Category bounds for nonnegative Ricci curvature manifolds with infinite fundamental group (Q2759034)

A space \(X\) has \(\text{cat}(X) = n\) whenever \(n\) is the smallest number of contractible open subsets of \(X\) covering \(X\). \(X\) has cup \((X)= k\) whenever \(k\) is the largest number of cohomology classes in H\(^\ast (X)\) with cup product different from \(0\). It is known that any compact Riemannian manifold \(M\) of non-negative Ricci curvature admits a finite covering \(\overline{M}\) which is isometric to a product of a flat torus \(T^r\) and a simply connected manifold \(N\). The author proves that in this case \(\text{cat}(M)\geq r + \text{cup}(N)\geq b_1(M) + \text{cup}(N)\) if only the fundamental group of \(M\) is infinite. If \(M\) is, moreover, cohomologically symplectic (i.e. admits a cohomology class \(\omega\in\)H\(^2(M, \mathbb Q)\) with \(\omega ^n\neq 0\), \(\dim M = 2n\)), then \(2\text{cat}(M) - \dim M\geq b_1(M)\).