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Transactions of the American Mathematical Society

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Four-Manifolds With Surface Fundamental Groups


Authors: Alberto Cavicchioli, Friedrich Hegenbarth and Dusan Repovs
Journal: Trans. Amer. Math. Soc. 349 (1997), 4007-4019
MSC (1991): Primary 57N65, 57R67, 57Q10
DOI: https://doi.org/10.1090/S0002-9947-97-01751-0
MathSciNet review: 1376542
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Abstract: We study the homotopy type of closed connected topological $4$-manifolds whose fundamental group is that of an aspherical surface $F$. Then we use surgery theory to show that these manifolds are $s$-cobordant to connected sums of simply-connected manifolds with an $\mathbb {S}^{2}$-bundle over $F$.


References [Enhancements On Off] (What's this?)

  • 1. H. J. Baues, Combinatorial Homotopy and $4$-dimensional Complexes, Walter de Gruyter, Berlin - New York, 1991. MR 92h:55008
  • 2. W. Browder, Surgery on Simply-connected Manifolds, Springer-Verlag, Berlin- Heidelberg- New York, 1972. MR 50:11272
  • 3. T.D. Cochran and N. Habegger, On the homotopy theory of simply-connected four-manifolds, Topology 29 (1990), 419-440. MR 91h:57006
  • 4. M.H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 84b:57006
  • 5. M.H. Freedman and F. Quinn, Topology of $4$-manifolds, Princeton Univ. Press, Princeton, New Jersey, 1990. MR 94b:57021
  • 6. J.A. Hillman, On $4$-manifolds homotopy equivalent to surface bundles over surfaces, Topology Appl. 40 (1991), 275-286. MR 92g:57031
  • 7. J.A. Hillman, On $4$-manifolds with universal covering spaces $\mathbb {S}^{2}\times \mathbb {R}^{2}$ or $\mathbb {S}^{3}\times \mathbb {R}$, Topology Appl. 52 (1993), 23-42. MR 95b:57020
  • 8. J.A. Hillman, The Algebraic Characterization of Geometric $4$-Manifolds, London Math. Soc. Lect. Note Ser. 198, Cambridge Univ. Press, Cambridge, 1994. MR 95m:57032
  • 9. Y. Matsumoto, Personal Communication, 1995.
  • 10. Y. Matsumoto, Diffeomorphism types of elliptic surfaces, Topology 25 (1986), 549-563. MR 88b:32061
  • 11. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 33:4922
  • 12. C.T.C. Wall, On simply-connected $4$-manifolds, J. London Math. Soc. 39 (1964), 141-149. MR 29:627
  • 13. C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, London-New York, 1970. MR 55:4217
  • 14. J.H.C. Whitehead, A certain exact sequence, Ann. of Math. (2) 52 (1950), 51-110. MR 12:43c
  • 15. J.H.C. Whitehead, Elements of Homotopy Theory, Springer-Verlag, Berlin- Heidelberg- New York, 1978. MR 80b:55001

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Additional Information

Alberto Cavicchioli
Affiliation: Dipartimento di Matematica, Università di Modena, Via Campi 213/B, 41100 Modena, Italy
Email: albertoc@unimo.it

Friedrich Hegenbarth
Affiliation: Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
Email: hegenbarth@vmimat.mat.unimi.it

Dusan Repovs
Affiliation: Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P. O. Box 64, Ljubljana 61111, Slovenia
Email: Dusan.repovs@uni-lj.si

DOI: https://doi.org/10.1090/S0002-9947-97-01751-0
Keywords: Four-manifolds, homotopy type, $s$-cobordism, surgery, surface fundamental groups
Received by editor(s): January 20, 1995
Received by editor(s) in revised form: February 6, 1996
Additional Notes: Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and partially supported by the Ministero per la Ricerca Scientifica e Tecnologica of Italy within the projects “Geometria Reale e Complessa” and “Topologia” and by the Ministry for Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-94.
Article copyright: © Copyright 1997 American Mathematical Society

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