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Triangulation of Fibre Bundles

Published online by Cambridge University Press:  20 November 2018

H. Putz
H. Putz*
Affiliation:
Temple University, Philadelphia
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In this paper we consider the following problem. Let (E, M, N, π) be a differentiable fibre bundle, where E is the total space, M the base space, N the fibre, and π: E → M the projection map. Then, given a Cr triangulation (ƒ, D) of M, can one obtain a Cr triangulation (F, K) of E such that the induced map ƒ–1πF: K → D is linear? R. Lashof and M. Rothenberg (3) have obtained this result for vector bundles.

Using methods quite different from theirs, we obtain a solution in the general case. The methods we use are the geometric methods developed by J. H. C. Whitehead. (7) in his triangulation of differentiable manifolds, as found in (5). In fact, our solution consists of generalizing his techniques in a fibre bundle setting.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 499 - 513
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Alexandroff, P. and Hopf, H., Topologie (Springer, 1935).Google Scholar
2. Eilenberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton, 1952).Google Scholar
3. Lashof, R. and Rothenberg, M., Microbundles and smoothing, Topology, 3, 4 (1965), 357–388.Google Scholar
4. Milnor, J., Differential topology (mimeographed notes, Princeton, 1958).Google Scholar
5. Munkres, J. R., Elementary differential topology (Princeton, 1963).Google Scholar
6. Steenrod, N., The topology of fibre bundles (Princeton, 1951).Google Scholar
7. Whitehead, J. H. C., On C 1-complexes, Ann. of Math., 41 (1940), 809–824.Google Scholar