Mathematische Annalen

, Volume 345, Issue 2, pp 453–489

The topology of moduli spaces of free group representations

Article
  • 114 Downloads

Abstract

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the \({{\rm SLm}(3, mathbb {C})}\)-character variety of a rank 2 free group is homotopic to an 8 sphere and the \({{\rm SLm}(2, mathbb {C})}\)-character variety of a rank 3 free group is homotopic to a 6 sphere.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308(1505), 523–615 (1983). MR MR702806 (85k:14006)Google Scholar
  2. 2.
    Agnihotri, S., Woodward, C.: Eigenvalues of products of unitary matrices and quantum Schubert calculus. Math. Res. Lett. 5(6), 817–836 (1998). MR MR1671192 (2000a:14066)Google Scholar
  3. 3.
    Baird, T.: Moduli spaces of flat G-bundles over nonorientable surfaces. Doctoral thesis, University of Toronto (2007)Google Scholar
  4. 4.
    Borel, A.: Linear algebraic groups W. A. Benjamin, Inc., New York-Amsterdam, pp. xi+398 (1969). MR MR0251042 (40 #4273)Google Scholar
  5. 5.
    Bratholdt, S., Cooper, D.: On the topology of the character variety of a free group. Rend. Istit. Mat. Univ. Trieste 32(suppl. 1), 45–53 (2002) [Dedicated to the memory of Marco Reni. MR MR1889465 (2003d:14072) (2001)]Google Scholar
  6. 6.
    Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36. Springer, Berlin (1998) [Translated from the 1987 French original, Revised by the authors. MR MR1659509 (2000a:14067)]Google Scholar
  7. 7.
    Bradlow S., García-Prada O., Gothen P.: Homotopy groups of moduli spaces of representations. Topology 47(4), 203–224 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Culler, M., Shalen, P.B.: Varieties of group representations and splittings of 3-manifolds. Ann. Math. (2) 117(1), 109–146 (1983). MR MR683804 (84k:57005)Google Scholar
  9. 9.
    Duistermaat, J.J., Kolk, J.A.C.: Lie Groups, Universitext. Springer, Berlin (2000). MR MR1738431 (2001j:22008)Google Scholar
  10. 10.
    Dolgachev, I.: Lectures on invariant theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003). MR MR2004511 (2004g:14051)Google Scholar
  11. 11.
    Desale, U.V., Ramanan, S.: Poincaré polynomials of the variety of stable bundles. Math. Ann. 216(3), 233–244 (1975). MR MR0379497 (52 #402)Google Scholar
  12. 12.
    Daskalopoulos, G., Wentworth, R.: Cohomology of \({SL(2, \mathbb{C})}\) character varieties of surface groups and the action of the Torelli group (2008) arXiv:0808.0131Google Scholar
  13. 13.
    Florentino, C.A.A.: Invariants of 2 × 2 matrices, irreducible \({{\rm SL}(2,{\mathbb{C}})}\) characters and the Magnus trace map. Geom. Dedicata 121, 167–186 (2006). MR MR2276242 (2007k:14093)Google Scholar
  14. 14.
    Goldman, W.M.: Trace coordinates on fricke spaces of some simple hyperbolic surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory II. EMS Publishing House, Zürich (2008)Google Scholar
  15. 15.
    Goldman, W.M.: Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications, xx+316 pp. The Clarendon Press, Oxford University Press, New York (1999). ISBN: 0-19-853793-X MR 30F45 51M10 57M50Google Scholar
  16. 16.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). MR MR1867354 (2002k:55001)Google Scholar
  17. 17.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55(1), 59–126 (1987). MR MR887284 (89a:32021)Google Scholar
  18. 18.
    Hausel, T., Rodriguez-Villegas, F.: Mixed hodge polynomials of character varieties, arXiv:math/ 0612668Google Scholar
  19. 19.
    Huebschmann, J.: Kähler spaces, nilpotent orbits, and singular reduction. Mem. Am. Math. Soc. 172(814), vi+96 (2004). MR MR2096203 (2006i:53112)Google Scholar
  20. 20.
