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 J¨ 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.

1

Contemporary Mathematics

Volume 535, 2011

1

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 J¨ 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.

1

Contemporary Mathematics

Volume 535, 2011

1