Bundle structures and lifting problems (Ex)
Given a (pointed) map of pointed topological space, we define the homotopy fiber as\footnote{Here is the space of paths starting at the base-point of .}
and we denote by : the projection.
Show the following statement:\\A map of pointed space has a lift
Tex syntax erroralong
if and only if is homotopic to the constant map.
Hint: This is a special case of Proposition 4.72 in [A. Hatcher, Algebraic Topology].
\subsection*{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 .
\begin{enumerate}
\item Show: There is a homotopy equivalenceTex 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. \item 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.
\end{enumerate}
Tex syntax erroracts free and transitively on the set
of spin structures of an oriented vector bundle (X a compact pointed space).
Now recall that\footnote{The group structure onTex syntax erroris due to the Exercise 3.2.}
Tex syntax error, where
Tex syntax errordenotes an Eilenberg-MacLane-space.
This looks similar to the statement of Theorem 3.45 in our talk.
So we first prove the statement about classification of spin structures. The warm-up is the classification of orientations: \begin{itemize}
\item The first Stiefel-Whitney class is a mapTex syntax error.
\item The homotopy fiber hofib() is . \item The projection is the map induced by .
\end{itemize}
Assume that the homotopy groups of are known. \begin{enumerate} \item Calculate the homotopy groups of using the fibration .\item Calculate the homotopy groups of using the fibration
Tex syntax error
\item Calculate the homotopy fibre of . \end{enumerate}
Now we can classify the orientations on a vector bundle . For this we need to know that the sequence
Tex syntax error
fits into the following diagram\footnote{It's non-trivial to see that the functor 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] {} (m-1-3) (m-1-3) edge node[auto] {} (m-1-5) (m-2-1) edge node[auto] {} (m-2-3) (m-2-3) edge node[auto] {} (m-2-5) (m-1-1) edge node[auto] {} (m-2-1) (m-1-3) edge node[auto] {} (m-2-3) (m-1-5) edge node[auto] {} (m-2-5); \end{tikzpicture} \end{center}
Hence there are group structures onTex syntax errorand (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of
Tex syntax erroron induced
by .
Definition 0.1.
A vector bundle is called \underline{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
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
- Given an interpretation of the group
Tex syntax error
.
Now there are similar results for spin structures (on oriented vector bundles).
Exercise 0.2. 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 covers of .
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 is the space of paths starting at the base-point of .}and we denote by : the projection.
Show the following statement:\\A map of pointed space has a lift
Tex syntax erroralong
if and only if is homotopic to the constant map.
Hint: This is a special case of Proposition 4.72 in [A. Hatcher, Algebraic Topology].
\subsection*{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 .
\begin{enumerate}
\item Show: There is a homotopy equivalenceTex 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. \item 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.
\end{enumerate}
Tex syntax erroracts free and transitively on the set
of spin structures of an oriented vector bundle (X a compact pointed space).
Now recall that\footnote{The group structure onTex syntax erroris due to the Exercise 3.2.}
Tex syntax error, where
Tex syntax errordenotes an Eilenberg-MacLane-space.
This looks similar to the statement of Theorem 3.45 in our talk.
So we first prove the statement about classification of spin structures. The warm-up is the classification of orientations: \begin{itemize}
\item The first Stiefel-Whitney class is a mapTex syntax error.
\item The homotopy fiber hofib() is . \item The projection is the map induced by .
\end{itemize}
Assume that the homotopy groups of are known. \begin{enumerate} \item Calculate the homotopy groups of using the fibration .\item Calculate the homotopy groups of using the fibration
Tex syntax error
\item Calculate the homotopy fibre of . \end{enumerate}
Now we can classify the orientations on a vector bundle . For this we need to know that the sequence
Tex syntax error
fits into the following diagram\footnote{It's non-trivial to see that the functor 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] {} (m-1-3) (m-1-3) edge node[auto] {} (m-1-5) (m-2-1) edge node[auto] {} (m-2-3) (m-2-3) edge node[auto] {} (m-2-5) (m-1-1) edge node[auto] {} (m-2-1) (m-1-3) edge node[auto] {} (m-2-3) (m-1-5) edge node[auto] {} (m-2-5); \end{tikzpicture} \end{center}
Hence there are group structures onTex syntax errorand (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of
Tex syntax erroron induced
by .
Definition 0.1.
A vector bundle is called \underline{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
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
- Given an interpretation of the group
Tex syntax error
.
Now there are similar results for spin structures (on oriented vector bundles).
Exercise 0.2. 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 covers of .