# Structure set (Ex)

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.

$\displaystyle \begin{array}{rcl} \mathcal{S}(M) \times \mathcal{E}(M) & \to & \mathcal{S}(M),\\ ([f:N\to M],[g]) &\mapsto& [g\circ f: N\to M].\end{array}$
Show that the set $\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$$\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$ is in bijection with the set of h-cobordism classes of manifolds homotopy equivalent to $M$$M$.
• Let $\mathcal{S}^s(M) = \{[(N,f) \text{ such that } f\colon N\simeq_s M]\}$$\mathcal{S}^s(M) = \{[(N,f) \text{ such that } f\colon N\simeq_s M]\}$ be the simple structure set of a closed manifold and let $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$ be the group of simple homotopy self-equivalences of $M$$M$. Note that $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$ acts on $\mathcal{S}^s(M)$$\mathcal{S}^s(M)$ by post composition:
$\displaystyle \begin{array}{rcl} \mathcal{S}^s(M) \times \mathcal{E}^s(M) & \to & \mathcal{S}^s(M),\\ ([f:N\to M],[g]) &\mapsto & [g\circ f: N\to M].\end{array}$
Show that the set $\mathcal{M}^s(M):= \mathcal{S}^s(M)/\mathcal{E}^s(M)$$\mathcal{M}^s(M):= \mathcal{S}^s(M)/\mathcal{E}^s(M)$ is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to $M$$M$.