Characteristic Topological Invariants

TL;DR

This paper introduces higher topological invariants for simplicial complexes, proving their properties and generalizations, including relations to Euler characteristic and sphere formulas, advancing the understanding of topological invariants.

Contribution

It provides a new proof that higher characteristics equal the first characteristic and generalizes sphere formulas for simplicial complexes.

Findings

Higher characteristics satisfy k-point Green function identities.

Total higher characteristics of spheres at even and odd simplices are equal.

w_m(G) equals w_1(G) for closed manifolds.

Abstract

The higher characteristics w_m(G) for a finite abstract simplicial complex G are topological invariants that satisfy k-point Green function identities and can be computed in terms of Euler characteristic in the case of closed manifolds, where we give a new proof of w_m(G)=w_1(G). Also the sphere formula generalizes: for any simplicial complex, the total higher characteristics of unit spheres at even dimensional simplices is equal to the total higher characteristic of unit spheres at odd dimensional simplices.