Back in the 1990s, Mrowka and Kronheimer investigated what happens when you excise a two-dimensional surface from a four-dimensional manifold. If the manifold itself is simply connected, what conditions must surfaces meet to guarantee that their complements must also be simply connected?
Kronheimer and Mrowka knew that some kinds of surfaces could have complements that weren’t simply connected. But their work seemed to indicate that another broad class of surfaces must always have simply connected complements.
For nearly three decades, nobody could find an example of a surface in that class whose complement was not simply connected. But in the fall of 2023, after coming across the problem, Ruberman thought he could. Instead of starting with a four-dimensional manifold and cutting out a surface, he began with a two-dimensional surface that had the necessary properties and built a manifold around it.
First, he fattened the surface into a four-dimensional blob. This four-dimensional blob had a three-dimensional boundary, just as a three-dimensional object like a ball has a two-dimensional boundary. Ruberman wanted to attach a carefully chosen four-dimensional manifold to the other side of the boundary, which would serve as the surface’s complement. If the gambit worked, then this manifold would have a complicated fundamental group, yet the fundamental group of everything taken together would be trivial. The newly constructed four-dimensional manifold would therefore be simply connected.
But to be able to glue everything together in the right way, he had to show that the fundamental group of the new addition satisfied all sorts of properties. “I had no idea how to do that,” Ruberman said.
Then in January, Hughes — a group theorist — gave a talk at Brandeis. Ruberman was in the audience. He recognized that Hughes might have the missing piece he was looking for. The two met up the following day, and within a few hours, they’d worked out the main ideas they needed. What Ruberman was missing “is something group theorists have been computing for 70, 80 years at this point,” Hughes said. “We’ve been at this forever.” By the end of the week, they had a completed proof.
“I knew some things, and he knew some things, and between the two of us, we knew enough to just do it,” Ruberman said.
Because of the way group theory gets used in the proof, “it’s a little bit unusual,” said Maggie Miller of the University of Texas, Austin. “It’s written a little bit different than most four-dimensional topologists would be comfortable with.”
The result is yet another example of how complicated four-dimensional topology can get. “There are more interesting embeddings of surfaces than we thought,” Hughes said. This makes it more difficult to classify manifolds, and harder to prove other kinds of results about them.
Nevertheless, in March, İnanç Baykur of the University of Massachusetts, Amherst, who organized last year’s list-making conference with Ruberman, announced the solution to another problem involving simply connected four-dimensional manifolds from the 1997 list.
It seems the topologists are cleaning house.
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