3-manifolds and their invariants

fundamental results on three manifolds

• Mostow-Prasad rigidity
• geometrization

maps between threefolds

• 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

Volume of knot complement

• Dedekind zeta funciton evaluated at 2 gives a number related to volume of 3-manifold
• 틀:수학노트
• 틀:수학노트

invariants

• Chern-Simons gauge theory and Witten's invariant
• Chern-Simons invariant
• 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)
• Kashaev's volume conjecture
• Ideal triangulations of 3-manifolds and the Bloch invariant
• Volume of hyperbolic threefolds and L-values and volume of knot complements
• Number fields and threefolds
• Reidemeister torsion

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• Topological quantum field theory(TQFT)
• quantum dilogarithm
• Chern-Simons invariant
• Gieseking's constant
• mathematics of x^3-x+1=0

encyclopedia

• http://en.wikipedia.org/wiki/Quantum_invariant
• http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)

books

• Saveliev, Nikolai. 1999. Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant. Walter De Gruyter Inc. http://www.amazon.com/Lectures-Topology-3-Manifolds-Introduction-Invariant/dp/3110162725
• Tomotada Ohtsuki Quantum Invariants

expositions

• Arithmetic properties of quantum invariants of manifolds http://www.mathnet.ru/php/presentation.phtml?presentid=3937&option_lang=rus Don Zagier
• Christian Blanchet, Vladimir Turaev Quantum Invariants of 3-manifolds
• Scott, Peter. 1983. “The Geometries of $3$-manifolds.” The Bulletin of the London Mathematical Society 15 (5): 401–487. doi:10.1112/blms/15.5.401.

articles

• 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.
• Kohno, Toshitake, and Toshie Takata. "Level-Rank Duality of Witten's 3-Manifold Invariants." Progress in algebraic combinatorics 24 (1996): 243. http://tqft.net/other-papers/knot-theory/Level-rank%20duality%20-%20Kohno,%20Takata.pdf
• 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