ELLIPTIC OPERATORS AND K-HOMOLOGY

If a differential operator D on a smooth Hermitian vector bundle S over a compact manifold M is symmetric, it is essentially self adjoint and so admits the use of functional calculus. If D is also elliptic, then the Hilbert space of square integrable sections of S with the canonical left C(M)-action and the operator chi(D) for chi a normalizing function is a Fredholm module, and its K-homology class is independent of chi. In this expository article, we provide a detailed proof of this fact following the outline in the book "Analytic K-homology" by Higson and Roe.