Algebraic & Geometric Topology

Configuration spaces and Vassiliev classes in any dimension

Alberto S Cattaneo, Paolo Cotta-Ramusino, and Riccardo Longoni

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Abstract

The real cohomology of the space of imbeddings of S1 into n, 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.

Article information

Source
Algebr. Geom. Topol., Volume 2, Number 2 (2002), 949-1000.

Dates
Received: 2 August 2002
Accepted: 12 October 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882748

Digital Object Identifier
doi:10.2140/agt.2002.2.949

Mathematical Reviews number (MathSciNet)
MR1936977

Zentralblatt MATH identifier
1029.57009

Subjects
Primary: 58D10: Spaces of imbeddings and immersions
Secondary: 55R80: Discriminantal varieties, configuration spaces 81Q30: Feynman integrals and graphs; applications of algebraic topology and algebraic geometry [See also 14D05, 32S40]

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

Citation

Cattaneo, Alberto S; Cotta-Ramusino, Paolo; Longoni, Riccardo. Configuration spaces and Vassiliev classes in any dimension. Algebr. Geom. Topol. 2 (2002), no. 2, 949--1000. doi:10.2140/agt.2002.2.949. https://projecteuclid.org/euclid.agt/1513882748


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References

  • D. Altschuler, L. Freidel: “Vassiliev knot invariants and Chern–Simons perturbation theory to all orders,” \cmp187, 261–287 (1997)
  • M. Alvarez, J. M. F. Labastida: “Analysis of observables in Chern–Simons perturbation theory,” \npb395, 198 (1993); and “Numerical knot invariants of finite type from Chern–Simons perturbation theory,” \npb433, 555 (1995)
  • S. Axelrod, I. M. Singer: “Chern–Simons perturbation theory,” in Proceedings of the XXth DGM Conference, edited by S. Catto and A. Rocha (World Scientific, Singapore, 1992), pp. 3–45; “Chern–Simons perturbation theory. II,” \jdg39, 173–213 (1994)
  • D. Bar-Natan: Perturbative Aspects of Chern–Simons Topological Quantum Field Theory, Ph.D. thesis, Princeton University, (1991); “Perturbative Chern–Simons theory,” \jktr4, 503–548 (1995)
  • D. Bar-Natan: “On the Vassiliev knot invariants,” Topology 34, 423–472 (1995)
  • R. Bott: “Configuration spaces and imbedding invariants,” Turkish J. Math. $\mathbf {20}$, no. 1, 1–17 (1996)
  • R. Bott, A. S. Cattaneo: “Integral invariants of 3–manifolds,” J. Diff. Geom. 48, 91–133 (1998)
  • R. Bott, A. S. Cattaneo: “Integral invariants of 3-manifolds. II,” J. Diff. Geom. 1–13 (1999)
  • R. Bott, C. Taubes: “On the self-linking of knots,” \jmp35, 5247–5287 (1994)
  • J-L. Brylinski: Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Mathematics, $\mathbf {107}$, Birkhäuser (1993)
  • A. S. Cattaneo, P. Cotta-Ramusino, J. Fröhlich, M. Martellini: “Topological BF theories in 3 and 4 dimensions,” \jmp36, 6137–6160 (1995)
  • A. S. Cattaneo, P. Cotta-Ramusino, R. Longoni: “Algebraic stuctures in graph homology,” in preparation
  • A. S. Cattaneo, P. Cotta-Ramusino, M. Rinaldi: “Loop and path spaces and four-dimensional $BF$ theories: connections, holonomies and observables,” \cmp204, 493–524 (1999)
  • A. S. Cattaneo, P. Cotta-Ramusino, C. Rossi: “Loop observables for BF theories in any dimension and the cohomology of knots,” \lmp51, 301–316 (2000)
  • A. S. Cattaneo, J. Fröhlich, B. Pedrini “Topological Field Theory Interpretation of String Topology”, to appear in Commun. Math. Phys. arXiv:math.GT/0202176,
  • A. S. Cattaneo, C. Rossi: “Higher-dimensional BF theories in the Batalin–Vilkovisky formalism: The BV action and generalized Wilson loops,” Commun. Math. Phys. 221, 591–657 (2001).
  • M. Chas, D. Sullivan “String Topology,” arXiv:math.GT/9911159
  • K.-T. Chen: “Iterated path integrals,” \bams83, 831–879 (1977)
  • J. Fröhlich, C. King: “The Chern–Simons theory and knot polynomials,” \cmp126, 167–199 (1989)
  • W. Fulton, R. MacPherson: “Compactification of configuration spaces,” Ann. Math. 139, 183–225 (1994)
  • E. Getzler, J. Jones, S. Petrack: “Differential forms on loop spaces and the cyclic bar complex,” Topology $\mathbf{30}$, 339–371 (1991)
  • T. G. Goodwillie, M. Weiss: “Embeddings from the point of view of immersion theory: Part II,” \gt3, 103–118 (1999)
  • E. Guadagnini, M. Martellini, M. Mintchev: “Chern–Simons field theory and link invariants,” \npb330, 575–607 (1989)
  • A. C. Hirshfeld, U. Sassenberg: “Explicit formulation of a third order finite knot invariant derived from Chern–Simons theory,” \jktr5, 805–847 (1996)
  • M. Kontsevich: “Vassiliev's knot invariants," \asm16, 137–150 (1993)
  • M. Kontsevich: “Feynman diagrams and low-dimensional topology,” First European Congress of Mathematics, Paris 1992, Volume II, Progress in Mathematics, $\mathbf {120}$, Birkhäuser (1994)
  • M. Kontsevich: “Deformation quantization of Poisson manifolds. I,” arXiv:q-alg/9709040
  • B. Ndombol, J.-C. Thomas: “On the cohomology algebra of free loop spaces,” Topology 41, 85–106 (2002).
  • S. Poirier: “Rationality results for the configuration space integral of knots,” arXiv:math.GT/9901028
  • D. Sinha: “On the topology of spaces of knots,” preprint
  • S. Smale: “The classification of immersions of spheres in Euclidean spaces,” Ann. Math. $\mathbf {69}$, 327–344 (1959)
  • D. Thurston: Integral Expressions for the Vassiliev Knot Invariants, AB thesis, Harvard University (1995).
  • V. Tourtchine: “Sur l'homologie des espaces de noeuds non-compacts,” arXiv:math.QA/0010017, “On the homology of the spaces of long knots” arXiv:math.QA/0105140
  • V. Tourtchine: private communication
  • V. Vassiliev: “Cohomology of knot spaces,” Adv. in Sov. Math.; Theory of Singularities and its Appl. (ed. V. I. Arnold) AMS, Providence, RI, 23–69 (1990)
  • V. Vassiliev: “Topology of two-connected graphs and homology of spaces of knots,” Differential and symplectic topology of knots and curves (S. L. Tabachnikov, ed.) AMS Transl. Ser. 2. Vol. 190. AMS, Providence, RI, 253–286 (1999)
  • E. Witten: “Quantum field theory and the Jones polynomial,” \cmp121, 351–399 (1989)