Classification of the congruence classes of $\mathbf{A}_n^5(n\geq 6)$ with 2-torsion free homology

TL;DR

This paper classifies certain high-connectivity, low-dimensional polyhedra with torsion-free homology using matrix problem techniques from algebra representation theory, advancing the understanding of their congruence classes.

Contribution

It introduces a classification of $ extbf{F}^5_{n(2)}$-polyhedra with 2-torsion free homology, applying matrix problem methods in homotopy theory for the first time.

Findings

Complete classification of the congruence classes for the specified polyhedra.

Application of matrix problem techniques to homotopy classification.

Identification of the structure of these polyhedra in terms of algebraic invariants.

Abstract

In this paper, we classify the congruence classes of $\mathbf{F}^5_{n(2)}$-polyhedra, i.e. $(n-1)$-connected, at most $(n+5)$-dimensional polyhedra with 2-torsion free homology. The proof relies on the matrix problem technique which was developed in the classification of representations of algebras and applied to homotopy theory by Baues and Drozd.