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Jordan's theorem for the diffeomorphism group of some manifolds


Author: Ignasi Mundet i Riera
Journal: Proc. Amer. Math. Soc. 138 (2010), 2253-2262
MSC (2010): Primary 57R50, 57S17
DOI: https://doi.org/10.1090/S0002-9939-10-10221-4
Published electronically: February 12, 2010
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Abstract: Let $ M$ be a compact connected $ n$-dimensional smooth manifold admitting an unramified covering $ \widetilde{M}\to M$ with cohomology classes $ \alpha_1,\dots,\alpha_n \in H^1(\widetilde{M};\mathbb{Z})$ satisfying $ \alpha_1\cup\dots\cup\alpha_n\neq 0$. We prove that there exists some number $ c$ such that: (1) any finite group of diffeomorphisms of $ M$ contains an abelian subgroup of index at most $ c$; (2) if $ \chi(M)\neq 0$, then any finite group of diffeomorphisms of $ M$ has at most $ c$ elements. We also give a new and short proof of Jordan's classical theorem for finite subgroups of $ \mathrm{GL}(n,\mathbb{C})$, of which our result is an analogue for $ \mathrm{Diff}(M)$.


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Additional Information

Ignasi Mundet i Riera
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: ignasi.mundet@ub.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10221-4
Received by editor(s): March 4, 2009
Received by editor(s) in revised form: October 12, 2009
Published electronically: February 12, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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