A strong non-immersion theorem for real projective spaces. (English) Zbl 0574.57014

Davis, Donald M.

Ann. Math. (2) 120, 517-528 (1984).

The following main result about non-immersability of real projective spaces improves upon many known ones and is very close to all others: Real projective space \(P^{2\ell}\) does not immerse into \({\mathbb{R}}^{4\ell-4d-2\alpha(\ell-d)}\), where d is the smallest nonnegative integer such that \(\alpha(\ell-d)\leq d+1\). The main ingredient of the proof, using BP-obstruction-theory as in the paper by L. Astey [Q. J. Math., Oxf. II. Ser. 31, 139-155 (1980; Zbl 0405.55020)], is a calculation of \(BP2^*\) of a product of two projective spaces, where \(BP2=BP<2>\) is the Baas-Sullivan version of the Brown- Peterson-spectrum.

Reviewer: M.Raussen

MSC:

57R42	Immersions in differential topology
55S40	Sectioning fiber spaces and bundles in algebraic topology
55P42	Stable homotopy theory, spectra
55N35	Other homology theories in algebraic topology
55N20	Generalized (extraordinary) homology and cohomology theories in algebraic topology

Keywords:

geometric dimension of bundles; Brown-Peterson spectrum; non- immersability of real projective spaces; BP-obstruction-theory

Citations:

Zbl 0429.55007; Zbl 0405.55020