Structure set (Ex)
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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 .