The corresponence between compact Hausdorff topological spaces and commutative unital $C^*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these categories. What is very easy: to define a structure of $C^*$-algebra (commutative and unital) on the algebra $C(X)$ where $X$ is compact Haudorff topological space. The real significance is the converse: namely, that each commutative unital $C^*$-algebra is of this form. This is the beginning of the whole story which culminates in the definition of spectral triple and Connes' reconstruction theorem. As I understood, the situation is the following:

-if a triple $(A,H,D)$ (where $A$ is unital) is a commutative spectral triple it does not follow that there is a compact orientable manifold $M$ such that $A=C^{\infty}(M)$

-if we assume some additional technical conditions (about regularity, dimension and so on) then there is a compact orientable manifold $M$ such that $A=C^{\infty}(M)$-this is the celebrated and deep theorem of Connes.

Furthermore, if I understood things correctly (see the discussion Noncommutative smooth manifolds) the main problem is to identyify the algebra $A$ as $C^{\infty}(M)$. Once you have that, it is less deep result (let us call it *baby reconstruction theorem*) that in this case there is also a hermitian vector bundle $E$ and essentially self adjoint elliptic operator $D_0$ such that $(A,H,D)=(C^{\infty}(M),L^2(M,E),D_0)$. After the formulation of reconstruction theorem, in the original paper of Connes there is also the following statement: each compact orientable smooth manifold arise in such way. I would like to focus on this part: what is known (for me at least), that once you have smooth, compact, spin$^c$ manifold it defines a spectral triple. So my first question is the following:

**Question 1** How it is *always* possible to obtain a commutative spectral triple from an arbitrary compact, smooth, orientable manifold (not necessary spin$^c$)?

Furthermore: let us look for $C^*$-algebra situation: then there is a canonical way to obtain the $C^*$-algebra structure on $C(X)$. If we combine Connes' reconstruction theorem with baby reconstruction we obtain that if we have a commutative spectral triple $(A,H,D)$ such that $A=C^{\infty}(M)$ where now $M$ is *spin$^c$* (compact, smooth) then in fact the hermitian bundle $E$ is in fact a spinor bundle (according to the cited discussion). But if the answer to the first question is affirmative then it is natural to ask:

**Question 2** What do we obtain if we start with an arbitrary smooth compact spin$^c$ manifold $M$: then form the spectral triple as in question 1 (as if $M$ would be not necessary spin$^c$) and after that apply reconstruction?

More general question would be:

**Question 3** Let $M$ be a smooth compact orientable manifolds. Is it somehow possible to parametrize all pairs $(H,D)$ such that $(A,H,D)$ is a spectral triple? How does it depends from the manifold $M$ (from the fact of being spin$^c$ etc.)

I would be grateful if anybody could clarify this issue for me.