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 