The Cohomology for Wu Characteristics

While Euler characteristic X(G)=sumx w(x) super counts simplices, Wu characteristics wk(G) = sum(x1,x2,...,xk) w(x1)...w(xk) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More general is the k-intersection number wk(G1,...Gk), where xi in Gi. We define interaction cohomology H^p(G1,...,Gk) compatible with wk and invariant under Barycentric subdivison. It allows to distinguish spaces which simplicial cohomology can not: it can identify algebraically the Moebius strip and the cylinder for example. The cohomology satisfies the Kuenneth formula: the Poincare polynomials pk(t) are ring homomorphisms from the strong ring to the ring of polynomials in t. The Dirac operator D=d+d^* defines the block diagonal Hodge Laplacian L=D² which leads to the generalized Hodge correspondence bp(G)=dim(H^p_k(G)) = dim(ker(Lp)) and Euler-Poincare wk(G)=sump (-1)^p dim(H^pk(G)) for Wu characteristic. Also, like for traditional simplicial cohomology, isospectral Lax deformation D' = [B(D),D], with B(t)=d(t)-d^*(t)-ib(t), D(t)=d(t)+d(t)^* + b(t) can deform the exterior derivative d. The Brouwer-Lefschetz fixed point theorem generalizes to all Wu characteristics: given an endomorphism T of G, the super trace of its induced map on k'th cohomology defines a Lefschetz number Lk(T). The Brouwer index iT,k(x1,...,xk) = productj=1^k w(xj) sign(T|xj) attached to simplex tuple which is invariant under T leads to the formula Lk(T) = sumT(x)=x iT,k(x). For T=Id, the Lefschetz number Lk(Id) is equal to the k'th Wu characteristic wk(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincare formula for Wu characteristic.