Bundle structures and lifting problems (Ex)

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<wikitex>;
<wikitex>;
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Given a (pointed) map $g:Y\to Z$ of pointed topological space, we define the homotopy fiber as\footnote{Here
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== 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 $Z$.}
$PZ$ 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\}
\mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\}
$$
$$
−
and we denote by $p$: $\mathrm{hofib}(g)\to Y$ the projection.
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and we denote by $p$: $\text{hofib}(g)\to Y$ the projection.
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Show the following statement:\\
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{{beginthm|Exercise}}
−
A map $f:X\to Y$ of pointed space has a lift $\bar f:X\to \rm{hofib}(g)$ along $p$
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Prove the following:
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if and only if $g\circ f$ is homotopic to the constant map.
+
−
Hint: This is a special case of Proposition 4.72 in [A. Hatcher, Algebraic Topology].
+
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.
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{{endthm}}
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{{beginrem|Hint}} This is a special case of \cite{Hatcher2002|Proposition 4.72}.
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{{endrem}}
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\subsection*{Classification of orientations and spin structures on vector bundles}
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== Classification of orientations and spin structures on vector bundles ==
Recall the Definition of Eilenberg-MacLane-spaces.
Recall the Definition of Eilenberg-MacLane-spaces.
−
We denote the space of pointed loops in a space $X$ by $\Omega X$.
+
We denote the space of pointed loops in a space $X$ by $\Omega X$.
−
\begin{enumerate}
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{{beginthm|Exercise}}
−
\item Show: There is a homotopy equivalence $\Omega K(n+1,\bbz/2\bbz)\simeq K(n,\bbz/2\bbz)$.\\
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* 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
+
+
'''Hint''': Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy
associated to the path-space-fibration
associated to the path-space-fibration
$$\Omega K(n+1,\bbz/2\bbz)\to P(K(n+1,\bbz/2\bbz))\to K(n+1,\bbz/2\bbz) \;.$$
$$\Omega K(n+1,\bbz/2\bbz)\to P(K(n+1,\bbz/2\bbz))\to K(n+1,\bbz/2\bbz) \;.$$
Recall that the path-space is contractible.
Recall that the path-space is contractible.
−
\item Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure
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* Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure
−
induced by composition of paths.\\
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induced by composition of paths.
−
Hint: This is similar to the group structure of the fundamental group.
+
−
\end{enumerate}
+
+
'''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
−
Maybe you've heard that the group $H^1(X,\bbz/2\bbz)$ acts free and transitively on the set
+
of spin structures of an oriented vector bundle $\xi\to X$ (X a compact pointed space).
of spin structures of an oriented vector bundle $\xi\to X$ (X a compact pointed space).
−
Now recall that\footnote{The group structure on $H^1(X,\bbz/2\bbz)$ is due to the Exercise 3.2.}
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Now recall that \ref{The group structure on $H^1(X,\bbz/2\bbz)$ is due to the Exercise 3.2.}
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$H^1(X,\bbz/2\bbz)\cong [X,K(1,\bbz/2\bbz)]$,
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$H^1(X,\bbz/2\bbz)\cong [X,K(1,\bbz/2\bbz)]$, where $K(1,\bbz/2\bbz)$ denotes an Eilenberg-MacLane-space.
−
where $K(1,\bbz/2\bbz)$ denotes an Eilenberg-MacLane-space.
+
−
+
−
This looks similar to the statement of Theorem 3.45 in our talk.
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So we first prove the statement about classification of spin structures.
So we first prove the statement about classification of spin structures.
The warm-up is the classification of orientations:
The warm-up is the classification of orientations:
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\begin{itemize}
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# The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\bbz/2\bbz)$.
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\item The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\bbz/2\bbz)$.
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# The homotopy fiber hofib($w_1$) is $BSO$.
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\item The homotopy fiber hofib($w_1$) is $BSO$.
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# The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$.
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\item The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$.
+
−
\end{itemize}
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Assume that the homotopy groups of $O$ are known.
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Assume that the homotopy groups of $O$ are known.
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# Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$.
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\begin{enumerate}
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# Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\bbz/2\bbz)\;.$$
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\item Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$.
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# Calculate the homotopy fibre of $p:BSO\to BO$.
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\item Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\bbz/2\bbz)\;.$$
+
−
\item Calculate the homotopy fibre of $p:BSO\to BO$.
+
−
\end{enumerate}
+
Now we can classify the orientations on a vector bundle $\xi\to X$.
Now we can classify the orientations on a vector bundle $\xi\to X$.
For this we need to know that the sequence
For this we need to know that the sequence
−
$$\rm{hofib}(p)\to BSO \to BO$$
+
$$\text{hofib}(p)\to BSO \to BO$$
fits into the following diagram\footnote{It's non-trivial
fits into the following diagram\footnote{It's non-trivial
to see that the functor $B$ can be applied to each of those spaces.}:
to see that the functor $B$ can be applied to each of those spaces.}:
−
\begin{center}
+
−
\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
+
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 $q$.
−
\matrix (m) [matrix of math nodes, row sep=3em,
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−
column sep=2.5em, text height=1.5ex, text depth=0.25ex]
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−
{ \text{hofib}(p) & & BSO & & BO \\
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\Omega B\rm{hofib}(p) & & \Omega BBSO & & \Omega BBO \\ };
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\path[->,font=\scriptsize]
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−
(m-1-1) edge node[auto] {$q$} (m-1-3)
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−
(m-1-3) edge node[auto] {$p$} (m-1-5)
+
−
(m-2-1) edge node[auto] {$\Omega B q$} (m-2-3)
+
−
(m-2-3) edge node[auto] {$\Omega B p$} (m-2-5)
+
−
(m-1-1) edge node[auto] {$\simeq$} (m-2-1)
+
−
(m-1-3) edge node[auto] {$\simeq$} (m-2-3)
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−
(m-1-5) edge node[auto] {$\simeq$} (m-2-5);
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\end{tikzpicture}
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−
\end{center}
+
−
Hence there are group structures on $[X,\rm{hofib}(p)]$ and $[X,BSO]$ (the latter one is the Whitney-sum
+
−
of vector bundles). Furthermore there is an action of $[X,\rm{hofib}(p)]$ on $[X,BSO]$ induced
+
−
by $q$.
+
{{beginthm|Definition}}
{{beginthm|Definition}}

