An \(\mathrm{SO}(3)\)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants. (English) Zbl 07000313
Memoirs of the American Mathematical Society 1226. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1421-4/print; 978-1-4704-4915-5/ebook). xiv, 238 p. (2018).
Preliminary review / Publisher’s description: We prove an analogue of the Kotschick-Morgan Conjecture in the context of \(\mathrm{SO}(3)\) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the \(\mathrm{SO}(3)\)-monopole cobordism. The main technical difficulty in the \(\mathrm{SO}(3)\)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible \(\mathrm{SO}(3)\) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of \(\mathrm{SO}(3)\) monopoles (Feehan and Leness, PU(2) monopoles. I. Regularity, Uhlenbeck compactness, and transversality, 1998). In this monograph, we prove – modulo a gluing theorem which is an extension of our earlier work in PU(2) monopoles. III: Existence of gluing and obstruction maps (arXiv:math/9907107) – that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. Our proofs that the \(\mathrm{SO}(3)\)-monopole cobordism yields both the Superconformal Simple Type Conjecture of Moore, Mariño, and Peradze (Superconformal invariance and the geography of four-manifolds, 1999; Four-manifold geography and superconformal symmetry, 1999) and Witten’s Conjecture (Monopoles and four-manifolds, 1994) in full generality for all closed, oriented, smooth four-manifolds with \(b_1=0\) and odd \(b^+\geq 3\) appear in Feehan and Leness, Superconformal simple type and Witten’s conjecture (arXiv:1408.5085) and \(\text SO(3)\) monopole cobordism and superconformal simple type (arXiv:1408.5307).
MSC:
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 57K40 | General topology of 4-manifolds |
| 57R57 | Applications of global analysis to structures on manifolds |
| 58D27 | Moduli problems for differential geometric structures |
| 58D29 | Moduli problems for topological structures |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 58J05 | Elliptic equations on manifolds, general theory |
| 58J20 | Index theory and related fixed-point theorems on manifolds |
Keywords:
anti-self-dual connections; classification of smooth four-manifolds; cobordisms; Donaldson invariants; gauge theory; gluing theory; intersection theory; Kotschick-Morgan conjecture; moduli spaces; monopoles; Seiberg-Witten invariants; stratified spaces; Uhlenbeck compactification; Witten’s conjectureReferences:
| [1] | Adams, Robert A.; Fournier, John J. F., Sobolev spaces, Pure and Applied Mathematics (Amsterdam) 140, xiv+305 pp. (2003), Elsevier/Academic Press, Amsterdam · Zbl 1098.46001 |
| [2] | Atiyah, M. F.; Hitchin, N. J.; Singer, I. M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362, 1711, 425-461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143 |
| [3] | Atiyah, M. F.; Jones, J. D. S., Topological aspects of Yang-Mills theory, Comm. Math. Phys., 61, 2, 97-118 (1978) · Zbl 0387.55009 |
| [4] | Atiyah, M. F.; Singer, I. M., Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. U.S.A., 81, 8, , Phys. Sci., 2597-2600 (1984) · Zbl 0547.58033 · doi:10.1073/pnas.81.8.2597 |
| [5] | Aubin, Thierry, Nonlinear analysis on manifolds. Monge-Amp\`“ere equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 252, xii+204 pp. (1982), Springer-Verlag, New York · Zbl 0512.53044 · doi:10.1007/978-1-4612-5734-9 |
| [6] | Borel, A.; et al., Intersection cohomology, Modern Birkh\`“auser Classics, x+234 pp. (2008), Birkh\'”auser Boston, Inc., Boston, MA |
