Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Exotic smooth structures on $3\mathbf{CP}^2 \char93 n\overline{\mathbf{CP}}^2$

Author(s): B. Doug Park
Journal: Proc. Amer. Math. Soc. 128 (2000), 3057-3065.
MSC (2000): Primary 57R55; Secondary 57R57, 53D05.
Posted: March 2, 2000
Errata: Proc. Amer. Math. Soc. 136 (2008), 1503-1503.
MathSciNet review: 1664426
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We construct exotic $3\mathbf{CP}^2 \char93 10 \overline{\mathbf{CP}}^2$and $3\mathbf{CP}^2 \char93 12{\overline{\mathbf{CP}}}^2$ as a corollary of recent results of I. Dolgachev and C. Werner concerning a numerical Godeaux surface. We also construct another exotic $3\mathbf{CP}^2 \char93 12 {\overline{\mathbf{CP}}}^2$using the surgery techniques of R. Fintushel and R. J. Stern. We show that these 4-manifolds are irreducible by computing their Seiberg-Witten invariants.

References:

[B]
R. Barlow: A simply connected surface of general type with $p_g =0$, Invent. Math. 79 (1985), 293-301. MR 87a:14033

[CG]
P. Craighero and R. Gattazzo: Quintic surfaces of ${\bf P}^3$having a nonsingular model with $q=p_g=0$, $P_2 \neq 0$, Rend. Sem. Mat. Univ. Padova 91 (1994), 187-198. MR 95g:14038

[DW]
I. Dolgachev and C. Werner: A simply connected numerical Godeaux surface with ample canonical class, preprint.

[FS1]
R. Fintushel and R. Stern: Immersed Spheres in 4-Manifolds and The Immersed Thom Conjecture, Turkish J. of Math. 19 (1995), 145-157. MR 96j:57036

[FS2]
R. Fintushel and R. Stern: Knots, Links and 4-manifolds, to appear in Invent. Math.

[FQ]
M.H. Freedman and F. Quinn: Topology of 4-Manifolds, Princeton University Press, 1990. MR 94b:57021

[G]
R.E. Gompf: A new construction of symplectic manifolds, Annals of Math. 142 (1995), 527-595. MR 96j:57025

[GS]
R.E. Gompf and A.I. Stipsicz: An Introduction to 4-Manifolds and Kirby Calculus, preprint.

[Ko]
D. Kotschick: On manifolds homeomorphic to ${\bf CP}^2 \char93 8{\overline {\bf CP}}^2$, Invent. Math. 95 (1989), 591-600. MR 90a:57047

[KM]
P.B. Kronheimer and T.S. Mrowka: The genus of embedded surfaces in the projective plane, Math. Res. Letters 1 (1994), 797-804. MR 96a:57073

[L]
K. Lamotke: The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), 15-51. MR 81m:14019

[M]
J.W. Morgan: The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Math. Notes 44, Princeton University Press, 1996. MR 97d:57042

[MST]
J.W. Morgan, Z. Szabó and C.H. Taubes: A Product Formula for the Seiberg-Witten Invariants and the Generalized Thom Conjecture, J. Diff. Geom. 44 (1996), 706-788. MR 97m:57052

[OS]
P. Ozsváth and Z. Szabó: The symplectic Thom conjecture, preprint.

[P]
B.D. Park: Exotic smooth structures on $3\mathbf{CP}^2\char93 n\overline{\mathbf{CP}}^2$, Part II, this issue.

[Sa]
D. Salamon: Spin Geometry and Seiberg-Witten Invariants, preprint.

[St]
A. Stipsicz: Donaldson invariants of certain symplectic manifolds, J. Reine Angew. Math. 465 (1995), 1-10. MR 96h:57030

[Sz1]
Z. Szabó: Irreducible four-manifolds with small Euler characteristics, Topology 35 (1996), 411-426. MR 97c:57021

[Sz2]
Z. Szabó: Exotic 4-manifolds with $b_2 ^+ =1$, Math. Res. Letters 3 (1996), 731-741. MR 97j:57049

[Ta]
C.H. Taubes: The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1 (1994), 809-822. MR 95j:57039

[Th]
W. Thurston: Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468. MR 53:6578

[Wi]
E. Witten: Monopoles and four-manifolds, Math. Res. Letters 1 (1994), 769-796. MR 96d:57035

[Y]
B.Z. Yu, Some computations of Donaldson's polynomials via flat connections, Thesis, U.C. Berkeley, 1994.

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57R55, 57R57, 53D05.

Retrieve articles in all Journals with MSC (2000): 57R55, 57R57, 53D05.

Additional Information:

B. Doug Park
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: bahnpark@math.princeton.edu

DOI: 10.1090/S0002-9939-00-05357-0
PII: S 0002-9939(00)05357-0
Received by editor(s): August 4, 1998
Received by editor(s) in revised form: November 2, 1998
Posted: March 2, 2000
Communicated by: Ronald A. Fintushel
Copyright of article: Copyright 2000, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia