Séminaire Géométrie Topologie Dynamique
Algorithmic homeomorphism of 3-manifolds is elementary recursive
June 2023
Intervenant : Greg Kuperberg
Institution : UC Davis
Heure : 14h00 - 15h00
Lieu : 2L8
The topic of this talk begins with the folklore result that the geometrization conjecture (now theorem of course) implies that there is an algorithm to determine whether two closed 3-manifolds are homeomorphic.   This corollary was one of Thurston's motivations for the geometrization conjecture.  I will outline an argument for it that does not use any later developments in normal surface theory or hyperbolic groups.  Once we know that the homeomorphism problem is computable, the next question is to find any explicit upper bound on the computational complexity of doing so.  I will briefly mention various complexity classes that could be relevant to recognition or homeomorphism problems in 3-manifold topology.  Finally, I will describe my result that the homeomorphism problem is elementary recursive, meaning that the complexity is bounded by a bounded tower of exponentials.  The most original idea is a self-refinement construction that can be used for triangulations of closed (or finite-volume) hyperbolic manifolds in any dimension.
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