    Huebschmann, J.: Singular Poisson-Kähler geometry of certain adjoint quotients. Geometry and topology of manifolds, Banach Center Publ., vol. 76, pp. 325–347. Polish Acad. Sci., Warsaw (2007). MR MR2346967Google Scholar
  21. 21.
    Jeffrey, L.C., Weitsman, J.: Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Comm. Math. Phys. 150(3), 593–630 (1992). MR MR1204322 (94g:58085)Google Scholar
  22. 22.
    Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984). MR MR766741 (86i:58050)Google Scholar
  23. 23.
    Kempf, G., Ness, L.: The length of vectors in representation spaces. Algebraic Geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen 1978). Lecture Notes in Math., vol. 732, pp. 233–243. Springer, Berlin (1979). MR MR555701 (81i:14032)Google Scholar
  24. 24.
    Knapp, A.W.: Lie groups beyond an introduction. Progress in Mathematics, 2nd edn., vol. 140. Birkhäuser Boston Inc., Boston (2002). MR MR1920389 (2003c:22001)Google Scholar
  25. 25.
    Lawton, S.: Generators, relations and symmetries in pairs of 3 ×  3 unimodular matrices. J. Algebra 313(2), 782–801 (2007). MR MR2329569Google Scholar
  26. 26.
    Lawton S.: Poisson geometry of \({{\rm SL}(3, \mathbb{C})}\)-character varieties relative to a surface with boundary. Trans. Am. Math. Soc. 361, 2397–2429 (2009)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Luft, E.: On contractible open topological manifolds. Invent. Math. 4, 192–201 (1967). MR MR0221486 (36 #4538)Google Scholar
  28. 28.
    Luna, D.: Sur certaines opérations différentiables des groupes de Lie. Am. J. Math. 97, 172–181 (1975). MR MR0364272 (51 #527)Google Scholar
  29. 29.
    Luna, D.: Fonctions différentiables invariantes sous l’opération d’un groupe réductif. Ann. Inst. Fourier (Grenoble) 26(1), 33–49 (1976). MR MR0423398 (54 #11377)Google Scholar
  30. 30.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 3rd edn., vol. 34. Springer, Berlin (1994). MR MR1304906 (95m:14012)Google Scholar
  31. 31.
    Mumford, D.: The red book of varieties and schemes, expanded ed. Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1999) [Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello. MR MR1748380 (2001b:14001)]Google Scholar
  32. 32.
    Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974). MR MR0402819 (53 #6633)Google Scholar
  33. 33.
    Nagata, M.: Invariants of a group in an affine ring. J. Math. Kyoto Univ. 3, 369–377 (1963/1964). MR MR0179268 (31 #3516)Google Scholar
  34. 34.
    Neeman, A.: The topology of quotient varieties. Ann. Math. (2) 122(3), 419–459 (1985). MR MR819554 (87g:14010)Google Scholar
  35. 35.
    Narasimhan, M.S., Seshadri, C.S.: Holomorphic vector bundles on a compact Riemann surface. Differential Analysis, pp. 249–250. Bombay Colloq., Oxford Univ. Press, London (1964). MR MR0182985 (32 #467)Google Scholar
  36. 36.
    Procesi, C.: The invariant theory of n × n matrices. Adv. Math. 19(3), 306–381 (1976). MR MR0419491 (54 #7512)Google Scholar
  37. 37.
    Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81(3), 539–554 (1985). MR MR807071 (87h:20078)Google Scholar
  38. 38.
    Schwarz, G.W.: The topology of algebraic quotients. Topological methods in algebraic transformation groups (New Brunswick, NJ 1988). Progr. Math., vol. 80, pp. 135–151. Birkhäuser Boston, Boston (1989). MR MR1040861 (90m:14043)Google Scholar
  39. 39.
    Shafarevich, I.R.: Basic Algebraic Geometry, vol. 2. Springer, Berlin (1994). MR MR1328834 (95m:14002)Google Scholar
  40. 40.
    Sikora, A.S.: Quantizations of character varieties and quantum knot invariants (2008) arXiv: 0807.0943v1Google Scholar
  41. 41.
    Sjamaar, R., Lerman, E.: Stratified symplectic spaces and reduction. Ann. Math. (2) 134(2), 375–422 (1991). MR MR1127479 (92g:58036)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisbonPortugal

Personalised recommendations