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A linear or multi-linear valuation on a finite abstract simplicial complexcan be expressed as an analytic index dim(ker(D)) -dim(ker(D^*)) of adifferential complex D:E -> F. In the discrete, a complex D can be calledelliptic if a McKean-Singer spectral symmetry applies as this impliesstr(exp(-t D^2)) is t-independent. In that case, the analytic index of D is thesum of (-1)^k b_k(D), where b_k(D) is the k'th Betti number, which by Hodge isthe nullity of the (k+1)'th block of the Hodge operator L=D^2. It can also bewritten as a topological index summing K(v) over the set of zero-dimensionalsimplices in G and where K is an Euler type curvature defined by G and D. Thiscan be interpreted as a Atiyah-Singer type correspondence between analytic andtopological index. Examples are the de Rham differential complex for the Eulercharacteristic X(G) or the connection differential complex for Wucharacteristic w_k(G). Given an endomorphism T of an elliptic complex, theLefschetz number X(T,G,D) is defined as the super trace of T acting oncohomology defined by E. It is equal to the sum i(v) over V which are containedin fixed simplices of T, and i is a Brouwer type index. This Atiyah-Bott resultgeneralizes the Brouwer-Lefschetz fixed point theorem for an endomorphism ofthe simplicial complex G. In both the static and dynamic setting, the proof isdone by heat deforming the Koopman operator U(T) to get the cohomologicalpicture 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 thevaluation to push str(U(T)) to V, getting curvature K(v) or the Brouwer typeindex i(v).

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