A polarized view of string topology

Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h∗(LM), where h∗ is a generalized homology theory that supports an orientation of M . We will show that these operations give h∗(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h∗(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q ≥ 1. The absence of a counit follows from the nonexistence of an operation associated to the disk, D, viewed as a cobordism from the circle to the empty set. We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h∗(LM) to be an appropriate homological proobject associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the “polarized Atiyah dual” of LM . An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces. Introduction LetM be a closed, oriented manifold of dimension n, and let LM be its free loop space. The “string topology” theory of Chas-Sullivan [3] describes a rich structure in the homology and equivariant homology of LM . The most basic operation is an intersection type product, ◦ : Hq(LM)×Hr(LM) −→ Hq+r−n(LM) that is compatible with both the intersection product in the homology of the manifold, and the Pontrjagin product in the homology of the based loop space, H∗(ΩM). Moreover this product The first author was partially supported by a grant from the NSF