Spivak normal fibration (Ex)
In the following exercises is a connected Poincaré complex of formal dimension and is a compact manifold of dimension .
Exercise 0.1. Let be a compact, connected, oriented, -dimensional manifold with boundary, embedded in . The collapse map is defined by
Let be the Hurewicz homomorphism, show that
Exercise 0.2. 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 0.3. 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]
Remark 0.4. Exercise 0.2 is a diffcult problem. It was solved in greater generality in [Klein2001a, Theorem I].