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The Mod Two Cohomology of the Moduli Space of Rank Two Stable Bundles on a Surface and Skew Schur Polynomials

Part of: Families, fibrations Differential topology

Published online by Cambridge University Press:  07 January 2019

Christopher W. Scaduto  and
Matthew Stoffregen
Christopher W. Scaduto
Affiliation:
Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794-3636, USA Email: [email protected]
Matthew Stoffregen
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA Email: [email protected]
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Abstract

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We compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

Keywords

stable bundlesmod two cohomologyskew schur polynomial

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 57R58: Floer homology
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 683 - 715
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author C. S. was supported by NSF grant DMS-1503100.

References

Atiyah, M. F. and Bott, R., The Yang-Mills equations over Riemann surfaces . Philos. Trans. Roy. Soc. London Ser. A 308(1983), no. 1505, 523–615. https://doi.org/10.1098/rsta.1983.0017.Google Scholar
Baranovskiĭ, V. Yu., The cohomology ring of the moduli space of stable bundles with odd determinant . Izv. Ross. Akad. Nauk Ser. Mat. 58(1994), no. 4, 204–210.Google Scholar
Chen, B. and Scaduto, C., Nilpotency in instanton homology, and the framed instanton homology of a surface times a circle. 2016. arxiv:1612.08690.Google Scholar
Feehan, Paul M. N. and Leness, Thomas G., PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces . J. Reine Angew. Math. 538(2001), 57–133.Google Scholar
Frøyshov, Kim A., An inequality for the h-invariant in instanton Floer theory . Topology 43(2004), no. 2, 407–432. https://doi.org/10.1016/S0040-9383(03)00049-1.Google Scholar
Gessel, Ira M. and Viennot, X. G., Determinants, paths, and plane partitions, 1989. http://people.brandeis.edu/∼gessel/homepage/papers/pp.pdf.Google Scholar
Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002.Google Scholar
Kirwan, Frances, The cohomology rings of moduli spaces of bundles over Riemann surfaces . J. Amer. Math. Soc. 5(1992), no. 4, 853–906. https://doi.org/10.1090/S0894-0347-1992-1145826-8.Google Scholar
King, A. D. and Newstead, P. E., On the cohomology ring of the moduli space of rank 2 vector bundles on a curve . Topology 37(1998), no. 2, 407–418. https://doi.org/10.1016/S0040-9383(97)00041-4.Google Scholar
Macdonald, I. G., Symmetric functions and Hall polynomials, Second edition. Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, 2015.Google Scholar
Muñoz, Vicente, Ring structure of the Floer cohomology of 𝛴 × S 1 . Topology 38(1999), no. 3, 517–528. https://doi.org/10.1016/S0040-9383(98)00028-7.Google Scholar
Newstead, P. E., Topological properties of some spaces of stable bundles . Topology 6(1967), 241–262. https://doi.org/10.1016/0040-9383(67)90037-7.Google Scholar
Newstead, P. E., Characteristic classes of stable bundles of rank 2 over an algebraic curve . Trans. Amer. Math. Soc. 169(1972), 337–345.Google Scholar
Siebert, Bernd and Tian, Gang, Recursive relations for the cohomology ring of moduli spaces of stable bundles . Turkish J. Math. 19(1995), no. 2, 131–144.Google Scholar
Stanley, Richard P., Two remarks on skew tableaux . Electron. J. Combin. 18(2011), no. 2, Paper 16, 8.Google Scholar
Thaddeus, Michael, Conformal field theory and the cohomology of the moduli space of stable bundles . J. Differential Geom. 35(1992), no. 1, 131–149. https://doi.org/10.4310/jdg/1214447808.Google Scholar
Thaddeus, Michael, An introduction to the topology of the moduli space of stable bundles on a Riemann surface . In: Geometry and physics, Lecture Notes in Pure and Appl. Math., 184. Dekker, New York, 1997, pp. 71–99.Google Scholar
Wall, C. T. C., Determination of the cobordism ring . Ann. of Math. (2) 72(1960), 292–311. https://doi.org/10.2307/1970136.Google Scholar
Zagier, Don, On the cohomology of moduli spaces of rank two vector bundles over curves . In: The moduli space of curves, Progr. Math., 129. Birkhäuser Boston, Boston, MA, 1995, pp. 533–563.Google Scholar