Date: Sun, 09 May 1999 18:57:07 +0200 From: "Michele A. Vaccaro" Subject: [Fwd: Message] I, Michele A. Vaccaro of Roma - Italy, with this message annonce the following proof of the Poincaré Conjecture. Title: Some characterizations of the 3-spheres Summary: In this paper we prove that the following five classes of 3-manifolds are coincident: A) - The 3-manifolds V³ for which exists a PL-map of the standard 3-sphere S³ on it with a special part, i.e. an open part of the 3-manifold homeomorphic with its inverse image in S³; B) - The 3-manifolds V³ for which exists a simplicial map of the standard 3-sphere S³ into V³, both suitably triangulated, with a special 3-simplex of V³ having one and only one prototype in s³; C) - The 3-manifolds V³ for which exists a simplicial map of the standard 3-sphere s³ onto V³, both suitably triangulated, in which every 3-simplex of V³ is special; D) - The homotopy 3-spheres, i.e. the 3-manifolds in which every loop can be extended in it to a singular disc; E) - The topological 3-spheres, i.e. the 3-manifolds homeomorphic to the standard 3-sphere S³ . We will prove such coincidences by proving all the inclusions of the sequence: B) < C) < E) < B) < A) < B) < C) < D) < B). It is obvious that the coincidence of the two classes D) and E) is a proof of the Poincaré Conjecture. The coincidence of the two classes A) and D) is a characterization of the homotopy 3-spheres. (It is independent from the inclusion C) < E).) THEOREM B) < C): It is proved by utilizing an operation that is similar to a "bulldozer", starting from the special 3-simplex and reaching every 3-simplex through 2-simplexes. THEOREM C) < E): It is proved by smoothing a particular simplicial map by systematically smoothing the inverse images, called nerves, of dual simplexes, until it becomes a homeomorphism. THEOREMS E) < B), B) < A) and A) < B) are trivially proved. THEOREM C) < D): It is proved by transferring any loop in the 3-manifold in a loop of the 3-sphere and mapping a singular 2-disc bounded by it anew into the 3-manifold. P.S. I will send at request a hard copy (26 pages) to everyone interested, to its postal address. --------------DD09CD2217B249DAAD160889 Content-Type: text/html; charset=us-ascii Content-Transfer-Encoding: 7bit Dear Don Davis;
   can you affix in the List the following message:
 
    I, Michele A. Vaccaro of Roma - Italy, with this message annonce the following proof of the Poincaré Conjecture.
    Title:  Some characterizations of the 3-spheres
    Summary: 
    In this paper we prove that the following five classes of 3-manifolds are coincident:
    A) - The 3-manifolds V³ for which exists a PL-map of the standard 3-sphere S³ on it with
a special part, i.e. an open part of the 3-manifold homeomorphic with its inverse image in S³;
    B) - The 3-manifolds V³ for which exists a simplicial map of the standard 3-sphere S³ into V³, both suitably triangulated, with a special 3-simplex of V³ having one and only one prototype in s³;
    C) - The 3-manifolds V³ for which exists a simplicial map of the standard  3-sphere s³ onto V³, both suitably triangulated, in which every  3-simplex of V³  is special;
    D) - The homotopy 3-spheres, i.e. the 3-manifolds in which every loop can be extended in it to a singular disc;
    E) - The topological 3-spheres, i.e. the 3-manifolds homeomorphic to the standard  3-sphere S³ .
    We will prove such coincidences by proving all the inclusions of the sequence:
        B) < C) < E) < B) < A) < B) < C) < D) < B).
    It is obvious that the coincidence of the two classes D) and E) is a proof of the Poincaré Conjecture.
    The coincidence of the two classes A) and D) is a characterization of the homotopy 3-spheres. (It is independent from the inclusion C) < E).)
    THEOREM B) < C): It is proved by utilizing an operation that is similar to a "bulldozer", starting from the special 3-simplex and reaching every 3-simplex through 2-simplexes.
    THEOREM C) < E):  It is proved by smoothing a particular simplicial map by systematically smoothing the inverse images, called nerves, of dual simplexes, until it becomes a homeomorphism.
    THEOREMS E) < B), B) < A) and A) < B) are trivially proved.
    THEOREM C) < D): It is proved by transferring any loop in the 3-manifold in a loop of the 3-sphere and mapping a singular 2-disc bounded by it anew into the 3-manifold.

    P.S. I will send at request a hard copy (26 pages) to everyone interested, to its postal address. --------------DD09CD2217B249DAAD160889-- --------------B5FF5B67B0A176FB28D042DA--