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