The Cohomology for Wu Characteristics

Abstract

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu characteristics w_k(G) = sum_(x_1,x_2,...,x_k) w(x_1)...w(x_k) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More general is the k-intersection number w_k(G_1,...G_k), where x_i in G_i. We define interaction cohomology H^p(G_1,...,G_k) compatible with w_k 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 p_k(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=D^2 which leads to the generalized Hodge correspondence b_p(G)=dim(H^p_k(G)) = dim(ker(L_p)) and Euler-Poincare w_k(G)=sum_p (-1)^p dim(H^p_k(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 L_k(T). The Brouwer index i_T,k(x_1,...,x_k) = product_j=1^k w(x_j) sign(T|x_j) attached to simplex tuple which is invariant under T leads to the formula L_k(T) = sum_T(x)=x i_T,k(x). For T=Id, the Lefschetz number L_k(Id) is equal to the k'th Wu characteristic w_k(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincare formula for Wu characteristic.