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<wikitex>;
== Lifting maps ==
Given a (pointed) map $g: Y\to Z$ of pointed topological spaces, we define the homotopy fibre of $g$ as
$$
\mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\},
$$
where $PZ$ is the space of paths starting at the base-point of $Z$. We denote by $p$: $\mathrm{hofib}(g)\to Y$ the projection.

{{beginthm|Exercise}}
Prove the following:

A map $f:X \to Y$ of pointed space has a lift  $\bar f:X\to \text{hofib}(g)$ along $p$ if and only if $g\circ f$ is homotopic to the constant map.
{{endthm}}
{{beginrem|Hint}} This is a special case of \cite{Hatcher2002|Proposition 4.72}.
{{endrem}}

== 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 $X$ by $\Omega X$.
{{beginthm|Exercise}}
* Show: There is a homotopy equivalence $\Omega K(n+1,\Zz/2\Zz)\simeq K(n,\Zz/2\Zz)$.

'''Hint''': Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy associated to the path-space-fibration 
$$\Omega K(n+1,\Zz/2 \Zz)\to P(K(n+1,\Zz/2 \Zz))\to K(n+1,\Zz/2 \Zz) \;.$$

Recall that the path-space is contractible.
* Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ 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 $H^1(X,\Zz/2\Zz)$ acts free and transitively on the set 
of spin structures of an oriented vector bundle $\xi\to X$ (X a compact pointed space).
Now recall that \ref{The group structure on $H^1(X,\Zz/2\Zz)$ is due to the Exercise 3.2.} 
$H^1(X,\Zz/2\Zz)\cong [X,K(1,\Zz/2\Zz)]$, where $K(1,\Zz/2\Zz)$ 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 $w_1:BO\to K(1,\Zz/2\Zz)$.
# The homotopy fiber hofib($w_1$) is $BSO$.
# The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$.

Assume that the homotopy groups of $O$ are known.
# Calculate the homotopy groups of $BO$ using the fibration $O\to EO\to BO$.
# Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\Zz/2\Zz)\;.$$
# Calculate the homotopy fibre of $p:BSO\to BO$.

Now we can classify the orientations on a vector bundle $\xi\to X$. 
For this we need to know that the sequence
$$\text{hofib}(p)\to BSO \to BO$$
fits into the following diagram\footnote{It's non-trivial 
to see that the functor $B$ can be applied to each of those spaces.}:

Hence there are group structures on $[X,\text{hofib}(p)]$ and $[X,BSO]$ (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of $[X,\text{hofib}(p)]$ on $[X,BSO]$ induced by $q$.

{{beginthm|Definition}}
A vector bundle is called ''orientable'' if its classifying map $X \to BO$ lifts along $p:BSO \to BO$.  An ''orientation'' is the choice of such a lift.
{{endthm}}

Let $X$ denote a compact pointed space and $\xi\to X$ a vector bundle on $X$.
#Use Exercise 3.1 to show that $\xi$ is orientable if and only if its first Stiefel-Whitney class vanishes.
#Show that the group $[X,\text{hofib}(p)]$ 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 $[X,\text{hofib}(p)]$.

Now there are similar results for spin structures (on oriented vector bundles).
{{beginthm|Exercise}}
Repeat Exercises 3.3 and 3.4 using the second Stiefel-Whitney class $w_2:BSO\to K(2,\Zz/2\Zz)$.  The homotopy fibre of $w_2$ is BSpin, where Spin is the colimit over $Spin(n)$, the universal cover of $SO(n)$.
{{endthm}}
</wikitex>
== References ==
{{#RefList:}}
[[Category:Exercises]]
[[Category:Exercises without solution]]