Cohomological estimates for $\cat(X,\xi)$

This paper studies the homotopy invariant $\cat(X,\xi)$ introduced in \cite{farbe2}. Given a finite cell-complex $X$, we study the function $\xi\mapsto \cat(X,\xi)$ where $\xi$ varies in the cohomology space $H^1(X;\R)$. Note that $\cat(X,\xi)$ turns into the classical Lusternik - Schnirelmann category $\cat(X)$ in the case $\xi=0$. Interest in $\cat(X,\xi)$ is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms.