Simplifying group actions

The equivariant blow-up construction can simplify the orbit structure of a G-manifold. For abelian G the action can be simplified to an action in which all isotropy subgroups are Z2-vector spaces and the codimension of the set of points having any isotropy subgroup is just the dimension of that subgroup as a Z2-vector space. Such actions are called nonsingular. Nonsingular actions have smooth quotient spaces (with corners). Moreover, the tangent bundle of a nonsingular action of an abelian group G on M can be written as a direct sum of the tangent bundle of the quotient manifold plus a sum of line bundles which are the extensions (to the whole of M) of the normal bundles of the various fixed point sets.