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I am going through some exercises in Hatcher's Algebraic Topology. You have a $\Delta$-complex obtained from $\Delta^3$ (a tetrahedron) and perform edge identifications $[v_0,v_1]\sim[v_1,v_3]$ and $[v_0,v_2]\sim[v_2,v_3]$. How can you show that this deformation retracts onto a Klein bottle?

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1 Answer 1

flatten the tetrahedron and draw it in the plane (triangle with a vertex inside and edges going out to the vertices of the triangle). if you cut it up a little, you're looking at the standard "rectangle-with-sides-identified" picture of the klein bottle. sorry for the terrible picture, mspaint hasnt changed since 3.x as far as i can tell...

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I remember from doing this exercise that the rectangle-with-sides-identified isn't quite the standard one (by which I mean pairs of opposite sides identified, one with a twist). The edge orientations that are specified by the delta-complex structure mean that you end up with something that needs a little cutting and gluing to see that it is your friendly ordinary klein bottle. –  NKS Jan 18 '12 at 16:50
    
Bit confused about how to go about the squishing of it. Whenever I try it doesn't get to the Klein bottle square. –  09867 Jan 18 '12 at 16:52
    
@NKS yes you do have to cut it up, my bad –  yoyo Jan 18 '12 at 19:12

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