On Weighted Simplicial Homology

Abstract

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. G\'{e}om. Diff\'{e}r. Cat\'{e}g. 31 (1990), pp. 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, $H_n^v(X)$, features a new boundary operator, $\partial_n^v$. In difference to Dawson, our approach is centered at a natural homomorphism $\theta$ of weighted chain complexes. The key object is $H^v_{n}(X/\theta)$, the weighted homology of a quotient of chain complexes induced by $\theta$, 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 $\kappa_n$- or $\mu_n$-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 $R=\mathbb{F}[[\pi]]$, where $\mathbb{F}$ 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 $\kappa_n$ and $\mu_{n+1}$ simplices.