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Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

Published online by Cambridge University Press:  24 October 2008

D. Kotschick  and
G. Matić
D. Kotschick
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland
G. Matić
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, U.S.A.
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Extract

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula

is the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 275 - 286
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

REFERENCES

[1]Catanese, F.. On the moduli spaces of surfaces of general type, J. Differential Geometry 19 (1984), 483–515.CrossRefGoogle Scholar
[2]Donaldson, S. K.. The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differential Geometry 26 (1987), 397–428.CrossRefGoogle Scholar
[3]Donaldson, S. K.. Complex curves and surgery. Publ. Math. IHES 68 (1989), 91–97.CrossRefGoogle Scholar
[4]Donaldson, S. K.. Polynomial invariants for smooth four-manifolds. Topology 29 (1990), 257–315.CrossRefGoogle Scholar
[5]Friedman, R. and Morgan, J. W.. Algebraic surfaces and 4-manifolds: some conjectures and speculations. Bull. (New Ser.) Am. Math. Soc. 18 (1988), 1–19.CrossRefGoogle Scholar
[6]Friedman, R. and Morgan, J. W.. Smooth 4-manifolds and complex surfaces (Springer-Verlag, 1994).CrossRefGoogle Scholar
[7]Hartshorne, R.. Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics 156 (Springer-Verlag, 1970).CrossRefGoogle Scholar
[8]Hsiang, W. C. and Szczarba, R. H.. On embedding surfaces in four-manifolds. Proc. Symp. Pure Math. 22 (1971), 97–103.CrossRefGoogle Scholar
[9]Itenberg, V. S.. Medium-dimension homologies of a submanifold of codimension two. Func. Analysis Appl. 8 (1974), 121–126.CrossRefGoogle Scholar
[10]Kervaire, M. A. and Milnor, J. W.. On 2-spheres in 4-manifolds. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651–1657.CrossRefGoogle ScholarPubMed
[11]Kirby, R.. Problems in low-dimensional topology. Proc. Symp. Pure Math. 32 (1978), 273–322.CrossRefGoogle Scholar
[12]Kotschick, D.. On manifolds homeomorphic to CP 2 # 8CP 2. Invent. math. 95 (1989), 591–600.CrossRefGoogle Scholar
[13]Kotschick, D.. SO(3)-invariants for 4-manifolds with b +2= 1. Proc. London Math. Soc. 63 (1991), 426–448.CrossRefGoogle Scholar
[14]Kronheimer, P. B.. The genus-minimizing property of algebraic curves. Bull. Amer. Math. Soc. (New Series) 29 (1993), 63–69.CrossRefGoogle Scholar
[15]Kronheimer, P. B. and Mrowka, T. S.. Gauge theory for embedded surfaces I, II. Topology (to appear).Google Scholar
[16]Lawson, T.. Smooth embeddings of 2-spheres in 4-manifolds. Expo. Math. 10 (1992), 289–309.Google Scholar
[17]Lee, R. and Wilczynski, D., (in preparation).Google Scholar
[18]Morgan, J. W. and Mrowka, T. S.. A note on Donaldson's polynomial invariants. Int. Math. Research Notices 10 (1992), 223–230.CrossRefGoogle Scholar
[19]Nori, M. V.. Zariski's conjecture and related problems. Ann. Sci. École Norm. Sup. 16 (1983), 305–344.CrossRefGoogle Scholar
[20]Rokhlin, V. A.. Two-dimensional submanifolds of four-dimensional manifolds. Fune. Analysis Appl. 6 (1971), 39–48.Google Scholar
[21]Rudolph, L.. Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv. 59 (1984), 592–599.CrossRefGoogle Scholar
[22]Rudolph, L.. Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (New Series) 29 (1993), 51–59.CrossRefGoogle Scholar
[23]Suciu, A. I.. Immersed spheres in CP 2 and -S 2 × S 2. Math. Zeitschrift 196 (1987), 51–57.CrossRefGoogle Scholar