Poincaré duality, spin structure and Connes' reconstruction theorem

Question

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?

Answer 1

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.