Spivak normal fibration (Ex)
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{{beginthm|Exercise}} | {{beginthm|Exercise}} | ||
− | Let $(M, \partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in $ | + | 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 |
$$c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else}\end{cases}$$ | $$c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else}\end{cases}$$ | ||
Let $h:\pi_n(M, \partial M)\to H_n(M, \partial M)$ be the [[Hurewicz homomorphism]], show that | Let $h:\pi_n(M, \partial M)\to H_n(M, \partial M)$ be the [[Hurewicz homomorphism]], show that |
Revision as of 07:39, 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 -sphere . 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].