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Duality theorems for foliations. (English) Zbl 0559.58022

Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 108-116 (1984).
[For the entire collection see Zbl 0534.00014.]
For a Riemannian foliation, the mean curvature form \(\kappa\) is the trace of the Weingarten operator of the leaves. Suppose this form is parallel, and the manifold is compact. Then the adjoint of the basic exterior derivative \(d_ B\) with respect to the global inner product of basic forms is the operator \(d_ B-\kappa \bigwedge\), the cohomology spaces of both operators are finite dimensional, and the global inner product induces a non-degenerate pairing of the two kinds of cohomology groups, in dimensions complementary with respect to the codimension of the foliation. In particular, if \(\kappa =0\), Poincaré duality holds for basic cohomology. If no assumptions are made on the Riemannian structure, there is still a duality between basic cohomology and the homology of transversal invariant currents. The results are carefully stated, but most proofs are deferred to a later paper. These results are the most geometric resolution of the problems presented by the reviewer’s duality ”theorem” [Am. J. Math. 81, 529-536 (1959; Zbl 0088.079)] and Carrière’s recent counterexample (unpublished thesis, Lille).
Reviewer: B.Reinhart

MSC:

58J10 Differential complexes
57R30 Foliations in differential topology; geometric theory
53C20 Global Riemannian geometry, including pinching