﻿New Approach to the Generalized Poincare Conjecture

Applied Mathematics
Vol. 4  No. 9 (2013) , Article ID: 37220 , 5 pages DOI:10.4236/am.2013.49183

New Approach to the Generalized Poincare Conjecture

Alexander A. Ermolitski

IIT-BSUIR, Minsk, Belarus

Email: ermolitski@mail.by

Copyright © 2013 Alexander A. Ermolitski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received April 18, 2013; revised May 18, 2013; accepted May 28, 2013

Keywords: Compact Smooth Manifolds; Riemannian Metric; Smooth Triangulation; Homotopy-Equivalence; Algorithms

ABSTRACT

Using our proof of the Poincare conjecture in dimension three and the method of mathematical induction a short and transparent proof of the generalized Poincare conjecture (the main theorem below) has been obtained. Main Theorem. Let Mn be a n-dimensional, connected, simply connected, compact, closed, smooth manifold and there exists a smooth finite triangulation on Mn which is coordinated with the smoothness structure of Mn. If Sn is the n-dimensional sphere then the manifolds Mn and Sn are homemorphic.

1. Introduction

We can fix some Riemannian metric g on a manifold Mn of dimension n which defines the length of arc of a piecewise smooth curve and the continuous function of the distance between two points. The topology defined by the function of distance (metric) is the same as the topology of the manifold Mn [1].

In Section 1, using a smooth triangulation considered in the main theorem and a Riemannian metric we construct an algorithm of extension of coordinate neighborhood. With the help of this algorithm we get that every compact, connected, closed manifold Mn of dimension n having the triangulation above can be represented as a union of a n-dimensional cell Cn and a connected union of some finite number of simplexes of the triangulation having dimension less or equal. A sufficiently small closed neighborhood of is called a geometric black hole [2]. Simplexes with boundaries can be retracted i.e. a decomposition can be obtained where contains less simplexes than does.

In Section 2, we consider the proof of the main theorem consisting of the realization of several algorithms. Using the method of mathematical induction and the algorithms we retract all the simplexes from to a point x0, therefore a decomposition is obtained and Mn is homeomorphic to the sphere.

2. On Algorithm of Extension of Coordinate Neighborhood

1) Let Mn be a connected, compact, closed and smooth manifold of dimension n and Cn be a cell (coordinate neighborhood) on Mn. A standard simplex ∆n of dimension n is the set of points defined by conditions

We consider the interval of a straight line connected the center of some face of ∆n and the vertex which is opposite to this face. It is clear that the center of ∆n belongs to the interval. We can decompose ∆n as a set of intervals which are parallel to that mentioned above. If the center of ∆n is connected by intervals with points of some face of ∆n then a subsimplex of ∆n is obtained. All the faces of ∆n considered, ∆n is seen as a set of all such subsimplexes. Let be some open neighborhood of ∆n in Rn. A diffeomorphism is called a singular nsimplex on the manifold Mn. Faces, edges, the center, vertexes of the simplex are defined as the images of those of ∆n with respect to.

The manifold Mn is triangulable [3]. It means that for any such a finite set of diffeomorphisms is defined that

a) Mn is a disjunct union of images;

b) if then for every i where is the linear mapping transferring the vertexes of the simplex in the vertexes of the simplex.

We suppose that there exists a smooth finite triangulation on Mn which is coordinated with the smoothness structure of Mn and fix the triangulation. Such triangulations exist for manifolds of dimension 2 or 3.

2) Let be some simplex of the fixed triangulation of the manifold Mn. We paint the inner part of the simplex white and the boundary of black. There exist coordinates on given by diffeomorphism. A subsimplex is defined by a black face and the center of. We connect with the center of the face and decompose the subsimplex as a set of intervals which are parallel to the interval. The face is a face of some simplex that has not been painted. We draw an interval between and the vertex of the subsimplex which is opposite to the face then we decompose as a set of intervals which are parallel to the interval. The set is a union of such broken lines every one from which consists of two intervals where the endpoint of the first interval coincides with the beginning of the second interval (in the face) the first interval belongs to and the second interval belongs to. We construct a homeomorphism (extension):. Let us consider a point and let x belong to a broken line consisting of two intervals the first interval is of a length of and the second interval is of a length of and let x be at a distance of s from the beginning of the first interval. Then we suppose that belongs to the same broken line at a distance of from the beginning of the first interval. It is clear that is a homeomorphism giving coordinates on. We paint points of white. Assuming the coordinates of points of white initial faces of subsimplex to be fixed we obtain correctly introduced coordinates on. The set is called a canonical polyhedron. We paint faces of the boundary black.

