Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets

Let G denote a finite group and n = 2 k 6 an even integer. Let X denote a simply connected, compact, oriented, smooth G -manifold of dimension n . Let L denote a union of connected, compact, neat submanifolds in X of dimension k . We invoke the hypothesis that L is a G -subcomplex of a G -equivariant smooth triangulation of X and contains the singular set of the action of G on X . If the dimension of the G -singular set is also k then the ordinary equivariant self-intersection form is not well defined, although the equivariant intersection form is well defined. The first goal of the paper is to eliminate the deficiency above by constructing a new, well defined, equivariant, self-intersection form, called the generalized (or doubly parametrized) equivariant self-intersection form. Its value at a given element agrees with that of the ordinary equivariant self-intersection form when the latter value is well defined. Let denote a finite family of immersions withtrivial normal bundle of k -dimensional, connected, closed, orientable, smooth manifolds into X . Assume that the integral (and mod 2) intersection forms applied to members of and to orientable (and nonorientable) k -dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on × and the generalized equivariant self-intersection form on is a necessary and sufficient condition that is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L . This property, when is a finite family of immersions of the k -dimensional sphere S k into X , is just what is needed for constructing an equivariant surgery theory for G -manifolds X as above whose G -singular set has dimension less than or equal to k . What is new for surgery theory is that the equivariant surgery obstruction is defined for an almost arbitrary singular set of dimension k and in particular, the k -dimensional components of the singular set can be nonorientable.