Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T20:27:59.560Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Antonelli, P. L., Structure theory for Montgomery-Samelson fiberings between manifolds. I, Can. J. Math. 21 (1969), 170–179 Google Scholar
2. Conner, P. E., On the impossibility of fibering certain manifolds by compact fibre, Michigan Math. J. 5 (1957), 249–255.Google Scholar
3. Gluck, H., Unknotting s1 in s4, Bull. Amer. Math. Soc. 69 (1963), 91–94 Google Scholar
4. Hu, S., Mappings of a normal space into an absolute neighborhood retract. Trans. Amer. Math. Soc. 64 (1948), 336–358 Google Scholar
5. Hu, S., Homotopy theory (Academic Press, New York, 1959).Google Scholar
6. Montgomery, D. and Samelson, H., Fiberings with singularities, Duke Math. J. 13 (1946), 51–56.Google Scholar
7. Newman, M. H. A., The Engulfing theorem for locally tame sets, Bull. Amer. Math. Soc. 72 (1966), 861–862.Google Scholar
8. Olum, P., Non-Abelian cohomology and Van Kampen's theorem, Ann. of Math. (2) 68 (1958), 658–668.Google Scholar
9. Stallings, J., On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490–503.Google Scholar
10. Timourian, J. G., Singular fiberings of manifolds, unpublished Ph.D. thesis, Syracuse University, 1967.Google Scholar