| [7] | Bredon, Glen E., Sheaf theory, Graduate Texts in Mathematics 170, xii+502 pp. (1997), Springer-Verlag, New York · Zbl 0874.55001 · doi:10.1007/978-1-4612-0647-7 |
| [8] | Bredon, Glen E., Topology and geometry, Graduate Texts in Mathematics 139, xiv+557 pp. (1997), Springer-Verlag, New York · Zbl 0934.55001 |
| [9] | R. Cohen, J.D.S. Jones, and G. Segal, Morse theory and classifying spaces, http://math.stanford.edu/ ralph/papers.html, December 1995. |
| [10] | Dold, Albrecht, Die Homotopieerweiterungseigenschaft \((={\rm HEP})\) ist eine lokale Eigenschaft, Invent. Math., 6, 185-189 (1968) · Zbl 0167.51604 · doi:10.1007/BF01404823 |
| [11] | Dold, Albrecht, Lectures on algebraic topology, Classics in Mathematics, xii+377 pp. (1995), Springer-Verlag, Berlin · Zbl 0872.55001 · doi:10.1007/978-3-642-67821-9 |
| [12] | Donaldson, S. K., Connections, cohomology and the intersection forms of \(4\)-manifolds, J. Differential Geom., 24, 3, 275-341 (1986) · Zbl 0635.57007 |
| [13] | Donaldson, S. K., Irrationality and the \(h\)-cobordism conjecture, J. Differential Geom., 26, 1, 141-168 (1987) · Zbl 0631.57010 |
| [14] | Donaldson, S. K., The orientation of Yang-Mills moduli spaces and \(4\)-manifold topology, J. Differential Geom., 26, 3, 397-428 (1987) · Zbl 0683.57005 |
| [15] | Donaldson, S. K.; Kronheimer, P. B., The geometry of four-manifolds, Oxford Mathematical Monographs, x+440 pp. (1990), The Clarendon Press, Oxford University Press, New York · Zbl 0820.57002 |
| [16] | P. M. N. Feehan, Discreteness for energies of Yang-Mills connections over four-dimensional manifolds, arXiv:1505.06995. |
| [17] | Feehan, Paul M. N., Geometry of the ends of the moduli space of anti-self-dual connections, J. Differential Geom., 42, 3, 465-553 (1995) · Zbl 0856.58007 |
| [18] | Feehan, Paul M. N., Generic metrics, irreducible rank-one PU(2) monopoles, and transversality, Comm. Anal. Geom., 8, 5, 905-967 (2000) · Zbl 0993.58003 · doi:10.4310/CAG.2000.v8.n5.a1 |
| [19] | Feehan, Paul M. N., Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections, Pacific J. Math., 200, 1, 71-118 (2001) · Zbl 1052.58014 · doi:10.2140/pjm.2001.200.71 |
| [20] | Feehan, P. M. N.; Kronheimer, P. B.; Leness, T. G.; Mrowka, T. S., \( \rm PU(2)\) monopoles and a conjecture of Mari\~no, Moore, and Peradze, Math. Res. Lett., 6, 2, 169-182 (1999) · Zbl 0967.57027 · doi:10.4310/MRL.1999.v6.n2.a5 |
| [21] | P. M. N. Feehan and T. G. Leness, Donaldson invariants and wall-crossing formulas. I: Continuity of gluing and obstruction maps, arXiv:math/9812060. |
| [22] | Feehan, Paul M. N.; Leness, Thomas G., \( \rm SO(3)\)-monopoles: the overlap problem. Geometry and topology of manifolds, Fields Inst. Commun. 47, 97-118 (2005), Amer. Math. Soc., Providence, RI · Zbl 1094.58003 |
| [23] | P. M. N. Feehan and T. G. Leness, The \(\text{SO}(3)\) monopole cobordism and superconformal simple type, arXiv:1408.5307. · Zbl 1430.53025 |
| [24] | P. M. N. Feehan and T. G. Leness, \(PU(2)\) monopoles. III: Existence of gluing and obstruction maps, arXiv:math/9907107. · Zbl 0983.57024 |
| [25] | P. M. N. Feehan and T. G. Leness, Superconformal simple type and Witten’s conjecture, arXiv:1408.5085. · Zbl 1328.57031 |
| [26] | Feehan, Paul M. N.; Leness, Thomas G., \({\rm PU}(2)\) monopoles and relations between four-manifold invariants, Topology Appl., 88, 1-2, 111-145 (1998) · Zbl 0931.58012 · doi:10.1016/S0166-8641(97)00201-0 |
| [27] | Feehan, Paul M. N.; Leness, Thomas G., \( \rm PU(2)\) monopoles. I. Regularity, Uhlenbeck compactness, and transversality, J. Differential Geom., 49, 2, 265-410 (1998) · Zbl 0998.57057 |
