Configuration spaces and Vassiliev classes in any dimension

Cattaneo, Alberto S¹ ; Cotta-Ramusino, Paolo² ; Longoni, Riccardo³

¹Mathematisches Institut, Universität Zürich–Irchel, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
²Dipartimento di Fisica, Università degli Studi di Milano & INFN Sezione di Milano, Via Celoria, 16, I-20133 Milano, Italy
³Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro, 5, I-00185 Roma, Italy

Algebraic and Geometric Topology, Tome 2 (2002) no. 2, pp. 949-1000

Résumé

The real cohomology of the space of imbeddings of S1 into ℝn, n > 3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.

Détail

DOI : 10.2140/agt.2002.2.949

Keywords: configuration spaces, Vassiliev invariants, de Rham cohomology of spaces of imbeddings, immersions, Chen's iterated integrals, graph cohomology

Bibliographie

[1] D Altschuler, L Freidel, Vassiliev knot invariants and Chern–Simons perturbation theory to all orders, Comm. Math. Phys. 187 (1997) 261

[2] M Alvarez, J M F Labastida, Numerical knot invariants of finite type from Chern–Simons perturbation theory, Nuclear Phys. B 433 (1995) 555

[3] S Axelrod, I M Singer, Chern–Simons perturbation theory, from: "Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991)", World Sci. Publ., River Edge, NJ (1992) 3

[4] D Bar-Natan, Perturbative Chern–Simons theory, J. Knot Theory Ramifications 4 (1995) 503

[5] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423

[6] R Bott, Configuration spaces and imbedding invariants, Turkish J. Math. 20 (1996) 1

[7] R Bott, A S Cattaneo, Integral invariants of 3–manifolds, J. Differential Geom. 48 (1998) 91

[8] R Bott, A S Cattaneo, Integral invariants of 3–manifolds II, J. Differential Geom. 53 (1999) 1

[9] R Bott, C Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994) 5247

[10] J L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics 107, Birkhäuser (1993)

[11] A S Cattaneo, P Cotta-Ramusino, J Fröhlich, M Martellini, Topological $BF$ theories in 3 and 4 dimensions, J. Math. Phys. 36 (1995) 6137

[12] A S Cattaneo, P Cotta-Ramusino, R Longoni, Algebraic structures on graph cohomology, J. Knot Theory Ramifications 14 (2005) 627

[13] A S Cattaneo, P Cotta-Ramusino, M Rinaldi, Loop and path spaces and four-dimensional $BF$ theories: connections, holonomies and observables, Comm. Math. Phys. 204 (1999) 493

[14] A S Cattaneo, P Cotta-Ramusino, C A Rossi, Loop observables for $BF$ theories in any dimension and the cohomology of knots, Lett. Math. Phys. 51 (2000) 301

[15] A S Cattaneo, J Fröhlich, B Pedrini, Topological field theory interpretation of string topology, Comm. Math. Phys. 240 (2003) 397

[16] A S Cattaneo, C A Rossi, Higher-dimensional $BF$ theories in the Batalin–Vilkovisky formalism: the BV action and generalized Wilson loops, Comm. Math. Phys. 221 (2001) 591

[17] M Chas, D Sullivan, String topology,

[18] K T Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831

[19] J Fröhlich, C King, The Chern–Simons theory and knot polynomials, Comm. Math. Phys. 126 (1989) 167

[20] W Fulton, R Macpherson, A compactification of configuration spaces, Ann. of Math. $(2)$ 139 (1994) 183

[21] E Getzler, J D S Jones, S Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991) 339

[22] T G Goodwillie, M Weiss, Embeddings from the point of view of immersion theory II, Geom. Topol. 3 (1999) 103

[23] E Guadagnini, M Martellini, M Mintchev, Wilson lines in Chern–Simons theory and link invariants, Nuclear Phys. B 330 (1990) 575

[24] A C Hirshfeld, U Sassenberg, Explicit formulation of a third order finite knot invariant derived from Chern–Simons theory, J. Knot Theory Ramifications 5 (1996) 805

[25] M Kontsevich, Vassiliev's knot invariants, from: "I. M. Gel'fand Seminar", Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 137

[26] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)", Progr. Math. 120, Birkhäuser (1994) 97

[27] M Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003) 157

[28] B Ndombol, J C Thomas, On the cohomology algebra of free loop spaces, Topology 41 (2002) 85

[29] S Poirier, Rationality results for the configuration space integral of knots,

[30] D Sinha, On the topology of spaces of knots, preprint

[31] S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. $(2)$ 69 (1959) 327

[32] D Thurston, Integral Expressions for the Vassiliev Knot Invariants, AB thesis, Harvard University (1995)

[33] V Tourtchine, On the homology of the spaces of long knots, from: "Advances in topological quantum field theory", NATO Sci. Ser. II Math. Phys. Chem. 179, Kluwer Acad. Publ. (2004) 23

[34] V Tourtchine, private communication

[35] V A Vassiliev, Cohomology of knot spaces, from: "Theory of singularities and its applications", Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 23

[36] V A Vassiliev, Topology of two-connected graphs and homology of spaces of knots, from: "Differential and symplectic topology of knots and curves", Amer. Math. Soc. Transl. Ser. 2 190, Amer. Math. Soc. (1999) 253

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