Representing nonnegative homology classes of by minimal genus smooth embeddings
Author:
BangHe Li
Journal:
Trans. Amer. Math. Soc. 352 (2000), 41554169
MSC (1991):
Primary 57R95, 57R40
Published electronically:
May 21, 1999
MathSciNet review:
1637082
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Abstract: For any nonnegative class in , the minimal genus of smoothly embedded surfaces which represent is given for , and in some cases with , the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus , we prove that it is true for with and for with .
 [D1]
S.
K. Donaldson, The orientation of YangMills moduli spaces and
4manifold topology, J. Differential Geom. 26 (1987),
no. 3, 397–428. MR 910015
(88j:57020)
 [D2]
S.
K. Donaldson, The SeibergWitten equations and
4manifold topology, Bull. Amer. Math. Soc.
(N.S.) 33 (1996), no. 1, 45–70. MR 1339810
(96k:57033), http://dx.doi.org/10.1090/S0273097996006258
 [FM]
Robert
Friedman and John
W. Morgan, On the diffeomorphism types of certain algebraic
surfaces. I, J. Differential Geom. 27 (1988),
no. 2, 297–369. MR 925124
(89d:57046)
 [FS]
Ronald
Fintushel and Ronald
J. Stern, Immersed spheres in 4manifolds and the immersed Thom
conjecture, Turkish J. Math. 19 (1995), no. 2,
145–157. MR 1349567
(96j:57036)
 [G1]
Hong
Zhu Gao, Representing homology classes of almost definite
4manifolds, Topology Appl. 52 (1993), no. 2,
109–120. MR 1241187
(94j:57020), http://dx.doi.org/10.1016/01668641(93)90030H
 [G2]
Danyan
Gan, Embedded 2spheres in indefinite 4manifolds, Chinese
Ann. Math. Ser. B 17 (1996), no. 3, 257–262. A
Chinese summary appears in Chinese Ann.\ Math.\ Ser.\ A 17 (1996), no.\ 4,
515. MR
1415253 (97j:57056)
 [HS]
W.
C. Hsiang and R.
H. Szczarba, On embedding surfaces in fourmanifolds,
Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin,
Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971,
pp. 97–103. MR 0339239
(49 #4000)
 [K1]
Kazunori
Kikuchi, Representing positive homology classes
of 𝐶𝑃²#2\overline{𝐶𝑃}² and
𝐶𝑃²#3\overline{𝐶𝑃}², Proc. Amer. Math. Soc. 117 (1993), no. 3, 861–869. MR 1131036
(93d:57065), http://dx.doi.org/10.1090/S0002993919931131036X
 [K2]
Kazunori
Kikuchi, Positive 2spheres in 4manifolds of signature
(1,𝑛), Pacific J. Math. 160 (1993),
no. 2, 245–258. MR 1233354
(94f:57027)
 [KM1]
Michel
A. Kervaire and John
W. Milnor, On 2spheres in 4manifolds, Proc. Nat. Acad. Sci.
U.S.A. 47 (1961), 1651–1657. MR 0133134
(24 #A2968)
 [KM2]
P.
B. Kronheimer and T.
S. Mrowka, The genus of embedded surfaces in the projective
plane, Math. Res. Lett. 1 (1994), no. 6,
797–808. MR 1306022
(96a:57073), http://dx.doi.org/10.4310/MRL.1994.v1.n6.a14
 [L]
Terry
Lawson, Smooth embeddings of 2spheres in 4manifolds,
Exposition. Math. 10 (1992), no. 4, 289–309. MR 1184697
(93m:57041)
 [Li]
Bang
He Li, Embeddings of surfaces in 4manifolds. I, II, Chinese
Sci. Bull. 36 (1991), no. 24, 2025–2029,
2030–2033. MR 1150851
(93d:57037)
 [LiL1]
BangHe
Li and TianJun
Li, Minimal genus smooth embeddings in
𝑆²×𝑆² and
𝐶𝑃²#𝑛\overline{𝐶𝑃}² with
𝑛≤8, Topology 37 (1998), no. 3,
575–594. MR 1604887
(99b:57059), http://dx.doi.org/10.1016/S00409383(97)000426
 [LiL2]
BangHe
Li and TianJun
Li, Minimal genus embeddings in 𝑆²bundles over
surfaces, Math. Res. Lett. 4 (1997), no. 23,
379–394. MR 1453068
(98h:57062), http://dx.doi.org/10.4310/MRL.1997.v4.n3.a7
 [LL1]
T.
