#### Building Bridges with Floer Homotopy Theory

SLMath - November 2023

It started with a walk. Lisa Traynor and Hiro Lee Tanaka liked to hike through the fire trails near the Simons Laufer Mathematical Sciences Institute (SLMath) during their lunch hours. Both were participants in the Floer homotopy theory semester program but came to the program from different edges of the field. Traynor, a professor at Bryn Mawr College, specializes in symplectic and contact structures on manifolds, while Tanaka of Texas State University focuses on homotopy theory. But eventually their walks became a path to a joint research project.

Traynor has been interested in knot invariants for many years. As a mathematical object, a knot is similar to a knot in your shoelaces, but the ends of the shoelaces are fused together to make a continuous loop. In the field of contact topology, one studies Legendrian knots, which are knots that satisfy an extra geometrical condition. An important problem is to understand when a Legendrian knot can be realized as the boundary edge of a surface that satisfies Lagrangian conditions imposed by a symplectic structure.

The same kind of knot can look different depending on how it is drawn in two dimensions, so mathematicians have developed tools called knot invariants to determine whether two different-looking pictures of knots actually represent different knots or are just different snapshots of the same knot. Knot invariants can be numbers, polynomials (expressions like \(x^2+2x+7\)), or more complicated mathematical structures. In recent years, knot theorists have used topological objects called homology groups to create new knot invariants. Meanwhile, in homotopy theory, researchers have developed a structure called a spectrum that, among other things, provides extra structure to homology groups. During the SLMath program, Traynor started wondering whether there was a spectrum invariant that could be associated to a Legendrian knot or a higher-dimensional Legendrian shape. “It was a natural question,” Traynor says. "Can you define an invariant spectrum for a Legendrian knot?”

*A Lagrangian filling of a Legendrian knot leads to a relationship between their spectra. Image: Lisa Traynor*

The same kind of knot can look different depending on how it is drawn in two dimensions, so mathematicians have developed tools called knot invariants to determine whether two different-looking pictures of knots actually represent different knots or are just different snapshots of the same knot. Knot invariants can be numbers, polynomials (expressions like \(x^2+2x+7\)), or more complicated mathematical structures. In recent years, knot theorists have used topological objects called homology groups to create new knot invariants. Meanwhile, in homotopy theory, researchers have developed a structure called a spectrum that, among other things, provides extra structure to homology groups. During the SLMath program, Traynor started wondering whether there was a spectrum invariant that could be associated to a Legendrian knot or a higher-dimensional Legendrian shape. “It was a natural question,” Traynor says. "Can you define an invariant spectrum for a Legendrian knot?”

“I tried to do it on my own for a couple of weeks,” Traynor says, but she felt like she didn’t know enough about homotopy theory to start applying it to her work. “Finally I realized I had to join forces with somebody.” Through the conversations she had been having with Tanaka on their lunchtime walks, she discovered that he was the right collaborator to bring on board.

“Lisa introduced me to a question that I immediately found appealing,” Tanaka says. When Traynor came to him with invariants based in homology theory, his homotopy theory background allowed him to recognize where the extra structure of spectra could come from. “I was given a picture of something, and I recognized that picture because I’d seen it before,” he says. Together, they were able to show that Traynor’s homologically-defined invariants could be extended to a spectrum, and when the Legendrian has a Lagrangian filling, the spectrum of the Legendrian reflects a spectrum that can be associated to the filling.

Tanaka and Traynor are still in the process of writing up their results to publish in a journal, and after that, they are excited about the new avenues of inquiry their work has opened. Tanaka sees their research as building a bridge between two mathematical islands: the geometric world of a particular class of manifolds and the very abstract notion of a spectrum in homotopy theory. “In the process of building a bridge between two islands, somehow we’ve ended up with a better perspective of both,” he says.

Floer homotopy theory is a broad field, and organizers of the SLMath semester program took care to invite researchers from a wide range of mathematical research areas falling within that field and to facilitate conversations between them. “There was such an effort to initiate dialogue between different fields,” Tanaka says. “Our project is just one example of how that succeeded.”

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*Thumbnail image: Nathalie Wahl*)