The \(\mathrm{SO}(3)\) monopole cobordism and superconformal simple type. (English) Zbl 1430.53025
The authors continue their research line developed in several papers (see the bibliography in this paper). See also the related paper by the same authors [Adv. Math. 356, Article ID 106821, 35 p. (2019; Zbl 1430.53026)].
Let \(X\) be a smooth, closed, connected orientable four-manifold with \(b^1 (X) = 0\) and \(b^+ (X) = 2 m + 1 \) with \(m\) integer and \(m \ge 1\), i.e., let \(X\) be standard.
All known standard four-manifolds have Seiberg-Witten simple type [P. Kronheimer and T. Mrowka, Monopoles and three-manifolds. Cambridge: Cambridge University Press (2007; Zbl 1158.57002)]. Marino, Moore and Peradze conjecture that all standard four-manifolds of simple Seiberg-Witten type also have superconformal simple type; for the meaning of this notion and the conjecture, see [M. Mariño et al., Math. Res. Lett. 6, No. 3–4, 429–437 (1999; Zbl 0974.57018); Commun. Math. Phys. 205, No. 3, 691–735 (1999; Zbl 0988.57019)].
In a previous paper by the authors et al. [Math. Res. Lett. 6, No. 2, 169–182 (1999; Zbl 0967.57027)], it was shown that \(X\) has superconformal simple type under a certain hypothesis. Here they focus on the following hypothesis: the local gluing map, defined in a previous work by the authors [“\(\mathrm{PU}(2)\) monopoles. III: Existence of gluing and obstruction maps”, Preprint, arXiv:math/9907107], gives a continuous parametrization of a neighborhood of \(M_s \times \Sigma\) in \(\widetilde{\mathcal{M}}_t\) for each smooth stratum \(\Sigma \subset \mathrm{Sym}^\ell (X)\).
The main result of the paper is the following theorem: Let \(X\) be a standard four-manifold of Seiberg-Witten simple type and assume the hypothesis above. Then \(X\) has superconformal simple type.
Let \(X\) be a smooth, closed, connected orientable four-manifold with \(b^1 (X) = 0\) and \(b^+ (X) = 2 m + 1 \) with \(m\) integer and \(m \ge 1\), i.e., let \(X\) be standard.
All known standard four-manifolds have Seiberg-Witten simple type [P. Kronheimer and T. Mrowka, Monopoles and three-manifolds. Cambridge: Cambridge University Press (2007; Zbl 1158.57002)]. Marino, Moore and Peradze conjecture that all standard four-manifolds of simple Seiberg-Witten type also have superconformal simple type; for the meaning of this notion and the conjecture, see [M. Mariño et al., Math. Res. Lett. 6, No. 3–4, 429–437 (1999; Zbl 0974.57018); Commun. Math. Phys. 205, No. 3, 691–735 (1999; Zbl 0988.57019)].
In a previous paper by the authors et al. [Math. Res. Lett. 6, No. 2, 169–182 (1999; Zbl 0967.57027)], it was shown that \(X\) has superconformal simple type under a certain hypothesis. Here they focus on the following hypothesis: the local gluing map, defined in a previous work by the authors [“\(\mathrm{PU}(2)\) monopoles. III: Existence of gluing and obstruction maps”, Preprint, arXiv:math/9907107], gives a continuous parametrization of a neighborhood of \(M_s \times \Sigma\) in \(\widetilde{\mathcal{M}}_t\) for each smooth stratum \(\Sigma \subset \mathrm{Sym}^\ell (X)\).
The main result of the paper is the following theorem: Let \(X\) be a standard four-manifold of Seiberg-Witten simple type and assume the hypothesis above. Then \(X\) has superconformal simple type.
