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Structure Theory for Montgomery-Samelson Fiberings between Manifolds, II

Published online by Cambridge University Press:  20 November 2018

Peter L. Antonelli
Peter L. Antonelli*
Affiliation:
Syracuse University, Syracuse, New York The University of Tennessee, Knoxville, Tennessee
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Let f: Mn→ Npbe the projection map of an MS-fibering of manifolds β with finite non-empty singular set Aand simply connected total space (see 1). Results of Timourian (10) imply that (n, p) = (4, 3), (8, 5) or (16, 9), while a theorem of Conner (2) yields that #(A), the cardinality of the singular set, is equal to the Euler characteristic of Mn. We give an elementary proof of this fact and, in addition, prove that #(A) is actually determined by bn/2(Mn), the middle betti number of Mn, or what is the same, by bn/2(Np – f(A)). It is then shown that β is topologically the suspension of a (Hopf) sphere bundle when Np is a sphere and bn/2(Mn) = 0. It follows as a corollary that β must also be a suspension when Mn is n/4-connected with vanishing bn/2. Examples where bn/2 is not zero are constructed and we state a couple of conjectures concerning the classification of such objects.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 180 - 186
Copyright
Copyright © Canadian Mathematical Society 1969

References

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