Computational complexity and 3-manifolds and zombies
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We show the problem of counting homomorphisms from the fundamental group of a homology 3-sphere M to a finite, non-abelian simple group G is #P-complete, in the case that G is fixed and M is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions, we can guarantee that every non-trivial homomorphism is a surjection. As a corollary, for any fixed integer m > 5, it is NP-complete to decide whether M admits a connected m-sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit C, we construct M so that evaluations of C with certain initialization and finalization conditions correspond to homomorphisms π_1(M) → G. An intermediate state of C likewise corresponds to a homomorphism π_1(Σ_g) → G, where Σ_g is a pointed Heegaard surface of M of genus g. We analyze the action on these homomorphisms by the pointed mapping class group MCG_*(Σ_g) and its Torelli subgroup Tor_*(Σ_g). By results of Dunfield-Thurston, the action of MCG_*(Σ_g) is as large as possible when g is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of Sp(2g,Z). Our results can be interpreted as a sharp classical universality property of an associated combinatorial (2+1)-dimensional TQFT.
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