Skip to main content
Log in

Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • [AK] Akbulut, S., Kirby, R.: Branched covers of surfaces in 4-manifolds. Math. Ann.252, 111–131 (1980)

    Google Scholar 

  • [A] Atiyah, M.: New invariants of 3 and 4 dimensional manifolds. In: Wells, R.O., Jr. (ed.) The Mathematical Heritage of Hermann Weyl (Proc. Symp. Pure Math. vol. 48, pp. 285–299) Providence, RI: Am. Math. Soc. 1988

    Google Scholar 

  • [APS1] Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc.77, 43–69 (1975)

    Google Scholar 

  • [APS2] Atiyah, M., Patodi, V., Singer, I.: Spectral asymmetry and Riemannian geometry. II. Math. Proc. Camb. Philos Soc.78, 405–432 (1975)

    Google Scholar 

  • [BPV] Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  • [Bv] Beauville, A.: Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersectiones completes. In: Complex Analysis and Algebraic Geometry. Grauer, H. (ed.) (Lect. Notes Math., vol. 1194, pp. 8–18) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  • [BO] Boileau, M., Otal, J.: Groupes des difféotopies de certaines variéte de Seifert. C.R. Acad. Sci.303, 19–22 (1986)

    Google Scholar 

  • [Br] Brieskorn, E.: Bespiele zur differentialtialtopologie von singularitaten. Invent. Math.2, 1–14 (1966)

    Google Scholar 

  • [CG] Casson, A., Gordon, C.: Cobordism of classical knots. In: Guillon, L., Marin, A. (eds), A la Recherche de la Topologie Perdue, pp. 181–197, Baston Bassel Stuttgast: Birkhäuser, 1986

    Google Scholar 

  • [Dv] Dolgachev, I.: Weighted projective varieties. In: Carrell, J.B. (ed.) Group Actions and Vector Fields. (Lect. Notes Math., vol. 956, pp. 34–71) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  • [D1] Donaldson, S.: An application of gauge theory to the topology of 4-manifolds. Differ. Geom.18, 269–316 (1983)

    Google Scholar 

  • [D2] Donaldson, S.: Anti-self-dual Yang-Mills connections on complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc.3, 1–26 (1985)

    Google Scholar 

  • [D3] Donaldson, S.: Connections, cohomology and the intersecton forms of 4-manifolds. J. Differ. Geom.24, 275–341 (1986)

    Google Scholar 

  • [D4] Donaldson, S.: Irrationality and theh-cobordism conjecture. J. Differ. Geom.26, 141–168 (1987)

    Google Scholar 

  • [D5] Donaldson, S.: The orientation of Yang-Mills moduli spaces and 4-dimensional topology. J. Differ. Geom.26, 397–428 (1987)

    Google Scholar 

  • [D6] Donaldson, S.: Polynomial invariants for smooth 4-manifolds. Topology28, 257–315 (1990)

    Google Scholar 

  • [DFK] Donaldson, S., Furuta, M., Kotschick, D.: Floer Homology Groups in Yang: Mills Theory. (preprint)

  • [DK] Donaldson, S., Kronheimer: The Geometry of Four-Manifolds. (Oxf. Math. Monogr.) Oxford: Oxford University Press 1990

    Google Scholar 

  • [E1] Ebeling, W.: The Momodromy Groups of Isolated Singularities of Complete Intersections. (Lect. Notes Math., vol. 1293) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  • [E2] Ebeling, W.: An example of two homeomorphic, nondiffeomorphic complete intersection surface. Invent. Math.99, 651–654 (1990)

    Google Scholar 

  • [EO1] Ebeling, W., Okonek, C.: Donaldson invariants, monodromy groups, and singularities. Int. Math.1, 233–250 (1990)

    Google Scholar 

  • [EO2] Ebeling, W., Okonek, C.: On the diffeomorphism groups of certain algebraic surfaces. Enseign. Math.37, 249–262 (1991)

    Google Scholar 

  • [FS1] Fintushel, R., Stern, R.: Instanton homology of Seifert fibered homology three sphere. Proc. Lond. Math. Soc.61, 109–137 (1990)

    Google Scholar 

  • [FS2] Fintushel, R., Stern, R.: 2-Torsion instanton invariants. J. Am. Math. Soc.6, 299–339 (1993)

    Google Scholar 

  • [FS3] Fintushel, R., Stern, R.: Using the Floer exact triangle to compute Donaldson invariants. (preprint)

  • [Fl1] Floer, A.: An instanton invariant for 3-manifolds. Commun. Math. Phys.118, 215–240 (1988)

    Google Scholar 

  • [Fl2] Floer, A.: Instanton homology, surgery, and knots. In: Donaldson, S., Thomas, C.B. (eds.) Geometry of Low-Dimensional Manifolds, I. (Lond. Math. Soc. Lect. Notes Ser., vol. 150, pp. 97–114) Cambridge: Cambridge University Press 1989

    Google Scholar 

  • [FU] Freed, D., Uhlenbeck, K.: Instantous and Four-Manifolds. (Math. Sci. Res. Inst. Ser. vol. 1) Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  • [Fr] Freedman, M.: The topology of four-dimensional manifolds. J. Differ. Geom.17, 357–454 (1982)

    Google Scholar 

  • [FM1] Friedman, R., Morgan, J.: On the diffeomorphism types of certain algebraic surfaces I, II. J. Differ. Geom.27, 297–369 (1988)

    Google Scholar 

  • [FM2] Friedman, R., Morgan, J.: Smooth Four-Manifolds and Complex Surfaces. (to appear)

