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Let $W=[0,1]\times[0,1]$, where $(0,y)\sim(1,1-y)$ for all $0\leq y\leq 1$. Calculate the homology groups of $W$

I am a little confused with this exercise, I do not know how many vertices there are, nor 1-simplices, nor 2-simplices, could someone tell me this and the orientations? I think with this I could calculate the homology groups. Thank you very much

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Pedro
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user402543
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    This is the Mobius strip. You could triangulate it, and use simplicial homology. Or you could observe that is is homotopy equivalent to the circle. – Angina Seng Jun 05 '18 at 04:30
  • See https://math.stackexchange.com/questions/2529034/simplex-triangulation-of-cylinder-and-mobius-strip/2529134 which has a picture of a triangulation of the Möbius strip. – Cheerful Parsnip Jun 05 '18 at 05:07

1 Answers1

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One possible solution is to "convert" $W/\sim$ to something simplier and better understood:

The space $M:=W/\sim$ is the Mobius strip. Consider $A\subseteq M$ given by $A:=\{[t,1/2]_{\sim}\ |\ t\in[0,1]\}$ and note that $[0,1]\to A$ given by $t\mapsto [t,1/2]_\sim$ is a quotient map which induces homeomorphism $S^1\to A$.

Now consider the function

$$F:M\times[0,1]\to M$$ $$F([x,y]_\sim, t)=[x, t\frac{1}{2}+(1-t)y]_\sim$$

This $F$ is a deformation retraction meaning $A$ is a deformation retract of $M$. In particular $M$ is homotopy equivalent to $A$ being homeomorphic to $S^1$. So homologies of $M$ are the same as homologies of $S^1$.

freakish
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