On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes

Abstract

A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) -dim(ker(D^*)) of a differential complex D:E -> F. In the discrete, a complex D can be called elliptic if a McKean-Singer spectral symmetry applies as this implies str(exp(-t D^2)) is t-independent. In that case, the analytic index of D is the sum of (-1)^k b_k(D), where b_k(D) is the k'th Betti number, which by Hodge is the nullity of the (k+1)'th block of the Hodge operator L=D^2. It can also be written as a topological index summing K(v) over the set of zero-dimensional simplices in G and where K is an Euler type curvature defined by G and D. This can be interpreted as a Atiyah-Singer type correspondence between analytic and topological index. Examples are the de Rham differential complex for the Euler characteristic X(G) or the connection differential complex for Wu characteristic w_k(G). Given an endomorphism T of an elliptic complex, the Lefschetz number X(T,G,D) is defined as the super trace of T acting on cohomology defined by E. It is equal to the sum i(v) over V which are contained in fixed simplices of T, and i is a Brouwer type index. This Atiyah-Bott result generalizes the Brouwer-Lefschetz fixed point theorem for an endomorphism of the simplicial complex G. In both the static and dynamic setting, the proof is done by heat deforming the Koopman operator U(T) to get the cohomological picture str(exp(-t D^2) U(T)) in the limit t to infinity and then use Hodge, and then by applying a discrete gradient flow to the simplex data defining the valuation to push str(U(T)) to V, getting curvature K(v) or the Brouwer type index i(v).