For a continuous map \(f:X\to Y\) between topological spaces the cohomological width of \(f\), width\(_*(f;R)\), is the maximal rank of the restriction homomorphisms \(H^*(X;R)\to H^*(f^{-1}(y);R)\), over all \(y\in Y\), in Čech cohomology with coefficients in a ring \(R\). The minimum of width\(_*(f;R)\) over all continuous maps \(f:X\to Y\) is the cohomological width of \(X\) over \(Y\), denoted by width\(_*(X/Y;R)\). In [Geom. Funct. Anal. 19, No. 3, 743–841 (2009; Zbl 1195.58010); ibid. 20, No. 2, 416–526 (2010; Zbl 1251.05039)] M. Gromov obtained lower bounds for the cohomological width in the case that \(X=T^n\) is a torus and \(Y\) a finite-dimensional simplicial complex. The present paper improves and generalizes the results of Gromov. In addition to \(X\) being a torus the author considers products \((S^p)^n\) of spheres. The first main theorem states that width\(_1(T^n/N;R)=n-\dim N\) where \(N\) is a manifold. This result is optimal as the projections \(T^n\to T^q\) show. The second main theorem states that width\(_p((S^p)^n/N);\mathbb{Q})\ge n-\dim N\), provided \(p\ge3\) is odd and \(n\le p-2\).