We describe the contents of the successive step of the algorithm of extension of coordinate neighborhood. Let us have a canonical polyhedron with white inner points (they have introduced white coordinates) and the black boundary. We look for such an n-simplex in, let it be that has such a black face, let it be that is simultaneously a face of some n-simplex, let it be, inner points of which are not painted. Then we apply the procedure described above to the pair,. As a result we have a polyhedron with one simplex more than has. Points of are painted white and the boundary is painted black. The process is finished in the case when all the black faces of the last polyhedron border on the set of white points (the cell) from two sides.

After that all the points of the manifold are painted in black or white, otherwise we would have that (the points of would be painted and those of would be not) with and being unconnected, which would contradict of connectivity of.

Thus, we have proved the following.

Theorem 1. Let be a connected, compact, closed, smooth manifold of dimension n. Then, , where is an п-dimensional cell and is a union of some finite number of -simplexes of the triangulation.

3) We consider the initial simplex of the triangulation and its center. Drawing intervals between the point and points of all the faces of we obtain a decomposition of as a set of the intervals. In 2) the homeomorphism: was constructed and evidently maps every interval above on a piecewise smooth broken line in Cn. We denote. is a connected and simply connected manifold if is that. Let, we define a homotopy in the following way

a) for every point;

b) if a point x belongs to the broken line in and the distance between x and its limit point is then is on the same broken line at a distance of from the point z.

It is clear that, and we have obtained the following.

Theorem 2. The spaces and are homotopyequivalent, in particular, the groups of singular homologies and are isomorphic for every k.

Corollary 2.1. The space is connected and if Mn is simply connected then is simply connected too.

Remark 1. The white coordinates are extended from the simplex in the simplex through the face hence has also the white coordinates. On the other hand there exist two linear structures (intervals, the center etc.) on induced from and respectively. Further, we set that the linear structure of is the structure induced from.

Remark 2. In the process of getting of in 2) we can construct a maximal tree L connecting by intervals all the centers of the n-simplexes of the triangulation via the centers of some white faces.

Conversely, if we have such a maximal tree L connecting by intervals all the centers of the n-simplexes of the triangulation via the centers of some faces (any from two possible centers of a face can be chosen) then we can extend white coordinates from any simplex on the maximal cell Cn as it was shown in 2). Thus, the maximal tree L defines the maximal cell C3 and white faces.

4) Definition 1.

a) A simplex is called free if it has at least one free face i.e. such a face that it is not a face of any other k-simplex from.

b) An edge is called semi-isolated if it is not an edge of any simplex from. A semi-isolated edge is called isolated if it is free.

Let us have a free simplex with some free face. We consider such a polyhedron that is the set of all n-simplexes having common point with.

Proposition 3. We can redistribute coordinates of white points of the polyhedron (retract the free simplex) i.e. construct the corresponding mapping in such a way that the following conditions are fulfilled:

a) all the points of are painted white i.e. have new white coordinatesb) white coordinates of points of boundary faces of the polyhedron are not changed.

Proof. a) We consider the unit disk D2 having the center in the origin of the coordinate system of and the radius.

We define a mapping by the following way:

• for any chord which is parallel to ,

It is clear that maps onto and on the boundary circle of.

b) We consider the unit disk having the center in the origin of the coordinate system and the semidisk, ,. By inductive hypothesis we assume that such a mapping has been constructed that maps onto and on the boundary of.

Further, we consider the unit disk in the coordinate system, the semidisk, and the family of disks, ,. We denote By inductive hypothesis there exists such the family of mappings that every maps onto and on the boundary of. Union of all gives the mapping, maps onto and on the boundary of.

Thus, the mapping is constructed for any by the method of mathematical induction.

c) It is clear that there exists such a homeomorphism that and. We define the mapping then is a required homeomorphism introducing new white coordinates in.

QED.

Remark 3. In is clear that the rebuilt complex is connected and simply connected because of a homotopy-equivalence.

5) We assume that in the process of painting free simplexes white by the Proposition 3 we get a representation, where K1 is the connected union of black edges of the triangulation. Since the process of painting free simplexes white does not influence simply connectivity of a space that has been obtained every step then K1 is a tree if the complex is simply connected. Painting isolated edges of K1 white by the Proposition 3 we have got unique black point x0 as result. Thus, we obtain a representation, where is an open geodesic ball with the center in x0 and of a radius. The manifold Mn is homeomorfic to the sphere Sn by the following lemma 4.

Lemma 4 [1]. If a topological manifold Mn is a union of two n-dimensional cells then Mn is homeomorfic to the sphere Sn.

3. Proof of the Main Theorem

The proof has a combinatorial nature and assumes the realization of a number of algorithms. We consider that step by step. The initial complex is assumed to be connected, simply connected and without free simplexes.

1) Proposition 5 (opening an input). Let be some n–simplex of the triangulation having a black face. Then can be repainted white to get a new decomposition, where is a new connected and simply connected complex.

