Bundle structures and lifting problems (Ex)

1 Lifting maps

Given a (pointed) map $g: Y\to Z$$<wikitex>; == Lifting maps == Given a (pointed) map $g: Y\to Z$ of pointed topological space, we define the homotopy fiber as\footnote{Here $PZ$ is the space of paths starting at the base-point of $Z$.} $$ \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\} $$ and we denote by $p$: $\text{hofib}(g)\to Y$ the projection. {{beginthm|Exercise}} Prove the following: A map $f:X \to Y$ of pointed space has a lift $\bar f:X\to \text{hofib}(g)$ along $p$ if and only if $g\circ f$ is homotopic to the constant map. {{endthm}} {{beginrem|Hint}} This is a special case of \cite{Hatcher2002|Proposition 4.72}. {{endrem}} == Classification of orientations and spin structures on vector bundles == Recall the Definition of Eilenberg-MacLane-spaces. We denote the space of pointed loops in a space $X$ by $\Omega X$. {{beginthm|Exercise}} * Show: There is a homotopy equivalence $\Omega K(n+1,\bbz/2\bbz)\simeq K(n,\bbz/2\bbz)$.\ '''Hint''': Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy associated to the path-space-fibration $$\Omega K(n+1,\bbz/2\bbz)\to P(K(n+1,\bbz/2\bbz))\to K(n+1,\bbz/2\bbz) \;.$$ Recall that the path-space is contractible. * Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure induced by composition of paths. '''Hint''': This is similar to the group structure of the fundamental group. Maybe you've heard that the group $H^1(X,\Zz/2\Zz)$ acts free and transitively on the set of spin structures of an oriented vector bundle $\xi\to X$ (X a compact pointed space). Now recall that \ref{The group structure on $H^1(X,\bbz/2\bbz)$ is due to the Exercise 3.2.} $H^1(X,\bbz/2\bbz)\cong [X,K(1,\bbz/2\bbz)]$, where $K(1,\bbz/2\bbz)$ denotes an Eilenberg-MacLane-space. So we first prove the statement about classification of spin structures. The warm-up is the classification of orientations: # The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\bbz/2\bbz)$. # The homotopy fiber hofib($w_1$) is $BSO$. # The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$. Assume that the homotopy groups of $O$ are known. # Calculate the homotopy groups of $BO$ using the fibration <script type="math/tex">g: Y\to Z$ of pointed topological space, we define the homotopy fiber as\footnote{Here $PZ$$PZ$ is the space of paths starting at the base-point of $Z$$Z$.}

$\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\}$ $$\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\}$$

and we denote by $p$$p$: $\text{hofib}(g)\to Y$$\text{hofib}(g)\to Y$ the projection.

Exercise 2.1. Prove the following:

A map $f:X \to Y$$f:X \to Y$ of pointed space has a lift $\bar f:X\to \text{hofib}(g)$$\bar f:X\to \text{hofib}(g)$ along $p$$p$ if and only if $g\circ f$$g\circ f$ 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 $X$$X$ by $\Omega X$$\Omega X$.

Exercise 4.1.

• Show: There is a homotopy equivalence

  Tex syntax error

  $\Omega K(n+1,\bbz/2\bbz)\simeq K(n,\bbz/2\bbz)$.\\

Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy

      associated to the path-space-fibration 

Tex syntax error

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

      Recall that the path-space is contractible.

• Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$$[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)$$H^1(X,\Zz/2\Zz)$ acts free and transitively on the set of spin structures of an oriented vector bundle $\xi\to X$$\xi\to X$ (X a compact pointed space).

Now recall that \ref{The group structure on

Tex syntax error

$H^1(X,\bbz/2\bbz)$ is due to the Exercise 3.2.}

Tex syntax error

$H^1(X,\bbz/2\bbz)\cong [X,K(1,\bbz/2\bbz)]$, where

Tex syntax error

$K(1,\bbz/2\bbz)$ 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

   Tex syntax error

   $w_1:BO\to K(1,\bbz/2\bbz)$.
2. The homotopy fiber hofib($w_1$$w_1$) is $BSO$$BSO$.
3. The projection $p:BSO\to BO$$p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$$SO\hookrightarrow O$.

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

1. Calculate the homotopy groups of $BO$$BO$ using the fibration $0\to EO\to BO$$0\to EO\to BO$.
2. Calculate the homotopy groups of $BSO$$BSO$ using the fibration

   Tex syntax error

   $$\displaystyle BSO\to BO\to K(1,\bbz/2\bbz)\;.$$
3. Calculate the homotopy fibre of $p:BSO\to BO$$p:BSO\to BO$.

