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The immersion conjecture for differentiable manifolds. (English) Zbl 0592.57022

This paper contains the final proof of the celebrated immersion conjecture for differential manifolds; to wit: If \(M^ n\) is a compact, \(C^{\infty}\), n-dimensional manifold, \(n>1\), then there exists a differentiable immersion of \(M^ n\) into \({\mathbb{R}}^{2n-\alpha (n)}\), where \(\alpha\) (n) is the number of ones in the dyadic expansion of n.
A scheme for proving this theorem had been developed and partially carried out by E. H. Brown and F. P. Peterson. They constructed a factoring of the stable normal maps \(M^ n\to BO\) through a space \(BO/I_ n\) reflecting all relations between Stiefel-Whitney classes of manifolds. This paper establishes a factoring of the map \(\rho\) : BO/I\({}_ n\to BO\) through BO(n-\(\alpha\) (n)) and hence, by Hirsch’s theorem, a proof of the conjecture. E. H. Brown jun. and F. P. Peterson [Comment. Math. Helv. 54, 405-430 (1979; Zbl 0415.55011)] had already found a lifting \(MO/I_ n\to MO(n-\alpha (n))\) on the Thom space level which is shown to ”de-Thomify” in the paper under review.
Although well-organized and carefully written, this paper is heavy reading. For a survey of the key ideas and the history of the proof of the immersion conjecture, one may consult the paper by the author [Proc. Int. Congr. Math., Warszawa 1983; Vol. 1, 627-639 (1984; Zbl 0574.57015)] or by J. Lannes [Sémin. Bourbaki, 34e année, Vol. 1981/82, Exp. No.594, Astérisque 92/93, 331-346 (1982; Zbl 0507.57015)].
Reviewer: M.Raußen

MSC:

57R42 Immersions in differential topology
55P42 Stable homotopy theory, spectra
55S40 Sectioning fiber spaces and bundles in algebraic topology
55S10 Steenrod algebra
55S45 Postnikov systems, \(k\)-invariants
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