Bundle structures and lifting problems (Ex)
m |
m |
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Line 21: | Line 21: | ||
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 $\Omega X$. | ||
{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
− | * Show: There is a homotopy equivalence $\Omega K(n+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 | '''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 | ||
− | $$\Omega K(n+1,\ | + | $$\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. | Recall that the path-space is contractible. | ||
* Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure | * Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure | ||
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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,\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). | 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,\ | + | Now recall that \ref{The group structure on $H^1(X,\Zz/2\Zz)$ is due to the Exercise 3.2.} |
− | $H^1(X,\ | + | $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. | 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: | ||
− | # The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\ | + | # 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 homotopy fiber hofib($w_1$) is $BSO$. | ||
# The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$. | # The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$. | ||
Line 45: | Line 45: | ||
Assume that the homotopy groups of $O$ are known. | Assume that the homotopy groups of $O$ are known. | ||
# Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$. | # Calculate the homotopy groups of $BO$ using the fibration $0\to EO\to BO$. | ||
− | # Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\ | + | # Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\Zz/2\Zz)\;.$$ |
# Calculate the homotopy fibre of $p:BSO\to BO$. | # Calculate the homotopy fibre of $p:BSO\to BO$. | ||
Revision as of 20:39, 28 March 2012
1 Lifting maps
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.
Exercise 2.1. Prove the following:
A map of pointed space has a lift along if and only if 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 by .
Exercise 4.1.
- Show: There is a homotopy equivalence .\\
Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy
associated to the path-space-fibration
Recall that the path-space is contractible.
- 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.
Maybe you've heard that the group acts free and transitively on the set of spin structures of an oriented vector bundle (X a compact pointed space). Now recall that \ref{The group structure on is due to the Exercise 3.2.} , where 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 .
- The homotopy fiber hofib() is .
- The projection is the map induced by .
Assume that the homotopy groups of are known.
- Calculate the homotopy groups of using the fibration .
- Calculate the homotopy groups of using the fibration
- Calculate the homotopy fibre of .
Now we can classify the orientations on a vector bundle . For this we need to know that the sequence
fits into the following diagram\footnote{It's non-trivial to see that the functor can be applied to each of those spaces.}:
Hence there are group structures on and (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of on induced by .
Definition 4.2.
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
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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
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.
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 . The homotopy fibre of is BSpin, where Spin is the colimit over , the universal covers of .
References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
and we denote by : the projection.
Exercise 2.1. Prove the following:
A map of pointed space has a lift along if and only if 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 by .
Exercise 4.1.
- Show: There is a homotopy equivalence .\\
Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy
associated to the path-space-fibration
Recall that the path-space is contractible.
- 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.
Maybe you've heard that the group acts free and transitively on the set of spin structures of an oriented vector bundle (X a compact pointed space). Now recall that \ref{The group structure on is due to the Exercise 3.2.} , where 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 .
- The homotopy fiber hofib() is .
- The projection is the map induced by .
Assume that the homotopy groups of are known.
- Calculate the homotopy groups of using the fibration .
- Calculate the homotopy groups of using the fibration
- Calculate the homotopy fibre of .
Now we can classify the orientations on a vector bundle . For this we need to know that the sequence
fits into the following diagram\footnote{It's non-trivial to see that the functor can be applied to each of those spaces.}:
Hence there are group structures on and (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of on induced by .
Definition 4.2.
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 4.3. 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
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001