# Poincaré duality, spin structure and Connes' reconstruction theorem

In the paper "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS", A.Connes proves two "reconstruction" theorems. Informally, they are like :

Theorem 1 : Commutative spectral triple $$(A,H,D)$$ + 5 conditions $$\Longrightarrow$$ there exists a compact oriented smooth manifold $$X$$ such that $$A \cong C^\infty(X).$$

Theorem 2 : Commutative spectral triple $$(A,H,D)$$ + 5 conditions + Poincaré duality $$\Longrightarrow$$ there exists a compact oriented smooth $$\textbf{spin}^\mathbf c$$ manifold $$X$$ such that $$A \cong C^\infty(X) \qquad H \cong L^2(X,S)$$ and $$D$$ is the dirac operator on the $$\text{spin}^c$$-bundle $$S$$.

Then, the Poincaré duality seems to be link with the existence of a $$\text{spin}^c$$ structure.

In other hand, in the paper "Noncommutative geometry and reality" by the same A.Connes, he says (p-6203)

We can thus assert that, in the simply connected case, a closed manifold is in a rather deep sense more or less the same thing as a homotopy type $$X$$ satisfying Poincaré duality in ordinary homology together with a preferred element $$\nu_x \in \operatorname{KO}_*(X)$$ which induces Poincare duality in KO theory tensored by $$\mathbb{Z}[1/2]$$.

Then, it seems (maybe I don't understand well?) that here, closed manifolds are equivalent to objects $$X$$ (path connected simplicial complex) which satisfy Poincaré duality for a suitable class $$\nu_x \in \operatorname{KO}_*(X)$$ (and so there doesn't seem to be any connection here with a spin structure).

Questions :

• Can you explain to me, what is purpose of the Poincaré duality axiom in the reconstruction theorem? It is to recover the spin structure ? It is for the manifold structure (as the second point seems to explain) ?
• If it is for the spin structure, can you explain to me how Poincaré duality and the spin structure are linked?
• With regards to your first question, Poincaré duality is completely and utterly irrelevant to the reconstruction theorem–consult this question and answer for a detailed picture of what the reconstruction theorem actually does. With regards to your second question, the Poincaré duality "axiom" is an abstract generalization of a highly non-trivial $K$-homological property of spin Dirac operators–it's a choice of real structure $J$ that picks out a spin structure. Commented Jun 26 at 16:26
• Actually, I think I see where your confusion comes from: Connes makes a throwaway observation after Theorem 1.2 that the extra hypothesis of Theorem 1.2 can be interpreted as a highly attenuated echo of the Poincaré duality "axiom" from the 1995 paper. If you aren't a specialist in index theory, just observe that the multiplicity mentioned in Theorem 1.2 will simply be the rank of the Clifford module bundle reconstructed by the general reconstruction theorem, so that the extra hypothesis is exactly the requirement that this Clifford module bundle be irreducible. Commented Jun 26 at 16:41
• Not just some Hermitian vector bundle, but a Hermitian Clifford module bundle---and $D_E$ will be a Dirac-type operator (in the most general sense where $D_E^2$ is a Laplace-type operator). If $E$ has the correct rank (as required in Theorem 1.2), then it must be an irreducible Hermitian Clifford module bundle, which is precisely a spinor bundle for a spin$^{\mathbb{C}}$ structure. Commented Jun 27 at 12:11
• If you then have a real structure $J$, it induces precisely the isomorphism of Hermitian Clifford modules $E \cong E^\prime$ you need to conclude that $E$: see this answer, though note that the discussion under point 1 is slightly out of date---I didn't yet understand that the "baby reconstruction theorem" necessarily recovered Hermitian Clifford module bundles and Dirac-type operators. Commented Jun 27 at 12:14
• Finally, the Poincaré duality that Connes considers in the 1995 paper is really an analogue for $K$-theory qua generalized cohomology theory of the usual Poincaré duality for singular or de Rham cohomology. In general, suppose $D_E$ is a self-adjoint Dirac-type operator on a Hermitian Clifford module bundle $E$ on a compact oriented Riemannian manifold $M$. Given Hermitian vector bundles with connection $(F_1,\nabla_1)$ and $(F_2,\nabla_2)$ on $M$, you can twist $D_E$ on each side to get a Fredholm operator on $L^2(M,F_1 \otimes E \otimes F_2)$ whose index you can take. Commented Jun 27 at 12:33

Let me summarize the discussion in the comments as a straightforward answer.