| [28] | Feehan, Paul M. N.; Leness, Thomas G., \( \rm PU(2)\) monopoles and links of top-level Seiberg-Witten moduli spaces, J. Reine Angew. Math., 538, 57-133 (2001) · Zbl 0983.57024 · doi:10.1515/crll.2001.069 |
| [29] | Seiberg, N.; Witten, E., Monopoles, duality and chiral symmetry breaking in \(N=2\) supersymmetric QCD, Nuclear Phys. B, 431, 3, 484-550 (1994) · Zbl 1020.81911 · doi:10.1016/0550-3213(94)90214-3 |
| [30] | Feehan, Paul M. N.; Leness, Thomas G., \( \rm SO(3)\) monopoles, level-one Seiberg-Witten moduli spaces, and Witten’s conjecture in low degrees, Proceedings of the 1999 Georgia Topology Conference (Athens, GA). Topology Appl., 124, 2, 221-326 (2002) · Zbl 1028.58012 · doi:10.1016/S0166-8641(01)00233-4 |
| [31] | Liu, Kefeng; Ma, Xiaonan, A remark on: “Some numerical results in complex differential geometry” [arxiv.org/abs/math/0512625] by S. K. Donaldson, Math. Res. Lett., 14, 2, 165-171 (2007) · Zbl 1175.32014 · doi:10.4310/MRL.2007.v14.n2.a1 |
| [32] | Feehan, Paul M. N.; Leness, Thomas G., \( \rm SO(3)\)-monopoles: the overlap problem. Geometry and topology of manifolds, Fields Inst. Commun. 47, 97-118 (2005), Amer. Math. Soc., Providence, RI · Zbl 1094.58003 |
| [33] | Feehan, Paul M. N.; Leness, Thomas G., Witten’s conjecture for many four-manifolds of simple type, J. Eur. Math. Soc. (JEMS), 17, 4, 899-923 (2015) · Zbl 1328.57031 · doi:10.4171/JEMS/521 |
| [34] | Freed, Daniel S.; Uhlenbeck, Karen K., Instantons and four-manifolds, Mathematical Sciences Research Institute Publications 1, xxii+194 pp. (1991), Springer-Verlag, New York · doi:10.1007/978-1-4613-9703-8 |
| [35] | Friedman, Robert; Morgan, John W., Smooth four-manifolds and complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 27, x+520 pp. (1994), Springer-Verlag, Berlin · Zbl 0817.14017 · doi:10.1007/978-3-662-03028-8 |
| [36] | Gompf, Robert E.; Stipsicz, Andr\'as I., \(4\)-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, xvi+558 pp. (1999), American Mathematical Society, Providence, RI · Zbl 0933.57020 · doi:10.1090/gsm/020 |
| [37] | Goresky, Mark; MacPherson, Robert, Intersection homology theory, Topology, 19, 2, 135-162 (1980) · Zbl 0448.55004 · doi:10.1016/0040-9383(80)90003-8 |
| [38] | Goresky, Mark; MacPherson, Robert, Intersection homology. II, Invent. Math., 72, 1, 77-129 (1983) · Zbl 0529.55007 · doi:10.1007/BF01389130 |
| [39] | Goresky, Mark; MacPherson, Robert, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 14, xiv+272 pp. (1988), Springer-Verlag, Berlin · Zbl 0639.14012 · doi:10.1007/978-3-642-71714-7 |
| [40] | Goresky, R. Mark, Triangulation of stratified objects, Proc. Amer. Math. Soc., 72, 1, 193-200 (1978) · Zbl 0392.57001 · doi:10.2307/2042563 |
| [41] | G\`“ottsche, Lothar, Modular forms and Donaldson invariants for \(4\)-manifolds with \(b_+=1\), J. Amer. Math. Soc., 9, 3, 827-843 (1996) · Zbl 0872.57027 · doi:10.1090/S0894-0347-96-00212-3 |
| [42] | G\`“ottsche, Lothar; Nakajima, Hiraku; Yoshioka, K\=ota, Instanton counting and Donaldson invariants, J. Differential Geom., 80, 3, 343-390 (2008) · Zbl 1172.57015 |
| [43] | G\`“ottsche, Lothar; Nakajima, Hiraku; Yoshioka, K\=ota, \(K\)-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Q., 5, 3, Special Issue: In honor of Friedrich Hirzebruch., 1029-1111 (2009) · Zbl 1192.14011 · doi:10.4310/PAMQ.2009.v5.n3.a5 |
| [44] | G\`“ottsche, Lothar; Nakajima, Hiraku; Yoshioka, K\=ota, Donaldson = Seiberg-Witten from Mochizuki”s formula and instanton counting, Publ. Res. Inst. Math. Sci., 47, 1, 307-359 (2011) · Zbl 1259.14046 · doi:10.2977/PRIMS/37 |
