Download Lecture 4(30.01.09) Universal Bundles

Lecture 4(30.01.09)
Universal Bundles
Let G be a Lie group and assume that it acts freely on a space EG with πk (EG ) = 0
for all k. Consider the corresponding principal G-bundle
G
pG : EG → BG .
We call it the universal G-bundle and the base space BG is the classifying space of G.
Theorem 1 Any principal G-bundle p : E → B over a CW complex B is isomorphic to
a pull back (EG , BG , G, pG ) under some continuous map f : B → BG . Two pull backs are
isomorphic if and only if the corresponding maps are homotopic. The corresponding pull
back map f is the classifying map of p : E → B.
Proof. According to Theorem 4 from previous lecture a principal G-bundle p : E → B is
isomorphic to a pull back of pG : EG → BG under a continuous map f : B → BG if and
only if there is an equivariant map ψ : E → f ∗ (EG ) of the total spaces of these bundles.
Therefore, it is enough to construct an equivariant map F : E → EG .
Let B be a CW -complex, B0 ⊂ B its CW -subcomplex. Consider a principal G-bundle
p : E → B and denote p−1 (B0 ) by E0 . Assume that F0 : E0 → EG is an equivariant
continuous map. We show that F0 can be extended to an equivariant continuous map
F : E → BG .
We will be using a familiar fact.
Exercise 1 Any principal G-bundle over a disc D n is isomorphic to a trivial bundle
Assume that Sk(n−1) [B] ⊂ B0 and Din is not in B0 . We extend F0 on Din . Let
φi : Din → B be the attaching map, φi (Sin ) ⊂ B0 . Consider the restriction of p : E → B
on Din . According to the exercise it is a trivial bundle Din × G. The restriction of F0 on
(n−1)
φi (Si
) ⊂ B0 induces an equivariant map:
F0 (x, g) = F0 (x, e)g ∈ EG ,
(n−1)
x ∈ Si
,
g ∈ G.
Consider a map F0 (x, e) : Sin−1 → EG . Because πn−1 (EG ) = 0 by definition of EG , this
map extends to a continuous map F (x, 1) : Din → EG . We then define
F (x, g) := F (x, e)g,
x ∈ Din ,
g ∈ G.
S
This map is equivariant and extends F0 on p−1 (B0 Din ) ⊃ E0 . We then proceed by
induction and get the first statement of the theorem if B0 = ∅.
The second statement follows from the arguments above if we replace B by B × I and
S
B0 by B × 0 B × 1.
Corollary 1 If BG is a CW -complex then it is uniquely defined up to homotopy equivalence.
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′
Proof. Indeed, if we have two principal G-bundles p : EG → BG and p : EG′ → BG
with
′
′
πi (EG ) = πi (EG ) = 0, with BG , BG being CW -complexes then applying the theorem we
get continuous maps
′
′
f : BG → BG
, g : BG
→ BG
such that f ∗ (EG′ ) = EG and g ∗(EG ) = EG′ . Thus, (f g)∗ (EG ) = EG and, therefore,
′
f g ∼ id. Similarly, gf ∼ id and BG ∼ BG
.
Remark 1 If we have any fibre bundle (E, B, F, p) with the structure Lie group G then
we can consider the corresponding principal G-bundle (E ′ , B, G, p). It is related to the
initial bundle as
E = E ′ × F/{(x ◦ g, y) ∼ (x, gy)},
x ∈ E, y ∈ F, g ∈ G ⊂ Homeo(F ).
Therefore, if f : B → BG is the classifying map of a principal G-bundle E ′ , then E is a
pull back under f of the universal F -bundle with the structure group G:
EG F = EG × F/{(x ◦ g, y) ∼ (x, gy)},
x ∈ EG, y ∈ F, g ∈ G ⊂ Homeo(F ).
Let us now construct a classifying space BG of a Lie group G. We start with G = U(n)
and consider a principal U(n)-bundle:
U (n)
p : Vnn+N ∼
= CGnn+N .
= U(n + N)/U(N) −→ U(n + N)/(U(N) ⊕ U(n)) ∼
The space Vnn+N is the complex Stiefel manifold (not a complex manifold) of unitary
n-frames in complex space C n+N is acted upon by U(n) where each element of U(n) is
considered as a unitary transformation of the complex subspace generated by the frame.
The factor space of this action is a proper complex manifold of n-dimensional subspaces in
C n+N . Embedding onto the first n + N coordinates C n+N ⊂ C n+N +1 induces embeddings
U(n + N) ⊂ U(n + N + 1);
Vnn+N ⊂ Vnn+N +1 ;
CGnn+N ⊂ CGnn+N
We therefore define
Vn∞ = lim Vnn+N ,
N →∞
n+N
BU(n) = CG∞
.
n = lim CGn
N →∞
Let us check that p : Vn∞ → BU(n) is the universal U(n)-bundle. We need to check that
πk (Vn∞ ) = 0. We again use EHSFB:
· · · πk (U(N)) → πk (U(n + N)) → πk (Vnn+N ) → πk−1 (U(N)) → πk−1 (U(n + N)) → · · ·
Last lecture we checked (again from EHSFB) that j : U(N) → U(n + N) induces isomorphisms of homotopy groups πk for all k < 2N. Therefore, πk (VnN +n ) = 0 for k < 2N.
Because,
πk (Vn∞ ) = limN →∞ πk (Vnn+N ),
we deduce that πk (Vn∞ ) = 0.
