On the cohomology ring of chains in $R^d$

TL;DR

This paper proves that for dimensions $d \\geq 3$, the cohomology rings of chain spaces in $\\mathbb{R}^d$ uniquely determine their topological structure.

Contribution

It establishes that the mod 2 cohomology rings completely characterize the topology of chain spaces in Euclidean spaces for $d \\geq 3$, a new result in algebraic topology.

Findings

Cohomology rings determine chain space topology for $d \\geq 3$

Mod 2 cohomology is sufficient for classification

Topological invariants of chain spaces are fully captured by cohomology rings

Abstract

We prove that the spaces of chains in $\bbr^d$ for $d\geq 3$ are determined by their (${\rm mod}2$)-cohomology rings.