In §1 of this chapter we describe a further extension of mock bundles, to the equivariant case – the theory dual to equivariant bordism – and in §2 give a general construction which includes power operations and characteristic classes. The remainder of the paper is concerned with the case of Z2-operations on pl cobordism. In §3 we expound the ‘expanded squares’ (‘expanded’ rather than the familiar ‘reduced’ because of our indexing convention for cohomology) and in §4 we give the relation with torn Dieck's operations [5], §5 describes the characteristic classes associated to Z2-block bundles and in §6 we give a result inspired by Quillen [3] which relates the total square of a mock bundle with the transfer of the euler class of its twisted normal bundle. This leads, in some cases, to the familiar connection between characteristic classes and squares. Finally in §7 we give an alternative definition of squares, based on trans versatility. This is like the ‘internal’ definition of the cup product (see II end of §4).

EQUIVARIANT MOCK BUNDLES

Let G be a finite group and X a polyhedron. By a G-action on X we mean a (pl) map G × X → X satisfying

1. (i) for all g1 g2 ∈ G and x ∈ X, g1(g2x) = (g1g2)x.

2. (ii) if e ∈ G is the identity then ex = x for all x ∈ X.

3. If X, Y are G polyhedra then a map f: X → Y is a G-map if f commutes with the G-action.