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I've been trying to solve a problem for a while.

Let $W$ be an $n$-manifold and $F:M\to M$ be a diffeomorphism. The suspension of $F$ is defined by taking $M\times [0,1]$ and identifying every point $(x,0)\in M$ with $(F(x),1)$, and is denoted $M_F$. Show that $M_F$ is an $(n+1)$-manifold.

Here's what I've done so far:

Assume $\mathcal{A}=\{(U_\alpha, \varphi_\alpha)\}$ is an $n$-atlas on $M$. I was able to show that $\mathcal{B}=\{(F(U_\alpha),\varphi\circ F^{-1})\}$ is an $n$-atlas on $M$ as well. Given this, the problem comes down to showing that $\{(U_\alpha, \varphi_\alpha)\}$ and $\{F(U_\alpha),\varphi_\alpha\circ F^{-1})\}$ are compatible, but I can't seem to show this. Any help would be great. Thanks.

Patrick Da Silva
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user124083
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  • By identifying you mean like a quotient topology? – Patrick Da Silva Jan 27 '14 at 02:36
  • The transition functions are going to look like $\varphi_{\alpha} \circ F^{-1} \circ \varphi_{\beta}^{-1}$ and $\varphi_{\beta} \circ (\varphi_{\alpha} \circ F^{-1})^{-1} = \varphi_{\beta} \circ F \circ \varphi_{\alpha}^{-1}$. These maps are smooth precisely because $F$ is a diffeomorphism ; thus you have nothing to do to show that the maps you described are compatible. – Patrick Da Silva Jan 27 '14 at 02:39
  • What is $W$? Is that a typo? – user89987 Jan 27 '14 at 13:31

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