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Degeneracy loci of families of Dirac operators


Author: Thomas G. Leness
Journal: Trans. Amer. Math. Soc. 364 (2012), 5995-6008
MSC (2010): Primary 53C07, 57R57, 58J05, 58J20, 58J52
DOI: https://doi.org/10.1090/S0002-9947-2012-05679-0
Published electronically: June 12, 2012
MathSciNet review: 2946940
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Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincaré dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $ \operatorname {SO}(3)$ connections on a closed four-manifold. This should be a useful tool in comparing gauge theoretic invariants of smooth four-manifolds.


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  • 1. M. F. Atiyah and J. D. S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), 97-118. MR 503187 (80j:58021)
  • 2. M. F. Atiyah and I. M. Singer, Index of elliptic operators. IV, Ann. Math. 93 (1971), 119-138. MR 0279833 (43:5554)
  • 3. S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257-315. MR 1066174 (92a:57035)
  • 4. S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Univ. Press, Oxford, 1990. MR 1079726 (92a:57036)
  • 5. P. M. N. Feehan, Generic metrics, irreducible rank-one $ {PU(2)}$ monopoles, and transversality, Comm. Anal. Geom. 8 (2000), 905-967. MR 1846123 (2002m:57041)
  • 6. P. M. N. Feehan, P. B. Kronheimer, T. G. Leness, and T. S. Mrowka, $ {PU(2)}$ monopoles and a conjecture of Mariño, Moore, and Peradze, Math. Res. Lett. 6 (1999), 169-182. MR 1689207 (2000f:57035)
  • 7. P. M. N. Feehan and T. G. Leness, Donaldson invariants and wall-crossing formulas. I: Continuity of gluing and obstruction maps, submitted to a print journal, math.DG/9812060 (v3).
  • 8. -, An $ {SO}(3)$-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants, www.fiu.edu/~lenesst/CobordismBook.pdf.
  • 9. P. M. N. Feehan and T. G. Leness, Witten's conjecture for many four-manifolds of simple type, arXiv:math/0609530.
  • 10. -, $ {PU(2)}$ monopoles. I: Regularity, Uhlenbeck compactness, and transversality, J. Differential Geom. 49 (1998), 265-410. MR 1664908 (2000e:57052)
  • 11. -, $ \rm {PU}(2)$ monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math. 538 (2001), 57-133. MR 1855754 (2002f:57067)
  • 12. -, $ \rm {PU}(2)$ monopoles. II. Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees, J. Reine Angew. Math. 538 (2001), 135-212. MR 1855755 (2002f:57068)
  • 13. -, $ \rm {SO}(3)$ monopoles, level-one Seiberg-Witten moduli spaces, and Witten's conjecture in low degrees, Proceedings of the 1999 Georgia Topology Conference (Athens, GA), vol. 124, 2002, pp. 221-326. MR 1936209 (2003i:57049)
  • 14. -, $ \rm {SO}(3)$-monopoles: The overlap problem, Geometry and topology of manifolds, Fields Inst. Commun., vol. 47, Amer. Math. Soc., Providence, RI, 2005, pp. 97-118. MR 2189928 (2006g:57058)
  • 15. D. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Springer, New York, 1991. MR 1081321 (91i:57019)
  • 16. R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer, Berlin, 1994. MR 1288304 (95m:57046)
  • 17. Lothar Göttsche, Hiraku Nakajima, and Kōta Yoshioka, Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 307-359. MR 2827729 (2012f:14085)
  • 18. U. Koschorke, Infinite-dimensional K-theory and characteristic classes of Fredholm bundle maps, Global Analysis (F. E. Browder, ed.), Proc. Symp. Pure Math., vol. 15, Amer. Math. Soc., Providence, RI, 1970, pp. 95-133. MR 0279838 (43:5559)
  • 19. D. Kotschick and J. W. Morgan, $ {SO(3)}$ invariants for four-manifolds with $ b^+=1$. II, J. Differential Geom. 39 (1994), 433-456. MR 1267898 (95g:57047)
  • 20. P. B. Kronheimer and T. Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, Cambridge University Press, Cambridge, 2007. MR 2388043 (2009f:57049)
  • 21. P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson's polynomial invariants, J. Differential Geom. 43 (1995), 573-734. MR 1338483 (96e:57019)
  • 22. T. G. Leness, Donaldson wall-crossing formulas via topology, Forum Math. 11 (1999), 417-457. MR 1699168 (2000g:57044)
  • 23. W.-M. R. Leung, On $ \text {spin}^c$ invariants of four-manifolds, Ph.D. thesis, Oxford University, 1995.
  • 24. I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995, With contributions by A. Zelevinsky. MR 1354144 (96h:05207)
  • 25. A. Malmendier and K. Ono, $ {SO}(3)$-Donaldson invariants of $ \bar {\mathbb{CP}}^2$ and mock theta functions, arXiv:08.08.1442.
  • 26. J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton Univ. Press, Princeton, NJ, 1974. MR 0440554 (55:13428)
  • 27. J. W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton Univ. Press, Princeton, NJ, 1996. MR 1367507 (97d:57042)
  • 28. P. S. Ozsváth, Some blowup formulas for $ {SU(2)}$ Donaldson polynomials, J. Differential Geom. 40 (1994), 411-447. MR 1293659 (95e:57054)
  • 29. V. Y. Pidstrigatch and A. N. Tyurin, Invariants of the smooth structure of an algebraic surface arising from the Dirac operator, Russian Acad. Sci. Izv. Math. 40 (1993), 267-351. MR 1180377 (93m:14036)
  • 30. C. H. Taubes, Self-dual connections on $ 4$-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), 517-560. MR 755237 (86b:53025)
  • 31. A. Teleman, Moduli spaces of $ {PU(2)}$-monopoles, Asian J. Math. 4 (2000), 391-435. MR 1797591 (2001m:57059)
  • 32. E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796. MR 1306021 (96d:57035)

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Additional Information

Thomas G. Leness
Affiliation: Department of Mathematics, Florida International University, Miami, Florida 33199
Email: lenesst@fiu.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05679-0
Received by editor(s): November 23, 2009
Received by editor(s) in revised form: March 4, 2011
Published electronically: June 12, 2012
Additional Notes: The author was supported in part by a Florida International University Faculty Research Grant and by NSF grant DMS #0905786.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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