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I haven't posted here in some time, and I am in need of the expertise of you fine folks. I am busy doing some work on spin geometry. Now, as you guys know, spin structures exist on manifolds if their second Stiefel-Whitney class vanishes. This class is an element of the second cohomology group. So, one needs to be versed in calculating cohomology groups in order to start working with these spin manifols. Much to my embarrassment I discovered that, although I know my way around homotopy, my homology knowledge is very lacking.

The go-to book for algebraic topology is Hatcher's great book, and it is from this book that my question comes. In it, after quite a lot of theoretical build-up, we reach the punch line that cohomology groups can be calculated via the split short exact sequence

0 -> Ext(H_{n-1}(X),G) -> H^{n}(X;G) -> Hom(H_{n}(X),G) -> 0

This formula appears on page 198. The part I'm having some issues with is the Ext. I don't get an intuitive sense for what it's doing from the discussion on the preceding pages.

If anyone is able/willing to shed some light on this for me, I would greatly appreciate it.

Thanks in advance!

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# ALgebraic Topology Query (Hatcher) - Not Homework

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