Vol. 30, No. 2, 1969

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Twisted cohomology and enumeration of vector bundles

Lawrence Louis Larmore

Vol. 30 (1969), No. 2, 437–457

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 6h2, 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.

Mathematical Subject Classification
Primary: 57.30
Received: 20 January 1968
Revised: 27 January 1969
Published: 1 August 1969
Lawrence Louis Larmore