# Thom spaces (Ex)

From Manifold Atlas

**Exercise 0.1.**
Let be -complexes and let be vector bundles
over respectively.
Denote by the product bundle over .
Find homeomorphisms

With the following exercises we work out the details of [Lück2001, page 58f].

**Exercise 0.2.**
Let be the universal oriented vector bundle of rank
and let : be a bundle map. Define

Show that for all we have .

**Exercise 0.3.**
Define

and

where : is the suspension homomorphism. Show that for all we have .

**Question 0.4.**
Can we do similar things for unoriented manifolds, manifolds with spin structure,...?