Introduction to Quantum Topology

Question 1

Discuss the motivating problem of counting homomorphisms between groups

Answer 1

We will consider F f.g. and G finite so that |Hom(F, G)| is finite

In particular we will consider homomorphisms from fundamental group of a closed manifold to finite group G

In case d=1, X = S^1 so pi_1 is just Z and Hom(Z, G) = |G|

In case d = 2, need to use triangulations…

pg 1-2

Question 2

Define: triangulation of a surface

Answer 2

A triangle in a surface is the image of a euclidean triangle under an embedding

A triangulation of surface S is a finite set t of triangles in S s.t.
1. The triangles of t cover S
2. The intersection of 2 distinct triangles is either empty, a vertex, or an edge

Thm. (Rado) Any compact surface has a triangulation
pg 3-4

Question 3

Discuss moves of triangulations

Answer 3

1. Ambient isotopy: consider a homeomorphism f:S –> S isotopic to identity <– i.e. there exists a homotopy H s.t. H( _ , t) is a homeomorphism for all t, H( _ , 0) = id and H( _ , 1) = f

We map the triangulation t –> f(t) = { f(triangle) : triangle in t}

2. Pachner 1-3. Barrycentrically subdivide 1 triangle
3. Pachner 2-2. Flip 2 adjacent triangles

Thm. (Pachner) Two triangulations of S are related by a finite sequence of isotopies, Pachner moves, and their inverses.

pg 7-8

Question 4

Discuss how to prove Euler characteristic is a topological invariant using moves on triangulations

Answer 4

See invariant under different moves + homeomorphisms

pg 8-9

Question 5

Discuss state sum invariants of closed surfaces from finite groups. Show topological invariant

Answer 5

V = C[G] <– group algebra of finite group
a in V (x) V (x) V
N in (V (x) V)*

a = 1/|G| sum_{g,h,k in G s.t. ghk = 1} g(x)h(x)k

N(g(x)h) = |G| delta_gh, 1

S triangulated by t.

Assign a to each triangle.

Contract components at each edge using N
pg 12 - 17

Question 6

Compute Z_G(S^2)

Answer 6

=|G|

pg 17

Question 7

How is Z_G(S) related to the number of homomorphisms from pi_1(S) to G?

Proof?

Answer 7

pg 17-18