Geometric Objects and Cohomology Operations

TL;DR

This paper introduces an incremental algorithm that computes cohomology operations, including cup products and secondary operations, for finite simplicial complexes, enhancing algebraic invariants analysis.

Contribution

It develops a novel incremental algorithm that preserves algebraic information for cohomology operations, extending computational tools beyond homology.

Findings

Algorithm computes cup products efficiently.

Effective evaluation of primary and secondary cohomology operations.

Implementation in Mathematica demonstrates practical applicability.

Abstract

Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a version of the incremental algorithm for computing homology which saves algebraic information, allowing us the computation of the cup product and the effective evaluation of the primary and secondary cohomology operations on the cohomology of a finite simplicial complex. We study the computational complexity of these processes and a program in Mathematica for cohomology computations is presented.