Title: A homotopy theoretic realization of string topology
Authors: Ralph L. Cohen and John D.S. Jones
AMS Classification numbers: 55N45 57R19 18D50
Addresses:
Cohen: Dept. of Mathematics, Stanford University, Stanford, CA 94305
Jones: Dept. of Mathematics, University of Warwick, Coventry CV4 7AL
England
Email: Cohen: ralph@math.stanford.edu
Jones: jdsj@maths.warwick.ac.uk
Let M be a closed, oriented manifold of dimension d. Let LM be the space
of smooth loops in M. Chas and Sullivan have recently defined a kind of
intersection product on the homology H_*(LM) of total degree -d. They
then investigated other structure that this product induces, including a
Lie algebra structure on H_*(LM), and an induced product on the
S^1 equivariant homology, H_*^{S^1}(LM) . These algebraic structures, as
well as others, came under the general heading of the ``String topology" of M.
In this paper we describe a realization of the Chas - Sullivan loop
product in terms of a ring spectrum structure on the Thom spectrum of a
certain virtual bundle over the loop space. We show that this ring
spectrum structure extends to an operad action of the the ``cactus o
perad", originally defined by Voronov, which is equivalent to operad of
framed disks in R^2. We then describe a cosimplicial model of this spectrum and,
by applying the singular cochain functor to this cosimplicial spectrum
we show that this ring structure can be interpreted as the cup product in
the Hochschild cohomology of the cochains, HH^*(C^*(M); C^*(M)).