Homology of Möbius Strip How does one calculate the homology groups of the Möbius strip?
I'm thinking of two methods.


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*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!

Take the usual paper strip that you would glue to make a Möbius strip.  If you want to get a simplicial complex, draw a line diagonally from the top corner on the left to the bottom corner on the right, so your paper strip is a union of two triangles.  
Your strip now has two triangular faces, five 1-simplices (the four edges of the paper strip, together with the line you drew to divide it into two triangles), and 4 vertices.  But when you glue you identify the two edges at either end of the strip, so for the Möbius band there are two faces, four 1-simplices, and 2 vertices.  
Now just write down the corresponding simplicial chain complex and compute away!
