A Boundary Operator for Simplices

TL;DR

This paper introduces a generalized boundary operator for simplices that replaces face simplices with internal combinations, potentially leading to new homology theories and offering novel constructions on the standard simplex.

Contribution

It presents a new variant of the boundary operator in simplicial homology, expanding the framework to include internal simplices and exploring their implications.

Findings

Potential for infinite non-isomorphic homology theories

New constructions on the standard simplex

Extension of classical boundary operator concepts

Abstract

We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary (n-1)-simplices of each n-simplex are replaced by combinations of internal (n-1)- simplices parallel to the faces. This construction may lead to an infinite class of extraordinary non-isomorphic homology theories. We show further some interesting constructions on the standard simplex.