# Structure set (Ex)

From Manifold Atlas

The exercise has two parts depending on whether we are talking about homotopy equivalences or *simple* homotopy equivalence. Both proofs are rather careful reading of definitions and share the same idea.

- Let be the structure set of a closed manifold and let be the group of homotopy self-equivalences of . Note that acts on by post composition: Show that the set is in bijection with the set of h-cobordism classes of manifolds homotopy equivalent to .

- Let be the simple structure set of a closed manifold and let be the group of simple homotopy self-equivalences of . Note that acts on by post composition: Show that the set is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to .