Bundle structures and lifting problems (Ex)

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	<wikitex>;		<wikitex>;
	== Lifting maps ==		== Lifting maps ==
−	Given a (pointed) map $g: Y\to Z$ of pointed topological space, we define the homotopy fiber as\footnote{Here	+	Given a (pointed) map $g: Y\to Z$ of pointed topological spaces, we define the homotopy fibre of $g$ as
−	$PZ$ 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 $p$: $\text{hofib}(g)\to Y$ the projection.	+	where $PZ$ is the space of paths starting at the base-point of $Z$. We denote by $p$: $\mathrm{hofib}(g)\to Y$ the projection.
	{{beginthm|Exercise}}		{{beginthm|Exercise}}
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	== Classification of orientations and spin structures on vector bundles ==		== 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 $\Omega X$.
	{{beginthm|Exercise}}		{{beginthm|Exercise}}
−	* Show: There is a homotopy equivalence $\Omega K(n+1,\Zz/2\Zz)\simeq K(n,\Zz/2\Zz)$.\\	+	* 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	+	$$\Omega K(n+1,\Zz/2 \Zz)\to P(K(n+1,\Zz/2 \Zz))\to K(n+1,\Zz/2 \Zz) \;.$$
−	$$\Omega K(n+1,\Zz/2\Zz)\to P(K(n+1,\Zz/2\Zz))\to K(n+1,\Zz/2\Zz) \;.$$	+
−	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 induced by composition of paths.
−	induced by composition of paths.	+
	'''Hint''': This is similar to the group structure of the fundamental group.		'''Hint''': This is similar to the group structure of the fundamental group.
		+	{{endthm}}
	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
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	# 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 $SO\hookrightarrow O$.
		+	{{beginthm|Exercise}}
	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 $O\to EO\to BO$.
−	# Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\to K(1,\Zz/2\Zz)\;.$$	+	# Calculate the homotopy groups of $BSO$ using the fibration $$BSO\to BO\stackrel{w_1}{\to} K(1,\Zz/2\Zz)\;.$$
	# Calculate the homotopy fibre of $p:BSO\to BO$.		# Calculate the homotopy fibre of $p:BSO\to BO$.
		+	{{endthm}}
−	Now we can classify the orientations on a vector bundle $\xi\to X$.	+	Now we can classify the orientations on a vector bundle $\xi: E\to X$.
	For this we need to know that the sequence		For this we need to know that the sequence
	$$\text{hofib}(p)\to BSO \to BO$$		$$\text{hofib}(p)\to BSO \to BO$$
	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.}:
		+	$$
		+	\xymatrix{
		+	\mathrm{hofib}(p) \ar[r]^q \ar[d]^{\simeq} & BSO \ar[r]^p \ar[d]^{\simeq} & BO \ar[d]^{\simeq} \\
		+	\Omega B\mathrm{hofib}(p) \ar[r]^{\Omega Bq} & \Omega BBSO \ar[r]^{\Omega Bp} & \Omega BBO }
		+	$$
	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 $q$.		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}}		{{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.	+	A vector bundle is called ''orientable'' if its classifying map $f: X \to BO$ lifts along $p:BSO \to BO$. An ''orientation'' is the choice of such a lift.
		+	The first Stiefel-Whitney class of $\xi$ is defined as the composition $w_1\circ f$, where $w_1$ is defined as above.
	{{endthm}}		{{endthm}}
−	Let $X$ denote a compact pointed space and $\xi\to X$ a vector bundle on $X$.	+	{{beginthm|Exercise}}
−	#Use Exercise 3.1 to show that $\xi$ is orientable if and only if its first Stiefel-Whitney class	+	Let $X$ denote a compact pointed space and $\xi: E\to X$ a vector bundle on $X$.
−	vanishes.	+	#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	+	#Show that the group $[X,\text{hofib}(p)]$ acts free and transitively on the set of homotopy classes of lifts.
−	homotopy classes of lifts.	+
	'''Hint:''' Use the homotopy-lifting property and Exercise 3.1		'''Hint:''' Use the homotopy-lifting property and Exercise 3.1
−	#Given an interpretation of the group $[X,\rm{hofib}(p)]$.	+	#Give an interpretation of the group $[X,\text{hofib}(p)]$.
	Now there are similar results for spin structures (on oriented vector bundles).		Now there are similar results for spin structures (on oriented vector bundles).
−
	{{beginthm|Exercise}}		{{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 $SO(n)$.	+	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 cover of $SO(n)$.
	{{endthm}}		{{endthm}}
	</wikitex>		</wikitex>
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	{{#RefList:}}		{{#RefList:}}
	[[Category:Exercises]]		[[Category:Exercises]]
		+	[[Category:Exercises with solution]]

---

Latest revision as of 17:53, 12 April 2012

[edit] 1 Lifting maps

Given a (pointed) map $<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 $\Omega K(n+1,\Zz/2\Zz)\simeq K(n,\Zz/2\Zz)$.\ '''Hint''': Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy associated to the path-space-fibration $$\Omega K(n+1,\Zz/2\Zz)\to P(K(n+1,\Zz/2\Zz))\to K(n+1,\Zz/2\Zz) \;.$$ 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: # 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 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 spaces, we define the homotopy fibre of $g$ as

where $PZ$ is the space of paths starting at the base-point of $Z$. We denote by $p$: $\mathrm{hofib}(g)\to Y$ the projection.

[edit] 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 $\Omega X$.

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 $SO\hookrightarrow O$.

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

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 $q$.

[edit] 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 $$BSO\to BO\to K(1,\Zz/2\Zz)\;.$$ # 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 spaces, we define the homotopy fibre of $g$ as

where $PZ$ is the space of paths starting at the base-point of $Z$. We denote by $p$: $\mathrm{hofib}(g)\to Y$ the projection.

[edit] 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 $\Omega X$.

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 $SO\hookrightarrow O$.

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

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 $q$.

[edit] References

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