# Spivak normal fibration (Ex)

From Manifold Atlas

In the following exercises is a connected PoincarĂ© 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 -sphere . 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 PoincarĂ© 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]