The stable moduli space of flat connections over a surface
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- by Daniel A. Ramras PDF
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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 QuillenâLichtenbaum conjectures in algebraic $K$âtheory. The proofs utilize Tyler Lawsonâs work in deformation $K$âtheory, and rely heavily on Yang-Mills theory and gauge theory.References
- Ralph Abraham. Lectures of Smale on differential topology. Mimeographed notes. Columbia University, Dept. of Mathematics, New York, 1964(?).
- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523â615. MR 702806, DOI 10.1098/rsta.1983.0017
- M. F. Atiyah and G. B. Segal, Equivariant $K$-theory and completion, J. Differential Geometry 3 (1969), 1â18. MR 259946
- Thomas Baird. The moduli space of flat $SU(2)$âbundles over a nonorientable surface. Q. J. Math., 61(2):141â170, 2010.
- Thomas Baird. Antiperfection of YangâMills Morse theory over a nonorientable surface in rank three. arXiv:0902.4581, 2009.
- Thomas Baird. Moduli of flat $SU(3)$âbundles over a Klein bottle. arXiv:0901.1604, 2009.
- Carl-Friedrich Bödigheimer, Splitting the KĂŒnneth sequence in $K$-theory, Math. Ann. 242 (1979), no. 2, 159â171. MR 537958, DOI 10.1007/BF01420413
- 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 (2000), no. 4, 851â871. MR 1760431, DOI 10.1016/S0040-9383(99)00044-0
- Gunnar Carlsson. Derived representation theory and the algebraic $K$-theory of fields. arXiv:0810.4826, 2003.
- Georgios D. Daskalopoulos, The topology of the space of stable bundles on a compact Riemann surface, J. Differential Geom. 36 (1992), no. 3, 699â746. MR 1189501
- 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
- W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 106â119. MR 928826, DOI 10.1007/BFb0083003
- A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
- B. Fine, P. Kirk, and E. Klassen, A local analytic splitting of the holonomy map on flat connections, Math. Ann. 299 (1994), no. 1, 171â189. MR 1273082, DOI 10.1007/BF01459778
- L. Gutzwiller and S. A. Mitchell, The topology of Birkhoff varieties, Transform. Groups 14 (2009), no. 3, 541â556. MR 2534799, DOI 10.1007/s00031-009-9060-2
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Nan-Kuo Ho and Chiu-Chu Melissa Liu, Yang-Mills connections on orientable and nonorientable surfaces, Mem. Amer. Math. Soc. 202 (2009), no. 948, viii+98. MR 2561624, DOI 10.1090/S0065-9266-09-00564-X
- Nan-Kuo Ho and Chiu-Chu Melissa Liu, Connected components of the space of surface group representations, Int. Math. Res. Not. 44 (2003), 2359â2372. MR 2003827, DOI 10.1155/S1073792803131297
- Nan-Kuo Ho and Chiu-Chu Melissa Liu, Connected components of spaces of surface group representations. II, Int. Math. Res. Not. 16 (2005), 959â979. MR 2146187, DOI 10.1155/IMRN.2005.959
- Nan-Kuo Ho and Chiu-Chu Melissa Liu, Yang-Mills connections on nonorientable surfaces, Comm. Anal. Geom. 16 (2008), no. 3, 617â679. MR 2429971
- Nan-Kuo Ho and Chiu-Chu Melissa Liu. Anti-perfect Morse stratification. arXiv:0808.3974, 2009.
- Johannes Huebschmann, Extended moduli spaces, the Kan construction, and lattice gauge theory, Topology 38 (1999), no. 3, 555â596. MR 1670408, DOI 10.1016/S0040-9383(98)00033-0
- 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 (2000), 129â183. MR 1770606, DOI 10.1515/crll.2000.054
- Serge Lang, Differential and Riemannian manifolds, 3rd ed., Graduate Texts in Mathematics, vol. 160, Springer-Verlag, New York, 1995. MR 1335233, DOI 10.1007/978-1-4612-4182-9
- Tyler Lawson, The product formula in unitary deformation $K$-theory, $K$-Theory 37 (2006), no. 4, 395â422. MR 2269819, DOI 10.1007/s10977-006-9003-9
- Tyler Lawson, The Bott cofiber sequence in deformation $K$-theory and simultaneous similarity in $\textrm {U}(n)$, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 379â393. MR 2475972, DOI 10.1017/S0305004108001928
- M. Levine. K-theory and motivic cohomology of schemes. UIUC $K$-theory preprint server, 1999. Updated version available at www.uni-due.de/~bm0032.
- M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441â512. MR 1806878, DOI 10.1112/S0024611501012692
- J. Peter May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975), no. 1, 155, xiii+98. MR 370579, DOI 10.1090/memo/0155
- D. McDuff and G. Segal, Homology fibrations and the âgroup-completionâ theorem, Invent. Math. 31 (1975/76), no. 3, 279â284. MR 402733, DOI 10.1007/BF01403148
- Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39â87. MR 750716, DOI 10.2307/2007071
- 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 (1981), no. 4, 457â472. MR 623962
- James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- Johan RĂ„de, On the Yang-Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992), 123â163. MR 1179335, DOI 10.1515/crll.1992.431.123
- Daniel A. Ramras, Excision for deformation $K$-theory of free products, Algebr. Geom. Topol. 7 (2007), 2239â2270. MR 2366192, DOI 10.2140/agt.2007.7.2239
- Daniel A. Ramras. Quillen-Lichtenbaum phenomena in the stable representation theory of crystallographic groups. Submitted. arXiv:1007.0406, 2010.
- Daniel A. Ramras. Stable Representation Theory of Infinite Discrete Groups. Ph.D. thesis, Stanford University, 2007.
- Daniel A. Ramras. On the YangâMills stratification for surfaces. To appear in Proc. AMS.
- Daniel A. Ramras, Yang-Mills theory over surfaces and the Atiyah-Segal theorem, Algebr. Geom. Topol. 8 (2008), no. 4, 2209â2251. MR 2465739, DOI 10.2140/agt.2008.8.2209
- Andreas Rosenschon and Paul Arne ĂstvĂŠr, The homotopy limit problem for two-primary algebraic $K$-theory, Topology 44 (2005), no. 6, 1159â1179. MR 2168573, DOI 10.1016/j.top.2005.04.004
- Stefan Schwede, $S$-modules and symmetric spectra, Math. Ann. 319 (2001), no. 3, 517â532. MR 1819881, DOI 10.1007/PL00004446
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Ătudes Sci. Publ. Math. 34 (1968), 105â112. MR 232393
- Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31â42. MR 648356
- Katrin Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), ZĂŒrich, 2004. MR 2030823, DOI 10.4171/004
- Don Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, BirkhĂ€user Boston, Boston, MA, 1995, pp. 533â563. MR 1363070, DOI 10.1007/978-1-4612-4264-2_{2}0
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
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2728597