Let be
an embedded closed connected exact Lagrangian submanifold in a connected cotangent
bundle .
In this paper we prove that such an embedding is, up to a finite covering space lift of
, a homology
equivalence. We prove this by constructing a fibrant parametrized family of ring spectra
parametrized by the
manifold . The homology
of will be the (twisted)
symplectic cohomology of .
The fibrancy property will imply that there is a Serre spectral sequence converging to the
homology of .
The fiber-wise ring structure combined with the intersection product on
induces a
product on this spectral sequence. This product structure and its relation to the intersection
product on is
then used to obtain the result. Combining this result with work of Abouzaid we arrive at the
conclusion that
is always a homotopy equivalence.
Department of Mathematics
Columbia University
2990 Broadway
New York, NY 10027
USA
and
Simons Center for Geometry and Physics
State University of New York
Stony Brook, NY 11794-3636
USA