“[Floer] Homology theory depends only on the topology of the manifold. [This] It was Flor’s incredible insight,” said Augustine Moreno of the Institute for Advanced Study.
divide by zero
Floer theory ended up being very useful in many areas of geometry and topology, including mirror symmetry and knot research.
“It’s a core tool for the subject,” Manolescu said.
But Flor’s theory does not fully solve the Arnold conjecture, because Flor’s method only works for one manifold.For the next two decades, symplectic geometers worked on massive community effort to overcome this obstacle. Ultimately, this work proves that the homogeneity of Arnold’s conjecture is computed using rational numbers. However, it did not solve the Arnold conjecture when counting holes using other number systems such as cyclic numbers.
The reason this work is not extended to the cyclic number system is that the proof involves dividing by a symmetric number of a particular object. This is always possible for rational numbers. But for periodic numbers, division is more finicky. If the number system loops back after five times—counting 0, 1, 2, 3, 4, 0, 1, 2, 3, 4—then the numbers 5 and 10 both equal zero. (This is similar to the way 13:00 is the same as 1 pm.) So dividing by 5 in this setup is the same as dividing by zero – which is forbidden in math. Obviously someone will have to develop new tools to circumvent this problem.
“If someone asks me what the technical factors are that hinder the development of Floer’s theory, the first thing that comes to mind is that we have to introduce these denominators,” Abouzaid said.
To extend Floer’s theory and prove Arnold’s conjecture with cycle numbers, Abouzaid and Blumberg needed to go beyond homology.
Climb the Topologist’s Tower
Mathematicians often view homology as the result of applying a specific recipe to a shape. In the 20th century, topologists began to study homology in their own terms, independent of the process used to create it.
“Let’s not think about the recipe. Let’s think about what this recipe will produce. What structure, what properties does this homologous group have?” Abu Zeid said.
Topologists look for other theories that satisfy the same fundamental properties as homology. These are called generalized homology theories. Using homology as a basis, topologists have constructed an increasingly complex tower of generalized homology theories, all of which can be used to classify space.
Floer homology reflects the underlying homology theory. But symplectic geometers have long wondered whether it would be possible to develop Floer’s version of topological theory higher up in the tower: a theory linking generalized homology to specific properties of space in an infinite-dimensional environment, as in Floer’s original theory. do that.
Flor himself never had the opportunity to try the work, and died in 1991 at the age of 34. But the mathematician continued to find ways to expand his ideas.
Benchmarking new theories
Now, after nearly five years of work, Abouzaid and Blumberg have achieved that vision. Their new paper develops a Floer version of Morava K– Then they used to prove the theory of Arnold’s conjecture for cyclic number systems.
“In a sense, this completes a cycle for us that has always been associated with Flor’s original work,” Keating said.