We show the problem of counting homomorphisms from the fundamental group of a homology @math -sphere @math to a finite, non-abelian simple group @math is #P-complete, in the case that @math is fixed and @math 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 @math , it is NP-complete to decide whether @math admits a connected @math -sheeted covering. Our construction is inspired by universality results in topological quantum computation. Given a classical reversible circuit @math , we construct @math so that evaluations of @math with certain initialization and finalization conditions correspond to homomorphisms @math . An intermediate state of @math likewise corresponds to a homomorphism @math , where @math is a pointed Heegaard surface of @math of genus @math . We analyze the action on these homomorphisms by the pointed mapping class group @math and its Torelli subgroup @math . By results of Dunfield-Thurston, the action of @math is as large as possible when @math is sufficiently large; we can pass to the Torelli group using the congruence subgroup property of @math . Our results can be interpreted as a sharp classical universality property of an associated combinatorial @math -dimensional TQFT.
One early, important UMTC is the (truncated) category @math of quantum representations of @math at a principal root of unity. This category yields the Jones polynomial for a link @math (taking @math , the first irreducible object) and the Jones-Witten-Reshetikhin-Turaev invariant of a closed 3-manifold. In separate papers, Freedman, Larsen, and Wang showed that @math and @math are both quantumly universal representations of @math and @math @cite_22 @cite_4 .
@cite_4 @cite_22
2962824725 2088353214 2080743555 1990891135
Freedman, Kitaev, and Wang (2002), and later Aharonov, Jones, and Landau (2009), established a quantum algorithm to "additively" approximate the Jones polynomial V(L;t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang (2002) established that the approximation is universal for quantum computation at a non- lattice, principal root of unity. We show that any value-distinguishing approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. Given the power to decide whetherjV(L;t)j b for fixed constants 0 1, T(G; x; y) is #P-hard to approximate within a factor of c even for planar graphs G. Along the way, we clarify and generalize both Aaronson's theorem and the Solovay- Kitaev theorem. The generalization of Jones polynomial of links to the case of graphs inR3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups. The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory We describe a correspondence between the Donaldson–Thomas invariants enumerating D0–D6 bound states on a Calabi–Yau 3-fold and certain Gromov–Witten invariants counting rational curves in a family of blowups of weighted projective planes. This is a variation on a correspondence found by Gross–Pandharipande, with D0–D6 bound states replacing representations of generalised Kronecker quivers. We build on a small part of the theories developed by Joyce–Song and Kontsevich–Soibelman for wall-crossing formulae and by Gross–Pandharipande–Siebert for factorisations in the tropical vertex group. Along the way we write down an explicit formula for the BPS state counts which arise up to rank 3 and prove their integrality. We also compare with previous “noncommutative DT invariants” computations in the physics literature.