“[Floer] Homology theory depends only on the topology of your variety. [This] it’s Floer’s incredible vision, ”said Agustín Moreno of the Institute for Advanced Study.
Division by Zero
Floer’s theory ended up being very useful in many areas of geometry and topology, including mirror symmetry and the study of knots.
“It’s the central tool on the subject,” Manolescu said.
But Floer’s theory did not completely solve Arnold’s conjecture because Floer’s method only worked on one type of variety. Over the next two decades, symplectic surveyors engaged in a massive community effort to overcome this obstruction. Finally, the work led to a test of Arnold’s conjecture where homology is calculated using rational numbers. But he did not solve Arnold’s conjecture when holes are counted using other number systems, such as cyclic numbers.
The reason why the work did not extend to cyclic numerical systems is that the test involved dividing by the number of symmetries of a specific object. This is always possible with rational numbers. But with cyclic numbers, the division is more complicated. If the number system returns after five (counting 0, 1, 2, 3, 4, 0, 1, 2, 3, 4), the numbers 5 and 10 are both zero. (This is similar to the way 13:00 equals 13:00.) As a result, dividing by 5 in this setting is the same as dividing by zero, which is forbidden in math. It was clear that someone would have to develop new tools to avoid this problem.
“If anyone asked me what are the technical things that hinder the development of Floer theory, the first thing that comes to mind is the fact that we need to introduce these denominators,” Abouzaid said.
To extend Floer’s theory and prove Arnold’s conjecture with cyclic numbers, Abouzaid and Blumberg had to look beyond homology.
Climbing the Topologist’s Tower
Mathematicians often think of homology as the result of applying a specific recipe to a shape. During the twentieth century, topologists began to look at homology in its own terms, regardless of the process used to create it.
“We don’t think about the recipe. Let’s think about what comes out of the recipe. What structure, what properties did this homology group have? “Abouzaid said.
Topologists sought other theories that satisfied the same fundamental properties as homology. These became known as generalized homology theories. With homology at its core, topologists built a tower of increasingly complicated theories of generalized homology, all of which can be used to classify spaces.
Floer’s homology reflects the theory of ground floor homology. But symplectic geometers have long wondered whether it is possible to develop Floer’s versions of topological theories above the tower: theories that connect generalized homology with specific features of a space in an environment of infinite dimensions, such as as did Floer’s original theory.
Floer never had a chance to try this job himself, dying in 1991 at the age of 34. But mathematicians continued to look for ways to expand their ideas.
Benchmarking a new theory
Now, after almost five years of work, Abouzaid and Blumberg have made that vision a reality. His new article develops a Floer version of Morava K-theory they then use to prove Arnold’s conjecture for cyclic number systems.
“There’s a sense that this completes a circle for us that relates to Floer’s original work,” Keating said.
