Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-07T19:40:42.101Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

References

1. Browder, W., Higher torsion in H-spaces, Trans. Amer. Math. Soc. 108 (1963), 353–375.Google Scholar
2. Brown, M., Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331–341.Google Scholar
3. Conner, P. E., On the impossibility of fiber ing certain manifolds by compact fibre, Michigan Math. J. 5 (1957), 249–255.Google Scholar
4. Conner, P. E. and Dyer, E., On singular fiberings by spheres, Michigan Math. J. 6 (1959), 303–311.Google Scholar
5. Dugundjie, J., Remark on homotopy inverses, Portugal. Math. 14 (1955), 39–41.Google Scholar
6. Eilenberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton Univ. Press, Princeton, N.J., 1952).Google Scholar
7. Gluck, H., Unknotting s1 in s4, Bull. Amer. Math. Soc. 69 (1963), 91–94 Google Scholar
8. Hu, S., Mappings of anormal space into an absolute neighborhood retract, Trans. Amer. Math. Soc. 64 (1948), 336–358.Google Scholar
9. Hu, S., Homotopy theory (Academic Press, New York, 1959).Google Scholar
10. Hu, S., Theory of retracts (Wayne State Univ. Press, Detroit, 1965).Google Scholar
11. Mahowald, M., On singular fiber ings, Can. J. Math. 15 (1963), 214–219.Google Scholar
12. Montgomery, D. and Samelson, H., Fiberings with singularities, Duke Math. J. 13 (1946), 51–56.Google Scholar
13. Newman, M. H. A., The Engulfing theorem for locally tame sets, Bull. Amer. Math. Soc. 72 (1966), 861–862.Google Scholar
14. Spanier, E., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
15. Stallings, J., On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490–503.Google Scholar
16. Steenrod, N., The topology of fibre bundles (Princeton Univ. Press, Princeton, N.J., 1950).Google Scholar
17. Steenrod, N., Cohomology operations, Ann. Math. Studies, No. 50 (Princeton Univ. Press, Princeton, N.J., 1962).Google Scholar
18. Timourian, J. G., Singular fiberings of manifolds, unpublished Ph.D. thesis, Syracuse University, 1967.Google Scholar
19. Whyburn, H., The mapping of Betti groups under interior transformations, Duke Math. J. 4 (1938), 1–8.Google Scholar