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In the present paper we give a
technique for completely enumerating real 4-plane bundles over a 4-dimensional
space, real 5-plane bundles over a 5-dimensional space, and real 6-plane bundles over
a 6-dimensional space. We give a complete table of real and complex vector
bundles over real projective space Pk, for k ≦ 5. Some interesting results
are:
(0.1.1.) Over P5, there are four oriented 4-plane bundles which could be the
normal bundle to an immersion of P5 in R9, i.e., have stable class 2h + 2,
where h is the canonical line bundle. Of these, two have a unique complex
structure.
(0.1.2.) Over P5 there is an oriented 4-plane bundle which we call C, which has
stable class 6h− 2, which has two distinct complex structures. D, the conjugate of C,
i.e., reversed orientation, has no complex structure.
(0.1.3) Over P5, there are no 4-plane bundles of stable class 5h − 1 or
7h − 3.
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