Spivak normal fibration (Ex)

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	<wikitex>;		<wikitex>;
−	In the following exercises $X$ be a connected Poincaré complex of formal dimension $n$.	+	In the following exercises $X$ is a connected Poincaré complex of formal dimension $n$ and $M$ is a compact manifold of dimension $n$.
		+
	{{beginthm|Exercise}}		{{beginthm|Exercise}}
		+	Let $(M, \partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in the $n$-sphere $S^n$. The collapse map $c:S^n\to M/\partial M$ is defined by
		+	$$c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else}\end{cases}$$
		+	Let $h:\pi_n(M, \partial M)\to H_n(M, \partial M)$ be the [[Hurewicz homomorphism]], show that
		+	$$h([c])=\pm [M,\partial M]$$
		+	{{endthm}}
		+
		+	{{beginthm|Exercise}} \label{ex1}
	Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$.		Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$.
	{{endthm}}		{{endthm}}
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	Determine the Spivak normal fibration of $E$ above in terms of $\xi$ and the Spivak normal fibration of $X$.		Determine the Spivak normal fibration of $E$ above in terms of $\xi$ and the Spivak normal fibration of $X$.
	{{endthm}}		{{endthm}}
		+
	Here are some hints for this problem: [[Tangent bundles of bundles (Ex)]], {{citeD|Wall1966a}}, {{citeD|Chazin1975}}		Here are some hints for this problem: [[Tangent bundles of bundles (Ex)]], {{citeD|Wall1966a}}, {{citeD|Chazin1975}}
−	{{beginthm|Exercise}}	+
−	Let $(M,\partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in $\mathbb{R}^n$. The collapse map $c:S^n\to M/\partial M$ is defined by$$c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else} \end{cases}$$Let $h:\pi_n(M/\partial M)\to H_n(M/\partial M)$ be the Hurewicz homomorphism, show that$$h([c])=\pm [M,\partial M]$$	+	{{beginrem|Remark}}
−	{{endthm}}	+	Exercise \ref{ex1} is a diffcult problem. It was solved in greater generality in {{citeD|Klein2001|Theorem I}}.
		+	{{endrem}}
	</wikitex>		</wikitex>
	<!-- == References ==		<!-- == References ==

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Latest revision as of 07:41, 30 May 2012

In the following exercises $<wikitex>; In the following exercises $X$ be a connected Poincaré complex of formal dimension $n$. {{beginthm|Exercise}} Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$. {{endthm}} Here is an interesting problem we now confront {{beginthm|Problem}} Determine the Spivak normal fibration of $E$ above in terms of $\xi$ and the Spivak normal fibration of $X$. {{endthm}} Here are some hints for this problem: [[Tangent bundles of bundles (Ex)]], {{citeD|Wall1966a}}, {{citeD|Chazin1975}} {{beginthm|Exercise}} Let $(M,\partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in $\mathbb{R}^n$. The collapse map $c:S^n\to M/\partial M$ is defined by$$c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\{\partial M\}&\text{else} \end{cases}$$Let $h:\pi_n(M/\partial M)\to H_n(M/\partial M)$ be the Hurewicz homomorphism, show that$$h([c])=\pm [M,\partial M]$$ {{endthm}} </wikitex> <!-- == References == {{#RefList:}} --> [[Category:Exercises]] [[Category:Exercises without solution]]X$ is a connected Poincaré complex of formal dimension $n$ and

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$M$ is a compact manifold of dimension $n$.

Here is an interesting problem we now confront

Here are some hints for this problem: Tangent bundles of bundles (Ex), [Wall1966a], [Chazin1975]