“[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight, ”he said Agustin Moreno from the Department of Advanced Studies.
Divide by zero
Flower theory ended up being wildly useful in many areas of geometry and topology, among others mirror symmetry and the study of knots.
“It’s the central tool in the subject,” Manolescu said.
But the Floer theory did not completely solve the Arnold conjecture, because Floer’s method only worked on one type of manifold. Over the next two decades, sympathetic geometries dealt with one massive community effort to overcome this obstacle. Eventually, the work led to a proof of the Arnold conjecture, in which the homology is calculated using rational numbers. But it did not solve the Arnold conjecture when holes are counted using other number systems, such as cyclic numbers.
The reason why the work did not extend to cyclic number systems is that the evidence involved dividing by the number of symmetries for a particular object. This is always possible with rational numbers. But with cyclical numbers, division is more picky. If the number system goes back after five – counts 0, 1, 2, 3, 4, 0, 1, 2, 3, 4 – then the numbers 5 and 10 both correspond to zero. (This is similar to the way 13:00 is the same as 13:00.) As a result, dividing by 5 in this setting is the same as dividing by zero – something that is forbidden in mathematics. It was clear that someone needed to develop new tools to work around this problem.
“If anyone asked me what are the technical things that prevent the Floer theory from evolving, the first thing that comes to mind is the fact that we have to introduce these denominators,” Abouzaid said.
To expand Floers’ theory and prove the Arnold conjecture by cyclical numbers, Abouzaid and Blumberg had to look beyond homology.
Ascension of the Topologist’s Tower
Mathematicians often think of homology as the result of applying a specific recipe to a form. During the 20th century, topologists began to look at homology on its own terms, regardless of the process used to create it.
“Let’s not think about the recipe. Let’s think about what comes out of the recipe. What structure, what characteristics did this homology group have? “Said Abouzaid.
Topologists sought other theories that met the same basic characteristics as homology. These became known as generalized theories of homology. With homology at the bottom, topologists built a tower of increasingly complex generalized homology theories, all of which can be used to classify spaces.
Flower homology reflects the theory of homology on the ground floor. But sympathetic geometries have long wondered whether it is possible to develop Floer versions of topological theories higher up in the tower: theories that connect the generalized homology with specific features of a space in an infinite-dimensional setting, like Floer’s original theory did.
Floer never had a chance to try this work for himself, and died in 1991 at the age of 34. But mathematicians continued to look for ways to expand his ideas.
Benchmarking a new theory
Now, after almost five years of work, Abouzaid and Blumberg have realized this vision. Their new paper develops a Floer version of Morava Ktheory, which they then use to prove the Arnold conjecture of cyclic number systems.
“There’s a sense in which this completes a circle for us that ties right back to Flo’s original work,” Keating said.