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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 onthe matrix problem technique which was developed in the classification ofrepresentations of algebras and applied to homotopy theory by Baues and Drozd.

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