Spivak normal fibration (Ex)

In the following exercises $X$$<wikitex>; In the following exercises $X$ is a connected Poincaré complex of formal dimension $n$ and $M$ is a compact manifold of dimension $n$. {{beginthm|Exercise}} Let $(M, \partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in the $n$-sphere $S^n$. The collapse map $c:S^n\to M/\partial M$ is defined by $$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]$$ {{endthm}} {{beginthm|Exercise}} \label{ex1} Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$. {{endthm}} Here is an interesting problem we now confront {{beginthm|Problem}} Determine the Spivak normal fibration of $E$ above in terms of $\xi$ and the Spivak normal fibration of $X$. {{endthm}} Here are some hints for this problem: [[Tangent bundles of bundles (Ex)]], {{citeD|Wall1966a}}, {{citeD|Chazin1975}} {{beginrem|Remark}} Exercise \ref{ex1} is a diffcult problem. It was solved in greater generality in {{citeD|Klein2001|Theorem I}}. {{endrem}} </wikitex> <!-- == References == {{#RefList:}} --> [[Category:Exercises]] [[Category:Exercises without solution]]X$ is a connected Poincaré complex of formal dimension $n$$n$ and $M$$M$ is a compact manifold of dimension $n$$n$.

Exercise 0.1. Let $(M, \partial M)$$(M, \partial M)$ be a compact, connected, oriented, $n$$n$-dimensional manifold with boundary, embedded in the $n$$n$-sphere $S^n$$S^n$. The collapse map $c:S^n\to M/\partial M$$c:S^n\to M/\partial M$ is defined by

$\displaystyle c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else}\end{cases}$ $$\displaystyle c(x)=\begin{cases} x&\text{if }x\in M-\partial M\\\{\partial M\}&\text{else}\end{cases}$$

Let $h:\pi_n(M, \partial M)\to H_n(M, \partial M)$$h:\pi_n(M, \partial M)\to H_n(M, \partial M)$ be the Hurewicz homomorphism, show that

$\displaystyle h([c])=\pm [M,\partial M]$ $$\displaystyle h([c])=\pm [M,\partial M]$$

Exercise 0.2. Let $\xi \colon E \to X$$\xi \colon E \to X$ be a spherical fibration $X$$X$ with homotopy fibre $S^k$$S^k$. Show that $E$$E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$$n + k$.

Here is an interesting problem we now confront

Problem 0.3. Determine the Spivak normal fibration of $E$$E$ above in terms of $\xi$$\xi$ and the Spivak normal fibration of $X$$X$.

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 [Klein2001, Theorem I].