Desingularizing Compact Lie Group Actions

This note surveys the well-known structure of G-manifolds and summarizes parts of two papers that have not yet appeared: [4], joint with J. Brüning and F. W. Kamber, and [8], joint with I. Prokhorenkov. In particular, from a given manifold on which a compact Lie group acts smoothly, we construct a sequence of manifolds on which the same Lie group acts, but with fewer levels of singular strata. Global analysis and geometric results on the simpler manifolds may be translated to results on the original manifold. Further, we show that by utilizing bundles over a G-manifold with singular strata, we may construct natural equivariant transverse Dirac-type operators that have properties similar to Dirac operators on closed manifolds. 1. Manifolds and compact Lie group actions Suppose that a compact Lie group G acts smoothly on a smooth, connected, closed manifold M . We assume that the action is effective, meaning that no g ∈ G fixes all of M . (Otherwise, replace G with G {g ∈ G : gx = x for all x ∈M} = ∅.) Choose a Riemannian metric for which G acts by isometries; average the pullbacks of any fixed Riemannian metric over the group of diffeomorphisms to obtain such a metric. Given such an action and x ∈ M , the isotropy subgroup Gx < G is defined to be {g ∈ G : gx = x}. The orbit Ox of a point x is defined to be {gx : g ∈ G}. Note that Ggx = gGxg −1, so the conjugacy class of the isotropy subgroup of a point is fixed along an orbit. The conjugacy class of the isotropy subgroups along an orbit is called the orbit type. On any such G-manifold, there are a finite number of orbit types, and there is a partial order on the set of orbit types. Given subgroups H and K of G that occur as isotropy subgroups, we say that [H] ≤ [K] if H is conjugate to a subgroup of K, and we say [H] < [K] if [H] ≤ [K] and [H] 6= [K]. We may enumerate the conjugacy classes of isotropy subgroups as [G0] , ..., [Gr−1] such that [Gi] ≤ [Gj] if and only if i ≤ j. It is well-known that the union M0 of the principal orbits (those with type [G0]) form an open dense subset M0 of the manifold M , and the other orbits are called singular. As a consequence, every isotropy subgroup H satisfies [G0] ≤ [H]; M0 is called the principal stratum. Let Mj denote the set of points of M of orbit type [Gj] for each j; the set Mj is called the stratum corresponding to [Gj]. A stratum Mj is called a most singular stratum if there does not exist a stratum Mk such that [Gj] < [Gk]. It is known that each stratum is a G-invariant submanifold of M , and in fact a most singular stratum is a closed (but not necessarily connected) submanifold. Also, for each j, the submanifold M≥j := ⋃ [Gk]≥[Gj ] Mk is a closed, G-invariant submanifold. Consider the following simple examples. Example 1.1. G = Z2 × Z2 acts on M = S ⊂ R by (1, 0) (x, y, z) = (x,−y, z) , (0, 1) (x, y, z) = (x, y,−z) . 1