Spivak normal fibration (Ex)

In the following exercises $X$$<wikitex>; In the following exercises $X$ be a connected Poincaré complex of formal dimension $n$. {{beginthm|Exercise}} Let $\xi \colon E \to X$ be a spherical fibration $X$ with homotopy fibre $S^k$. Show that $E$ is homotopy equivalent to a Poincaré complex of formal dimension $n + k$. {{endthm}} Here is an interesting problem we now confront {{beginthm|Problem}} Determine the Spivak normal fibration of $E$ above in terms of $\xi$ and the Spivak normal fibration of $X$. {{endthm}} Here are some hints for this problem: [[Tangent bundles of bundles (Ex)]], {{citeD|Wall1966a}}, {{citeD|Chazin1975}} {{beginthm|Exercise}} Let $(M,\partial M)$ be a compact, connected, oriented, $n$-dimensional manifold with boundary, embedded in $\mathbb{R}^n$. The collapse map $c:S^n\to M/\partial M$ is defined by$$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]$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]X$ be a connected Poincaré complex of formal dimension $n$$n$.

Exercise 0.1. 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 0.2. 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]

Exercise 0.3.

Let $(M,\partial M)$$(M,\partial M)$ be a compact, connected, oriented, $n$$n$-dimensional manifold with boundary, embedded in $\mathbb{R}^n$$\mathbb{R}^n$. The collapse map $c:S^n\to M/\partial M$$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

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