Unlocking the Mysteries of 3-Manifolds and Zombies with Eric Samperton 🧟‍♂️

Computation in geometric topology
14 December 2017

Abstract: Let G be a finite group, and let M be a three-manifold. I’ll discuss the computational complexity of the problem of counting homomorphisms of pi_1(M) to G. When G is nonabelian simple, we show that the problem is #P-complete. This conclusion holds even if we only consider integer homology three-spheres, or knot complements. The structure of the proof is inspired by topological quantum computing, except nothing is quantum. The main tools and ideas are Aut(G)-equivariant reversible circuits, joint surjectivity lemmas in group theory, mapping class group actions on G-representation sets, and stabilization results for G-covers a la Livingston and Dunfield-Thurston.