We describe here the theory of mock bundles. This is a bundle theory giving rise to a cohomology theory which in the simplest case is pl cobordism. In this interpretation, all the usual products have simple definitions, and the Thorn isomorphism and duality theorems have short transparent proofs. Another feature is that mock bundles can be composed yielding a cohomology transfer. The theory also provides a short proof of the pl transversality theorem [12; 1. 2], In a final section, classifying ∆-sets are constructed. The construction is similar to Quinn's [7; §1].

MOCK BUNDLES AS A COHOMOLOGY THEORY

Let K be a ball complex. A q-mock bundle ξq/K with base K and total space consists of a pl projection pξ: Eξ → ∣K∣ such that, for each is a compact pl manifold of dimension q + dim σ, with boundary We denote by ξ(σ), and call it the block over σ.

The empty set is regarded as a manifold of any dimension; thus ξ(σ) may be empty for some cells σ ∈ K. Therefore, q could be negative, and then ξ(σ) = ø if dim σ< -q. The empty bundle ø/K has the empty set for total space, and is a q-mock bundle for all q ∈ Z.

Figure 2 shows a 1-mock bundle over the union of two 1-simplexes.

Mock bundles ξ, η/K are isomorphic, written ξ ≅ η if there is a pl homeomorphism h: Eξ → Eη which respects blocks; h(ξ(σ)) = η(σ) for each σ ∈ K.