E. Thomas [19] introduced the
notion of span of a differentiable manifold (or of a vector bundle). The notion of
span can be extended in an obvious way to PL-microbundles, topological
microbundles and spherical fibrations. In the case of a vector bundle or a
microbundle the dimension of the fibre will be referred to as its rank. A
spherical fibration with fibre homotopically equivalent to Sk−1 will be said to be
of rank k. In this paper we study stably trivial objects of rank k over a
CW-complex of dimension ≦ k from each of the above collections. Then we
determine the span of such stably trivial objects over CW-complexes of
a “special type” yielding generalizations of the Bredon-Kosinski, Thomas
theorem on the span of a closed differentiable π-manifold [3], [19]. Though
originally PL-microbundles were defined only over simplicial complexes, in
this paper by a PL-microbundle of rank k over a CW-complex X we mean
an element of the set [X,BPL(k)] of homotopy classes of maps of X into
BPL(k).
