Structure set (Ex)

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Revision as of 21:12, 25 August 2013

Let $\mathcal{S} = \{[f:M\simeq N]\}$ $<wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]\mathcal{S} = \{[f:M\simeq N]\}$ 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 $M$ $M$ .

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Structure set (Ex)

Revision as of 21:12, 25 August 2013

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+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 $M$.
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 == References ==