# Structures on M x I (Ex)

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Let $M$$; Let M be a closed oriented topological manifold of dimension n \geq 5 and let \tilde \pi_0 \textup{Homeo}_+(M) denote the pseudo-isotopy group of orientation preserving self homeomorphisms of M: two self homeomorphisms h_0 and h_1 are pseudo-isotopic if then extend to a homeomorphism F \colon M \times I \cong M \times I: the group operation is composition. Let \tilde \pi_0 \textup{SHomeo}_+(M) \subset \tilde \pi_0 \textup{Homeo}_+(M) be the subgroup of equivalences classes which are homotopic to the identity. {{beginthm|Exercise}} Using the fact that the topological surgery exact sequence is a long exact sequence of ''abelian'' groups show that \tilde \pi_0 \textup{SHomeo}_+(M) is abelian. {{endthm}} [[Category:Exercises]] [[Category:Exercises without solution]]M$ be a closed oriented topological manifold of dimension $n \geq 5$$n \geq 5$ and let $\tilde \pi_0 \textup{Homeo}_+(M)$$\tilde \pi_0 \textup{Homeo}_+(M)$ denote the pseudo-isotopy group of orientation preserving self homeomorphisms of $M$$M$: two self homeomorphisms $h_0$$h_0$ and $h_1$$h_1$ are pseudo-isotopic if then extend to a homeomorphism $F \colon M \times I \cong M \times I$$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)$$\tilde \pi_0 \textup{SHomeo}_+(M)$ is abelian.