# Alternative (Oriented) Singular Cochains and the Modified Cup Product

## TL;DR

This paper introduces the alternative singular cochain complex, demonstrating its natural splitting from singular cochains and defining a modified cup product analogous to the wedge product in differential forms, with implications for topology and differential equations.

## Contribution

It shows that alternative singular cochains form a summand of singular cochains with a natural splitting and introduces a modified cup product with properties similar to the wedge product.

## Key findings

- Alternative cochains are summands of singular cochains.
- Cohomology splits into alternative and zero parts for compact spaces.
- A modified cup product with wedge-like properties is defined.

## Abstract

A special subcomplex of the singular chain complex for a topological space, historically called oriented singular chain complex is used here with the new name "alternative" singular chain complex. It was already known that this subcomplex and so its dual complex are chain homotopy equivalent to singular chains and cochains respectively and thus have the same homology and cohomology. Here, in addition to revisiting some aspects of this subcomplex, it is shown that alternative singular cochains (dual of alternative singular chains) with coefficients in rational or real numbers are indeed summands of singular cochains through a natural splitting. It is shown that this natural splitting also hold for cohomologies: At any order, the singular cohomology splits into the alternative cohomology and another summand which is zero if the considered topological space is compact. Also in this case similar to the wedge product for differential forms, a modified cup product can be defined with the same algebraic properties as in the wedge product in differential forms. This provides an idea to investigate some topological and structure-free aspects of nonlinear global differential equations on manifolds..