  We prove that $$ \cat X\le \frac{\cd(\pi_1(X))+\dim X}{2}$$ for every CW
complex $X$ where $\cd(\pi_1(X))$ denotes the cohomological dimension of the
fundamental group of $X$. We obtain this as a corollary of the inequality $$
\cat X\le\frac{\cat (u_X)+\dim X}{2} $$ where $u_X:X\to B\pi_1(X)$ is a
classifying map for the universal covering of $X$.
