Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-07-07T17:58:54.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds

Published online by Cambridge University Press:  17 April 2009

Victor Brunsden
Victor Brunsden
Affiliation:
Department of Mathematics, Penn State Altoona, 3000 Ivyside Park, Altoona PA 16601, United States of America e-mail: vwb2@psu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 2 , April 1999 , pp. 271 - 295
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Abraham, R., Lectures of Smale on differential topology, (mimeographed lecture notes) (Columbia University, New York, NY, 1961).Google Scholar
[2]Bourbaki, N., Topological vector spaces Chapter 1–5 (Springer-Verlag, Berlin, Heidelberg, New York, 1987).CrossRefGoogle Scholar
[3]Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 1980).Google Scholar
[4]Deimling, K., Non-linear functional analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1985).CrossRefGoogle Scholar
[5]Eells, J., ‘On the geometry of function spaces’, in Symposia de topologia algebraica (Universidad Nacional Autonóma de México, Mexico City, 1958).Google Scholar
[6]Ebin, D.G. and Marsden, J.E., ‘Groups of diffeomorphisms and the motion of incompressible fluid’, Annals of Math. 92 (1970), 102–163.CrossRefGoogle Scholar
[7]Fleming, P., ‘Structural stability and group cohomology’, Trans. Amer. Math. Soc. 275 (1983), 791–809.CrossRefGoogle Scholar
[8]Lang, S., Differential and Riemannian manifolds, 3rd ed. (Springer-Verlag, Berlin, Heidelber, New York, 1995).CrossRefGoogle Scholar
[9]Matsushima, Y. and Murakami, S., ‘On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds’, Ann. of Math. 78 (1963), 365–416.CrossRefGoogle Scholar
[10]Omori, H., ‘On the group of diffeomorphisms on a compact manifold’, Proc. Sympos. Pure Math. 15 (1970), 167–184.CrossRefGoogle Scholar
[11]Palais, R., ‘Equivalence of nearby differentiable actions of a compact group’, Bull. Amer. Math. Soc. 67 (1961), 362–364.CrossRefGoogle Scholar
[12]Robinson, D.J.R., A course in the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[13]Stowe, D., ‘The stability of the stationary set of a group action’, Proc. Amer. Math. Soc. 79 (1980), 139.CrossRefGoogle Scholar
[14]Weil, A., ‘Remarks on the cohomology of groups’, Ann. of Math. 80 (1964), 149.CrossRefGoogle Scholar
[15]Zimmer, R., ‘Actions of semisimple groups and discrete subgroups’, in Proceedings of the International Congress of Mathematicians, (Berkeley CA, 1986) (American Mathematical Society, Providence R.I., 1987), pp. 1247–1258.Google Scholar