nLab
hyperbolic 3-manifold

Contents

Context

Riemannian geometry

Manifolds and cobordisms

Contents

Idea

A Riemannian manifold which is both a 3-manifold and a hyperbolic manifold is a hyperbolic 3-manifold.

Equivalently this is a Riemannian manifold which is isometric to the quotient space 3/Γ\mathbb{H}^3/ \Gamma of hyperbolic 3-space by the action of a torsion-free discrete group Γ\Gamma acting by isometries.

Properties

Volume conjecture

See volume conjecture.

Adjust the following text

There is a curious relation of volumes of hyperbolic 3-manifolds to the action functional of Chern-Simons theory/Dijkgraaf-Witten theory (volume conjecture).

Let GG be a Lie group and c:BGB 3U(1)\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^3 U(1) a cocycle in degree-3 (generalized) Lie group cohomology. Write G\flat G for the underlying discrete group and c:BGB 3U(1)\flat \mathbf{c} \colon \mathbf{B} \flat G \to \mathbf{B}^3 \flat U(1) for the induced cocycle in ordinary (discrete) group cohomology, [c]H Grp 3(G disc,U(1) disc)[\flat \mathbf{c}] \in H^3_{Grp}(G_{disc},U(1)_{disc}).

Then for Σ\Sigma a closed manifold of dimension 3, a map (of smooth infinity-groupoids) ΣBG\Sigma \to \mathbf{B}\flat G is a flat GG-principal connection on Σ\Sigma and the composite

[Σ,BG][Σ,c][Σ,B 3U(1)] ΣU(1) [\Sigma, \mathbf{B}\flat G] \stackrel{[\Sigma, \flat \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 \flat U(1)] \stackrel{\int_{\Sigma}}{\to} U(1)

is the action functional for GG-Chern-Simons theory on Σ\Sigma restricted to GG-flat connections, or equivalently is the action functional of G\flat G-Dijkgraaf-Witten theory.

Now for G=SL(n,)G = SL(n,\mathbb{C}) the complex special linear group and hence for Chern-Simons theory with complex gauge group, it turns out that the imaginary part of this flat Chern-Simons/Dijkgraaf-Witten invariant of 3-manifolds always has an expression as a combination of volumes of hyperbolic 3-manifolds.

Examples

References

See also

Last revised on May 22, 2019 at 11:45:45. See the history of this page for a list of all contributions to it.