ON WEIGHTED SIMPLICIAL HOMOLOGY

We develop a framework for computing the homology of weighted simplicial complexes with coefficients in a discrete valuation ring. A weighted simplicial complex, (X,v), introduced by Dawson (Cah. Topol. Géom. Différ. Catég. 31 (1990), 229–243), is a simplicial complex, X, together with an integer-valued function, v, assigning weights to simplices, such that the weight of any of faces are monotonously increasing. In addition, weighted homology, Hnv(X), features a new boundary operator, ∂nv. In difference to Dawson, our approach is centered at a natural homomorphism 𝜃 of weighted chain complexes. The key object is Hnv(X∕𝜃), the weighted homology of a quotient of chain complexes induced by 𝜃, appearing in a long exact sequence linking weighted homologies with different weights. We shall construct bases for the kernel and image of the weighted boundary map, identifying n-simplices as either κn- or μn-vertices. Long exact sequences of weighted homology groups and the bases, allow us to prove a structure theorem for the weighted simplicial homology with coefficients in a ring of formal power series R=𝔽[[π]], where 𝔽 is a field. Relative to simplicial homology new torsion arises and we shall show that the torsion modules are connected to a pairing between distinguished κn and μn+1 simplices.