Structure set (Ex)

From Manifold Atlas
(Difference between revisions)
Jump to: navigation, search
(Created page with "<wikitex>; ... </wikitex> == References == {{#RefList:}} Category:Exercises Category:Exercises without solution")
m
(3 intermediate revisions by 3 users not shown)
Line 1: Line 1:
<wikitex>;
<wikitex>;
...
+
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 $\mathcal{S}(M) = \{[(N,f) \text{ such that } f\colon N\simeq M]\}$ 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 $M$.
</wikitex>
</wikitex>
== References ==
== References ==
{{#RefList:}}
{{#RefList:}}
[[Category:Exercises]]
[[Category:Exercises]]
[[Category:Exercises without solution]]
+
[[Category:Exercises with solution]]

Latest revision as of 19:42, 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 \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:
    \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) 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:
    \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) is in bijection with the set of diffeomorphism classes of manifolds homotopy equivalent to M.

[edit] References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox