Geometry & Topology Monographs 14 (2008) 477–504

DOI: 10.2140/gtm.2008.14.477

arXiv: 0904.1861

Abstract

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 C²–foliation F one has cat F≤ dim F by our earlier work, [Topology 42 (2003) 603-627; Theorem 5.2], this implies that cat F = dim F for K(π,1)–foliations of class C² 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 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(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 C¹–foliations of tangential category at most 1 on closed manifolds are locally trivial homotopy sphere bundles. Thus among 2–dimensional C²–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.

Keywords

Lusternik–Schnirelmann category, foliations, classifying space, topological groupoid, K(π,1)

Mathematical Subject Classification

Primary: 57R30

Secondary: 55M30, 57R32

References

Publication

Received: 31 May 2006
Revised: 26 July 2006
Accepted: 31 May 2007
Published: 29 April 2008

Authors