J. Li and A.
Liu, General wall crossing formula, Math. Res. Lett.
2 (1995), no. 6, 797–810. MR 1362971
(96m:57053), http://dx.doi.org/10.4310/MRL.1995.v2.n6.a11
 [LL2]
T.
J. Li and A.
Liu, Symplectic structure on ruled surfaces and a generalized
adjunction formula, Math. Res. Lett. 2 (1995),
no. 4, 453–471. MR 1355707
(96m:57052), http://dx.doi.org/10.4310/MRL.1995.v2.n4.a6
 [MST]
John
W. Morgan, Zoltán
Szabó, and Clifford
Henry Taubes, A product formula for the SeibergWitten invariants
and the generalized Thom conjecture, J. Differential Geom.
44 (1996), no. 4, 706–788. MR 1438191
(97m:57052)
 [Ro]
V. Rohlin, Twodimensional submanifolds of fourdimensional manifolds, Functional Analysis Appl. 6 (1972), 136138.
 [Ru]
Daniel
Ruberman, The minimal genus of an embedded surface of nonnegative
square in a rational surface, Turkish J. Math. 20
(1996), no. 1, 129–133. MR 1392668
(97k:57036)
 [T1]
Clifford
Henry Taubes, The SeibergWitten invariants and symplectic
forms, Math. Res. Lett. 1 (1994), no. 6,
809–822. MR 1306023
(95j:57039), http://dx.doi.org/10.4310/MRL.1994.v1.n6.a15
 [T2]
Clifford
Henry Taubes, More constraints on symplectic forms from
SeibergWitten invariants, Math. Res. Lett. 2 (1995),
no. 1, 9–13. MR 1312973
(96a:57075), http://dx.doi.org/10.4310/MRL.1995.v2.n1.a2
 [T3]
Clifford
Henry Taubes, The SeibergWitten and Gromov invariants, Math.
Res. Lett. 2 (1995), no. 2, 221–238. MR 1324704
(96a:57076), http://dx.doi.org/10.4310/MRL.1995.v2.n2.a10
 [T4]
, Talks given at Harvard.
 [Ue]
M. Ue, Embedded surfaces in blowing up rational and elliptic surfaces, (abstract for `` Art of LowDimensional Topology III"), Preprint of the Division of Mathematics, Kyoto Univ.
 [Wa]
C.
T. C. Wall, Classification problems in differential topology. II.
Diffeomorphisms of handlebodies, Topology 2 (1963),
263–272. MR 0156354
(27 #6278)
 [Wi]
Edward
Witten, Monopoles and fourmanifolds, Math. Res. Lett.
