# Structures on M x I (Ex)

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− | Let $M$ be a closed oriented topological manifold | + | 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- | + | $\tilde \pi_0 \textup{Homeo}_+(M)$ 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 be a closed oriented topological manifold of dimension and let denote the pseudo-isotopy group of orientation preserving self homeomorphisms of : two self homeomorphisms and are pseudo-isotopic if then extend to a homeomorphism : the group operation is composition. Let

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 is abelian.