Putting all the pieces together, Connes's reconstruction theorem lets you conclude the following:

1. A commutative spectral triple $$(\mathcal{A},H,D)$$ of dimension $$p$$ recovers an essentially unique compact oriented Riemannian $$p$$-manifold $$M$$, Hermitian Clifford module bundle $$E$$ on $$M$$, and self-adjoint Dirac-type operator $$D_E$$ on $$E$$, such that $$(\mathcal{A},H,D) \cong (C^\infty(M),L^2(M,E),D_E)$$.
2. Recall that $$\lambda$$ is the faithful trace defined by equation 5 in Connes's reconstruction theorem paper. Then $$H \cong L^2(\mathcal{A}^{\prime\prime},\lambda)^r$$ as Hilbert space representations of the von Neumann algebra $$\mathcal{A}^{\prime\prime}$$ iff the vector bundle $$E$$ has constant rank $$r$$. Thus, by standard Clifford-algebraic lore, $$E$$ is an irreducible Clifford module bundle (i.e., the spinor bundle for some spin$$^\mathbb{C}$$ structure on $$M$$) iff $$r=2^{\lfloor p/2\rfloor}$$—this is what Connes's Theorem 1.2 is really saying.
3. Suppose that $$E$$ is indeed the spinor bundle for some spin$$^\mathbb{C}$$ structure on $$M$$. Then it is the spinor bundle for a spin structure on $$M$$ iff $$(\mathcal{A},H,D)$$ can be equipped with a real structure of $$KO$$-dimension $$p \bmod 8$$, in which case $$D_E$$ is the spin Dirac operator on $$M$$ with this spin structure up to perturbation by a self-adjoint bundle endomorphism.

So, what does Poincaré duality have to do with anything? In the setting above, the Dirac-type operator $$D_E$$ induces a pairing $$\mu : K^0(M) \times K^0(M) \to \mathbb{Z}$$ as follows. Given Hermitian vector bundles with connection $$(F_1,\nabla_1)$$ and $$(F_2,\nabla_2)$$ on $$M$$, twist $$D_E$$ on either side to get an elliptic operator $$1 \otimes_{\nabla_1} D_E \otimes_{\nabla_2} 1$$ on $$F_1 \otimes E \otimes F_2$$; then $$\mu([F_1],[F_2])$$ is the Fredholm index of this elliptic operator. It is a non-trivial result that this pairing is non-degenerate when $$E$$ is the spinor bundle of a spin$$^{\mathbb{C}}$$ structure (e.g., when $$E$$ is the spinor bundle of a spin structure and $$D_E$$ is the spin Dirac operator), yielding an analogue of Poincaré duality for $$K$$-theory. This now motivated Connes to formalize an abstract notion of $$K$$-theoretic Poincaré duality that may or may not be satisfied by any given real spectral triple. Because non-degeneracy of the pairing is a non-trivial consequence of $$E$$ being the spinor bundle for a spin$$^{\mathbb{C}}$$ structure, Connes views the irreducibility hypothesis of Theorem 1.2 as an echo of his abstract $$K$$-theoretic Poincaré duality.

• So it's possible that a commutative spectral triples does not satisfy the Poincaré duality ? So, how can it represent a closed oriented manifold?
– eomp
Commented Jun 27 at 15:41
• I'm not handy enough with index theory to be able to give you an example, though you might try playing around with $d+d^\ast$ on a non-spin$^{\mathbb{C}}$ manifold. What I can say is that neither the reconstruction theorem nor the "baby reconstruction theorem" require any form of $K$-theoretic Poincaré duality as a hypothesis, and that the extra hypothesis in Connes's Theorem 1.2 is best viewed as a characterization of having the spinor bundle of a spin$^{\mathbb{C}}$ structure (which then, in turn, guarantees $K$-theoretic Poincaré duality). Commented Jun 27 at 17:01
• Just be aware that the NCG literature is extremely messy when it comes to statements of the reconstruction theorem and its implications. In particular, qutie a few sources just carry forward Connes's full set of proposed axioms from 1995 (including Poincaré duality) without actually checking against the final 2008 reconstruction theorem paper. Commented Jun 27 at 17:02
• One last comment for clarity: in the context of spectral triples, $K$-theoretic Poincaré duality isn't a property of the manifold but rather of the Dirac-type operator (or abstract generalization thereof). It's just that when you have a compact oriented Riemannian spin manifold, you have a canonical choice of Dirac-type operator, the spin Dirac operator, whose index theory (e.g., intersection form) has clear topological significance. The theory of spectral triples has deep historical roots in index theory, so the example of the spin Dirac operator is enormously influential. Commented Jun 27 at 17:11