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Transactions of the American Mathematical Society

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The stable moduli space of flat connections over a surface


Author: Daniel A. Ramras
Journal: Trans. Amer. Math. Soc. 363 (2011), 1061-1100
MSC (2010): Primary 58D27, 55N15; Secondary 53C07, 55P42
DOI: https://doi.org/10.1090/S0002-9947-2010-05218-3
Published electronically: September 21, 2010
MathSciNet review: 2728597
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Abstract: We compute the homotopy type of the moduli space of flat, unitary connections over an aspherical surface, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface $ M^g$, we show that this space has the homotopy type of the infinite symmetric product of $ M^g$, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic $ K$-theory. The proofs utilize Tyler Lawson's work in deformation $ K$-theory, and rely heavily on Yang-Mills theory and gauge theory.


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  • 1. Ralph Abraham.
    Lectures of Smale on differential topology.
    Mimeographed notes. Columbia University, Dept. of Mathematics, New York, 1964(?).
  • 2. M. F. Atiyah and R. Bott.
    The Yang-Mills equations over Riemann surfaces.
    Philos. Trans. Roy. Soc. London Ser. A, 308(1505):523-615, 1983. MR 702806 (85k:14006)
  • 3. M. F. Atiyah and G. B. Segal.
    Equivariant $ K$-theory and completion.
    J. Differential Geometry, 3:1-18, 1969. MR 0259946 (41:4575)
  • 4. Thomas Baird.
    The moduli space of flat $ SU(2)$-bundles over a nonorientable surface. Q. J. Math., 61(2):141-170, 2010.
  • 5. Thomas Baird.
    Antiperfection of Yang-Mills Morse theory over a nonorientable surface in rank three.
    arXiv:0902.4581, 2009.
  • 6. Thomas Baird.
    Moduli of flat $ SU(3)$-bundles over a Klein bottle.
    arXiv:0901.1604, 2009.
  • 7. Carl-Friedrich Bödigheimer.
    Splitting the Künneth sequence in $ K$-theory.
    Math. Ann., 242(2):159-171, 1979. MR 0537958 (80k:55013)
  • 8. S. E. Cappell, R. Lee, and E. Y. Miller.
    The action of the Torelli group on the homology of representation spaces is nontrivial.
    Topology, 39(4):851-871, 2000. MR 1760431 (2001g:57034)
  • 9. Gunnar Carlsson.
    Derived representation theory and the algebraic $ K$-theory of fields.
    arXiv:0810.4826, 2003.
  • 10. Georgios D. Daskalopoulos.
    The topology of the space of stable bundles on a compact Riemann surface.
    J. Differential Geom., 36(3):699-746, 1992. MR 1189501 (93i:58026)
  • 11. S. K. Donaldson and P. B. Kronheimer.
    The geometry of four-manifolds.
    Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1990.
    Oxford Science Publications. MR 1079726 (92a:57036)
  • 12. W. Dwyer and A. Zabrodsky.
    Maps between classifying spaces.
    In Algebraic topology, Barcelona, 1986, volume 1298 of Lecture Notes in Math., pages 106-119. Springer, Berlin, 1987. MR 928826 (89b:55018)
  • 13. A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May.
    Rings, modules, and algebras in stable homotopy theory, volume 47 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI, 1997.
    With an appendix by M. Cole. MR 1417719 (97h:55006)
  • 14. B. Fine, P. Kirk, and E. Klassen.
    A local analytic splitting of the holonomy map on flat connections.
    Math. Ann., 299(1):171-189, 1994. MR 1273082 (95e:58031)
  • 15. Luke Gutzwiller and Stephen A. Mitchell.
    The topology of Birkhoff varieties. Transform Groups, 14(3), 541-556, 2009. MR 2534799
  • 16. Allen Hatcher.
    Algebraic topology.
    Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
  • 17. Nan-Kuo Ho and Chiu-Chu Melissa Liu.
    Yang-Mills connections on orientable and nonorientable surfaces. Mem. Amer. Math. Soc., 202, 2009. MR 2561624
  • 18. Nan-Kuo Ho and Chiu-Chu Melissa Liu.
    Connected components of the space of surface group representations.
    Int. Math. Res. Not., (44):2359-2372, 2003. MR 2003827 (2004h:53116)
  • 19. Nan-Kuo Ho and Chiu-Chu Melissa Liu.
    Connected components of spaces of surface group representations. II.
    Int. Math. Res. Not., (16):959-979, 2005. MR 2146187 (2006b:53108)
  • 20. Nan-Kuo Ho and Chiu-Chu Melissa Liu.
    Yang-Mills connections on nonorientable surfaces.
    Comm. Anal. Geom., 16(3):617-679, 2008. MR 2429971 (2009k:53054)
  • 21. Nan-Kuo Ho and Chiu-Chu Melissa Liu.
    Anti-perfect Morse stratification.
    arXiv:0808.3974, 2009.
  • 22. Johannes Huebschmann.
    Extended moduli spaces, the Kan construction, and lattice gauge theory.
