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
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 $S^1$.