From: Linus Kramer Newsgroups: sci.math.research Subject: Re: Vector bundles on S^n Date: Thu, 06 Nov 1997 16:11:36 +0100 [This was a multi-part message in MIME format; streamlined -- djr] Silvirio Marques wrote: > > How many isomorphism classes of vector bundles are on the spheres > S^n ? > > Any references ? > > Thank's > > Goncalo S. Marques -- Linus Kramer Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg Germany E-mail: kramer@mathematik.uni-wuerzburg.de http://www.mathematik.uni-wuerzburg.de/~kramer --------------5C9A037C7205B9C154A4DD0 Difficult question. The k-plane bundles over S^n are classified by homotopy classes of maps from S^n into the classifying space BO(k), which is essentially the same as the nth homotopy group of BO(k), which is the same as the (n-1)st homotopy group of O(k). There is also a more direct way to see this: such a bundle is obtained by glueing together a trivial bundle on the northern hemisphere and a trivial bundle on the southern hemisphere along the equator, which is an S^{n-1}. Steenrod's book on fiber bundles is a good reference and contains explicit calculations in low dimension. The homotopy groups of O(k) beyond a certain dimension are not known; in the stable range k >= n+2 these groups are known explicitly by Bott periodicity. Husemoller's book on fiber bundles is a good reference for that. In the 'Handbook of Algebraic Topology' you can find a list is of the first 15 homotopy groups of the classical groups. Regards, Linus Kramer