THE COMBINATORICS OF WEIGHTED COHOMOLOGY

Source abstract

We introduce weighted simplicial cohomology with coefficients in ℚ[[π]]. This cohomology is derived from a weighted coboundary operator incorporating simplex weights from ℚ[[π]]. We provide a particular bipartition for the set of n-simplices into μn and κn-simplices and then we establish a four-term exact sequence relating the torsion module of weighted cohomology with regular cohomology having coefficients in ℚ[[π]]. Furthermore, we prove a structure theorem for the torsion, expressing its invariant factors as ratios of weights of distinguished (μn−1,κn)-simplex-pairs. We then employ this result to interpret the long homology sequence arising from a natural map connecting weighted and regular cohomology over ℚ[[π]]. Secondly we leverage weighted homology by a bipartition into μn- and κn-simplices with its torsion expressed via a pairing (κn,μn−1). We show that cohomological torsion is described by a pairing of the form (μn−1,μn), which gives rise to an isomorphism between weighted cohomological torsion and Hom(Im ∂nv,R).