Computing intersection numbers and bases of cohomology groups for triangulated closed three-dimensional manifolds

TL;DR

This paper presents efficient algorithms for computing intersection numbers and cohomology bases in triangulated closed 3-manifolds using simplicial homology and cohomology modulo 2, advancing computational topology methods.

Contribution

It introduces new algorithms for intersection number computation and cohomology basis construction in 3-manifolds, improving computational efficiency.

Findings

Developed two algorithms for intersection numbers of cycles

Constructed cohomology bases from homology groups

Enhanced computational methods for 3-manifold topology

Abstract

We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional cycles are developed. By means of these algorithms it is possible to construct a basis of cohomology group from the homology group of two cycles of complementary dimensions.