The diffeomorphism type of certain 𝑆³-bundles over 𝑆⁴

Sanchez, Marc; Wilhelm, Frederick

doi:10.1090/S0002-9939-01-06380-8

The diffeomorphism type of certain $S^{3}$-bundles over $S^{4}$
HTML articles powered by AMS MathViewer

by Marc Sanchez and Frederick Wilhelm PDF

Proc. Amer. Math. Soc. 130 (2002), 1139-1143 Request permission

Abstract:

In this note we show that the unit tangent bundle of $S^{4}$ is diffeomorphic to the total space of a certain principal $S^{3}$-bundle over $S^{4}$, solving a problem of James and Whitehead.

References

D. Crowley and C. Escher, A classification of $S^{3}$–bundles over $S^{4},$ preprint.
Jean Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma _{4}=0)$, Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin-New York, 1968 (French). MR 0229250, DOI 10.1007/BFb0060395
Herman Gluck, Frank Warner, and Wolfgang Ziller, The geometry of the Hopf fibrations, Enseign. Math. (2) 32 (1986), no. 3-4, 173–198. MR 874686
Detlef Gromoll and Wolfgang Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. (2) 100 (1974), 401–406. MR 375151, DOI 10.2307/1971078
K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Annals of Math. 152 (2000) 331-367.
Allen E. Hatcher, A proof of the Smale conjecture, $\textrm {Diff}(S^{3})\simeq \textrm {O}(4)$, Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR 701256, DOI 10.2307/2007035
Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482, DOI 10.1007/978-1-4757-2261-1
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Peter Petersen and Frederick Wilhelm, Examples of Riemannian manifolds with positive curvature almost everywhere, Geom. Topol. 3 (1999), 331–367. MR 1714915, DOI 10.2140/gt.1999.3.331
A. Rigas, Some bundles of non-negative curvature, Math. Ann. 232 (1978), no. 2, 187–193. MR 464254, DOI 10.1007/BF01421406
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
F. Wilhelm, Exotic spheres with lots of positive curvatures, J. Geom. Anal. 11 (2001) 161-186.
F. Wilhelm, An exotic sphere with positive curvature almost everywhere, J. Geom. Anal., to appear.