2

Using Poincare duality, I believe that if $X$ and $Y$ are two orientable $n$-manifolds with $X\simeq Y$, then we should have $H^i_c(X)\cong H_{n-i}(X)\cong H_{n-i}(Y)\cong H^i_c(Y)$ for all $i$. However, in a comment to the question linked at the bottom, the user asserts that the torus with a point removed and a sphere with three points removed have different cohomology with finite support, despite having the same dimension and being homotopy equivalent.

Have I misunderstood the statement of Poincare Duality for compactly supported cohomology, and if so can someone explain how the commenter computed these groups for the sphere with three points removed and the torus with one point removed?

Torus minus a point homeomorphic to sphere with three points?

Alex Menzia
  • 43

1 Answers1

2

His calculation of the compactly supported cohomology of the sphere minus 3 points is incorrect, it’s $H^1_c=\mathbb {Z}^2$ since it’s compactification is the quotient of three points on a sphere. The way to answer the original question is to just see that the one point compactifications of the spaces are not homeomorphic (one is a manifold and one is not).

You are correct they should have the same compactly supported cohomology by Poincaré duality.

Connor Malin
  • 11,661