    Topology, 38(3):555-596, 1999. MR 1670408 (2000c:57066)
  • 23. Sören Illman.
    Existence and uniqueness of equivariant triangulations of smooth proper $ G$-manifolds with some applications to equivariant Whitehead torsion.
    J. Reine Angew. Math., 524:129-183, 2000. MR 1770606 (2001j:57032)
  • 24. Serge Lang.
    Differential and Riemannian manifolds, volume 160 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, third edition, 1995. MR 1335233 (96d:53001)
  • 25. Tyler Lawson.
    The product formula in unitary deformation $ K$-theory.
    $ K$-Theory, 37(4):395-422, 2006. MR 2269819 (2007k:19007)
  • 26. Tyler Lawson.
    The Bott cofiber sequence in deformation $ K$-theory and simultaneous similarity in $ U(n)$.
    Math. Proc. Cambridge Philos. Soc., 146(2):379-393, 2009. MR 2475972 (2010c:19002)
  • 27. M. Levine.
    K-theory and motivic cohomology of schemes.
    UIUC $ K$-theory preprint server, 1999. Updated version available at www.uni-due.de/~bm0032.
  • 28. M. A. Mandell, J. P. May, S. Schwede, and B. Shipley.
    Model categories of diagram spectra.
    Proc. London Math. Soc. (3), 82(2):441-512, 2001. MR 1806878 (2001k:55025)
  • 29. J. Peter May.
    Classifying spaces and fibrations.
    Mem. Amer. Math. Soc., 1(1, 155):xiii+98, 1975. MR 0370579 (51:6806)
  • 30. D. McDuff and G. Segal.
    Homology fibrations and the ``group-completion'' theorem.
    Invent. Math., 31(3):279-284, 1975/76. MR 0402733 (53:6547)
  • 31. Haynes Miller.
    The Sullivan conjecture on maps from classifying spaces.
    Ann. of Math. (2), 120(1):39-87, 1984. MR 750716 (85i:55012)
  • 32. P. K. Mitter and C.-M. Viallet.
    On the bundle of connections and the gauge orbit manifold in Yang-Mills theory.
    Comm. Math. Phys., 79(4):457-472, 1981. MR 623962 (83f:81056)
  • 33. James R. Munkres.
    Topology. (Second Edition).
    Prentice-Hall Inc., Englewood Cliffs, N.J., 2000. MR 0464128 (57:4063)
  • 34. Johan Råde.
    On the Yang-Mills heat equation in two and three dimensions.
    J. Reine Angew. Math., 431:123-163, 1992. MR 1179335 (94a:58041)
  • 35. Daniel A. Ramras.
    Excision for deformation $ K$-theory of free products.
    Algebr. Geom. Topol., 7:2239-2270, 2007. MR 2366192 (2008j:19007)
  • 36. Daniel A. Ramras. Quillen-Lichtenbaum phenomena in the stable representation theory of crystallographic groups. Submitted. arXiv:1007.0406, 2010.
  • 37. Daniel A. Ramras.
    Stable Representation Theory of Infinite Discrete Groups.
    Ph.D. thesis, Stanford University, 2007.
  • 38. Daniel A. Ramras.
    On the Yang-Mills stratification for surfaces. To appear in Proc. AMS.
  • 39. Daniel A. Ramras.
    Yang-Mills theory over surfaces and the Atiyah-Segal theorem.
    Algebr. Geom. Topol., 8(4):2209-2251, 2008. MR 2465739 (2009k:55008)
  • 40. Andreas Rosenschon and Paul Arne Østvær.
    The homotopy limit problem for two-primary algebraic $ K$-theory.
    Topology, 44(6):1159-1179, 2005. MR 2168573 (2006d:19001)
  • 41. Stefan Schwede.
    $ S$-modules and symmetric spectra.
    Math. Ann., 319(3):517-532, 2001. MR 1819881 (2001m:55050)
  • 42. Graeme Segal.
    Classifying spaces and spectral sequences.
    Inst. Hautes Études Sci. Publ. Math., (34):105-112, 1968. MR 0232393 (38:718)
  • 43. Karen K. Uhlenbeck.
    Connections with $ L\sp{p}$ bounds on curvature.
    Comm. Math. Phys., 83(1):31-42, 1982. MR 0648356 (83e:53035)
  • 44. Katrin Wehrheim.
    Uhlenbeck compactness.
    EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich, 2004. MR 2030823 (2004m:53045)
  • 45. Don Zagier.
    On the cohomology of moduli spaces of rank two vector bundles over curves.
    In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 533-563. Birkhäuser Boston, Boston, MA, 1995. MR 1363070 (97g:14010)

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Additional Information

Daniel A. Ramras
Affiliation: Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nash- ville, Tennessee 37240
Address at time of publication: Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Department 3MB, Las Cruces, New Mexico 88003-8001
Email: daniel.a.ramras@vanderbilt.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05218-3
Received by editor(s): November 1, 2008
Received by editor(s) in revised form: October 7, 2009
Published electronically: September 21, 2010
Additional Notes: This work was partially supported by an NSF graduate fellowship and by NSF grants DMS-0353640 (RTG) and DMS-0804553.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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