The stable geometric dimension of vector bundles over real projective spaces

Davis, Donald M.; Gitler, Sam; Mahowald, Mark

doi:10.1090/S0002-9947-1981-0628445-8

The stable geometric dimension of vector bundles over real projective spaces
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by Donald M. Davis, Sam Gitler and Mark Mahowald

Trans. Amer. Math. Soc. 268 (1981), 39-61

DOI: https://doi.org/10.1090/S0002-9947-1981-0628445-8

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Correction: Trans. Amer. Math. Soc. 280 (1983), 841-843.

Abstract:

An elementary argument shows that the geometric dimension of any vector bundle of order ${2^e}$ over $R{P^n}$ depends only on $e$ and the residue of $n \bmod 8$ for $n$ sufficiently large. In this paper we calculate this geometric dimension, which is approximately $2e$. The nonlifting results are easily obtained using the spectrum $bJ$. The lifting results require $bo$-resolutions. Half of the paper is devoted to proving Mahowald’s theorem that beginning with the second stage $bo$-resolutions act almost like $K({Z_2})$-resolutions.

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