Structure set (Ex)

Let $\mathcal{S} = \{[f:M\simeq N]\}$$<wikitex>; Let $\mathcal{S} = \{[f:M\simeq N]\}$ be the structure set of a closed manifold and let $\mathcal{E}(M)$ be the group of homotopy self-equivalences of $M$. Note that $\mathcal{E}(M)$ acts on $\mathcal{S}(M)$ by post composition: $$ \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}$$ Show that the set $\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$ is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to $M$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]\mathcal{S} = \{[f:M\simeq N]\}$ be the structure set of a closed manifold and let $\mathcal{E}(M)$$\mathcal{E}(M)$ be the group of homotopy self-equivalences of $M$$M$. Note that $\mathcal{E}(M)$$\mathcal{E}(M)$ acts on $\mathcal{S}(M)$$\mathcal{S}(M)$ by post composition:

$\displaystyle \begin{array}{rcl} \mathcal{S}(M) \times \mathcal{E}(M) & \to & \mathcal{S}(M),\\ ([f:N\to M],[g]) &\mapsto& [g\circ f: N\to M].\end{array}$ $$\displaystyle \begin{array}{rcl} \mathcal{S}(M) \times \mathcal{E}(M) & \to & \mathcal{S}(M),\\ ([f:N\to M],[g]) &\mapsto& [g\circ f: N\to M].\end{array}$$

Show that the set $\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$$\mathcal{M}(M):= \mathcal{S}(M)/\mathcal{E}(M)$ is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to $M$$M$.