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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)].

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 |