Generalized Poincaré’s conjecture in dimensions greater than four.

*(English)*Zbl 0099.39202This paper contains the proofs of the results stated in the Note reviewed above [Bull. Am. Math. Soc. 66, 373–375 (1960; Zbl 0099.39201)]. But it also contains some new results. For instance it is proved that a compact contractible, \(C^{\infty}\) manifold, of even dimension, with simply connected boundary, is diffeomorphic to a disk (provided the dimension \(\geq 6\)). This implies uniqueness of differential structure and Hauptvermutung for disks in these dimensions. In the same dimensions, a strong form of Hauptvermutung for spheres is proved. Since this paper was written, other very important results an the same line were obtained by the author. Very roughly speaking one can say that by Smale’s work the difference between algebraic and differential topology in high dimension was outdone.

Reviewer: V. Poenaru