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Computational complexity and $3$–manifolds and zombies

Greg Kuperberg and Eric Samperton

Geometry & Topology 22 (2018) 3623–3670

We show the problem of counting homomorphisms from the fundamental group of a homology 3–sphere M to a finite, nonabelian simple group G is almost parsimoniously #P–complete, when G is fixed and M is the computational input. In the reduction, we guarantee that every nontrivial homomorphism is a surjection. As a corollary, any nontrivial information about the number of nontrivial homomorphisms is computationally intractable assuming standard conjectures in computer science. In particular, deciding if there is a nontrivial homomorphism is NP–complete. Another corollary is that for any fixed integer m 5, it is NP–complete to decide whether M admits a connected m–sheeted covering.

Given a classical reversible circuit C, we construct M so that evaluations of C with certain initialization and finalization conditions correspond to homomorphisms π1(M) G. An intermediate state of C likewise corresponds to homomorphism π1(Σg) G, where Σg is a Heegaard surface of M of genus g. We analyze the action on these homomorphisms by the pointed mapping class group MCG(Σg) and its Torelli subgroup Tor(Σg). Using refinements of results of Dunfield and Thurston, we show that the actions of these groups are as large as possible when g is large. Our results and our construction are inspired by universality results in topological quantum computation, even though the present work is nonquantum.

One tricky step in the construction is handling an inert “zombie” symbol in the computational alphabet, which corresponds to a trivial homomorphism from the fundamental group of a subsurface of the Heegaard surface.

NP–hardness, \#P–hardness, $3$–manifold invariants
Mathematical Subject Classification 2010
Primary: 20F10, 57M27, 68Q17
Received: 18 July 2017
Revised: 28 January 2018
Accepted: 12 February 2018
Published: 23 September 2018
Proposed: Ian Agol
Seconded: Walter Neumann, Rob Kirby
Greg Kuperberg
Department of Mathematics
University of California, Davis
Davis, CA
United States
Eric Samperton
Department of Mathematics
University of California, Davis
Davis, CA
United States