Equivariant Intersection Theory and Surgery Theory for Manifolds with Middle Dimensional Singular Sets
Bak, Anthony
Bak
Anthony
Morimoto, Masaharu
Morimoto
Masaharu
Let G denote a finite group and n = 2k >= 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 oil 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 file latter value is well defined. Let F denote a finite family of immersions with trivial normal bundle of k-dimensional, connected, closed, orientahle, smooth manifolds into X. Assume that the integral (and mod 2) intersection forms applied to members of F and to orientable (and nonorientable) k-dimensional members of L are trivial. Then the vanishing of the equivariant intersection form on F x F and the generalized equivariant self-intersection form oil F is a necessary and sufficient condition that F is regularly homotopic to a family of disjoint embeddings, each of which is disjoint from L. This property, when F is a finite family of immersions of the k-dimensional sphere S-k into X, is just what is needed for constructing all 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.
2
3
507-600
507-600
Cambridge University Press (CUP)
2008