Now we can classify the orientations on a vector bundle $\xi\to X$$\xi\to X$. For this we need to know that the sequence

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

fits into the following diagram\footnote{It's non-trivial to see that the functor $B$$B$ can be applied to each of those spaces.}:

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

Definition 4.2.

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

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

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

       vanishes.

1. Show that the group

   Tex syntax error

   $[X,\rm{hofib}(p)]$ 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

   Tex syntax error

   $[X,\rm{hofib}(p)]$.

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 $w_2:BSO\to K(2,\Zz/2\Zz)$$w_2:BSO\to K(2,\Zz/2\Zz)$. The homotopy fibre of $w_2$$w_2$ is BSpin, where Spin is the colimit over $Spin(n)$$Spin(n)$, the universal covers of $SO(n)$$SO(n)$.

References

• [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001

\to EO\to BO$. # Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\bbz/2\bbz)\;.$$ # 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 \underline{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 $[X,\rm{hofib}(p)]$ acts free and transitively on the set of homotopy classes of lifts. '''Hint:''' Use the homotopy-lifting property and Exercise 3.1 #Given an interpretation of the group $[X,\rm{hofib}(p)]$. 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 covers of $SO(n)$. {{endthm}} == References == {{#RefList:}} [[Category:Exercises]]g: Y\to Z of pointed topological space, we define the homotopy fiber as\footnote{Here $PZ$$PZ$ is the space of paths starting at the base-point of $Z$$Z$.}

$\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\}$ $$\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\}$$

and we denote by $p$$p$: $\text{hofib}(g)\to Y$$\text{hofib}(g)\to Y$ the projection.

Exercise 2.1. Prove the following:

A map $f:X \to Y$$f:X \to Y$ of pointed space has a lift $\bar f:X\to \text{hofib}(g)$$\bar f:X\to \text{hofib}(g)$ along $p$$p$ if and only if $g\circ f$$g\circ f$ 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 $X$$X$ by $\Omega X$$\Omega X$.

Exercise 4.1.

• Show: There is a homotopy equivalence

  Tex syntax error

  $\Omega K(n+1,\bbz/2\bbz)\simeq K(n,\bbz/2\bbz)$.\\

Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy

      associated to the path-space-fibration 

Tex syntax error

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

      Recall that the path-space is contractible.

• Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$$[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)$$H^1(X,\Zz/2\Zz)$ acts free and transitively on the set of spin structures of an oriented vector bundle $\xi\to X$$\xi\to X$ (X a compact pointed space).

Now recall that \ref{The group structure on

Tex syntax error

$H^1(X,\bbz/2\bbz)$ is due to the Exercise 3.2.}

Tex syntax error

$H^1(X,\bbz/2\bbz)\cong [X,K(1,\bbz/2\bbz)]$, where

Tex syntax error

$K(1,\bbz/2\bbz)$ 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

   Tex syntax error

   $w_1:BO\to K(1,\bbz/2\bbz)$.
2. The homotopy fiber hofib($w_1$$w_1$) is $BSO$$BSO$.
3. The projection $p:BSO\to BO$$p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$$SO\hookrightarrow O$.

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

1. Calculate the homotopy groups of $BO$$BO$ using the fibration $0\to EO\to BO$$0\to EO\to BO$.
2. Calculate the homotopy groups of $BSO$$BSO$ using the fibration

   Tex syntax error

   $$\displaystyle BSO\to BO\to K(1,\bbz/2\bbz)\;.$$
3. Calculate the homotopy fibre of $p:BSO\to BO$$p:BSO\to BO$.

Now we can classify the orientations on a vector bundle $\xi\to X$$\xi\to X$. For this we need to know that the sequence

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

fits into the following diagram\footnote{It's non-trivial to see that the functor $B$$B$ can be applied to each of those spaces.}:

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

Definition 4.2.

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

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

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

       vanishes.

1. Show that the group

   Tex syntax error

   $[X,\rm{hofib}(p)]$ 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

   Tex syntax error

   $[X,\rm{hofib}(p)]$.

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 $w_2:BSO\to K(2,\Zz/2\Zz)$$w_2:BSO\to K(2,\Zz/2\Zz)$. The homotopy fibre of $w_2$$w_2$ is BSpin, where Spin is the colimit over $Spin(n)$$Spin(n)$, the universal covers of $SO(n)$$SO(n)$.

References

• [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001