Milnor-Wood inequalities for products

We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times {\mathrm{\Sigma }}^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface $\mathrm{\Sigma }$ for $k$ sufficiently large.