Bundle structures and lifting problems (Ex)

Exercise 2.1. 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.

Hint 2.2. This is a special case of [Hatcher2002, Proposition 4.72].

Exercise 4.1.

• 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 

$$\displaystyle \Omega K(n+1,\Zz/2\Zz)\to P(K(n+1,\Zz/2\Zz))\to K(n+1,\Zz/2\Zz) \;.$$

      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:

1. The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\Zz/2\Zz)$.
2. The homotopy fiber hofib($w_1$) is $BSO$.
3. The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$.

Assume that the homotopy groups of $O$ are known.

1. Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$.
2. Calculate the homotopy groups of $BSO$ using the fibration
   $$\displaystyle BSO\to BO\to K(1,\Zz/2\Zz)\;.$$
3. 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

$$\displaystyle \text{hofib}(p)\to BSO \to BO$$

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 $q$.

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

Let $X$ denote a compact pointed space and $\xi\to X$ a vector bundle on $X$.

1. Use Exercise 3.1 to show that $\xi$ is orientable if and only if its first Stiefel-Whitney class

       vanishes.

1. 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

1. Given an interpretation of the group $[X,\text{hofib}(p)]$.

Now there are similar results for spin structures (on oriented vector bundles).

Exercise 2.1. 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.

Hint 2.2. This is a special case of [Hatcher2002, Proposition 4.72].

Exercise 4.1.

• 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 

$$\displaystyle \Omega K(n+1,\Zz/2\Zz)\to P(K(n+1,\Zz/2\Zz))\to K(n+1,\Zz/2\Zz) \;.$$

      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:

1. The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\Zz/2\Zz)$.
2. The homotopy fiber hofib($w_1$) is $BSO$.
3. The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$.

Assume that the homotopy groups of $O$ are known.

1. Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$.
2. Calculate the homotopy groups of $BSO$ using the fibration
   $$\displaystyle BSO\to BO\to K(1,\Zz/2\Zz)\;.$$
3. 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

$$\displaystyle \text{hofib}(p)\to BSO \to BO$$

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 $q$.

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

Let $X$ denote a compact pointed space and $\xi\to X$ a vector bundle on $X$.

1. Use Exercise 3.1 to show that $\xi$ is orientable if and only if its first Stiefel-Whitney class

       vanishes.

1. 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

1. Given an interpretation of the group $[X,\text{hofib}(p)]$.

Now there are similar results for spin structures (on oriented vector bundles).