Bundle structures and lifting problems (Ex)
1 Lifting maps
Given a (pointed) map of pointed topological spaces, we define the homotopy fibre of
as
![\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\},](/images/math/e/4/e/e4e25669332b236810c8e95e4836d200.png)
where is the space of paths starting at the base-point of
. We denote by
:
the projection.
Exercise 2.1. Prove the following:
A map of pointed space has a lift
along
if and only if
is homotopic to the constant map.
Hint 2.2. This is a special case of [Hatcher2002, Proposition 4.72].
2 Classification of orientations and spin structures on vector bundles
Recall the Definition of Eilenberg-MacLane-spaces.
We denote the space of pointed loops in a space by
.
Exercise 4.1.
- Show: There is a homotopy equivalence
.
Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy associated to the path-space-fibration
![\displaystyle \Omega K(n+1,\Zz/2 \Zz)\to P(K(n+1,\Zz/2 \Zz))\to K(n+1,\Zz/2 \Zz) \;.](/images/math/9/1/e/91eaadb0bfff1bd930b6a561ec133987.png)
Recall that the path-space is contractible.
- Show that the set of homotopy classes of pointed maps
has a group structure induced by composition of paths.
Hint: This is similar to the group structure of the fundamental group.
Maybe you've heard that the group acts free and transitively on the set
of spin structures of an oriented vector bundle
(X a compact pointed space).
Now recall that \ref{The group structure on
is due to the Exercise 3.2.}
, where
denotes an Eilenberg-MacLane-space.
So we first prove the statement about classification of spin structures. The warm-up is the classification of orientations:
- The first Stiefel-Whitney class is a map
.
- The homotopy fiber hofib(
) is
.
- The projection
is the map induced by
.
Assume that the homotopy groups of are known.
- Calculate the homotopy groups of
using the fibration
.
- Calculate the homotopy groups of
using the fibration
- Calculate the homotopy fibre of
.
Now we can classify the orientations on a vector bundle .
For this we need to know that the sequence
![\displaystyle \text{hofib}(p)\to BSO \to BO](/images/math/8/9/9/899e7f231e60bdc8accf783f446b5f67.png)
fits into the following diagram\footnote{It's non-trivial
to see that the functor can be applied to each of those spaces.}:
Hence there are group structures on and
(the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of
on
induced by
.
Definition 4.2.
A vector bundle is called orientable if its classifying map lifts along
. An orientation is the choice of such a lift.
Let denote a compact pointed space and
a vector bundle on
.
- Use Exercise 3.1 to show that
is orientable if and only if its first Stiefel-Whitney class vanishes.
- Show that the group
acts free and transitively on the set of homotopy classes of lifts.
Hint: Use the homotopy-lifting property and Exercise 3.1
- Give an interpretation of the group
.
Now there are similar results for spin structures (on oriented vector bundles).
Exercise 4.3.
Repeat Exercises 3.3 and 3.4 using the second Stiefel-Whitney class . The homotopy fibre of
is BSpin, where Spin is the colimit over
, the universal cover of
.
References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001