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On the Strong Homotopy Lie-Rinehart Algebra of a Foliation
It is well known that a foliation F of a smooth manifold M gives rise to a
rich cohomological theory, its characteristic (i.e., leafwise) cohomology.
Characteristic cohomologies of F may be interpreted, to some extent, as
functions on the space P of integral manifolds (of any dimension) of the
characteristic distribution C of F. Similarly, characteristic cohomologies with
local coefficients in the normal bundle TM/C of F may be interpreted as vector
fields on P. In particular, they possess a (graded) Lie bracket and act on
characteristic cohomology H. In this paper, I discuss how both the Lie bracket
and the action on H come from a strong homotopy structure at the level of
cochains. Finally, I show that such a strong homotopy structure is canonical up
to isomorphisms.Comment: 41 pages, v2: almost completely rewritten, title changed; v3:
presentation partly changed after numerous suggestions by Jim Stasheff,
mathematical content unchanged; v4: minor revisions, references added. v5:
(hopefully) final versio
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