This construction immediately gives a description of the classifying space for any
closed subgroup of U(n) and as such covers lots of the most important examples. Indeed,
if H ⊂ U(n) is a closed subgroup then it acts freely on Vn∞ with factor space BH .
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Exercise 2 Check all kinds of naturalities in the construction of BH with respect to group
homomorphisms and other group operations.
There is a more general construction which is applicable to all topological groups. The
idea is to start with the simplest G-bundle and then move further always constructing
G-bundles with more and more trivial homotopy groups.
We start with G-bundle E0 over a join. In general, even π0 (G) 6= 0. To make it equal
to zero we consider two copies of G and make a going:
E1 = G ∗ G = {(t, g, s, h) ∈ [0, 1] × G × [0, 1] × G| t + s = 1}/ ∼
where
(0, g1, 1, h) ∼ (0, g2 , 1, h),
(1, g, 0, h1) ∼ (1, g, 0, h2)
The group G acts diagonally on G ∗ G:
f ◦ (t, g, s, h) = (t, f g, s, f h),
f ∈ G.
This space is connected, i.e. π0 (G ∗ G) = 0. Similarly, we take E2 = G ∗ G ∗ G,...
Ek = G ∗ G · · · G with the corresponding diagonal action of G. There is an obvious
embedding of Ek ⊂ Ek+1 onto the first k-coordinate and it is equivariant with respect to
the action of G. We thus define
EG = lim G
∗ G ∗{z· · · ∗ G}
k→∞ |
k
The diagonal action of G makes it into a principal G-bundle and we only need to check
that πk (EG ) = 0 for any k. This is because Ek is k − 1-connected and to verify the later
statement we proof a lemma.
Lemma 1 The space X ∗ Y is homotopy equivalent to ΣZ for some space Z
Proof. We have
X ∗ Y = ((CX) × Y )
((CX) × Y
[
CY )
[
[
(X × (CY )) =
(X × (CY )
S
[
CX) = P1
[
P2 ,
S
where P1 = (CX×Y ) CY , P2 = (X×(CY )) CX. Each space P1 and P2 is contractible,
T
S
S
P1 P2 is P3 = (X × Y ) CX CY . Therefore, P1 and P2 are homotopy equivalent to
CP3 , and their union is homotopy equivalent to ΣP3 .
This lemma implies that Ek is homotopy equivalent to a k-fold suspension over some
space and, thus, is k − 1-connected.
Theorem 2 For any countable, connected simplicial complex X in the weak topology
(closed sets are those with closed intersections with every simplex), there exists a universal bundle with base space X, where the bundle space and group being countable CW complexes.
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Proof. This is a modification of the example of a contractible Serre fibration of all paths
on X. Let Sn be the space of all sequences (xn , xn−1 , · · · , x0 ) of points in X such that
each pair xi , xi−1 lie in a common simplex of X. This can be topologised as a subset of
X n+1 = X × X · · · × X. Let S be the topological sum of all the Sn . We now identified
points of S under the following equivalence relation:
(xn , · · · , xi , · · · , x0 ) ∼ (xn , · · · , x̂i , · · · , x0 )
whenever either xi = xi−1 or xi+1 = xi−1 (the ”hat” symbol x̂ denotes deletion). The
space obtain under this identification is S̃ = S/ ∼. For each point (xn , · · · , x0 ) of S let
[xn , · · · , x0 ] denote the corresponding point of S̃. Let v0 be the fixed point (vertex) of X.
As a bundle Ẽ we take the subset of S̃ consisting of all [xm , · · · , x0 ] with x0 = v0 . The
projection p : Ẽ → X is defined
p([xn , · · · , x1 , v0 ]) = xn .
Thus the fibre G̃ = p−1 (v0 ) consists of all [xn , · · · , x0 ] with xn = x0 = v0 .
It is more or less obvious that p is continuous and with a little bit of more effort you
can induce a CW -complex structure on S̃ from that on S.
I therefore check that it is a principal bundle and that the total space is contractible.
First we define a product of two elements [xn , · · · , x0 ] and [ym , · · · , y0] of S̃ whenever
x0 = ym :
[xn , · · · , x0 ] ∗ [ym , · · · , y0 ] = [xn , · · · , x0 , ym , · · · , y0 ]
It is a well defined associative operation (when it is defined). It is also continuous. Every
element [xn , · · · , x0 ] has an inverse
[xn , · · · , x0 ]−1 = [xn , · · · , x0 ],
[xn , · · · , x0 ]−1 · [xn , · · · , x0 ] = [x0 , x0 ]
Since the product of any two elements of G̃ is defined we thus have constructed a structure
of a topological group on G̃ with the identity element [v0 , v0 ].
We can easily introduce a principal G̃-bundle structure on p : Ẽ → X by looking at a
star neighbourhood of the jth vertex vj of X. For each Vj choose a fixed element
ej = [vj , xn−1 , · · · , x1 , v0 ]
of p−1 (vj ). Define the trivialisation
φj : Vj × G̃ → p−1 (Vj ),
by φj (x, g) = [x, vj ] ∗ ej ∗ g. It is elementary to check that this structure satisfies all the
conditions of principal G-bundle.
In order to complete the proof of the theorem it is only necessary to prove that Ẽ is
contractible. The idea is to consider an obvious contraction of subspace En ⊂ Sn which
first contract the first coordinate on to the second, the the second to the third etc, and
then to check that all subsets of degenerate sequences are also contractible.
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