# Conference - Loop spaces in geometry and topology

# Titres et résumés

### Mohammed Abouzaid

Title: Symplectic cohomology and loop homology

Abstract: The string topology of Chas-Sullivan produces operations on the homology of the free loop space of orientable manifolds, and analogous structures are known to exist on the symplectic cohomology of their cotangent bundles. Work of Kragh has indicated that these groups are only isomorphic under the assumption that the manifold is Spin. The goal of this minicourse is to deﬁne twisted versions of loop space homology and symplectic cohomology for an arbitrary closed smooth manifold which are isomorphic to each other. I will also explain how to build Batalin Vilkovisky structures on the two sides which are isomorphic. Reference: http://arxiv.org/abs/1312.3354

### Denis Auroux

Title: A plethora of Lagrangian tori

Abstract: Over the last decade, Lagrangian Floer homology and Fukaya categories have led to significant progress in our understanding of Lagrangian submanifolds, especially in cotangent bundles. And yet, recent results suggest that even the simplest symplectic manifolds contain plenty of "exotic" Lagrangian tori, and the classification up to Hamiltonian isotopy remains quite mysterious. We will review various examples (some of which have appeared in Renato Vianna's thesis).

### Somnath Basu

Title: The closed geodesic problem for four manifolds

Abstract: We will explain why a generic metric on a smooth four manifolds (with second Betti number at least three) has the exponential growth property, i.e., the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. Time permitting, we shall explain related topological consequences.

### Alexander Berglund

Title: Loop spaces and Koszul algebras

Abstract: I will begin by briefly reviewing Koszul duality for algebras over operads and discuss the relation between Koszulness and formality. Then I will explain how Koszul algebras can be used to construct tractable algebraic models for computing the homology of free and iterated loop spaces.

### Ralph Cohen

Title: Calabi-Yau categories, string topology, and the Floer field theory of the cotangent bundle.

Abstract: I will describe joint work with Sheel Ganatra, in which we prove an equivalence between two chain complex valued topological field theories: the String Topology of a manifold M, and the Floer field theory of its cotangent bundle. This expands upon results of Viterbo, Abbondandolo and Schwarz, Abouzaid, and others. We use recent work of Kontsevich and Vlassopolous which describes two duality conditions among A-\infty-algebras and categories and show how they give rise to topological field theories. These "Calabi-Yau" conditions are related to the two dimensional cobordism hypothesis viewpoint of Lurie. We use this perspective to prove the equivalence of the theories above. We then show how Koszul duality affects the Calabi-Yau condition, and how it gives rise to a duality relationship between field theories.

### Octav Cornea

Title: Some properties of the Grothendieck group of the derived Fukaya category.

Abstract: The derived Fukaya category is a triangulated category encoding symplectic rigidity properties of the Lagrangian submanifolds of a fixed symplectic manifold. In this talk, based on joint work with Paul Biran, I will discuss some properties of the associated Grothendieck group that follow from the study of Lagrangian cobordism.

### Kenji Fukaya

Title: Cyclic homology in Lagrangian Floer theory and pseudo-holomorphic curve

Abstract: Cyclic homology of A-infinity algebras or categories appearing in Lagrangian Floer thoery is expected to be related to the S^1-equivariant quantum cohomology or symplectic homology of the ambient symplectic manifold. It is also important to generalize Lagrangian Floer theory including pseudo-holomorphic curves from more general bordered Riemann surfaces than a disk. I will explain these issues and some of the techniques toward establishing such results.

### Kathryn Hess

Title: Cosimplicial-Simplicial models for the free loop space

Abstract: In this mini-course I will introduce various simplicial, cosimplicial, and chain complex models for free loop spaces and their structure maps and describe the relationships among these models.

Simplicial and cosimplicial models

- The cyclic nerve
- The simplicial Hochschild construction
- The simplicial coHochschild construction
- The cocyclic model of Jones

Chain complex models

- Preliminaries on the bar and cobar constructions
- The Hochschild complex
- The coHochschild complex
- The cyclic complex
- The cocyclic complex
- Hochschild cohomology

### Richard Hepworth

Title: String topology of classifying spaces.

Abstract: Let G be a compact Lie group. Form the classifying space BG. Then form the space LBG of all loops in the classifying space (the strings). Finally, take the homology H_*(LBG). What is its structure? One answer, due to Chataur and Menichi, is that it is part of a "homological conformal field theory", which is an algebraic structure governed by surfaces and their diffeomorphisms. A more recent answer, due to Anssi Lahtinen and myself, is that it is part of an "h-graph field theory" where the surfaces and diffeomorphisms are replaced by much looser homotopy-theoretical versions.
These lectures will go through the definitions and constructions in the case where G is finite, with lots of worked examples and so on. Lecture notes will appear on my homepage in due course, hopefully at least two weeks before the workshop.

http://homepages.abdn.ac.uk/r.hepworth/pages/

### Nancy Hingston

Title: Geodesics and the structure of the free loop space

Abstract: The Chas-Sullivan product is a product on the homology of the free loop space LM of a compact oriented manifold M, the loop homology of M. If we fix a metric on M, the critical points of the length function on LM are the closed geodesics on M in the given metric. Morse theory gives a link between the geometry of closed geodesics, and the algebraic structure on the loop homology M given by the Chas-Sullivan product.

