## 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

### MathOverflow Questions:

Is Cohen immersion conjecture (theorem) known for vector bundles?

### 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

### Citations:

Zbl 0415.55011; Zbl 0574.57015; Zbl 0507.57015
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