On the connection between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms (Q2638598)

Let X be a CW-complex with one cell in dimension 0 and 4n and with k cells in dimension 2n. The Hilton-Hopf invariant of the attaching map of the top cell determines the Hilton-Hopf quadratic form of X. Equivalently this quadratic form is determined by the cup product in cohomology. The topological genus of the suspension \(\Sigma\) X and the (weak) algebraic genus of the quadratic form are determined by the genus of X. The author constructs two series of examples which show that the converse is not always true, i.e., if X, Y are CW-complexes of the above form, then their quadratic forms may have the same weak genus, \(\Sigma\) X and \(\Sigma\) Y may have the same topological genus and nevertheless X and Y are not of the same topological genus. The construction of these counterexamples is possible if the torsion subgroup of the homotopy group \(\pi_{4n- 1}({\mathbb{S}}^{2n})\) is nontrivial resp. is not cyclic.