Spivak normal fibration (Ex)

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In the following exercises X is a connected Poincaré complex of formal dimension n and M is a compact manifold of dimension n.

Exercise 0.1. 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

\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) be the Hurewicz homomorphism, show that

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

Exercise 0.2. 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.

Here is an interesting problem we now confront

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

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