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Periodic Report Summary 1 - HFFUNDGRP (New connections in low-dimensional topology: Relating Heegaard Floer homology and the fundamental group)

Low-dimensional topology is a central area of 21st century research mathematics that has enjoyed a period of intense activity in recent years. On the one hand, steady progress in understanding the deep connections between group theory and three-manifold topology has led to the resolution of long-standing conjectures; on the other, the introduction of modern homological invariants (e.g. Heegaard Floer homology, Khovanov homology) have contributed new insight and perspective on old problems in low-dimensional topology while establishing a vibrant field of research exhibiting new ties to physics. The project HFFUNDGRP aims to treat a major open problem that is positioned at the nexus of these two areas:

Problem. Establish a relationship between the fundamental group of a three-manifold and its Heegaard Floer theory.

The work of Agol and Wise provides the most recent confirmation that the fundamental group of a three-manifold captures subtle aspects of the geometry and topology of the space in question. The fundamental group is a (if not, the) central algebraic object of study in low-dimensional topology. In turn, the field is generative of an extremely rich set of problems, interesting tools introduced in their resolution, and a surprisingly rich algebraic structure at its foundation.

Heegaard Floer homology presents a new tool -- with origins in physics and gauge theory -- for investigating orientable three-manifolds that is both extremely powerful and inherently different from the fundamental group. While there are certainly hints that some aspects of the fundamental group might be captured by Heegaard Floer homology, perhaps the most surprising potential connection is formulated in my joint work with Boyer and Gordon:

Conjecture. An irreducible three-manifold is an L-space if and only if its fundamental group is not left-orderable.

L-spaces are rational homology spheres with simplest-possible Heegaard Floer homology. Given that this family of three-manifolds comes up frequently in application, it has been asked if there is some characterisation of L-spaces that is topological -- one that does not involve the definition of Heegaard Floer homology. The resolution of the above conjecture would provide an answer to this question that is particularly interesting.

The other side of the conjecture involves an auxiliary structure on a group: a left-order is a (strict, total) order on the elements of the group that is invariant under multiplication on the left. There has been interest in left-orderable groups in low-dimensional topology that predates the introduction of Heegaard Floer homology, and there is a long-standing interaction between foliations of three-manifolds and order structures on the fundamental group

Understanding and, ultimately, resolving the above conjecture is the central goal of the project HFFUNDGRP. As a direct consequence of this project, through a collaborative effort of the PI with J. Hanselman (UT Austin), J. Rasmussen (Cambridge), and S. Rasmussen (Cambridge), the conjecture is now known to hold for a large and important class of three-manifolds called graph manifolds.

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