Spivak normal fibration (Ex)

In the following exercises $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ is a connected Poincaré complex of formal dimension $n$$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].