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

In this paper, we classify the congruence classes of ${\mathbf{F}}_{n(2)}^5$-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.