# String Topology Mini-workshop

**August 1–5, 2016**

IBS Center for Geometry and Physics, Pohang, South Korea

## Speakers

- Daniela E. Santander (Free University of Berlin)
- Kate Poirier (New York City College of Technology)
- Manuel Rivera (University of Miami)
- Nissim Ranade (Stony Brook University)

## Organizer

- Gabriel C. Drummond-Cole (IBS-CGP)

## Venue

- CGP Main Hall
- Information Research Lab #143, POSTECH

## Talk Schedule

Time | Aug. 1 (Mon) | Aug. 2 (Tue) | Aug. 3 (Wed) | Aug. 4 (Thu) | Aug. 5 (Fri) |
---|---|---|---|---|---|

11:00 – 12:00 | Free discussion |
N. Ranade | D. E. Santander | D. E. Santander | Free discussion |

16:00 – 17:00 | K. Poirier | M. Rivera | M. Rivera | K. Poirier | Free discussion |

## Title & Abstract

- Speaker
- Daniela E. Santander (Free University of Berlin)
- Title
- Sullivan diagrams and homological stability
- Abstact
- In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams. I will describe two stabilization maps for Sullivan diagrams one with respect to genus and one with respect to punctures and describe how some components of this complex have homological stability with respect to these maps. I will also give some computational results for small genus and number of punctures.

- Speaker
- Kate Poirier (New York City College of Technology)
- Talk 1 Title
- Chain-level string topology operations
- Abstact
- String topology studies algebraic invariants of manifolds arising from intersecting loops in the manifolds. Traditionally, the algebraic structure is phrased in terms of an action of the homology of the moduli space of Riemann surfaces on the homology of the free loop space of the manifold. It is expected that this action should be induced by an action of the chains on a compactification of moduli space on the chains of the free loop space. In this talk, we report on recent joint work with Drummond-Cole and Rounds constructing a space of operations on the chains of the free loop space which describes part of this action.
- Talk 2 Title
- String diagrams and directed graphs
- Abstact
- In ongoing joint work with Drummond-Cole and Rounds, we show that the space of string diagrams acts on the chains of the free loop space of a manifold. Previously, Tradler and Zeinalian showed that a chain complex of directed graphs acts on the Hochschild complex of a V-infinity algebra. It is thought that these actions should be a topological and algebraic version of the same story. In this talk, we report on current joint work with Tradler on the first step describing the relationship between these two actions. In particular, we describe a map relating the two spaces of operations.

- Speaker
- Manuel Rivera (University of Miami)
- Title
- Chain level transversality for string topology coproduct
- Abstact
- I will describe a geometric chain level formulation for a “secondary" coproduct on a suitable chain model for the free loop space of a manifold. This operation- which combines a 1-parameter family of self-intersections on a family of loops- was originally described by Goresky and Hingston at the level of the (relative) homology by using a finite dimensional approximation of Morse for the free loop space. The operation is also analogue to a coproduct described by Abbondandolo and Schwarz on (a version of) the symplectic Floer homology of the cotangent bundle.

To have a better grasp of the properties of this coproduct and to relate it to constructions in symplectic topology it is convenient to describe explicitly the transversality perturbations made at the chain level to obtain an operation parametrized by a nice geometric object. I will explain why this process is more subtle for this operation than for other string topology operations (such as the Chas-Sullivan loop product) and will outline how Dingyu Yang and I have achieved this in work in progress using the formalism of De Rham chains. There is also a rich algebraic theory behind this secondary coproduct and its compatibilities with other operations. Time permitting, I will describe some of the algebraic theory as well.

- Speaker
- Nissim Ranade (Stony Brook University)
- Title
- String topology operations and the diagonal map
- Abstact
- We will examine how the string topology loop product relates algebraically to the diagonal map on spaces. We will use algebraic models for manifolds developed be Cameron Crowe and appropriate algebraic models for the figure-8 space to understand these operations better.