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.
Our approach to th:main (and that of Krovi and Russell for their results) is inspired by quantum computation and topological quantum field theory. Every unitary modular tensor category (UMTC) @math yields a unitary 3-dimensional topological quantum field theory @cite_37 @cite_31 @cite_42 . The topological quantum field theory assigns a vector space @math , or , to every oriented, closed surface. It also assigns a state space @math to every oriented, closed surface with @math boundary circles, where @math is an object in @math interpreted as the color" of each boundary circle. Each state space @math has a projective action of the mapping class group @math . (In fact the unpointed mapping class group @math acts, but we will keep the basepoint for convenience.) These mapping class group actions then extend to invariants of 3-manifolds and links in 3-manifolds.
@cite_37 @cite_42 @cite_31
2080743555 2043706520 2052952091 2164171842
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 I describe the noncommutative Batalin-Vilkovisky geometry as- sociated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. I show, in particular, that the algebras over the Feyn- man transform of a twisted modular operad P are in one-to-one correspondence with solutions to quantum master equation of Batalin-Vilkovisky geometry on the affineP manifolds. As an application I give a construction of character- istic classes with values in the homology of the quotient of Deligne-Mumford moduli spaces. These classes are associated naturally with solutions to the quantum master equation on affineS(t) manifolds, where S(t) is the twisted modular Det operad constructed from symmetric groups, which generalizes the cyclic operad of associative algebras. Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense that a quantum circuit com- putation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere.