TITLE: Homology of Möbius Strip
QUESTION [16 upvotes]: How does one calculate the homology groups of the Möbius strip?
I'm thinking of two methods.

Use cellular homology. I tried to draw a delta-complex structure of the Möbius strip but I'm not sure if I'm right? I basically have a rectangle with opposite ends identified, so vertices (top-down) on the left are $a$ and $b$. On the right, they are $b$ and $a$. I know that there is an edge $e$ that joins from $b$ to $a$ and this is for both the left and right side of the rectangle. But this is not yet a delta complex structure, so I feel like I need to draw another edge across the rectangle from bottom left to top right, connecting $b$ to $b$. But how are these remaining 3 edges labelled? If I label the top and bottom edge, I think I'm creating another surface - the torus, RP2 or Klein bottle and so that's not the way to go. Does it mean this method is not possible?
Use $H_n(X^k,X)$ somehow, where $X^k$ is the $k$ skeleton of $X$. But I don't quite know how to proceed with this.

Any pointers in the right direction is greatly appreciated. Thanks!

REPLY [7 votes]: A slick way of calculating the homology groups here is to notice that the Möbius strip is homotopy equivalent to $S^1$. Then you can apply Corollary 2.11 (Section 2.1) of Hatcher that:
The maps $\ f_{\ast} : H_n(X) \rightarrow H_n(Y)$ induced by a homotopy equivalence $ \ f : X \rightarrow Y$ are isomorphisms for all $n$.
This should make the task much easier.