Vol. 67, No. 1, 1976

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Homotopy theoretic consequences of N. Levitt’s obstruction theory to transversality for spherical fibrations

Gregory Wayne Brumfiel and John W. Morgan

Vol. 67 (1976), No. 1, 1–100

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 to a certain subcomplexes . The fibre of the map 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[12] orientation of the Thom space , mixed with certain cohomology classes of , H4+1 (Tξ,Z(2)) and 𝒦H4∗−1(Tξ,Z2). These obstructions are then also interpretable as the obstructions to lifting in the fibration sequence G∕TOP BSTOP BSG.

Mathematical Subject Classification
Primary: 57B10, 57B10
Secondary: 55G35, 57D65
Received: 30 August 1974
Published: 1 November 1976
Gregory Wayne Brumfiel
John W. Morgan
Department of Mathematics
Columbia University
2990 Broadway
New York NY 10027-0029
United States