A K(π,1)–foliation is one for which the universal covers of all
leaves are contractible (thus all leaves are K(π,1)'s for some
π). In the first part of the paper we show that the tangential
Lusternik–Schnirelmann category cat F of a
K(π,1)–foliation F on a manifold M is bounded from
below by t  codim F for any t with H_{t}(M;A)≠ 0 for
some coefficient group A. Since for any C^{2}–foliation
F one has cat F≤ dim F by our
earlier work, [Topology 42 (2003) 603627; Theorem 5.2], this implies
that cat F = dim F for K(π,1)–foliations of
class C^{2} on closed manifolds.
For K(π,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 pth 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(π,1)–foliations of this type with cat F<
dim F. Other, more interesting examples of
K(π,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.
