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This question is similar to but not exactly the same as the old question (the issue of torsion).

In Exercise 11.8 of Dan Freed's notes he claims that the following pairing being nondegenerate is equivalent to Poincare duality $$\bar{I}_M: \text{Free}\,H^{2k}(M;\mathbb{Z}) \times \text{Free}\,H^{2k}(M;\mathbb{Z}) \longrightarrow \mathbb{Z}$$

My question is, how should we interpret the following possibly degenerate (when $H^{2k}(M;\mathbb{Z})$ has torsion) pairing using Poincare duality? $$I_M: H^{2k}(M;\mathbb{Z}) \times H^{2k}(M;\mathbb{Z}) \longrightarrow \mathbb{Z}$$ Does it invalidate the duality (of course not)?

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  • I wouldn't call that pairing degenerate. I would call a pairing of a finitely generated abelian group with itself nondegenerate if it's nondegenerate on the free quotient (when you mod out by torsion). Your pairing kills torsion, so I'm not sure what the question is. –  Aug 02 '16 at 23:09
  • @MikeMiller: by degenerate what I mean is if you put in a class $c_1$ belonging to the torsion subgroup then $I_M(c_1,c_2) = 0$ for any cohomology class $c_2$, then we cannot identify it as a linear functional of the cohomology group and in turn a homology class (the goal is to write it in the standard form of Poincare duality $H_{2k}(M;\mathbb{Z}) \cong H^{2k}(M;\mathbb{Z})$) – PhysicsMath Aug 02 '16 at 23:49
  • Yeah, I don't really understand Freed's point. But if you can show that the pairing is nondegenerate with coefficients in any field $\Bbb F$, you've certainly proven Poincare duality. –  Aug 04 '16 at 16:55
  • @MikeMiller: thanks Mike! Maybe torsion is the obstruction to extending the nondegeneracy to arbitrary field and that is the reason why Freed chooses the pairing on the free part not the entire cohomology group. – PhysicsMath Aug 05 '16 at 00:21
  • That's true, but he also just says that - the pairing necessarily kills torsion, so he quotients out by torsion. –  Aug 05 '16 at 00:22
  • @MikeMiller: all right Mike! Let us move to more meaningful questions! Thanks a lot for your attention! – PhysicsMath Aug 05 '16 at 00:49

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