Abstract
The purpose of this note is to prove that under certain conditions, if a compact Lie group G acts on a space M, then there will exist uncountably many topologically distinct actions of G on M. Montgomery and Yang have recently shown in [3] that SI can act on S7 in countably many topologically distinct ways. Their methods are different from the ones employed in this paper. In both cases, the actions are distinguished by the orbit spaces. However, Montgomery and Yang distinguish the orbit spaces, which are in fact differentiable manifolds, by global properties. In this note, the orbit spaces are distinguished by local properties, and hence they are not manifolds. The notation and terminology are standard. If G acts on M, and x is an element of M, then G. will denote the isotropy subgroup, G/G. the quotient subgroup, G(x) the orbit of x, M/G the orbit space, and PG the projection map from M to M/G. If A is an arc in the interior of a k-cell D, then D/A will denote the space obtained by identifying A to a point.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