I will "review" Morse theory, then introduce the Chas Sullivan product and the “dual” cohomology product, and the fundamental inequality for the "minimax" critical level cr(X) of a homology class X on LM: cr(X∗Y) ≤ cr(X) + cr(Y).

There is a dual inequality for loop cohomology (with opposite sign!): cr(x∗y) ≥ cr(x) + cr(y).

I will discuss what the geometry of geodesics tells us about algebraic structure on LM, and what algebra tells us about geometry. Some theorems from the past are naturally stated in terms of loop products. For manifolds such as spheres and projective spaces for which there is a metric with all geodesics closed, the loop homology and cohomology rings are nontrivial, and closely linked to the geometry. For spheres we obtain significant (and so far mysterious) constraints on the critical levels of loop homology classes: for any fixed metric on a sphere there are positive constants A and B so that for each homology class X we have A deg(X) - B ≤ cr(X) ≤ A deg(X) + B.

References:

[1] R. Bott, On the iteration of closed geodesics and the Sturm intersection theory, Comm. Pure Appl. Math. 9 (1956), 171-206.

[2] M. Chas and D. Sullivan, String topology, preprint, math.GT/9911159 (1999).

[3] R. L. Cohen, J. D. S. Jones, and J. Yan. The loop homology algebra of spheres and projective spaces. In Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), volume 215 of Progr. Math., pages 77--92. Birkh¨auser, Basel, 2004.

[4] M. Goresky and N. Hingston. Loop products and closed geodesics. Duke Math. J., 150(1):117--209, 2009.

[5] N. Hingston and A. Oancea. The space of paths in complex projective space with real boundary conditions. arXiv: 1311.7292, 2013.

[6] N.Hingston and H-B Rademacher, Resonance for loop homology of spheres. J. Differential Geom. 93 (2013), no. 1, 133--174.

[7] W. Klingenberg, Lectures on Closed Geodesics, Grundlehren der mathematischen Wissenschaften 230, Springer Verlag, Berlin, 1978.

[8]. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, Princeton N.J., 1963.

### Thomas Kragh

Tittle: A simple construction of the Fukaya, Seidel, Smith - spectral sequence.

Abstract: In a paper by Fukaya, Seidel, Smith in 2007 they used a spectral sequence on A-infinity categories, induced by a filtration given by certain Lefschetz thimbles, to prove that nearby Lagrangians (spin) in a simply connected (spin) cotangent bundle are homology equivalent to the zero section. In this talk I will present a much more direct and simple construction of a similar spectral sequence and sketch an alternate proof of this. Then if time permits I will also discus how this can be generalized as Abouzaid generalized Fukaya, Seidel and Smiths proof to the non simply connected (and non spin) cases.

### Janko Latschev

Title: Nonexact Liouville embeddings and symplectic homology

Abstract: I will explain how string topology type operations on symplectic homology give rise to obstructions to embedding Liouville domains into each other. The underlying argument was first outlined by Fukaya for Lagrangian embeddings, and has its roots in Gromov's original proof that there are no exact Lagrangian submanifolds in C^n. This is joint work in progress with Kai Cieliebak.

### Dennis Sullivan

Title: Algebraic Models of Manifolds

Abstract: It appears that some algebraic structures associated to compact manifolds through mapping spaces like loop spaces depend only on the homotopy type of the manifold [rel boundary]. One example is the string bracket for closed three manifolds generalizing the Goldman bracket for surfaces. Since the homotopy type is a strong invariant for three manifolds the string bracket is still quite useful for example in describing the form of Thurston's geometrizationd [Chas -Gadgil largely completing a story beginning in Abbaspour's Thesis].

At the moment it is not clear whether the generalization of the Turaev cobracket to the string cobracket is homotopy invariant.This is closely related to the same question for a coproduct studied with Chas and students and independently by Goresky and Hingston.

In a slightly different and more delicate setting Basu has convinced me one can prove the corresponding coproduct really depends on the homeomorphism type of certain three manifolds.

The area where a lot of homotopy invariance is understood is related to string topology operations associated to the chains on the open moduli space of riemann surfaces. This might be termed Noncompactified String Topology. The undecided questions about the cobracket and the coproduct is part of an area which is sometimes called Compactified String Topology. Finally, the results of Basu realizing structures that detect more than homotopy type restrict attention to natural subsets of the mapping spaces and is referred to as Stratified String Topology.

Underlying all of these theories and their distinctions is the fundamental question of characterizing by algebraic structures on "Chain Complexes with Poincare Duality" Compact manifolds up to homotopy, up to homeomorphism and up to diffeomorphism.

Even a clear conceptual answer to this question is outstanding for simply connected manifolds in characteristic zero and for large dimensions.

This answer in characteristic zero might be useful in areas outside topology. One example would be studying finite dimensional algorithms that could arise using this algebraic characterization of manifolds. These algorithms would describe processes that take place in space-time like 3D fluid motion. In the middle 90's the relation of the fluid PDE's to linking and to the reconnection of closed vortex lines motivated the pictures that began the study of String Topology.

### Craig Westerland

Title: Homology of stabilized moduli of Lefschetz fibrations

Abstract: This talk is about the space of relatively minimal Lefschetz fibrations over surfaces X with at most one node in each fibre. We study the homology of these spaces as the number of nodal fibres tends to infinity, and relate the stable homology to the homology of the function space of maps from X to a variant of the Deligne-Mumford compactification of M_g. Restricting our answer to H_0 yields a form of Auroux's stable classification of Lefschetz vibrations.