ (X+ ; KU) of a space X.
The algebraic tool to calculate the K-homology is the Atiyah-Hirzebruch spectral sequence
whose second term is given by
⇢
Hp (X+ , Z) q even
2 ⇠
Ep,q = Hp (X+ ; ⇡q (KU)) ⇠
.
=
0
q odd
3
The first non-trivial di↵erential is d3 = Sq 2 : Ep,q
! Ep3
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3,q+2 .
Example 23.12. Let X be a topological space and ⇠ ! X be a real vector bundle. We
define the associated Thom spectrum by
X ⇠ := ⌃1 Cofib((⇠ \ X) ! X)h .
This is meant to a model for a construction in spaces. As an illustration of the usage
of the language of spaces we write this construction in that language. We start with
the action of GL(n, R) on Rn . We apply the quotient construction to the G-equivariant
diagram
/ Rn
Rn \ {0}
$
and get
⇤
~
f
⇤
n
%
/ n
y
BGL(n, R)
the complement zero section of the universal n-dimensional vector bundle. For a map of
spaces ⇠ : X ! BGL(n, R) we define the Thom spectrum by
X ⇠ := ⌃1 Cofib(X ⇥BGL(n,R) f ) .
A map f : Y ! X induces a map of Thom spectra Y f
inclusion of a point in X gives a map ⌃dim(⇠) S ! X ⇠ .
⇤⇠
! X ⇠ . In particular, the
For all n 2 N we have equivalences
⌃
n
X⇠
✏n
' X⇠ .
We can define KO0 (X) as the group of stable isomorphism classes of vector bundles on
X. Therefore ⌃ dim(⇠) X ⇠ only depends up to equivalence on the KO-theory class of ⇠.
We thus can consider the Thom spectrum X ⇠ for a class ⇠ 2 KO0 (X).
Using this we can define e.g. the Thom spectrum
MSO := BSO⇠ ,
where ⇠ 2 KO0 (BSO) is the class of the tautological bundle. By the Pontrjagin-Thom
construction the homotopy groups ⇡n (MSO) are the bordism groups of n-dimensional
oriented manifolds. More generally, we have a cohomology theory X 7! H ⇤ (X; MSO)
and a homology theory X 7! H⇤ (X; MSO).
In general it is difficult to calculate the homotopy groups of a Thom spectrum. Often one
considers multiplicative cohomology theories represented by E 2 CAlg(Sp).
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The Adams spectral sequence is a machine which tries to calculate the homotopy groups
of a spectrum A starting from its E-homology H⇤ (A; E) ⇠
= ⇡⇤ (A ^ E). In good cases its
E2 -terms has an algebraic description as
E2 ⇠
= ExtE⇤ E (⇡⇤ (E), H⇤ (A; E)) .
An orientation of a vector bundle ⇠ for E is a class or 2 H dim(⇠) (X ⇠ ; E) whose restriction
to every point x 2 X a unit in ⇡0 (E). If the vector bundle ⇠ is oriented for E, then we
have a Thom isomorphism
H⇤+dim(⇠) (X; E) ⇠
= H⇤ (X ⇠ ; E) .
This allows to calculate the first input into the Adams spectral sequence.
1. The tautological bundle on BO(n) is oriented for HZ/2Z.
2. The tautological bundle on BSO(n) is oriented for HZ.
3. The tautological bundle on BSpinc (n) is oriented for KU.
4. The tautological bundle on BSpin(n) is oriented for KO.
In the last two cases the orientations are called Atiyah-Bott-Shapiro orientations.
For example, in the cases of MSO, MU, MSpin and MSpinc the Adams spectral
sequence works successfully with E = HZ/pZ and E = HQ and finally gives a complete
understanding of the homotopy groups.
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Example 23.13. A map ✓ : X ! BO(n) represents a vector bundle also denoted by ✓.
The Madsen-Tillman spectrum associated to this datum is defined as
MT✓ := X
✓
.
In simple cases like X = BSpin(n) or X = BSO(n) one can calculate ⇡n (MT✓) using
the methods indicated above.
The main goal of the seminar is to construct an equivalence
BC✓ ' ⌦1 ⌃MT✓ .
Remark 23.14. In this remark we sketch how (7) can be refined to an 1-loop map.
Let Fin+ be the category of finite pointed sets. We get a functor
Fin+ ! S⇤ /BO(n) ,
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F 7! F ^ X+ .
(7)
The Thom spectrum construction can be considered as a functor S⇤ /BO(n) ! Sp. Therefore by composition we get the functor
MT✓ : Fin+ ! Sp .
This is actually an object of the full subcategory of Sp ✓ SpFin+ of -spectra E 2 SpFin+
characterized by
E[n]+ '
n
Y
E[1]+
and a condition of being ”group-like” .
i=1
It refines ⌃MT✓ to a grouplike
We have an equivalence
spectrum, and its 1-loop space to a grouplike -space.
Sp
which under ⌦1 goes to
' CGrp(Sp ) ,
S ' CGrp(S) .
This says that the 1-loop space structure on ⌦1 ⌃MT✓ is equivalent to the 1-loop space
structure on ⌦1 ⌃MT✓ derived from the -space structure ⌦1 ⌃ MT✓.
On the other hand, using the functoriality of the bordism category in ✓ we can consider
the object BC✓ 2 S Fin+ . Since the equivalence (7) is natural in ✓ it refines to an equivalence in S Fin+ . Consequently, BC✓ is a group like -space, and (7) is an 1-loop map,
if we equip BC✓ with the 1-loop space structure coming from the -space structure. 2
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