Revision as of 20:38, 28 March 2012

1 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.}

\displaystyle \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.

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].

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 by \Omega X.

Exercise 4.1.

  • Show: There is a homotopy equivalence 
    Tex syntax error
    .\\

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

      associated to the path-space-fibration 

Tex syntax error
      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 
Tex syntax error
 is due to the Exercise 3.2.} 
Tex syntax error
, where 
Tex syntax error
 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
    .
  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 
    Tex syntax error
  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.

Definition 4.2. 

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 
    Tex syntax error
     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
    .

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). The homotopy fibre of w_2 is BSpin, where Spin is the colimit over Spin(n), the universal covers of SO(n).

References

\to EO\to BO$. \item Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\bbz/2\bbz)\;.$$ \item Calculate the homotopy fibre of $p:BSO\to BO$. \end{enumerate} Now we can classify the orientations on a vector bundle $\xi\to X$. For this we need to know that the sequence $$\rm{hofib}(p)\to BSO \to BO$$ fits into the following diagram\footnote{It's non-trivial to see that the functor $B$ can be applied to each of those spaces.}: \begin{center} \begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}] \matrix (m) [matrix of math nodes, row sep=3em, column sep=2.5em, text height=1.5ex, text depth=0.25ex] { \text{hofib}(p) & & BSO & & BO \ \Omega B\rm{hofib}(p) & & \Omega BBSO & & \Omega BBO \ }; \path[->,font=\scriptsize] (m-1-1) edge node[auto] {$q$} (m-1-3) (m-1-3) edge node[auto] {$p$} (m-1-5) (m-2-1) edge node[auto] {$\Omega B q$} (m-2-3) (m-2-3) edge node[auto] {$\Omega B p$} (m-2-5) (m-1-1) edge node[auto] {$\simeq$} (m-2-1) (m-1-3) edge node[auto] {$\simeq$} (m-2-3) (m-1-5) edge node[auto] {$\simeq$} (m-2-5); \end{tikzpicture} \end{center} Hence there are group structures on $[X,\rm{hofib}(p)]$ and $[X,BSO]$ (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of $[X,\rm{hofib}(p)]$ 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 is the space of paths starting at the base-point of Z.}

\displaystyle \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.

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].

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 by \Omega X.

Exercise 4.1.

  • Show: There is a homotopy equivalence 
    Tex syntax error
    .\\

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

      associated to the path-space-fibration 

Tex syntax error
      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 
Tex syntax error
 is due to the Exercise 3.2.} 
Tex syntax error
, where 
Tex syntax error
 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
    .
  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 
    Tex syntax error
  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.

Definition 4.2. 

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 
    Tex syntax error
     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
    .

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). The homotopy fibre of w_2 is BSpin, where Spin is the colimit over Spin(n), the universal covers of SO(n).

References

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