| [45] | G\`“ottsche, Lothar; Zagier, Don, Jacobi forms and the structure of Donaldson invariants for \(4\)-manifolds with \(b_+=1\), Selecta Math. (N.S.), 4, 1, 69-115 (1998) · Zbl 0924.57025 · doi:10.1007/s000290050025 |
| [46] | Graber, T.; Pandharipande, R., Localization of virtual classes, Invent. Math., 135, 2, 487-518 (1999) · Zbl 0953.14035 · doi:10.1007/s002220050293 |
| [47] | Greenberg, Marvin J.; Harper, John R., Algebraic topology, Mathematics Lecture Note Series 58, xi+311 pp. (loose errata) pp. (1981), Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass. · Zbl 0498.55001 |
| [48] | Hatcher, Allen, Algebraic topology, xii+544 pp. (2002), Cambridge University Press, Cambridge · Zbl 1044.55001 |
| [49] | Joyce, Dominic, On manifolds with corners. Advances in geometric analysis, Adv. Lect. Math. (ALM) 21, 225-258 (2012), Int. Press, Somerville, MA · Zbl 1317.58001 |
| [50] | Kervaire, Michel A., Relative characteristic classes, Amer. J. Math., 79, 517-558 (1957) · Zbl 0173.51201 · doi:10.2307/2372561 |
| [51] | Kotschick, D., \({\rm SO}(3)\)-invariants for \(4\)-manifolds with \(b^+_2=1\), Proc. London Math. Soc. (3), 63, 2, 426-448 (1991) · Zbl 0699.53036 · doi:10.1112/plms/s3-63.2.426 |
| [52] | Kotschick, D.; Morgan, J. W., \({\rm SO}(3)\)-invariants for \(4\)-manifolds with \(b^+_2=1\). II, J. Differential Geom., 39, 2, 433-456 (1994) · Zbl 0828.57013 |
| [53] | Kronheimer, P. B.; Mrowka, T. S., Embedded surfaces and the structure of Donaldson’s polynomial invariants, J. Differential Geom., 41, 3, 573-734 (1995) · Zbl 0842.57022 |
| [54] | Kronheimer, P. B.; Mrowka, T. S., Witten’s conjecture and property P, Geom. Topol., 8, 295-310 (2004) · Zbl 1072.57005 · doi:10.2140/gt.2004.8.295 |
| [55] | Kuranishi, M., New proof for the existence of locally complete families of complex structures. Proc. Conf. Complex Analysis, Minneapolis, 1964, 142-154 (1965), Springer, Berlin · Zbl 0144.21102 |
| [56] | Lang, Serge, Algebra, Graduate Texts in Mathematics 211, xvi+914 pp. (2002), Springer-Verlag, New York · Zbl 0984.00001 · doi:10.1007/978-1-4613-0041-0 |
| [57] | Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise, Spin geometry, Princeton Mathematical Series 38, xii+427 pp. (1989), Princeton University Press, Princeton, NJ · Zbl 0688.57001 |
| [58] | Leness, Thomas, Blow-up formulae for \({\rm SO}(3)\)-Donaldson polynomials, Math. Z., 227, 1, 1-26 (1998) · Zbl 0893.57018 · doi:10.1007/PL00004365 |
| [59] | Leness, Thomas G., Donaldson wall-crossing formulas via topology, Forum Math., 11, 4, 417-457 (1999) · Zbl 0923.57008 · doi:10.1515/form.1999.008 |
| [60] | Leness, Thomas G., The semi-algebraicity of the Uhlenbeck compactifications of \(S^4\) instanton moduli spaces, Differential Geom. Appl., 26, 1, 52-62 (2008) · Zbl 1181.53027 · doi:10.1016/j.difgeo.2007.11.003 |
| [61] | Li, Wei-Ping; Qin, Zhenbo; Wang, Weiqiang, The cohomology rings of Hilbert schemes via Jack polynomials. Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38, 249-258 (2004), Amer. Math. Soc., Providence, RI · Zbl 1104.14002 |
| [62] | Lieb, Elliott H.; Loss, Michael, Analysis, Graduate Studies in Mathematics 14, xxii+346 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0966.26002 · doi:10.1090/gsm/014 |
| [63] | Maciocia, Antony, Metrics on the moduli spaces of instantons over Euclidean \(4\)-space, Comm. Math. Phys., 135, 3, 467-482 (1991) · Zbl 0734.53025 |
