# Structure set (Ex)

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Marek Kaluba (Talk | contribs) (Corrected formulation of the exercise; Added the simple version) |
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− | Let $\mathcal{S} = \{[f:M\ | + | 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. |

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+ | * Let $\mathcal{S}(M) = \{[(N,f) \text{ such that } f\colon N\simeq M]\}$ 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 h-cobordism classes of manifolds homotopy equivalent to $M$. | ||

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

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== References == | == References == |

## Revision as of 18:07, 27 August 2013

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 .