- #1
Jamma
- 432
- 0
Hi there.
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 a singular chain 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.
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 a singular chain 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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