| [64] | Mari\~no, Marcos; Moore, Gregory; Peradze, Grigor, Four-manifold geography and superconformal symmetry, Math. Res. Lett., 6, 3-4, 429-437 (1999) · Zbl 0974.57018 · doi:10.4310/MRL.1999.v6.n4.a5 |
| [65] | Mari\~no, Marcos; Moore, Gregory; Peradze, Grigor, Superconformal invariance and the geography of four-manifolds, Comm. Math. Phys., 205, 3, 691-735 (1999) · Zbl 0988.57019 · doi:10.1007/s002200050694 |
| [66] | Mather, John, Notes on topological stability, Bull. Amer. Math. Soc. (N.S.), 49, 4, 475-506 (2012) · Zbl 1260.57049 · doi:10.1090/S0273-0979-2012-01383-6 |
| [67] | May, J. P., A concise course in algebraic topology, Chicago Lectures in Mathematics, x+243 pp. (1999), University of Chicago Press, Chicago, IL · Zbl 0923.55001 |
| [68] | Milnor, John W.; Stasheff, James D., Characteristic classes, vii+331 pp. (1974), Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo · Zbl 1079.57504 |
| [69] | Mochizuki, Takuro, Donaldson type invariants for algebraic surfaces, Lecture Notes in Mathematics 1972, xxiv+383 pp. (2009), Springer-Verlag, Berlin · Zbl 1177.14003 · doi:10.1007/978-3-540-93913-9 |
| [70] | Moore, Gregory; Witten, Edward, Integration over the \(u\)-plane in Donaldson theory, Adv. Theor. Math. Phys., 1, 2, 298-387 (1997) · Zbl 0899.57021 · doi:10.4310/ATMP.1997.v1.n2.a7 |
| [71] | Morgan, John W., The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, viii+128 pp. (1996), Princeton University Press, Princeton, NJ · Zbl 0846.57001 |
| [72] | J. W. Morgan and T. S. Mrowka, The gluing construction for anti-self-dual connections over manifolds with long tubes, preprint, October 13, 1994, 115 pages. |
| [73] | J. W. Morgan and T. S. Mrowka, A note on Donaldson’s polynomial invariants, Internat. Math. Res. Notices (1992), 223-230. \MR{1191573 (93m:57032)} · Zbl 0787.57011 |
| [74] | Mrowka, Tomasz Stanislaw, A local Mayer-Vietoris principle for Yang-Mills moduli spaces, 70 pp. (1988), ProQuest LLC, Ann Arbor, MI |
| [75] | Munkres, James R., Elements of algebraic topology, ix+454 pp. (1984), Addison-Wesley Publishing Company, Menlo Park, CA · Zbl 0673.55001 |
| [76] | Nakajima, Hiraku; Yoshioka, K\=ota, Lectures on instanton counting. Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38, 31-101 (2004), Amer. Math. Soc., Providence, RI · Zbl 1080.14016 |
| [77] | Nakajima, Hiraku; Yoshioka, K\=ota, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math., 162, 2, 313-355 (2005) · Zbl 1100.14009 · doi:10.1007/s00222-005-0444-1 |
| [78] | Nicolaescu, Liviu I., Notes on Seiberg-Witten theory, Graduate Studies in Mathematics 28, xviii+484 pp. (2000), American Mathematical Society, Providence, RI · Zbl 0978.57027 · doi:10.1090/gsm/028 |
| [79] | Ozsv\'ath, Peter S., Some blowup formulas for \({\rm SU}(2)\) Donaldson polynomials, J. Differential Geom., 40, 2, 411-447 (1994) · Zbl 0846.57014 |
| [80] | Peng, Xiao Wei, Asymptotic behavior of the \(L^2\)-metric on moduli spaces of Yang-Mills connections, Math. Z., 220, 1, 127-158 (1995) · Zbl 0836.53018 · doi:10.1007/BF02572606 |
| [81] | Peng, Xiao-Wei, Asymptotic behavior of the \(L^2\)-metric on moduli spaces of Yang-Mills connections. II, Math. Z., 222, 3, 425-449 (1996) · Zbl 0866.53015 · doi:10.1007/PL00004263 |
| [82] | Pflaum, Markus J., Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics 1768, viii+230 pp. (2001), Springer-Verlag, Berlin · Zbl 0988.58003 |
| [83] | V. Ya. Pidstrigatch and A. N. Tyurin, Localisation of Donaldson invariants along the Seiberg-Witten classes, arXiv:dg-ga/9507004. |
| [84] | 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.. Russian Acad. Sci. Izv. Math., 56 40, 2, 267-351 (1993) · Zbl 0796.14024 · doi:10.1070/IM1993v040n02ABEH002167 |
