Geodesic
Geometric interpretation of simplicial formulas for the Chern–Simons invariant
Marché, Julien1
1Centre de Mathématiques Laurent Schwartz, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 805-827
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We give a direct interpretation of Neumann’s combinatorial formula for the Chern–Simons invariant of a 3–manifold with a representation in PSL(2, ℂ) whose restriction to the boundary takes values in upper triangular matrices. Our construction does not involve group homology or Bloch group but is based on the construction of an explicit flat connection for each tetrahedron of a simplicial decomposition of the manifold.

DOI : 10.2140/agt.2012.12.805
Classification : 57M27, 58J28
Keywords: Chern–Simons, triangulation, simplicial formula

Marché, Julien  1

1 Centre de Mathématiques Laurent Schwartz, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France 
@article{10_2140_agt_2012_12_805,
     author = {March\'e, Julien},
     title = {Geometric interpretation of simplicial formulas for the {Chern{\textendash}Simons} invariant},
     journal = {Algebraic and Geometric Topology},
     pages = {805--827},
     year = {2012},
     volume = {12},
     number = {2},
     doi = {10.2140/agt.2012.12.805},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.805/}
}
TY  - JOUR
AU  - Marché, Julien
TI  - Geometric interpretation of simplicial formulas for the Chern–Simons invariant
JO  - Algebraic and Geometric Topology
PY  - 2012
SP  - 805
EP  - 827
VL  - 12
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.805/
DO  - 10.2140/agt.2012.12.805
ID  - 10_2140_agt_2012_12_805
ER  - 
%0 Journal Article
%A Marché, Julien
%T Geometric interpretation of simplicial formulas for the Chern–Simons invariant
%J Algebraic and Geometric Topology
%D 2012
%P 805-827
%V 12
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2012.12.805/
%R 10.2140/agt.2012.12.805
%F 10_2140_agt_2012_12_805
Marché, Julien. Geometric interpretation of simplicial formulas for the Chern–Simons invariant. Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 805-827. doi: 10.2140/agt.2012.12.805

[1] S Baseilhac, R Benedetti, Classical and quantum dilogarithmic invariants of flat PSL(2, C)–bundles over 3–manifolds, Geom. Topol. 9 (2005) 493

[2] S S Chern, J Simons, Characteristic forms and geometric invariants, Ann. of Math. 99 (1974) 48

[3] M Culler, Lifting representations to covering groups, Adv. in Math. 59 (1986) 64

[4] J L Dupont, The dilogarithm as a characteristic class for flat bundles, from: "Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)" (1987) 137

[5] J L Dupont, C K Zickert, A dilogarithmic formula for the Cheeger–Chern–Simons class, Geom. Topol. 10 (2006) 1347

[6] D S Freed, Classical Chern–Simons theory I, Adv. Math. 113 (1995) 237

[7] W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557

[8] R M Kashaev, Coordinates for the moduli space of flat PSL(2, R)–connections, Math. Res. Lett. 12 (2005) 23

[9] P Kirk, E Klassen, Chern–Simons invariants of 3–manifolds decomposed along tori and the circle bundle over the representation space of T2, Comm. Math. Phys. 153 (1993) 521

[10] W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3–manifolds, from: "Topology ’90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 243

[11] W D Neumann, Hilbert’s 3rd problem and invariants of 3–manifolds, from: "The Epstein birthday schrift", Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 383

[12] W D Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 8 (2004) 413

[13] E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351

[14] T Yoshida, The η–invariant of hyperbolic 3–manifolds, Invent. Math. 81 (1985) 473

Cité par Sources :