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Structure Theory for Montgomery-Samelson Fiberings between Manifolds. I

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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In (12), Montgomery and Samelson conjectured that an MS-fibering of polyhedra with total space an n-sphere must have a homology sphere as its singular set. Mahowald (11) has shown that, indeed, an orientable fibering with n ≧ 4 must have a Z2-cohomology sphere as its singular set, while Conner and Dyer (4) have shown this for n arbitrary provided the fiber itself is a Z2-cohomology sphere. We show that if the singular set is tame, then it is a Z-homology sphere if the fiber is also one. This result together with those of Stallings (15), Gluck (7), and Newman and Connell (13) are applied in the case where the singular sets are locally flat and tame. It is shown (Theorem 5.2) that MS-fiberings of spheres on spheres, with closed connected manifold fibers and singular sets, are topologically just suspensions of (Hopf) sphere bundles. In a subsequent publication, the case where the singular sets are finite shall be considered. The reader is invited to consult (3) and (18) in this case.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 170 - 179
Copyright
Copyright © Canadian Mathematical Society 1969

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