Structure set (Ex)
From Manifold Atlas
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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. | 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 $M$. | * 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 $M$. | ||
* 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 $M$. | * 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 == |
Latest revision as of 20:42, 28 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
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. 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 toTex syntax error
.
- Let be the simple structure set of a closed manifold and let be the group of simple homotopy self-equivalences of
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. Note that acts on by post composition:Show that the set is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent toTex syntax error
.