Spivak normal fibration (Ex)
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− | In the following exercises $X$ | + | In the following exercises $X$ is a connected Poincaré complex of formal dimension $n$ and $M$ is a compact manifold of dimension $n$. |
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{{beginthm|Exercise}} | {{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}} | ||
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+ | {{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}} | ||
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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| | + | |
− | + | {{beginthm|Remark}} | |
+ | Exercise \ref{ex1} is a diffcult problem. It was solved in greater generality in {{citeD|Klein2001a|Theorem I}}. | ||
{{endthm}} | {{endthm}} | ||
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</wikitex> | </wikitex> | ||
<!-- == References == | <!-- == References == |
Revision as of 07:33, 30 May 2012
In the following exercises is a connected Poincaré complex of formal dimension and is a compact manifold of dimension .
Exercise 0.1. Let be a compact, connected, oriented, -dimensional manifold with boundary, embedded in . The collapse map is defined by
Let be the Hurewicz homomorphism, show that
Exercise 0.2. Let be a spherical fibration with homotopy fibre . Show that is homotopy equivalent to a Poincaré complex of formal dimension .
Here is an interesting problem we now confront
Problem 0.3. Determine the Spivak normal fibration of above in terms of and the Spivak normal fibration of .
Here are some hints for this problem: Tangent bundles of bundles (Ex), [Wall1966a], [Chazin1975]
Remark 0.4. Exercise 0.2 is a diffcult problem. It was solved in greater generality in [Klein2001a, Theorem I].