1 (1994), no. 6, 769–796. MR 1306021
(96d:57035), http://dx.doi.org/10.4310/MRL.1994.v1.n6.a13
 [D1]
 S. Donaldson, The orientation of YangMills moduli spaces and manifold topology, J. Differential Geom. 26 (1987), 397428. MR 88j:57020
 [D2]
 , The SeibergWitten equations and manifold topology, Bull. (N.S.) Amer. Math. Soc. 33 (1996), 4570. MR 96k:57033
 [FM]
 R. Friedman and J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces, I, J. Differential Geom. 27 (1988), 297369. MR 89d:57046
 [FS]
 R. Fintushel and R. J. Stern, Immersed spheres in 4manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995), 145157. MR 96j:57036
 [G1]
 H. Z. Gao, Representing homology classes of 4manifolds, Topology and App. 52(2) (1993), 109120. MR 94j:57020
 [G2]
 D. Y. Gan, Embedded 2spheres in indefinite 4manifolds, Chinese Ann. Math. Ser. B 17 (1996), 257262. MR 97j:57056
 [HS]
 W. C. Hsiang and R. Szczarba, On embedding surfaces in fourmanifold, Proc. Symp. Pure Math., vol. 22, Amer. Math. Soc., Providence, RI, 1970, pp. 97103. MR 49:4000
 [K1]
 K. Kikuchi, Representing positive homology classes of and , Proc. Amer. Math. Soc. 117 (1993), 861969. MR 93d:57065
 [K2]
 , Positive spheres in manifolds of signature , Pacific J. Math. 160 (1993), 245258. MR 94f:57027
 [KM1]
 M. Kervaire and W. Milnor, On spheres in manifolds, Proc. Nat. Acad. Sci. USA 47 (1961), 16511657. MR 24:A2968
 [KM2]
 P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Research Letters 1 (1994), 797808. MR 96a:57073
 [L]
 T. Lawson, Smooth embeddings of spheres in manifolds, Expo. Math. 10 (1992), 289309. MR 93m:57041
 [Li]
 B. H. Li, Embeddings of surfaces in manifolds, Chinese Science Bulletin 36 (1991), 20252033. MR 93d:57037
 [LiL1]
 B. H. Li and T. J. Li, Minimal genus smooth embeddings in and with , Topology 37 (1998), 575594. MR 99b:57059
 [LiL2]
 , Minimal genus embeddings of surfaces in bundles over surfaces, Math. Research Letters 4 (1997), 379384. MR 98h:57062
 [LL1]
 T. J. Li and A. Liu, General Wall crossing formula, Math. Research Letters 2 (1995), 797810. MR 96m:57053
 [LL2]
 , Symplectic structures on ruled surfaces and generalized adjunction formula, Math. Research Letters 2 (1995), 453471. MR 96m:57052
 [MST]
 J. Morgan, Z. Szabo and C. H. Taubes, A product formula for the SeibergWitten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996), 706788. MR 97m:57052
 [Ro]
 V. Rohlin, Twodimensional submanifolds of fourdimensional manifolds, Functional Analysis Appl. 6 (1972), 136138.
 [Ru]
 D. Ruberman, The minimal genus of an embedded surface of nonnegative square in a rational surface, Turkish J. Math. 20 (1996), 129133. MR 97k:57036
 [T1]
 C. H. Taubes, The SeibergWitten invariants and symplectic forms, Math. Research Letters 1 (1994), 809822. MR 95j:57039
 [T2]
 , More constraints on symplectic forms from SeibergWitten invariants, Math. Research Letters 2 (1995), 913. MR 96a:57075
 [T3]
 , The SeibergWitten invariants and The Gromov invariants, Math. Research Letters 2 (1995), 221238. MR 96a:57076
 [T4]
 , Talks given at Harvard.
 [Ue]
 M. Ue, Embedded surfaces in blowing up rational and elliptic surfaces, (abstract for `` Art of LowDimensional Topology III"), Preprint of the Division of Mathematics, Kyoto Univ.
 [Wa]
 C. T. C. Wall, Diffeomorphisms of manifolds, J. London Math. Soc. 39 (1964), 131140. MR 27:6278
 [Wi]
 E. Witten, Monopoles and FourManifolds, Math. Research Letters 1 (1994), 769796. MR 96d:57035
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Additional Information
BangHe Li
Affiliation:
Institute of Systems Science, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China
Email:
libh@iss06.iss.ac.cn
DOI:
http://dx.doi.org/10.1090/S0002994799024228
PII:
S 00029947(99)024228
Received by editor(s):
March 25, 1998
Published electronically:
May 21, 1999
Additional Notes:
The author is supported partially by the Tianyuan Foundation of Peoples Republic of China
Article copyright:
© Copyright 2000
American Mathematical Society
