Bordism and Involutions

Abstract

Let X be a topological space with A c X a subspace, and let z-: (X, A) (X, A) be an involution; i.e., a continuous map z-: X X with square the identity and such that MA c A. Combining the notions of bordism (Atiyah [1]) and of differentiable periodic maps (Conner and Floyd [4]), one may define bordism groups of the involution (X, A, z). Specifically, a (free) equivariant bordism class of (X, A, z) is an equivalence class of triples (M, A, f) with M a compact differentiable manifold with boundary, A: M e M a differentiable (fixed-point free) involution on M, and f: (M, AM) (X, A) a continuous equivariant map [zf = fte] sending AM into A. Two triples (M, a, f ) and (M', a', f') are equivalent, or bordant, if there is a 4-tuple (W, V, v, g) such that W and V are compact differentiable manifolds with boundary, a V = AM U AM'and a W = M U M' U V/DM U AM' a V, v: (W, V) (W, V) is a differentiable (fixed-point free) involution restricting to , on M and ,t' on M', and g: (W, V) (X, A) is a continuous equivariant map [zg = gv] restricting to f on M and f ' on M'. The disjoint union of triples induces an operation on the set of (free) equivariant bordism classes of (X, A, z) making this set into an abelian group. This is a graded group, where the grading is given by the dimension of the manifold M, and one lets 9Z,(X, A, z) be the group of equivariant bordism classes of (X, A, z), and one lets 9Z*(X, A, z) be the group of free equivariant bordism classes of (X, A, 4). If A is empty, one writes W*(X, z) and 9M(X, z) for these groups. Letting X be a point, with A empty and z the identity map, this reduces to the situation studied by Conner and Floyd [4], with 9Z*(pt, 1) = I*(Z2) and 9Z*(pt, 1) = 9*(Z2) in their notation. The main purpose of this paper is to compute the groups 9Z*(X, z) and 9Z*(X, z) and to explore their interrelationships. As always, however, the study of an involution (X, z) is really the study of a pair (X, FT, z), with FT the fixed-point set of z, and one is forced to consider pairs in order to study the absolute case. To study more general pairs tends to force a study of 4tuples, and this additional complication will naturally be avoided. The major portion of this paper is a geometric analysis of bordism of in-