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Let $(M,g)$ be a (not-necessarily compact) oriented, connected Riemannian manifold. Lets consider the pairing

$$\Omega^{k}(M)\times\Omega^{d-k}_{c}(M)\to\mathbb{R}, (\alpha,\beta)\mapsto\int_{M}\alpha\wedge\beta$$

It is well-known that this pairing induces a well-defined pairing on cohomology

$$H^{k}(M)\times H^{d-k}_{c}(M)\to\mathbb{R}, ([\alpha],[\beta])\mapsto\int_{M}\alpha\wedge\beta\quad\quad\quad (\ast)$$

Poincaré duality states that this pairing is non-degenerate (in both entries I suppose (?)). Now, my question is, what are the precise assumption for this to hold?

  1. In the book Manifolds and Differential Geometry by J. M. Lee, only non-degenaracy in the first entry is proven, i.e. that ($\ast$) viewed as a map $H^{k}(M)\to (H^{d-k}_{c}(M))^{\ast}$ is an isomorphism provided $(M,g)$ admits a finite good cover.
  2. On the other hand, in Connections, Curvature, and Cohomology 1 by W. Grueb, it is mentioned that the finite good cover assumption is actually only needed for the other side, i.e. to show that ($\ast$) viewed as a map $H^{d-k}_{c}(M)\to (H^{k}(M))^{\ast}$ is an isomorphism...

Does anyone know the precise statement or has some idea where to find it?

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1 Answer 1

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I found an answer myself:

Let us consider the pairing

$$\mathrm{PD}(\cdot,\cdot):H^{k}(M)\times H^{d-k}_{c}(M)\to\mathbb{R}$$

defined by

$$\mathrm{PD}([\alpha],[\beta])=\int_{M}\alpha\wedge\beta$$

Now, the statement of Poincaré duality is the following:

If $(M,g)$ admits a finite good cover, then $\mathrm{PD}$ is (strongly) non-degenerate in both entries, i.e. both maps $[\alpha]\mapsto ([\beta]\mapsto\mathrm{PD}([\alpha],[\beta]))$ and $[\beta]\mapsto ([\alpha]\mapsto\mathrm{PD}([\alpha],[\beta]))$ are isomorphisms, i.e. $H^{k}(M)\cong (H_{c}^{d-k}(M))^{\ast}$ and $H_{c}^{d-k}(M)=(H^{k}(M))^{\ast}$

Now, it turns out, that without the finite good cover assumption most of this remains true:

  • Without the finite good cover assumption, $\mathrm{PD}$ is always at least weakly non-degenerate in both entries, i.e. the maps $[\alpha]\mapsto ([\beta]\mapsto\mathrm{PD}([\alpha],[\beta]))$ and $[\beta]\mapsto ([\alpha]\mapsto\mathrm{PD}([\alpha],[\beta]))$ are injective.
  • Furthermore, the first map, is actually still surjective, i.e. we always get an isomorphism $H^{k}(M)\cong (H_{c}^{d-k}(M))^{\ast}$. However, what fails in general without the finite good cover assumption is the surjectivity of the second map, i.e. in general it won't be true that also $H_{c}^{d-k}(M)\cong (H^{k}(M))^{\ast}$.

References:

For the proof that $H^{k}(M)\cong (H_{c}^{d-k}(M))^{\ast}$ without the finite good cover assumption:

  • W. Greub, S. Halperin, and R. Vanstone. Connections, Curvature and Cohomology. Volume 1. Academic Press, New York, 1972.

For a discussion of why surjectivity of the second map fails without this assumption and hence the failure of $H_{c}^{d-k}(M)\cong(H^{k}(M))^{\ast}$, see:

  • R. Bott and L. W. Tu. Differential Forms in Algebraic Topology. Springer, New York, 1982.
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    $\begingroup$ It is Poincaré, not Poincarè $\endgroup$
    – Didier
    Oct 19, 2023 at 12:17

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