A polarized view of string topology

TL;DR

This paper explores string topology operations on the loop space of a manifold within a generalized homology framework, revealing a Frobenius algebra structure and proposing a polarized Atiyah-dual concept for broader surface operations.

Contribution

It introduces a new perspective on string topology via a polarized Atiyah-dual construction, extending the algebraic structure to include operations from surfaces with multiple boundaries.

Findings

h_*(LM) forms a unital, commutative Frobenius algebra without a counit.

Operations exist for surfaces with p incoming and q outgoing boundaries, q > 0.

Homological obstructions relate to the construction of a disk operation, leading to the concept of a polarized prospectrum.

Abstract

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 >0. The absence of a counit follows from the nonexistence of an operation associated to the disk, D^2, 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 pro-object 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.