Skip to main content
SpringerLink
Log in

The topology of moduli spaces of free group representations

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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. Baird, T.: Moduli spaces of flat G-bundles over nonorientable surfaces. Doctoral thesis, University of Toronto (2007)

  4. Borel, A.: Linear algebraic groups W. A. Benjamin, Inc., New York-Amsterdam, pp. xi+398 (1969). MR MR0251042 (40 #4273)

  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)]

  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)]

  7. Bradlow S., García-Prada O., Gothen P.: Homotopy groups of moduli spaces of representations. Topology 47(4), 203–224 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  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. Duistermaat, J.J., Kolk, J.A.C.: Lie Groups, Universitext. Springer, Berlin (2000). MR MR1738431 (2001j:22008)

  10. Dolgachev, I.: Lectures on invariant theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003). MR MR2004511 (2004g:14051)

  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. Daskalopoulos, G., Wentworth, R.: Cohomology of \({SL(2, \mathbb{C})}\) character varieties of surface groups and the action of the Torelli group (2008) arXiv:0808.0131

  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. 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)

  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 57M50

  16. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). MR MR1867354 (2002k:55001)

  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)

  18. Hausel, T., Rodriguez-Villegas, F.: Mixed hodge polynomials of character varieties, arXiv:math/ 0612668

  19. Huebschmann, J.: Kähler spaces, nilpotent orbits, and singular reduction. Mem. Am. Math. Soc. 172(814), vi+96 (2004). MR MR2096203 (2006i:53112)

  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 MR2346967

  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. Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, vol. 31. Princeton University Press, Princeton (1984). MR MR766741 (86i:58050)

  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)

  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)

  25. Lawton, S.: Generators, relations and symmetries in pairs of 3 ×  3 unimodular matrices. J. Algebra 313(2), 782–801 (2007). MR MR2329569

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Luft, E.: On contractible open topological manifolds. Invent. Math. 4, 192–201 (1967). MR MR0221486 (36 #4538)

    Google Scholar 

  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. 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. 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)

  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)]

  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. Nagata, M.: Invariants of a group in an affine ring. J. Math. Kyoto Univ. 3, 369–377 (1963/1964). MR MR0179268 (31 #3516)

  34. Neeman, A.: The topology of quotient varieties. Ann. Math. (2) 122(3), 419–459 (1985). MR MR819554 (87g:14010)

    Google Scholar 

  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)

  36. Procesi, C.: The invariant theory of n × n matrices. Adv. Math. 19(3), 306–381 (1976). MR MR0419491 (54 #7512)

  37. Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81(3), 539–554 (1985). MR MR807071 (87h:20078)

    Google Scholar 

  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)

  39. Shafarevich, I.R.: Basic Algebraic Geometry, vol. 2. Springer, Berlin (1994). MR MR1328834 (95m:14002)

  40. Sikora, A.S.: Quantizations of character varieties and quantum knot invariants (2008) arXiv: 0807.0943v1

  41. Sjamaar, R., Lerman, E.: Stratified symplectic spaces and reduction. Ann. Math. (2) 134(2), 375–422 (1991). MR MR1127479 (92g:58036)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Carlos Florentino & Sean Lawton

Authors
  1. Carlos Florentino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sean Lawton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sean Lawton.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Florentino, C., Lawton, S. The topology of moduli spaces of free group representations. Math. Ann. 345, 453–489 (2009). https://doi.org/10.1007/s00208-009-0362-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-009-0362-4

Keywords

Search

Navigation