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I'm looking at Poincare duality, and there's something extremely wrong with the way I'm looking at one or more of the concepts and I need to figure out which.

When dealing with non-compact manifolds, you can fix Poincare duality by looking at something called "cohomology with compact support". It mentions it on page 7 here:

http://www.math.upenn.edu/~ghrist/EAT/EATchapter6.pdf

It mentions there that you can restrict your cochains (singular, simplicial, cellular) to those which evaluate only on some compact subspace.

My confusion is that I don't understand how the boundary of asingularchain will necessarily be a singular chain with compact support (I can easily see it for the other two).

Take, for example, the 0-cochain which evaluates to 1 on a specific point and 0 elsewhere. The boundary of this chain (I think) will be the cochain which evaluates to 1 on all lines which start at the point and to -1 on all lines which end at the point.

So in that light I don't see how we get back something which is of compact support.

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# Cohomology with compact support

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