Structures on M x I (Ex)

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Let $M$ be a closed oriented topological manifold is dimension $n \geq 5$ and let
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Let $M$ be a closed oriented topological manifold of dimension $n \geq 5$ and let
$\tilde \pi_0 \textup{Homeo}_+(M)$ denote the pseudo-isotopy group of orientation preserving self homeomorphisms of $M$: two self homeomorphisms $h_0$ and $h_1$ are pseudo-isotopy if then extend to a homeomorphism $F \colon M \times I \cong M \times I$: the group operation is composition. Let
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$\tilde \pi_0 \textup{Homeo}_+(M)$ denote the pseudo-isotopy group of orientation preserving self homeomorphisms of $M$: two self homeomorphisms $h_0$ and $h_1$ are pseudo-isotopic if then extend to a homeomorphism $F \colon M \times I \cong M \times I$: the group operation is composition. Let
$$ \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M) $$
$$ \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M) $$
be the subgroup of equivalences classes which are homotopic to the identity.
be the subgroup of equivalences classes which are homotopic to the identity.

Revision as of 10:34, 30 May 2012

Let M be a closed oriented topological manifold of dimension n \geq 5 and let \tilde \pi_0 \textup{Homeo}_+(M) denote the pseudo-isotopy group of orientation preserving self homeomorphisms of M: two self homeomorphisms h_0 and h_1 are pseudo-isotopic if then extend to a homeomorphism F \colon M \times I \cong M \times I: the group operation is composition. Let

\displaystyle  \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M)

be the subgroup of equivalences classes which are homotopic to the identity.

Exercise 0.1. Using the fact that the topological surgery exact sequence is a long exact sequence of abelian groups show that \tilde \pi_0 \textup{SHomeo}_+(M) is abelian.

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