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?

realstructure $J$ that picks out a spin structure. $\endgroup$4more comments