Commutative Spectral Triples & The Spectral Reconstruction Theorem A Master

Given a unital and commutative algebra A associated to a spectral triple, we show how a differentiable structure is constructed on the spectrum of such an algebra whenever the spectral triple satisfies eight so-called “axioms”, in such a way that A ∼= C∞(M). This construction is the celebrated “reconstruction theorem” of Alain Connes [14], [21]. We discuss two spin manifolds, the circle and the 4-sphere, and show how several key properties of these manifolds relate to mathematical concepts and constructions used in the reconstruction theorem. In addition, we review the theory of Fredholm modules, cyclic homology, and noncommutative integrals, which are used as tools in the reconstruction theorem.