The Cohomology for Wu Characteristics

While the Euler characteristic χ(G) = ω1(G) = ∑ x ω(x) super counts simplices, the Wu characteristics ωk(G) = ∑ x1∼x2···∼xk ω(x1) · · ·ω(xk) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More generally, one can define the k-intersection number ωk(G1, . . . Gk), which is the same sum but where xi ∈ Gi. For every k ≥ 1 we define a cohomology H k (G1, . . . , Gk) compatible with ωk. It is invariant under Barycentric subdivison. This interaction cohomology allows to distinguish spaces which simplicial cohomology can not: for k = 2, it can identify algebraically the Möbius strip and the cylinder. The vector spaces H k (G) are defined by explicit exterior derivatives dk which generalize the incidence matrices for simplicial cohomology. The cohomology satisfies the Kuenneth formula: for every k, the Poincaré polynomials pk(t) are ring homomorphisms from the strong ring to the ring of polynomials in t. The case k = 1 for Euler characteristic is the familiar simplicial cohomology H 1 (G) = H (G). On any interaction level k, there is now a Dirac operator D = dk+d ∗ k. The block diagonal Laplacian L = D leads to the generalized Hodge correspondence bp(G) = dim(H p k (G)) = dim(ker(Lp)) and Euler-Poincaré ωk(G) = ∑ p(−1)dim(H p k (G)) for Wu characteristic and more generally ωk(G1, . . . Gk) = ∑ p(−1)dim(H p k (G1, . . . , Gk)). Also, like for traditional simplicial cohomology, an isospectral Lax deformation Ḋ = [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 which belongs to the interaction cohomology. Also 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) = ∏k j=1 ω(xj)sign(T |xj) attached to simplex tuple which is invariant under T leads to the formula Lk(T ) = ∑ T (x)=x iT,k(x). For T = Id, the Lefschetz number Lk(Id) is equal to the k’th Wu characteristic ωk(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincaré formula for Wu characteristic. Also this generalizes to the case, where automorphisms Tk act on Gk: there is a Lefschetz number Lk(T1, . . . , Tk) and indices ∏k j=1 ω(xj)sign(Tj |xj). For k = 1, it is the known Lefschetz formula for Euler characteristic. While Gauss-Bonnet for ωk can be seen as a particular case of a discrete Atiyah-Singer result, the Lefschetz formula is an particular case of a discrete Atiyah-Bott result. But unlike for Euler characteristic k = 1, the elliptic differential complexes for k > 1 are not yet associated to any constructions in the continuum.