Algebraic filling inequalities and cohomological width

For any continuous map f:X→Y and y∈Y the preimage f^{-1}(y) is a subset of X and we can consider the cohomological restriction homomorphism H^k(X;Z)→H^k(f^{-1}(y);Z). Gromov introduced the notion of cohomological width which is defined as min_{f:X→Y} max_{y∈Y} rk[H^k(X;Z)→H^k(f^{-1}(y);Z)]. We give new lower bounds for this quantity, when X is a product of projective spaces and Y is the real line and when X is a torus, an essential manifold with free abelian fundamental group or a product of higher-dimensional spheres and Y a manifold