“[Floer] similarity theory depends only on the topology of your manifold. [This] is Florer’s amazing insight,” says Agustin Moreno of the Institute for Advanced Study.
Divide by 0
Florer theory has become extremely useful in many areas of geometry and topology, including mirror symmetry and the study of knots.
“It’s the central tool of this theme,” says Manolescu.
But Floer theory does not completely solve the Arnold conjecture because Floer’s method only works on one type of manifold. Over the next two decades, geosynthetics participated in a great community effort to overcome this obstacle. The work eventually led to a proof of the Arnold conjecture in which the calculation is similar using rational numbers. But it doesn’t solve the Arnold conjecture when holes are counted using other number systems, like cyclic numbers.
The reason the study did not extend to cyclic number systems was the evidence concerning division by the symmetric number of a particular object. This is always possible with rational numbers. But with cyclical numbers, division is more difficult. If the numbering system reverts back after five — counting 0, 1, 2, 3, 4, 0, 1, 2, 3, 4 — then the numbers 5 and 10 are both equivalent to zero. (This is similar to how 13:00 is the same as 1 PM.) Therefore, dividing by 5 in this setting is the same as dividing by 0 – which is forbidden in math. Obviously someone will have to develop new tools to solve this problem.
“If someone asks me what technical things are preventing the development of Floer theory, the first thing that comes to mind is that we have to come up with these denominators,” says Abouzaid.
To extend Floer’s theory and prove Arnold’s conjecture with periodic numbers, Abouzaid and Blumberg needed to look beyond similarities.
Climb the Topologist tower
Mathematicians often think of similarity as the result of applying a particular formula to a shape. During the 20th century, topologists began to consider similarity in its own terms, independent of the process used to create it.
“Let’s not think about the formula. Think about what comes out of the formula. What structure, what properties did this homologous group have? ‘ said Abouzaid.
Topologists have searched for other theories that satisfy the same basic properties as similarity. These are called general similarity theories. With basic similarity, topologists have built an increasingly complex pyramid of general similarity theories, all of which can be used to classify spaces.
Floor homology reflects ground floor homology theory. But geosynthetics have long wondered if it is possible to develop Floer versions of topological theories higher up the tower: theories that connect general homology to specific characteristics of a space in an infinite-dimensional context, just as Floer’s original theory did.
Floer never had a chance to try the work himself, he died in 1991 at the age of 34. But mathematicians continued to find ways to extend his ideas.
Benchmarking a new theory
Now, after nearly five years of work, Abouzaid and Blumberg have realized this vision. Their new paper develops Morava’s version of Flora KY– the theory they later used to prove Arnold’s conjecture for periodic number systems.
“There’s a sense in which this completes a circle for us, linking all the way back to Florer’s original work,” says Keating.
