In this paper we develop an
obstruction theory for the problem of determining whether a bundle, E, over a
compact polyhedron, B, with non-compact Hilbert cube manifold fibers admits a
boundary in the sense that there exists a compact bundle Ē over B with Q-manifold
fibers and a sliced Z-set, A ⊂Ē, such that Ē= A ∪ E. Included in the work
is a new result on fibered weak proper homotopy equivalences, a theorem
on proper liftings of homotopies, and the development of a sliced shape
theory whose equivalences are shown to classify our boundaries through a
tie to Q-manifold theory via a sliced version of Chapman’s Complement
Theorem.
