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Mapping torus is an (m+1)-manifold

  1. Nov 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Let X be an m-manifold. Let M(f) be the space obtained from [itex]X\times [0,1][/itex] by gluing the ends together using [itex](x,0)\sim (f(x),1)[/itex]. 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 [itex]\{ (U_\alpha,\varphi_\alpha) \}[/itex], my first instinct was to define coordinate charts by [itex] \psi_\alpha : U_\alpha \times [0,1] \to \mathbb{R}^{m+1}; (x,t) \to (\varphi_\alpha(x),t)[/itex] 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 [itex]1<t< 1[/itex] in the usual way and to separately consider a point on the gluing edge [itex](x,0) \sim (f(x),1)[/itex]. There is a coordinate chart [itex]\varphi : U \subset X \to \mathbb{R}^n[/itex] where U is an open nbhd of x. This gives a coordinate chart at f(x) by [itex](f(U),\varphi\circ f^{-1})[/itex]. Now consider the subset of [itex]X \times [0,1][/itex] given by [itex]W = [0,\epsilon) \times U \cup (1-\epsilon,1]\times f(U)[/itex]. 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?

  2. jcsd
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