Reviewer: Giuseppe Gaeta (Milano)
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 57R57 | Applications of global analysis to structures on manifolds |
| 58J05 | Elliptic equations on manifolds, general theory |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 58J52 | Determinants and determinant bundles, analytic torsion |
Keywords:
\(\mathrm{SO}(3)\) monopoles; Donaldson invariants; gauge theory; smooth four-manifolds; Seiberg-Witten invariants; Witten conjectureReferences:
| [1] | Donaldson, S. K., The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differential Geom., 26, 397-428 (1987), MR 910015 (88j:57020) · Zbl 0683.57005 |
| [2] | Donaldson, S. K., Polynomial invariants for smooth four-manifolds, Topology, 29, 3, 257-315 (1990), MR 1066174 · Zbl 0715.57007 |
| [3] | Donaldson, S. K.; Kronheimer, P. B., The Geometry of Four-Manifolds (1990), Oxford University Press: Oxford University Press New York · Zbl 0820.57002 |
| [4] | Feehan, P. M.N., Generic metrics, irreducible rank-one PU(2) monopoles, and transversality, Comm. Anal. Geom., 8, 5, 905-967 (2000), MR 1846123 · Zbl 0993.58003 |
| [5] | Feehan, P. M.N.; Kronheimer, P. B.; Leness, T. G.; Mrowka, T. S., \(PU(2)\) monopoles and a conjecture of Mariño, Moore, and Peradze, Math. Res. Lett., 6, 2, 169-182 (1999), MR 1689207 · Zbl 0967.57027 |
| [6] | Feehan, P. M.N.; Leness, T. G., Donaldson invariants and wall-crossing formulas. I: continuity of gluing and obstruction maps |
| [7] | Feehan, P. M.N.; Leness, T. G., Gluing maps for \(SO(3)\) monopoles and invariants of smooth four-manifolds, and |
| [8] | Feehan, P. M.N.; Leness, T. G., \(PU(2)\) monopoles. III: existence of gluing and obstruction maps |
| [9] | Feehan, P. M.N.; Leness, T. G., \(PU(2)\) monopoles. I. Regularity, Uhlenbeck compactness, and transversality, J. Differential Geom., 49, 265-410 (1998), MR 1664908 (2000e:57052) · Zbl 0998.57057 |
| [10] | Feehan, P. M.N.; Leness, T. G., \(PU(2)\) monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math., 538, 57-133 (2001), MR 1855754 · Zbl 0983.57024 |
| [11] | Feehan, P. M.N.; Leness, T. G., \(PU(2)\) monopoles. II. Top-level Seiberg-Witten moduli spaces and Witten’s conjecture in low degrees, J. Reine Angew. Math., 538, 135-212 (2001), MR 1855755 · Zbl 0983.57025 |
| [12] | Feehan, P. M.N.; Leness, T. G., \(SO(3)\) monopoles, level-one Seiberg-Witten moduli spaces, and Witten’s conjecture in low degrees, (Proceedings of the 1999 Georgia Topology Conference. Proceedings of the 1999 Georgia Topology Conference, Athens, GA, vol. 124 (2002)), 221-326, MR 1936209 · Zbl 1028.58012 |
| [13] | Feehan, P. M.N.; Leness, T. G., \(SO(3)\)-monopoles: the overlap problem, (Geometry and Topology of Manifolds. Geometry and Topology of Manifolds, Fields Inst. Commun., vol. 47 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 97-118, MR 2189928 · Zbl 1094.58003 |
| [14] | Feehan, P. M.N.; Leness, T. G., Witten’s conjecture for many four-manifolds of simple type, J. Eur. Math. Soc. (JEMS), 17, 4, 899-923 (2015), MR 3349302 · Zbl 1328.57031 |
| [15] | Feehan, P. M.N.; Leness, T. G., An \(SO(3)\)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants, Mem. Amer. Math. Soc., 256, 1226 (2018), MR 3897982 · Zbl 07000313 |
| [16] | Feehan, P. M.N.; Leness, T. G., Superconformal simple type and Witten’s conjecture, Adv. Math. (2019), in press · Zbl 1430.53026 |
| [17] | Fintushel, R.; Park, J.; Stern, R. J., Rational surfaces and symplectic 4-manifolds with one basic class, Algebr. Geom. Topol., 2, 391-402 (2002), MR 1917059 · Zbl 0991.57031 |
| [18] | Fintushel, R.; Stern, R. J., The canonical class of a symplectic 4-manifold, Turkish J. Math., 25, 1, 137-145 (2001), MR 1829084 · Zbl 0977.57032 |
| [19] | Friedman, R.; Morgan, J. W., Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 27 (1994), Springer-Verlag: Springer-Verlag Berlin, MR 1288304 · Zbl 0817.14017 |
| [20] | Frøyshov, K. A., Compactness and Gluing Theory for Monopoles, Geometry & Topology Monographs, vol. 15 (2008), Geometry & Topology Publications: Geometry & Topology Publications Coventry, MR 2465077 (2010a:57050) · Zbl 1207.57044 |
| [21] | Gompf, R. E.; Stipsicz, A. I., 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20 (1999), American Mathematical Society: American Mathematical Society Providence, RI, MR 1707327 · Zbl 0933.57020 |
