- Davies, Juhász, Lackenby, Tomasev, The signature and cusp geometry of hyperbolic knots. - Nature article. - Links: - Geometrization - Hilbert symbol - Chern-Simons invariant - Dehn invariant

Three-manifold

Take a look at Machlachlan and Reid’s book “The Arithmetic of Hyperbolic 3-Manifolds”. #resources

Since finite volume hyperbolic structures are unique whenever an \(n\)-manifold (\(n\geq 3\)) has them, any invariants of the hyperbolic structure are invariants of the manifold. Hyperbolic manifolds are \(K(\pi,1)\) spaces, so they’re not just diffeo/homeomorphism invariants, but invariants of the homotopy-type.

Rohklin invariant : a \({\mathbf{Z}}/2\) invariant \(r\) for \({\mathbf{Z}}\operatorname{HS}^3\)
Casson invariant : a \({\mathbf{Z}}\) invariant \(c\) for \({\mathbf{Z}}\operatorname{HS}^3\) where \(c\operatorname{mod}2 = r\).

Links to this page

unresolved links output

Hyperbolic manifolds in Unsorted/Three-manifolds MOC

Rohklin invariant in Unsorted/Three-manifolds MOC
uniformization

In 3 dimensions, there are 8 geometries. The geometrization proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.
spherical manifold

Unsorted/Three-manifolds MOC with finite fundamental group are spherical.
character variety

See 3-manifold.
Weinstein

#why-care Theorem: Every compact 3-manifold arises as a combination of (2 different variants of) Weinstein surgeries on \(S^3\).
The Galois action on symplectic K-theory

etale homotopy of \({\mathcal{O}}_K\) for \(K\) a global field : a punctured 3-manifold
Poincare homology sphere

A 3-manifold and is the only \(X\in {\mathbf{Z}}\operatorname{HS}^n\) with finite \(\pi_1\).
Mostow rigidity

3-manifold
Lagrangian Floer homology

Major applications in knot theory and 3-manifolds, gives an algorithm for computing knot genus.
Geometrization

3-manifolds: Thurston’s Geometrization - Define a Geometric structure: a diffeo \(M\cong \tilde M/\Gamma\) where \(\Gamma\) is a discrete Lie group acting freely/transitively on \(X\). - Oriented prime 3-manifolds can be decomposed into geometric “pieces” of 8 possible types: - Spherical \(\sim S^3\) - Euclidean \(\sim {\mathbf{R}}^3\) - Hyperbolic \(\sim {\mathbb{H}}^3\) - \(S^2\times{\mathbf{R}}\) - \({\mathbb{H}}^2\times{\mathbf{R}}\) - \(\tilde{{\operatorname{SL}}(2, {\mathbf{R}})}\) - “Nil” - “Sol” - Proved by Perelman 2003, Ricci flow with surgery. - 4-manifolds: classified in the topological category by surgery, but not in the smooth category - Hard! Will examine special cases of Calabi-Yau - Open part of Poincare conjectures. - Dimension \(\geq 5\): surgery theory, diffeomorphic \(\iff\) s-cobordant|
Chern Simons and TFTs MSRI Fall 2021

Chern-Simons invariant: obstruction to immersing a 3-manifold in \({\mathbf{R}}^4\) conformally.

Three-manifolds MOC

- Davies, Juhász, Lackenby, Tomasev, The signature and cusp geometry of hyperbolic knots. - Nature article. - Links: - Geometrization - Hilbert symbol - Chern-Simons invariant - Dehn invariant

Three-manifold