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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The automorphism group of a compact group action


Author: W. D. Curtis
Journal: Trans. Amer. Math. Soc. 203 (1975), 45-54
MSC: Primary 58D05
DOI: https://doi.org/10.1090/S0002-9947-1975-0368066-5
MathSciNet review: 0368066
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Abstract: This paper contains results on the structure of the group, $ \operatorname{Diff} _G^r(M)$, of equivariant $ {C^r}$-diffeomorphisms of a free action of the compact Lie group $ G$ on $ M$. $ \operatorname{Diff} _G^r(M)$ is shown to be a locally trivial principal bundle over a submanifold of $ {\operatorname{Diff} ^r}(X),X$ the orbit manifold. The structural group of this bundle is $ {E^r}(G,M)$, the set of equivariant $ {C^r}$-diffeomorphisms which induce the identity on $ X$. $ {E^r}(G,M)$ is shown to be a submanifold of $ {\operatorname{Diff} ^r}(M)$ and in fact a Banach Lie group $ (r < \infty )$.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0368066-5
Keywords: Diffeomorphism group, equivariant diffeomorphism, Banach Lie group, principal bundle, spray, free action
Article copyright: © Copyright 1975 American Mathematical Society