The main goal of this paper
is a detailed analysis of the problem of imposing a topological bundle structure on a
spherical fibre space over a simply connected base. The method involves a careful
study of the notion of fibre homotopy transversality due to N. Levitt. The point is, a
topological disc bundle satisfies strong transversality properties for maps from
manifolds to the associated Thom space. These properties can be formulated at least
for spherical fibre spaces. Thus, obstructions to transversality can be interpreted
as obstructions to imposing a topological bundle structure on a spherical
fibre space. It turns out that over a simply connected base the obstructions
to transversality coincide exactly with the obstructions to a topologicaI
structure.
The obstructions to transversality for a spherical fibre space ξ can be interpreted
as obstructions to a deformation of the identity map on the Thom space Tξ to a
certain subcomplexes Wξ. The fibre of the map Wξ → Tξ is a space with
a suitable iterated loop space homotopy equivalent to G∕TOP. The total
obstruction to transversality becomes the obstruction to a KO ⊗Z[1∕2]
orientation of the Thom space Tξ, mixed with certain cohomology classes of Tξ,
ℒ∈ H4∗+1(Tξ,Z(2)) and 𝒦∈ H4∗−1(Tξ,Z∕2). These obstructions are then
also interpretable as the obstructions to lifting in the fibration sequence
G∕TOP → BSTOP → BSG.