| [22] | Göttsche, L., Modular forms and Donaldson invariants for 4-manifolds with \(b_+ = 1\), J. Amer. Math. Soc., 9, 827-843 (1996), MR 1362873 (96k:57029) · Zbl 0872.57027 |
| [23] | Göttsche, L.; Nakajima, H.; Yoshioka, K., Donaldson = Seiberg-Witten from Mochizuki’s formula and instanton counting, Publ. Res. Inst. Math. Sci., 47, 307-359 (2011), MR 2827729 (2012f:14085) · Zbl 1259.14046 |
| [24] | Kotschick, D.; Morgan, J. W., \(SO(3)\)-invariants for 4-manifolds with \(b_2^+ = 1\). II, J. Differential Geom., 39, 2, 433-456 (1994), MR 1267898 · Zbl 0828.57013 |
| [25] | Kronheimer, P. B.; Mrowka, T. S., Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom., 41, 573-734 (1995), MR 1338483 (96e:57019) · Zbl 0842.57022 |
| [26] | Kronheimer, P. B.; Mrowka, T. S., Monopoles and Three-Manifolds (2007), Cambridge University Press: Cambridge University Press Cambridge, MR 2388043 (2009f:57049) · Zbl 1158.57002 |
| [27] | Mariño, M.; Moore, G. W.; Peradze, G., Four-manifold geography and superconformal symmetry, Math. Res. Lett., 6, 3-4, 429-437 (1999), MR 1713141 · Zbl 0974.57018 |
| [28] | Mariño, M.; Moore, G. W.; Peradze, G., Superconformal invariance and the geography of four-manifolds, Comm. Math. Phys., 205, 3, 691-735 (1999), MR 1711332 · Zbl 0988.57019 |
| [29] | McDuff, D.; Wehrheim, K., Smooth Kuranishi atlases with isotropy, Geom. Topol., 21, 5, 2725-2809 (2017), MR 3687107 · Zbl 1410.53082 |
| [30] | McDuff, D.; Wehrheim, K., The topology of Kuranishi atlases, Proc. Lond. Math. Soc. (3), 115, 2, 221-292 (2017), MR 3684105 · Zbl 1384.53065 |
| [31] | McDuff, D.; Wehrheim, K., The fundamental class of smooth Kuranishi atlases with trivial isotropy, J. Topol. Anal., 10, 1, 71-243 (2018), MR 3737511 · Zbl 1381.53160 |
| [32] | Mochizuki, T., Donaldson Type Invariants for Algebraic SurfacesTransition of Moduli Stacks, Lecture Notes in Mathematics, vol. 1972 (2009), Springer-Verlag: Springer-Verlag Berlin, MR 2508583 · Zbl 1177.14003 |
| [33] | Morgan, J. W., The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds, Mathematical Notes, vol. 44 (1996), Princeton University Press: Princeton University Press Princeton, NJ, MR 1367507 · Zbl 0846.57001 |
| [34] | J.W. Morgan, T.S. Mrowka, The gluing construction for anti-self-dual connections over manifolds with long tubes, preprint, October 13, 1994, 115 pages.; J.W. Morgan, T.S. Mrowka, The gluing construction for anti-self-dual connections over manifolds with long tubes, preprint, October 13, 1994, 115 pages. |
| [35] | Mrowka, T. S., A Local Mayer-Vietoris Principle for Yang-Mills Moduli Spaces (1988), University of California: University of California Berkeley, CA, MR 2637291 |
| [36] | Nicolaescu, L. I., Notes on Seiberg-Witten Theory, Graduate Studies in Mathematics, vol. 28 (2000), American Mathematical Society: American Mathematical Society Providence, RI, MR 1787219 (2001k:57037) · Zbl 0978.57027 |
| [37] | Ozsváth, P. S., Some blowup formulas for \(SU(2)\) Donaldson polynomials, J. Differential Geom., 40, 411-447 (1994), MR 1293659 (95e:57054) · Zbl 0846.57014 |
| [38] | Pidstrigach, V. Ya.; Tyurin, A. N., Localisation of Donaldson invariants along the Seiberg-Witten classes · Zbl 0796.14024 |
| [39] | Pidstrigach, V. Ya.; Tyurin, A. N., The smooth structure invariants of an algebraic surface defined by the Dirac operator, Izv. Ross. Akad. Nauk Ser. Mat.. Izv. Ross. Akad. Nauk Ser. Mat., Russian Acad. Sci. Izv. Math., 40, 2, 267-351 (1993), MR 1180377 (93m:14036) · Zbl 0796.14024 |
| [40] | Spanier, E. H., Algebraic Topology (1995), Springer-Verlag: Springer-Verlag New York, corrected reprint of the 1966 original · Zbl 0810.55001 |
| [41] | Taubes, C. H., Self-dual Yang-Mills connections on non-self-dual 4-manifolds, J. Differential Geom., 17, 139-170 (1982), MR 658473 (83i:53055) · Zbl 0484.53026 |
| [42] | Taubes, C. H., Self-dual connections on 4-manifolds with indefinite intersection matrix, J. Differential Geom., 19, 517-560 (1984), MR 755237 (86b:53025) · Zbl 0552.53011 |
| [43] | Taubes, C. H., A framework for Morse theory for the Yang-Mills functional, Invent. Math., 94, 327-402 (1988), MR 958836 (90a:58035) · Zbl 0665.58006 |
| [44] | Teleman, A., Moduli spaces of \(PU(2)\)-monopoles, Asian J. Math., 4, 2, 391-435 (2000), MR 1797591 · Zbl 0982.58008 |
| [45] | Witten, E., Monopoles and four-manifolds, Math. Res. Lett., 1, 769-796 (1994), MR 1306021 (96d:57035) · Zbl 0867.57029 |
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