Title: 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).