The geography problem for irreducible spin four-manifolds

Authors:
B. Doug Park and Zoltán Szabó

Journal:
Trans. Amer. Math. Soc. **352** (2000), 3639-3650

MSC (2000):
Primary 57R15, 57R57; Secondary 57N65, 58D27

DOI:
https://doi.org/10.1090/S0002-9947-00-02467-3

Published electronically:
March 15, 2000

MathSciNet review:
1653371

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Abstract | References | Similar Articles | Additional Information

We study the geography problem for smooth irreducible simply-connected spin four-manifolds. For a large class of homotopy types, we exhibit both symplectic and non-symplectic representatives. We also compute the Seiberg-Witten invariants of all the four-manifolds we construct.

**[D]**S.K. Donaldson: An application of gauge theory to four-dimensional topology,*J. Diff. Geom.***18**(1983), 279-315. MR**85c:57015****[FS1]**R. Fintushel and R. Stern: Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds,*Invent. math.***117**(1994), 455-523. MR**95f:57040****[FS2]**R. Fintushel and R. Stern: Knots, links and 4-manifolds,*Invent. Math.***134**(1998), 363-400. CMP**99:03****[FS3]**R. Fintushel and R. Stern: Rational blowdowns of smooth 4-manifolds,*J. Diff. Geom.***46**(1997), 181-235. MR**98j:57047****[G1]**R.E. Gompf: Nuclei of elliptic surfaces,*Topology***30**(1991), 479-511. MR**92f:57042****[G2]**R.E. Gompf: A new construction of symplectic manifolds,*Annals of Math.***142**(1995), 527-595. MR**96j:57025****[GM]**R.E. Gompf and T.S. Mrowka: Irreducible 4-manifolds need not be complex,*Annals of Math.***138**(1993), 61-111. MR**95c:57053****[Gr]**M. Gromov:*Partial Differential Relations*, Springer-Verlag, Berlin, 1986. MR**90a:58201****[Ki]**R. Kirby: Problems in low-dimensional topology,*Geometric Topology*, AMS/IP*Stud. Adv. Math.***2.2**, 35-473, Amer. Math. Soc., 1997. CMP**98:01****[KM]**P.B. Kronheimer and T.S. Mrowka: The genus of embedded surfaces in the projective plane,*Math. Res. Lett.***1**(1994), 797-804. MR**96a:57073****[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****[MSz]**J.W. Morgan and Z. Szabó: Embedded genus-2 surfaces in four-manifolds,*Duke Math. J.*,**89**(1997), 577-602. MR**98k:57048****[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****[P]**J. Park: Ph.D. Thesis, Michigan State University, 1995.**[St]**A. Stipsicz: Elementary computations of Donaldson series,*Turkish J. of Math.***19**(1995), 159-166. MR**96k:57031****[Sz1]**Z. Szabó: Simply-connected irreducible 4-manifolds with no symplectic structures,*Invent. Math.***132**(1998), 457-466. MR**99f:57033****[Sz2]**Z. Szabó: Exotic 4-manifolds with ,*Math. Res. Lett.***3**, (1996), 731-741. MR**97j:57049****[Sz3]**Z. Szabó: Exotic smooth structures on ,*in preparation*.**[Sz4]**Z. Szabó: Irreducible four-manifolds with small Euler characteristics,*Topology***35**(1996), 411-426. MR**97c:57021****[T1]**C.H. Taubes: The Seiberg-Witten invariants and symplectic forms,*Math. Res. Letters***1**(1994), 809-822. MR**95j:57039****[T2]**C.H. Taubes: More constraints on symplectic manifolds from Seiberg-Witten invariants,*Math. Res. Lett.***2**, (1995), 9-13. MR**96a:57075****[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. Lett.***1**(1994), 769-796. MR**96d:57035**

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Additional Information

**B. Doug Park**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Email:
bahnpark@math.princeton.edu

**Zoltán Szabó**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Email:
szabo@math.princeton.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02467-3

Keywords:
Geography,
spin,
symplectic,
Seiberg-Witten

Received by editor(s):
April 1, 1998

Published electronically:
March 15, 2000

Additional Notes:
The second author was supported in part by NSF Grant DMS-970435.

Article copyright:
© Copyright 2000
American Mathematical Society