Products and Lusternik-Schnirelmann category of classifying spaces (Q1279708)

Let \(X=Y\times Z\) be a simply connected finite CW-complex. Using the theory of minimal models, the author proves that the Lyusternik-Shnirel'man category of the classifying space \(B\text{ aut }X\) is not finite if the LS category of \(B\) is finite. If \(\pi_\ast(\Omega X)\otimes{\mathbb Q}\) contains a free Lie ideal of finite codimension, then \(\pi_\ast(B\text{ aut }X)\otimes{\mathbb Q}\) grows exponentially.