Degeneracy loci of families of Dirac operators

Leness, Thomas

doi:10.1090/S0002-9947-2012-05679-0

Degeneracy loci of families of Dirac operators
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by Thomas G. Leness PDF

Trans. Amer. Math. Soc. 364 (2012), 5995-6008 Request permission

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.

References

M. F. Atiyah and J. D. S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), no. 2, 97–118. MR 503187, DOI 10.1007/BF01609489
M. F. Atiyah and I. M. Singer, The index of elliptic operators. IV, Ann. of Math. (2) 93 (1971), 119–138. MR 279833, DOI 10.2307/1970756
S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315. MR 1066174, DOI 10.1016/0040-9383(90)90001-Z
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
Paul M. N. Feehan, Generic metrics, irreducible rank-one PU(2) monopoles, and transversality, Comm. Anal. Geom. 8 (2000), no. 5, 905–967. MR 1846123, DOI 10.4310/CAG.2000.v8.n5.a1
P. M. N. Feehan, P. B. Kronheimer, T. G. Leness, and T. S. Mrowka, $\rm PU(2)$ monopoles and a conjecture of Mariño, Moore, and Peradze, Math. Res. Lett. 6 (1999), no. 2, 169–182. MR 1689207, DOI 10.4310/MRL.1999.v6.n2.a5
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).
—, An ${SO}(3)$-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants, www.fiu.edu/~lenesst/CobordismBook.pdf.
P. M. N. Feehan and T. G. Leness, Witten’s conjecture for many four-manifolds of simple type, arXiv:math/0609530.
Paul M. N. Feehan and Thomas G. Leness, $\rm PU(2)$ monopoles. I. Regularity, Uhlenbeck compactness, and transversality, J. Differential Geom. 49 (1998), no. 2, 265–410. MR 1664908
Paul M. N. Feehan and Thomas G. Leness, $\rm PU(2)$ monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math. 538 (2001), 57–133. MR 1855754, DOI 10.1515/crll.2001.069
Paul M. N. Feehan and Thomas G. Leness, $\rm PU(2)$ 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, DOI 10.1515/crll.2001.064
Paul M. N. Feehan and Thomas G. Leness, $\rm SO(3)$ monopoles, level-one Seiberg-Witten moduli spaces, and Witten’s conjecture in low degrees, Proceedings of the 1999 Georgia Topology Conference (Athens, GA), 2002, pp. 221–326. MR 1936209, DOI 10.1016/S0166-8641(01)00233-4
Paul M. N. Feehan and Thomas G. Leness, $\rm SO(3)$-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
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1991. MR 1081321, DOI 10.1007/978-1-4613-9703-8
Robert Friedman and John W. Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 27, Springer-Verlag, Berlin, 1994. MR 1288304, DOI 10.1007/978-3-662-03028-8
Lothar Göttsche, Hiraku Nakajima, and K\B{o}ta Yoshioka, Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 307–359. MR 2827729, DOI 10.2977/PRIMS/37
Ulrich Koschorke, Infinite dimensional $K$-theory and characteristic classes of Fredholm bundle maps, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 95–133. MR 0279838
D. Kotschick and J. W. Morgan, $\textrm {SO}(3)$-invariants for $4$-manifolds with $b^+_2=1$. II, J. Differential Geom. 39 (1994), no. 2, 433–456. MR 1267898, DOI 10.4310/jdg/1214454879
Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom. 41 (1995), no. 3, 573–734. MR 1338483, DOI 10.4310/jdg/1214456482
Thomas G. Leness, Donaldson wall-crossing formulas via topology, Forum Math. 11 (1999), no. 4, 417–457. MR 1699168, DOI 10.1515/form.1999.008
W.-M. R. Leung, On $\text {spin}^c$ invariants of four-manifolds, Ph.D. thesis, Oxford University, 1995.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
A. Malmendier and K. Ono, ${SO}(3)$-Donaldson invariants of $\bar {\mathbb {CP}}^2$ and mock theta functions, arXiv:08.08.1442.
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
John W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, vol. 44, Princeton University Press, Princeton, NJ, 1996. MR 1367507
Peter S. Ozsváth, Some blowup formulas for $\textrm {SU}(2)$ Donaldson polynomials, J. Differential Geom. 40 (1994), no. 2, 411–447. MR 1293659
V. Ya. Pidstrigach and A. N. Tyurin, The smooth structure invariants of an algebraic surface defined by the Dirac operator, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 2, 279–371 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 2, 267–351. MR 1180377, DOI 10.1070/IM1993v040n02ABEH002167
Clifford Henry Taubes, Self-dual connections on $4$-manifolds with indefinite intersection matrix, J. Differential Geom. 19 (1984), no. 2, 517–560. MR 755237
Andrei Teleman, Moduli spaces of $\textrm {PU}(2)$-monopoles, Asian J. Math. 4 (2000), no. 2, 391–435. MR 1797591, DOI 10.4310/AJM.2000.v4.n2.a10
Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769–796. MR 1306021, DOI 10.4310/MRL.1994.v1.n6.a13