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Deformation Theory Seminar

Wednesday, September 6, 2017 - 2:00pm

Jim Stasheff

U Penn


University of Pennsylvania


Bring or send suggestions for future speakers, including your self.

Foliations, even if regular, can have a topologically very bad quotient space of leaves. An alternative would be  a homotopy quotient and appropriate versions of multi-vector fields and  forms thereon.  Historically, this first occurred in 1983 with the work of Batalin-Fradkin-Vilkovisky on the foliation of a constraint surface in a symplectic manifold W by a set of first-class constraints. A key ingredient of the BFV construction is a resolution  $K_{A/I}$ of $A/I$  for the `constraint' ideal $I$ in the commutative algebra  $A = C^{\infty} (W)$ together with  a Chevalley-Eilenberg  complex $\kii$ for $I/I^{\ 2}$ as a Lie algebra.

This led to the strong homotopy version of Lie-Rinehart algebras, introduced by Kjeseth (1996)He constructed such an $\infty$-Lie-Rinehart algebra structure on $(K_{A/I},\kii)$.. Much later, such algebras were used to study general regular foliations  by Huebschmann (2003) and then Vitagliano (2012).

On the other hand,  singular foliations have been  studied beginning around 2006, primarily from a holonomy point of view. Most recently, singular foliations were studied   in Lavau's thesis (2016) from a homotopy point of view closer to that of BFV.

Our goal is to clarify the relation between these various constructions `up to homotopy'