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I need to find a bijection and inverse of the following:

$X = \{ (x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4 \}$ with its subspace topology in $\mathbb{R}^2$

$Y = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1$ and $ 0 \leq z \leq 1 \}$ with its subspace topology in $\mathbb{R}^3$

Then show they are homeomorphic. Not sure where to start.

Gerry Myerson
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user64013
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2 Answers2

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Hint: $X$ is an annulus, and $Y$ is a cylinder. Imagine "flattening" a cylinder by forcing its bottom edge inwards and its top edge outwards; it will eventually become an annulus. This is the idea of the map; you should work out the actual expressions describing it on your own.

Here is a gif animation I made with Mathematica to illustrate the idea:

enter image description here

F[R_][t_, x_, y_] := {(R + x) Cos[t], (R + x) Sin[t], y}

BentCylinder[R_, r_, s_, t_, z_] := F[R][t, r + s*Sin[z], s*Cos[z]]

BendingCylinder[R_, r_, z_] := 
 ParametricPlot3D[
  BentCylinder[R, r, s, t, z], {s, -r, r}, {t, 0, 2 Pi}, Mesh -> None,
   Boxed -> False, Axes -> None, PlotStyle -> Red, 
  PlotRange -> {{-10, 10}, {-10, 10}, {-5, 5}}, PlotPoints -> 50]

Export["animation.gif", 
 Table[BendingCylinder[6, 2, z], {z, 0, Pi/2, 0.02*Pi}], 
 "DisplayDurations" -> {0.25}]
Zev Chonoles
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  • Oh ya. I see what you mean by that. I completely forgot that the function I have produces an annulus. Visually, it makes complete sense to me. Its the actual writing it down part I never seem to get. Whats the bijection then? I am not seeing it right away. – user64013 Feb 27 '13 at 06:14
  • Perhaps it'd be easier to figure out the function that takes the top edge of the cylinder to the outer edge of the annulus, then find the function taking the bottom edge of the cylinder to the inner edge of the annulus, and then "interpolate" them. – Zev Chonoles Feb 27 '13 at 06:19
  • Wait a minute. Could I just put parameters on each shape. The cylinder would have parameters $(cos(a),sin(a),b)$ and the annulus would have parameters $(acos(x),asin(x))$. – user64013 Feb 27 '13 at 06:29
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You can use the similar idea of Stereographic Projection. Take a point above the cylinder along the axix, say $P:=(0,0,2)$. Now join $P$ by a straight line to each point of the cylinder say $p$ and extend it till it hits the $x-y$ plane at point, say $p^\prime$. Then the map $p\mapsto p^\prime$ maps the cylinder to an annulus. Now scale that annulus properly to get your desired annulus.

pritam
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