Vol. 60, No. 2, 1975

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ISSN: 0030-8730
Span and stably trivial bundles

Kalathoor Varadarajan

Vol. 60 (1975), No. 2, 277–287

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 Sk1 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).

Mathematical Subject Classification
Primary: 57D25, 57D25
Secondary: 55F25, 57A55, 57C50
Received: 21 August 1973
Published: 1 October 1975
Kalathoor Varadarajan