Introduction to Quantum Topology Flashcards
Discuss the motivating problem of counting homomorphisms between groups
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
Define: triangulation of a surface
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
Discuss moves of triangulations
- 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}
- Pachner 1-3. Barrycentrically subdivide 1 triangle
- 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
Discuss how to prove Euler characteristic is a topological invariant using moves on triangulations
See invariant under different moves + homeomorphisms
pg 8-9
Discuss state sum invariants of closed surfaces from finite groups. Show topological invariant
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
Compute Z_G(S^2)
=|G|
pg 17
How is Z_G(S) related to the number of homomorphisms from pi_1(S) to G?
Proof?
pg 17-18
