Spivak normal fibration (Ex)
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{{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$$h([c])=\pm [M,\partial M]$$ | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 15:45, 29 March 2012
In the following exercises be a connected Poincaré complex of formal dimension .
Exercise 0.1. 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 0.2. 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]
Exercise 0.3.
Let be a compact, connected, oriented, -dimensional manifold with boundary, embedded in . The collapse map is defined byLet be the Hurewicz homomorphism, show that