3-manifolds and their invariants

fundamental results on three manifolds

maps between aspherical 3 manifolds
aspherical threefolds = second and higher homotopy groups vanish
JSJ decomposition http://en.wikipedia.org/wiki/JSJ_decomposition
- cutting M into
  - Seifert fibered pieces ~ non hyperbolic pieces
  - atoroidal pieces ~ hyperbolic pieces
Thurston's geometrization
- S^3, E\times S^2, Sol
- E^3, E\times H^2, SL_2
- H^3, Nil

Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
Bloch-Wigner dilogarithm is involved

Prove
$$ \begin{align} \frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt & =\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7})) \\ & = \frac{2}{\sqrt{7}}(Cl(2\pi /7)+Cl(4\pi/7)-Cl(6\pi/7)) \end{align} $$
a log tangent integral

Turaev-Viro invariant (related to 6j symbols)
- Kauffman and Line 'The Temperley Lie algebra recoupling theory and invariants of 3-manifolds"
- Turaev-Viro "state sum invariants of 3-manifolds and quantum 6j-symbols)
Chern-Simons invariant
Kashaev's volume conjecture
Triangulations and the Bloch group
Volume of hyperbolic threefolds and L-values and volume of knot complements
Number fields and threefolds
Reidemeister torsion

Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links J.M. Borwein, D.J. Broadhurst, 1998
Gliozzi, F., and R. Tateo. 1995. Thermodynamic Bethe Ansatz and Threefold Triangulations. hep-th/9505102 (May 17). doi:doi:10.1142/S0217751X96001905. http://arxiv.org/abs/hep-th/9505102.
Three-manifolds and the Temperley-Lieb algebra W. B. R. Lickorish, 1991
Hyperbolic manifolds and special values of Dedekind zeta-functions Don Zagier, Inventiones Mathematicae, Volume 83, Number 2 / 1986년 6월