Tangential LS-category of K(pi,1)-foliations

A K(pi,1)-foliation is one for which the universal covers of all leaves are contractible (thus all leaves are K(pi,1)'s for some pi). In the first part of the paper we show that the tangential Lusternik--Schnirelmann category cat F of a K(pi,1)-foliation F on a manifold M is bounded from below by t-codim F for any t with H_t(M;A) nonzero for some coefficient group A. Since for any C^2-foliation F one has cat F <= dim F by our Theorem 5.2 of [Topology 42 (2003) 603-627], this implies that cat F = dim F for K(pi,1)-foliations of class C^2 on closed manifolds. For K(pi,1)-foliations on open manifolds the above estimate is far from optimal, so one might hope for some other homological lower bound for cat F. In the second part we see that foliated cohomology will not work. For we show that the p-th foliated cohomology group of a p-dimensional foliation of positive codimension is an infinite dimensional vector space, if the foliation is obtained from a foliation of a manifold by removing an appropriate closed set, for example a point. But there are simple examples of K(pi,1)-foliations of this type with cat F < dim F. Other, more interesting examples of K(pi,1)-foliations on open manifolds are provided by the finitely punctured Reeb foliations on lens spaces whose tangential category we calculate. In the final section we show that C^1-foliations of tangential category at most 1 on closed manifolds are locally trivial homotopy sphere bundles. Thus among 2-dimensional C^2-foliations on closed manifolds the only ones whose tangential category is still unknown are those which are 2-sphere bundles which do not admit sections.