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**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.

\[ \chi(M^n; Q; f_1,\dots, f_x, s)=M^n\cup D_1^n\cup D_2^n\cup \dots \cup D_k^n \]

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 |

### Citations:

Zbl 0099.39202
Full Text:
DOI

### 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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