  • [FMM] Friedman, R., Morgan, J., Moishezon, B.: On theC invariance of the canonical class of certain algebraic surfaces. Bull. Am. Math. Soc.17, 283–286 (1987)

    Google Scholar 

  • [G1] Gompf, R.: Nuclei of elliptic surfaces. Topology30, 479–511 (1991)

    Google Scholar 

  • [G2] Gompf, R.: Sums of elliptic surfaces. J. Differ. Geom.34 93–114 (1991)

    Google Scholar 

  • [GM1] Gompf, R., Mrowka, T.: Irreducible four manifolds need not be complex. Ann. Math. (to appear)

  • [GM2] Gompf, R., Mrowka, T.: In preparation

  • [HK] Hambleton, I., Kreck, M.: On the classification of topological 4-manifolds with finite fundamental group. Math. Ann.280, 85–104 (1988)

    Google Scholar 

  • [Hr] Harer, J.: On handlebody structure for hypersurfaces inC 3 andCP 3. Math. Ann.238, 51–58 (1978)

    Google Scholar 

  • [HKK] Harer, J., Kas, A., Kirby, R.: Handlebody decompositions of complex surface. Mem. Am. Math. Soc. Soc.62 (1986)

  • [Kl1] Klassen, E.: Representations of knot groups in SU (2) Trans. Am. Math. Soc.326, 795–828 (1991)

    Google Scholar 

  • [Kl2] Klassen, E.: Representations in SU(2) of the fundamental groups of the Whitehead link and doubled knots. Forum Math.5, 93–109 (1993)

    Google Scholar 

  • [Kd1] Kodaira, K.: On compact analytic surfaces I, II, II. Ann. Math.71, 111–152 (1960);77, 563–626 (1963);78, 1–40 (1963)

    Google Scholar 

  • [Kd2] Kodaira, K.: On the structure of compact complex analytic surfaces I, II, IV. Am. J. Math.86, 781–798 (1964);88, 682–721 (1966);90, 1048–1066 (1968)

    Google Scholar 

  • [Kd3] Kodaira, K.: On homotopy K3 surfaces. In: Essays on Topology and Related Topics, Mémoires dédiés a Georges deRham, pp. 58–69. Berlin, Heidelberg New York: Springer 1970

    Google Scholar 

  • Kotschick, D.: On manifolds homeomorphic to\(CP^2 \# \overline {CP} ^2 \). Invent. Math.95, 51–600 (1989)

    Google Scholar 

  • [Kr] Kronheimer, P.: Instanton invariants and flat connections on the Kummer surface. Duke Math. J.64, 229–241 (1991)

    Google Scholar 

  • [LM] Lockhart, R., McOwen, R.: Elliptic differential operators on noncompact manifolds. Ann. Sc. Norm. Super. Pisa12, 409–448 (1985)

    Google Scholar 

  • [Mt] Matumoto, T.: On the diffeomorphisms of a K3 surface. In: Nagota, M. et al (eds.) Algebraic and Topological Theries. Kinosaki 1984, pp. 616–621. Tokyo: Kinokuniya 1986

    Google Scholar 

  • [Mn1] Moishezon, B.: Complex Surfaces and Connected Sums of Projective Planes (Lect. Notes Math. vol. 603) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  • [Mn2] Moishezon, B.: Analogues of Lefschetz theorems for linear systems with isolated singularities. J. Differ. Geom.31, 47–72 (1990)

    Google Scholar 

  • [MMR] Morgan, J., Mrowka, T., Ruberman, D.: TheL 2-moduli space and a vanishing theorem for Donaldson polynomial invariants. (Preprint)

  • [Mr] Mrowka, T.: A local Mayer-Vietoris principle for Yang-Mills moduli spaces. Ph.D. Thesis, Berkeley (1988)

  • [OV1] Okonek, C., Van de Ven A. Stable vector bundles and differentiable structures on certain elliptic surfaces. Invent. Math.86, 357–370 (1986)

    Google Scholar 

  • [OV2] Okonek, C., Van de Ven A.I-type invariants associated to PU(2) bundles and the differentiable structure of Barlow's surface. Invent. Math.95, 601–614 (1989)

    Google Scholar 

  • [P] Persson, U.: An introduction to the geography of surfaces of general type. In: Block, S.J. (ed.) Algebraic Geometry Bowdon 1985. (Proc. Symp. Pure Math. vol. 46, pp. 195–220) Providence, RI: Am. Math. Soc. 1987

    Google Scholar 

  • [Sa] Salvetti, M.: On the number of nonequivalent differentiable structures on 4-manifolds. Manuscr. Mth.63, 157–171 (1989)

    Google Scholar 

  • [T] Taubes, C.:L 2-Moduli Spaces on 4-Manifolds with Cylindrical Ends. Hong Kong: International Press 1993

    Google Scholar 

  • [U] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31–42 (1982)

    Google Scholar 

  • [V] Viro, O.: Compact four-dimensional exotica with small homology. Leningr. Math. J.1, 871–880 (1990)

    Google Scholar 

  • [W] Wall, C.T.C.: On the orthogonal groups of unimodular quadratic forms. Math. Ann.147, 328–338 (1962)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Oblatum 9-I-1993 & 16-XI-1993

The first author was partially supported NSF Grant DMS9102522 and the second author by NSF Grant DMS9002517

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fintushel, R., Stern, R.J. Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds. Invent Math 117, 455–523 (1994). https://doi.org/10.1007/BF01232253

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01232253

Navigation