# Homework Help: Mapping torus is an (m+1)-manifold

1. Nov 9, 2007

### noospace

1. The problem statement, all variables and given/known data

Let X be an m-manifold. Let M(f) be the space obtained from $X\times [0,1]$ by gluing the ends together using $(x,0)\sim (f(x),1)$. Show that if M is an m-manifold then M(f) is an (m+1)-manifold.

3. The attempt at a solution

Since X has an atlas $\{ (U_\alpha,\varphi_\alpha) \}$, my first instinct was to define coordinate charts by $\psi_\alpha : U_\alpha \times [0,1] \to \mathbb{R}^{m+1}; (x,t) \to (\varphi_\alpha(x),t)$ but then we have to worry about the end-points. Right now I'm attempting to understand the solutions.

The idea behind the solutions is to map points with $1<t< 1$ in the usual way and to separately consider a point on the gluing edge $(x,0) \sim (f(x),1)$. There is a coordinate chart $\varphi : U \subset X \to \mathbb{R}^n$ where U is an open nbhd of x. This gives a coordinate chart at f(x) by $(f(U),\varphi\circ f^{-1})$. Now consider the subset of $X \times [0,1]$ given by $W = [0,\epsilon) \times U \cup (1-\epsilon,1]\times f(U)$. The claim is that this maps homeomorphically onto its image in M(f) but I don't see why. Can anyone help me understand this?

Thanks.