Resolution of smooth group actions
Pierre Albin and Richard Melrose
Abstract. A refined form of the ‘Folk Theorem’ that a smooth action by
a compact Lie group can be (canonically) resolved, by iterated blow up, to
have unique isotropy type is proved in the context of manifolds with corners.
This procedure is shown to capture the simultaneous resolution of all isotropy
types in a ‘resolution structure’ consisting of equivariant iterated fibrations
of the boundary faces. This structure projects to give a similar resolution
structure for the quotient. In particular these results apply to give a canonical
resolution of the radial compactification, to a ball, of any finite dimensional
representation of a compact Lie group; such resolutions of the normal action
of the isotropy groups appear in the boundary fibers in the general case.
Introduction
Borel showed that if the isotropy groups of a smooth action by a compact Lie
group, G, on a compact manifold, M, are all conjugate then the orbit space, G\M,
is smooth. Equivariant objects on M, for such an action, can then be understood
directly as objects on the quotient. In the case of a free action, which is to say
a principal G-bundle, Borel showed that the equivariant cohomology of M is then
naturally isomorphic to the cohomology of G\M. In a companion paper, [1], this is
extended to the unique isotropy case to show that the equivariant cohomology of
M reduces to the cohomology of G\M with coefficients in a flat bundle (the Borel
bundle). In this paper we show how, by resolution, a general smooth compact group
action on a compact manifold is related to an action with unique isotropy type on a
resolution, canonically associated to the given action, of the manifold to a compact
manifold with corners.
The resolution of a smooth Lie group action is discussed by Duistermaat and
Kolk [7] (which we follow quite closely), by Kawakubo [11] and by Wasserman
[13] but goes back at least as far as anich [10], Hsiang [9], and Davis [6]. See
also the discussion by Br¨ uning, Kamber and Richardson [5] which appeared after
the present work was complete. In these approaches there are either residual finite
group actions, particularly reflections, as a consequence of the use of real projective
blow up or else the manifold is repeatedly doubled. Using radial blow up, and hence
working in the category of manifolds with corners, such problems do not arise.
2010 Mathematics Subject Classification. Primary 58D19, 57S15.
The first author was partially supported by an NSF postdoctoral fellowship and NSF grant
DMS-0635607002 and the second author received partial support under NSF grant DMS-1005944.
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Contemporary Mathematics
Volume 535, 2011
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