| [85] | L. Qin, On the associativity of gluing, arXiv:1107.5527v1. · Zbl 1402.57023 |
| [86] | Ruan, Yongbin, Virtual neighborhoods and the monopole equations. Topics in symplectic \(4\)-manifolds, Irvine, CA, 1996, First Int. Press Lect. Ser., I, 101-116 (1998), Int. Press, Cambridge, MA · Zbl 0939.57024 |
| [87] | D. A. Salamon, Spin geometry and Seiberg-Witten invariants, unpublished book, available at \url{math.ethz.ch/ salamon/publications.html}. · Zbl 0865.57019 |
| [88] | Schwartz, M.-H., Lectures on stratification of complex analytic sets, Tata Institute of Fundamental Research Lectures on Mathematics, No. 38, iii+77+iv pp. (1966), Tata Institute of Fundamental Research, Bombay · Zbl 0169.41404 |
| [89] | Seiberg, N.; Witten, E., Electric-magnetic duality, monopole condensation, and confinement in \(N=2\) supersymmetric Yang-Mills theory, Nuclear Phys. B, 426, 1, 19-52 (1994) · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [90] | Selick, Paul, Introduction to homotopy theory, Fields Institute Monographs 9, xxii+188 pp. (1997), American Mathematical Society, Providence, RI · Zbl 0883.55001 |
| [91] | Sivek, Steven, Donaldson invariants of symplectic manifolds, Int. Math. Res. Not. IMRN, 6, 1688-1716 (2015) · Zbl 1320.57037 |
| [92] | Spanier, Edwin H., Algebraic topology, xvi+528 pp. ([1995?]), Springer-Verlag, New York · Zbl 0810.55001 |
| [93] | Spivak, Michael, A comprehensive introduction to differential geometry. Vol. V, v+661 pp. (1975), Publish or Perish, Inc., Boston, Mass. · Zbl 1213.53001 |
| [94] | Taubes, Clifford Henry, Self-dual Yang-Mills connections on non-self-dual \(4\)-manifolds, J. Differential Geom., 17, 1, 139-170 (1982) · Zbl 0484.53026 |
| [95] | Taubes, Clifford Henry, Self-dual connections on \(4\)-manifolds with indefinite intersection matrix, J. Differential Geom., 19, 2, 517-560 (1984) · Zbl 0552.53011 |
| [96] | Taubes, Clifford Henry, A framework for Morse theory for the Yang-Mills functional, Invent. Math., 94, 2, 327-402 (1988) · Zbl 0665.58006 · doi:10.1007/BF01394329 |
| [97] | Taubes, Clifford Henry, The stable topology of self-dual moduli spaces, J. Differential Geom., 29, 1, 163-230 (1989) · Zbl 0669.58005 |
| [98] | Taubes, Clifford Henry, \(L^2\) moduli spaces on 4-manifolds with cylindrical ends, Monographs in Geometry and Topology, I, iv+205 pp. (1993), International Press, Cambridge, MA · Zbl 0830.58004 |
| [99] | Teleman, Andrei, Moduli spaces of \({\rm PU}(2)\)-monopoles, Asian J. Math., 4, 2, 391-435 (2000) · Zbl 0982.58008 · doi:10.4310/AJM.2000.v4.n2.a10 |
| [100] | tom Dieck, Tammo, Transformation groups, De Gruyter Studies in Mathematics 8, x+312 pp. (1987), Walter de Gruyter & Co., Berlin · Zbl 0611.57002 · doi:10.1515/9783110858372.312 |
| [101] | Wehrheim, Katrin, Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing. Proceedings of the Freedman Fest, Geom. Topol. Monogr. 18, 369-450 (2012), Geom. Topol. Publ., Coventry · Zbl 1279.37028 · doi:10.2140/gtm.2012.18.369 |
| [102] | Wieczorek, Wojciech, The Donaldson invariant and embedded \(2\)-spheres, J. Reine Angew. Math., 489, 15-51 (1997) · Zbl 0890.57030 · doi:10.1515/crll.1997.489.15 |
| [103] | Witten, Edward, Monopoles and four-manifolds, Math. Res. Lett., 1, 6, 769-796 (1994) · Zbl 0867.57029 · doi:10.4310/MRL.1994.v1.n6.a13 |
| [104] | Yang, Hongjie, Transition functions and a blowup formula for Donaldson polynomials, 120 pp. (1992), ProQuest LLC, Ann Arbor, MI |
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