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I don't see the point in using homology and cohomology with coefficients in the field $\mathbb{Z}/2\mathbb{Z}$.

Can you provide some examples for why this is useful?

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    One quick reason: a manifold is always Z/2Z orientable. This allows results like Poincare duality to work.2011-06-23
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    Thanks! Where can I find a reference for this? I tried in Bott-Tu (Differential Forms etc.) and Irapidly noticed that the third book of Fomenko "Modern Geometry" spends lots of words about Z/2 cohomology, but I'm lacking a precise place.2011-06-23
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    Have you checked in Massey's Algebraic Topology book, or Bredon's Geometry and Topology book?2011-06-23
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    Another (more technical) thing is that it's easier to work in char 2, because you can ignore all signs — which makes many computations much, much easier (say, try to compute homology — or just Euler char — of a real Grassmannian...).2011-06-23
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    I asked a similar question - why do we use different coefficient's at all - and there are some interesting responses here: http://math.stackexchange.com/questions/37148/applications-of-universal-coefficient-theorem Have a look at Section 3.3 of [Hatcher's Book](http://www.math.cornell.edu/~hatcher/AT/AT.pdf) for a nice discussion of Poincare Duality and $\mathbb{Z}/2\mathbb{Z}$ coefficients.2011-06-23
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    Thanks a lot to everybody, I found a couple of good examples in your answers: is there a way to "accept" a comment?2011-06-25

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