Lusternik-Schnirelmann category and products of local spaces (Q846035)

If cat\((X)\) denotes the Lusternik-Schnirelmann category of the topological space \(X\), it is a well know fact that cat\((X\times Y)\leq\)cat\((X)+\)cat\((Y)\) and strict inequality may appear even if one of the spaces is a sphere. It is also known by a deep result of Félix, Halperin and Lemaire that, whenever \(X\) and \(Y\) are rational Poincaré duality, simply connected CW-complexes of finite type over \(\mathbb Q\), then equality holds, i.e., cat\((X\times Y)=\)cat\((X)+\)cat\((Y)\). In this paper, the author obtains the same result when dealing with \(p\)-local \(n\)-connected complexes located in the ``Anick range''. Explicitly: Let \(X\) and \(Y\) be \(p\)-local \(n\)-connected CW-complexes of finite type over \({\mathbb Z}_{(p)}\) such that \(\dim X+\dim Y\leq \min (n+2p-3,np-1)\). Assume also that \(H_q(\Omega X;{\mathbb Z}_{(p)})\) and \(H_q(\Omega X;{\mathbb Z}_{(p)})\) are free \(R\)-modules (with \(R\) a quotient ring of \({\mathbb Z}_{(p)}\)) for \(0<q<\dim X+\dim Y\). Then, cat\((X\times Y)=\)cat\((X)+\)cat\((Y)\). To this end, the author overcomes carefully all obstacles to adapt the process which leads to the same result in the classical rational case: from the original interpretation of the rational Lusternik-Schnirelmann category in terms of models due to Félix and Halperin and the reformulation of the Hess Theorem in the local case, to the final proof of Félix, Halperin and Lemaire.