Mapping torus is an (m+1)-manifold

In summary: Since \psi is injective and surjective, it suffices to show that it is continuous and its inverse is continuous. To show that \psi is continuous, we note that the preimage of an open set in M(f) under \psi is an open set in W, since \psi is a bijection. Similarly, the preimage of an open set in W under \psi^{-1} is an open set in M(f), so \psi^{-1} is continuous. Therefore, \psi is a homeomorphism.In summary, we can define a coordinate chart on the subset W of X \times [0,1] and show that it maps homeomorphically onto its image in M(f
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Homework Statement



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.

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?

Thanks.
 
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  • #2


Sure, let's break it down step by step:

1. First, we define a map \psi: W \to M(f) by \psi(t,x) = (x,t).

2. Next, we show that \psi is injective. Suppose \psi(t_1,x_1) = \psi(t_2,x_2). Then, we have (x_1,t_1) = (x_2,t_2) in M(f), which means that either t_1 = t_2 or x_1 = x_2. If t_1 = t_2, then (x_1,t_1) = (x_2,t_2) implies that x_1 = x_2, so we have (x_1,t_1) = (x_2,t_2) in X \times [0,1]. But this contradicts the fact that (x_1,0) \sim (f(x_1),1) and (x_2,0) \sim (f(x_2),1) in M(f). Similarly, if x_1 = x_2, then (x_1,t_1) = (x_2,t_2) implies that t_1 = t_2, again contradicting the fact that (x_1,0) \sim (f(x_1),1) and (x_2,0) \sim (f(x_2),1) in M(f). Therefore, \psi is injective.

3. Next, we show that \psi is surjective. Let (y,s) \in M(f). Then, either s \neq 0 or y \notin f(U). If s \neq 0, then (y,s) \in (1-\epsilon,1]\times f(U) \subset W, so there exists some (x,t) \in W such that \psi(x,t) = (y,s). If y \notin f(U), then y \in X \setminus f(U), so (y,s) \in (0,\epsilon) \times X \setminus f(U) \subset W. Again, there exists some (x,t) \in W such that \psi(x,t) = (y,s). Therefore, \psi is surjective.

4. Finally, we show that \psi is a
 

FAQ: Mapping torus is an (m+1)-manifold

1. What is a mapping torus?

A mapping torus is a mathematical construct that is created by taking a space and attaching a cylinder to it in a specific way. This results in a new space that is topologically equivalent to a torus, hence the name "mapping torus".

2. How is a mapping torus related to manifolds?

A mapping torus can be thought of as a type of manifold, specifically a (m+1)-manifold where m is the dimension of the original space. This means that the mapping torus has m+1 dimensions and can be smoothly deformed into a torus.

3. What is the significance of the (m+1)-manifold in relation to mapping tori?

The (m+1)-manifold is significant because it allows us to extend the original space into a higher dimension while still preserving its topological properties. This is useful in many mathematical and scientific applications, such as in the study of knots and surfaces.

4. How is the dimension of a mapping torus determined?

The dimension of a mapping torus is determined by the dimension of the original space, which is denoted by m, and the number of times the cylinder is attached, denoted by n. The resulting dimension is (m+1)n, so if the original space is a 2-dimensional surface and the cylinder is attached twice, the resulting mapping torus will be a 6-dimensional manifold.

5. Can mapping tori be used to study higher-dimensional spaces?

Yes, mapping tori can be used to study higher-dimensional spaces by attaching higher-dimensional objects, such as cubes or spheres, instead of just cylinders. This can lead to a better understanding of the topological properties of these higher-dimensional spaces.

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