A Proof That Some Spaces Can’t Be Cut

A well-written and mathematically accurate article from Quanta Magazine (no wonder, it’s published by the Simons Foundation):




The question is deceptively simple: Given a geometric space — a sphere, perhaps, or a doughnut-like torus — is it possible to divide it into smaller pieces? In the case of the two-dimensional surface of a sphere, the answer is clearly yes. Anyone can tile a mosaic of triangles over any two-dimensional surface. Likewise, any three-dimensional space can be cut up into an arbitrary number of pyramids.

But what about spaces in higher dimensions? Mathematicians have long been interested in the general properties of abstract spaces, or manifolds, which exist in every dimension. Could every four-dimensional manifold survive being sliced into smaller units? What about a five-dimensional manifold, or one with an arbitrarily large number of dimensions?

Subdividing a space in this way, a process known as triangulation, is a basic tool that topologists can use to tease out the properties of manifolds. And the triangulation conjecture, which posits that all manifolds can be triangulated, is one of the most famous problems in topology.

Ciprian Manolescu remembers hearing about the triangulation conjecture for the first time as a graduate student at Harvard University in the early 2000s. Though Manolescu was considered a phenomenon when he entered Harvard as an undergraduate — he had distinguished himself as the only person, then or since, to notch three consecutive perfect scores in the International Mathematical Olympiad — trying to prove a century-old conjecture isn’t the sort of project that a wise student takes on for a doctoral thesis. Manolescu instead wrote a well-regarded dissertation on the separate topic of Floer homology and spent most of the first decade of his professional career giving the triangulation conjecture little thought. “It sounded like an unapproachable problem, so I didn’t pay much attention to it,” he wrote recently an email.

Others kept working on the problem, however, clawing toward a solution that remained stubbornly out of reach. Then in late 2012, Manolescu, now a professor at the University of California, Los Angeles, had an unexpected realization: The theory he’d constructed in his thesis eight years earlier was just what was needed to clear the final hurdle that had tripped up every previous attempt to answer the conjecture.

Building on this insight, Manolescu quickly proved that not all manifolds can be triangulated. In doing so, he not only elevated himself to the top of his field, but created a tool with enormous potential to answer other long-standing problems in topology.


More at the links above.

(Yay for former IMO contestant! Many IMO participants go on to illustrious careers in mathematics and related fields.)

Manolescu’s result makes precise the “intuitive” notion that higher-dimension spaces are fundamentally different from the 2 & 3 dimensional spaces we are familiar with.

Thanks Irwan Iqbal for sending me this news.

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