Vector bundles of low geometric dimension over real projective spaces

Abstract

We construct stably nontrivial vector bundles of dimension 5 and also stably nontrivial vector bundles of dimension 6 on real projective spaces of arbitrarily high dimensions. Consequently, we answer a problem posed by J. F. Adams.

Projective product spaces

“…r r r r r r P P P P P P P P P P P P P P P q P P P P P P P P P P P P P P P q 15 suggests the possibility that the S ni with n i odd might be product factors of P n . The following result shows the limited extent to which this is true.…”

“…An immediate corollary of (3.5) is the following. Geometric dimension of multiples of the Hopf bundle over real projective spaces, sometimes called the generalized vector field problem, has been studied in many papers such as [2,6,7,14,15]. One consequence of Theorem 3.4 is that every case of the generalized vector field problem is solving an immersion question for some manifold.…”

“…The second type of geometric dimension result on which we focus is vector bundles of low geometric dimension. These were first studied by Adams in [1], and Lam and Randall provide the current status in [14], some of which is described in the following theorem. This has the following immediate consequence for imm(P n ).…”

“…([1],[14]) Assume n ≥ 18.• If 0 ≤ d ≤ 4, then gd(kξ n ) = d if and only if k ≡ d mod 2 φ(n) . • If ν(k) = φ(n) − 1 and n ≡ 7 mod 8 or if ν(k) = φ(n) −2 and n ≡ 2 or 4 mod 8, then gd(kξ n ) = 5.…”