## The generalized Poincaré conjecture in higher dimensions.(English)Zbl 0099.39201

This note is a short outline of a forthcoming paper of the author [Ann. Math (2) 74, 391–406 (1961; Zbl 0099.39202)]. The reviewer believes that this paper and the subsequent papers of the author on related topics represent one of the important moments in the history of differentiable topology. Consider a closed $$C^{\infty}$$ manifold $$M$$. A $$C^{\infty}$$ function on $$M$$ will be called “nice” if it possesses only nondegenerate critical points, and for each one $$f(p) = \lambda(p)$$ ($$\lambda$$ being the index). It is proved that on every closed $$C^{\infty}$$ manifold there exist nice functions. It seems very likely that nice functions will replace much of the use of triangulations in differential topology. Let now $$M^n$$ be a compact $$C^{\infty}$$ manifold, $$Q$$ a component of $$\partial M^n$$. Be $$f_i: \partial D^s_i\times D^{n-s}_i \to Q$$ $$(i = 1,\dots, k)$$ ($$D^r = r$$-dimensional disk) imbeddings with joint images. We put:
$\chi(M^n; Q; f_1,\dots, f_x, s)=M^n\cup D_1^n\cup D_2^n\cup \dots \cup D_k^n$
where $$D_i^n = D_i^s\times D_i^{n-s}$$. The author proves the following “handlebody-theorem”.
“Let $$n \geq 2 s + 2$$ and if $$s = 1$$, $$n\geq 5$$, $H = \chi (D^n:\partial D^n; \varphi_1,\dots, \varphi_k, s);~ V = \chi(H; \partial H; f_1,\dots, f_r,s+1),~ \pi_s(V)=0.$ Also if $$s=1$$ it is assumed that $$\pi_1(\chi(H;\partial H; f_1,\dots, f_{r-k},2))=1$$.
Then: $$V =\chi (D^n, \partial D^n, F_1,\dots, F_{r-k},s + 1).$$”
Using the existence of nice functions and the handlebody theorem the following results are proved:
a) Let $$M^n$$ be a closed $$C^{\infty}$$ manifold which is $$(m - 1)$$-connected and $$n \geq (2m, 5)$$. Then there is a nice function $$f$$ an $$M$$; with type numbers satisfying: $$M_0 = M_n = 1$$, $$M_i = 0$$ ($$0 < i < m$$, $$n-m< i < n)$$.
b) If two homotopy-spheres of dimensions $$2m - 1$$ ($$m\neq 2$$) are $$J$$-equivalent, in the sense of Thom, they are diffeomorphic.
c) If $$M^n$$ is a closed differentiable ($$C^{\infty}$$) manifold which is a homotopy sphere, $$n > 5$$, $$M^n$$ is homeomorphic to $$S^n$$. (This is the famous generalized Poincaré conjecture.)
d) There exists a manifold (combinatorial) with no differential structure at all.
e) The groups $$\Gamma^{2m+1}$$ are finite.
These are only author’s first results. In subsequent papers he obtains much stronger theorems.
Reviewer: V . Poenaru

### MSC:

 57R60 Homotopy spheres, Poincaré conjecture 57R55 Differentiable structures in differential topology

Zbl 0099.39202
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### References:

 [1] Barry Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59 – 65. · Zbl 0086.37004 [2] J. Milnor, Differentiable manifolds which are homotopy spheres, (mimeographed) Princeton University, 1959. · Zbl 0106.37001 [3] Stephen Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43 – 49. · Zbl 0100.29701
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