Mathematische Annalen

, Volume 322, Issue 2, pp 413–420 | Cite as

Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness

  • Wilderich Tuschmann
Original article

Abstract.

The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds:

1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension \(m<7\) which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension \(m\ge 7\) there exist counterexamples to the preceding statement.

2. For any given numbers C and D and any dimension m there exist for each natural number \(k\ge 2\) up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature \(\vert K \vert\le C\) and diameter \(\le D\).

Mathematics Subject Classification (1991):53C20, 53C21, 53C23, 57N99, 57R57 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Wilderich Tuschmann
    • 1
  1. 1.Max-Planck-Institute für Mathematik in den Naturwissenschaften, Inselstr. 22–26, 04103 Leipzig, Germany (e-mail: tusch@mis.mpg.de) DE

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