Proof. The face is the common face of n-simplexes and. We cansel the white painting of points of and paint the n-simplexe black. Repainting of black brings to a gap of the maximal tree L (see the Remark 2) on n subtrees or less where the center of belongs to. Further, we extend white coordinates from into through the face as it was shown in 2), 1 and connect the centers of, , by intervals. Those centers belong to the subtree. Other faces of are black and they are simultaneously some faces of other n-simplexes.

We consider the following cases.

a) or we have no a gap. The black faces of remain black.

b) We have got k subtrees where the subtrees define cells called dead ends. We repaint the closures of the dead ends black. Further, we are looking for a black face of which is simultaneously a face of other n-simplex with the center from. This face remains black. For every subtree we consider a n-simplex with the center from that has a common black face with. We extend white coordinates from through along the subtree as it was shown in 2), 1 and repaint inner points of this face and points of the corresponding dead end white. Further, we connect by intervals the centers of with the centers of and the other simplex connecting and.

After repainting all the dead ends white we obtain a new maximal tree L defining a new maximal cell. Retracting all the free simplexes by the Proposition 3 a new rebuilt complex is obtained which is connected and simply connected because of homotopyequivalence.

QED.

Remark 4. A broken line has been obtained in the proof above which connects by intervals the centers of, ,. This broken line is a part of the subtree of the maximal tree L. Let n-simplexes and have a common face having the white inter part and has no common points with the maximal tree. Then we can connect the centers of, , by the broken line by the method considered in the proof above.

2) We assume the following inductive hypothesis:

The generalized Poincare conjecture (the main theorem) can be proved by the method considered in [4] for dimension n–1 i.e. the representation can be obtained by the algorithm from 2), 1 and by the Propositions 3, 4, 5.

It is obvious for (see 5), 1) It is proved for in [4].

We choose a small ball with the center in a vertex x0 which is diffeomorphic to a small ball in and call a trace of k-simplex with a vertex in x0 its intersection with the sphere (smooth manifold) which is the boundary of. The sphere is supposed to be transversal to all the k-simplexes with the vertex x0. Such a sphere exists because of the smoothness of the triangulation of Mn [5,6]. All other vertexes of the triangulation are supposed to be out of. The ball can be chosen in such a vay that every edge with the endpoint x0 has only one point of the intersection with and every k-simplex with the vertex x0 has only one connected component of. Let be the set of black k-simplexes with x0 as their vertex and.

There exists one to one correspondence between the set of simplexes having a vertex (endpoint) x0 and the set of their traces on therefore all steps of the algorithm below bring to the corresponding steps on the sphere and the converse is true. In particular, a process of the construction of a maximal tree on the sphere (see the Remark 2) brings to the construction of a tree L1 connecting by intervals all the centers of the n-simplexes with x0 as their vertex via the centers of some white their faces. Every such the face has x0 as its vertex.

Proposition 6. The complex can be rebuilt in such a vay that contains only one 1-simplex.

Proof. We consider the smooth triangulation of induced by all the simplexes with the vertex and apply to this triangulation the algorithm from 2), 1 taking any -simplex as initial one where is the trace of with a vertex. Let be the trace on of with a vertex x0 where has a common face with. We repaint black and apply to it the proposition 5 (the remark 4) obtaining the canonical polyhedron on. Further, we iterate the algorithm. Every step of the algorithm on implies the transformation of and by the proposition 5 (the remark 4). The maximal tree on and the corresponding subtree have been constructed in the end. Further, free black simplexes on and the corresponding free simplexes from can be annihilated by the propositions 3, 4, 5. By the inductive hypothesis only one black point remains on in the end. This point is the trace of an edge which is isolated.

QED.

Remark 5. It is clear that if we paint black one inner vertex in the canonical polyhedron then we get two black points on in the end of the algorithm.

3) We consider a small ball with the center and the boundary which is similar to. The centers of all the n-simplexes having as their edge belong to the subtree L1 and the union of all the traces of this n-simplexes on forms the canonical polyhedron on having one black inner vertex (the trace of isolated edge). We apply the Proposition 6 (the Remark 5) to the and. As a result consists of two semi-isolated edges and.

Further, we iterate the process getting a broken line

and for consists of two black semi-isolated edges and. We remark that the process of the annihilation of black simplexes in cannot bring to an appearance of a black simplex having a generic point with. Really, otherwise such a black simplex gives an opportunity to connect the endpoints and of the semi-isolated edge by a black curve which is different from. As a result a black loop with the semi-isolated edge as its part has been obtained and the loop is not contractible that is a contradiction to the simply connectivity of.

The complex is connected therefore the broken line contains all the possible black vertexes from at some step of the algorithm i.e. we come to 5, 1.

By the method of mathematical induction the main theorem is true for every

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