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I'm studying cohomology group in A basic course in Algebraic Topology - Massey . My question is : " what is exactly geometric interpretation of cochains and cocycles ? . In his book , he assigns CW-complex as an example but I really don't understand what he have written . Help me figure out , thank you !

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Gankedbymom
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  • While this is a good question, it's probably difficult to convey a good answer in a short space. I might suggest trying a different book to learn the material if Massey's seems too advanced. Allen Hatcher has an algebraic topology textbook which is freely available online which might help you. – Cheerful Parsnip May 10 '17 at 14:04
  • Hmm , but Allan Hatcher have written a difference definition with definition which I have learnt . So I think it's difficult for me . – Gankedbymom May 10 '17 at 14:19
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    I understand. Well, if you want to stick to Massey, perhaps we can help you better if you can come up with a more specific question. What is it about the CW-complex example that puzzles you? – Cheerful Parsnip May 10 '17 at 16:17
  • I have editted my post . You can see he show four properties , but I don't understand what he tried to explain . As a corollary I don't want to read the rest of that section . – Gankedbymom May 10 '17 at 17:11
  • this may help https://math.stackexchange.com/questions/521265/understanding-cohomology-with-compact-support?rq=1 – Andres Mejia May 10 '17 at 17:14

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If you have a simplicial complex (as opposed to a cell complex) you can think of chains as linear combinations of simplices and the boundary operator assigns the sum of $n-1$ dimensional faces to each $n$-simplex, with alternating signs. For cochains, you can also think of them as linear combinations of $n$-simplices, but the coboundary is now the sum of $n+1$-dimensional simplices which have a given $n$ simplex as a face, with certain signs.

Let's take an easy example of a line segment $\tau$ connecting two points $A,B$. Then as a chain $\partial\tau =B-A$, assuming it is oriented from $A$ to $B$, but for cochains we have $\delta B=\tau$ and $\delta A=-\tau$.

I think Massey is probably going through the same construction with cell complexes, but in that case figuring out the coefficients of the boundary operator is a little trickier.

Cheerful Parsnip
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