Weighted Simplicial Complexes and Weighted Simplicial Maps
Shiquan Ren, Chengyuan Wu · arXiv (Cornell University) · 2021
Weighted simplicial complexes are simplicial complexes with assigned values (called weights) on the simplices. Twisting the boundary maps by the weights, we can construct weighted homology for weighted simplicial complexes. The weighted homology with integral coefficients depends on the weights; while the linear structure of the weighted homology with field coefficients only detects the nullity of the weights. In this paper, we equip quadratic forms or inner products on the chain complexes and the weighted homology for weighted simplicial complexes. We use the quadratic forms or inner products to study weighted simplicial complexes as well as weighted simplicial maps.