Inventiones mathematicae

, Volume 122, Issue 1, pp 509–529 | Cite as

4-Manifold topology I: Subexponential groups

  • Michael H. Freedman
  • Peter Teichner
Article

Abstract

The technical lemma underlying the 5-dimensional topologicals-cobordism conjecture and the 4-dimensional topological surgery conjecture is a purely smooth category statement about locating π1-null immersions of disks. These conjectures are theorems precisely for those fundamental groups (“good groups”) where the π1-null disk lemma (NDL) holds. We expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations: (1) extension and (2) direct limit. The finitely generated groups in this class are amenable and no amenable group is known to lie outside this class.

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References

  1. [C] A. Casson: Three lectures on new infinite constructions in 4-dimensional manifolds. Notes prepared by L. Guillou. Prepublications Orsay 81 T 06 (1974)Google Scholar
  2. [D] S.K. Donaldson: An application of Gauge theory to the topology of 4-manifolds. J. Diff. Geom.18, 279–315 (1983)Google Scholar
  3. [F1] M.H. Freedman: The topology of 4-dimensional manifolds. J. Diff. Geom.17, 357–453 (1982)Google Scholar
  4. [F2] M.H. Freedman: The disk theorem for four-dimensional manifolds. Proc. ICM Warsaw, 647–663 (1983)Google Scholar
  5. [FQ] M.H. Freedman, F. Quinn: The topology of 4-manifolds. Princeton Math. Series 39, Princeton, NJ (1990)Google Scholar
  6. [G] R.I. Grigorchuk: Degrees of growth of finitely generated groups, and the theory of invariant means. Math. USSR Izvestiya25, No. 2 259–300 (1985)Google Scholar
  7. [L] X.S. Lin: On equivalence relations of links in 3-manifolds. Preprint (1985)Google Scholar
  8. [M1] J. Milnor: Link Groups. Ann. Math.59, 177–195 (1954)Google Scholar
  9. [M2] J. Milnor: A note on curvature and fundamental group. J. Diff. Geom.2, 1–7 (1968)Google Scholar
  10. [Sto] R. Stong: Four-manifold topology and groups of polynomial growth. Pac. J. Math.157, 145–150 (1993)Google Scholar
  11. [St] J. Stallings: Homology and central series of groups. J. Algebra2, 1970–1981 (1965)Google Scholar
  12. [V] M. Vaughan-Lee: The restricted Burnside problem. London Math. Soc. Monographs New Series 8, Clarendon Press, 1993Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Michael H. Freedman
    • 1
  • Peter Teichner
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaSan Diego, La JollaUSA
  2. 2.Fachbereich MathematikUniversität MainzMainzGermany

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