Since the beginning of the research in digital image processing, the question of the definition of a sound topological framework arose. Though in the two dimensional case a solution was easy to find, its generalization to higher dimensions seemed very hard. This is easy to see from the vast amount of theories under consideration by the community. The central goal was to find preconditions so that a discrete analog to the theorem of Jordan-Brouwer–the generalization of the Jordan curve theorem to arbitrary dimensions–is satisfied. The Jordan curve theorem states that every closed curve in the plane, that is simple, i.e. has no crossings, separates the plane in exactly two regions: Its inside and its outside and is itself the boundary of both of these sets.
In this paper, we restrict the discrete setting to lattices for . This is sufficient, since it is possible to embed other settings in these sets and is very suitable in geometric terms as Albrecht Hübler shows for the 2-dimensional case . The main focus is on the adjacency relations with which we add structure to the . As figure 1 shows, the validity of a discrete “Jordan curve” theorem depends on the adjacency relations we apply to the points. It is in general not sufficient to use the same adjacencies for the white and black points, i.e. the background and the foreground, as this picture shows and so we have to deal with pairs of such relations. The pairs of adjacencies that make it possible to define a correct discrete notion of a simple closed curve are called “good pairs”. In this paper we will solve the problem of defining a discrete -manifold in , which is the -dimensional analog to a simple closed curve, and characterize the good pairs in all dimensions greater than 2.
The first person to give an idea for the 2-dimensional case was A. Rosenfeld  in 1973. E. Khalimsky  studied very special topological adjacency relations, which were suitable in any dimension, during the early 1980s. In the 1990s, G.T. Herman  proposed a framework with very general neighborhood relations, he tried to make the Jordan curve property to a property of pairs of points. Unfortunately the approach does not resemble the intuition given by the Euclidean case. Anyway, the theory showed a promising method, how to generalize the concept of a good pair to higher dimensions. Later, in 2003, M. Khachan et al.  brought together the theory of pairs of the form in arbitrary dimensions . They were the first ones using the notion of the simplicial complex, a basic structure known from algebraic topology. This approach led to deeper insights, but it was bound to the very special adjacency-relations used by the authors.
T.Y. Kong  renewed the question of finding a general theory for the problem of topologization of in 2001. The Approach we a taking is a new definition of good pairs in dimensions greater than 2 and a new general concept of a -manifold in . For a given -manifold in we are able to construct a simplicial complex , that preserves the topological properties of , if is endowed with a good pair of adjacencies. The complex then has nice topological properties–it is a so called Pseudomanifold (as studied by P.S. Alexandrov ) and therefore, we are able to embed it in -dimensional Euclidean space in a natural way. By doing this we are defining a real manifold if and only if the adjacencies on were chosen correctly and the real version of the Jordan-Brouwer Theorem can be applied.
The paper is organized as follows: In section 2 we give basic definitions to make precise the topological and graph theoretic terms. In section 3 we give the definition of a digital manifold and study its basic properties. Also we give a new and general definition of a good pair. In section 4 we introduce the Theorem of Jordan-Brouwer. Together with the notions of digital manifolds and good pairs, we are able to construct a simplicial complex with the same topological properties as the digital manifold. Having this tools established, we can give the general proof of the discrete variant of the Theorem of Jordan-Brouwer in arbitrary dimensions in section 5. The proof consists of three parts, all involving heavy case differentiations of technical nature. We end the paper with Conclusions in section 6.
2 Basic Definitions
First of all, we review some definitions from the field of simplicial complexes. Our goal is to construct these objects from subsets of .
2.1 Simplices and Simplicial Complexes
Let be affine independent points. The set
is the (open) simplex with vertices . We also write . The number is the dimension of . Sometimes, for brevity, we call just -Simplex.
Let be simplices, then is called face of , in terms: , if the vertices of are also vertices of . The relation means and .
A simplicial complex in is a finite set of simplices in with the following properties
For any and is .
For any with holds
Let be a simplicial complex in . The geometric realization of is the set
A simplicial complex is homogenous -dimensional, if every simplex in is face of a -simplex in . A homogenous -dimensional simplicial complex is strongly connected, if for any two -simplices and exists a sequence of -simplices , such that and share a common -face for every . The complex is non-degenerated, if every -simplex is face of exactly two -simplices.
A simplicial complex is a combinatorial -pseudomanifold without border, if
is homogenous -dimensional,
is strongly connected.
To establish structure on the points of the set we have to define some kind of connectivity relation.
Given a set , a relation is called adjacency if it has the following properties:
is finitary: .
is connected under .
Every finite subset of has at most one infinite connected component as complement.
A set is called connected if for any two points in exist points and a positive integer such that , and for all .
In the text we will consider pairs of adjacencies on the set . In this pair represents the adjacency on a set , while represents the adjacency on .
Let be the set of all translations on the set . The generators of induce a adjacency in a natural way:
Two points of are called proto-adjacent, in terms , if there exists a such that or .
We can use the standard base of as the generators of , since all sums of integer multiples of this base is a point in .
Another important adjacency on is .
For every and all the set is connected under .
In the rest of the text let and be two adjacencies on such that for any holds
The set is connected under .
Let be any two points in . We need to show that there exists a -path from to . Let and . We prove by induction on . In the case the points and only differ in one coordinate by 1, since all terms in the sum are positive and integral. It follows respectively and thus . In the case we look for the smallest index such that . The point
is by definition -adjacent to and
By the induction hypothesis exists a -path
and is the path we are looking for.
A point is called simple for the pair if the following properties hold:
and contain the same number of -connected components.
and contain the same number of -connected components.
2.3 The Separation Property
We call the set
the -dimensional standard cube in . The set can be embedded in different ways in . A general -cube in is defined by a translation of a standard cube.
Indeed, we can construct any -cube from one point with generators in the following way:
The dimension of is then . We use this construction in the next definition.
Let , and be a -cube, . The complement of is in not separated by under the pair , if for every -component of and every -subcube of the following is true:
If is such that has maximal cardinality among all sets of this form, and the sets and are both nonempty and lie in one common -component of , then holds
In the following, we only consider the case when has at most one -component. This can be justified by viewing any other -component besides the one considered as part of the background, since there is no -connection anyway. This property also gets important if we study the construction of the simplicial complex.
A set has the separation property under a pair , if for every -cube , as in the definition 7 the set is in not separated by
The meaning of the separation property is depicted in the figure 4.
Let be a -cube for that has exactly two -components in . Then we can bound the number of points in as follows:
The upper bound follows from the existence of at least two points in and the observation that a -cube has vertices.
The lower bound stems from the case that one of the two -components is a singleton. Because then at least al the neighbors of this one point need to be in . There are such points.
A -component of in a -cube with elements has at least
-neighbors in .
First, we fill a -cube with the points such that is a minimal integer and contains all the points. This cube can be translated in in ways. Therefore, each of the points in has -neighbors in . Finally, we can also have neighbors in .
Let be a set that has the separation property and for every cube let be -connected. Let be any -component of . Then in every subcube at most one -component of is contained in .
This lemma guarantees that one -components of any cannot be split into two or more components in any subcube of .
Assume for contradiction that is a -cube that contains a -subcube such that two -components of lie in one -component of . In are at least two -components. For any -subcube of exist certain generators and such that
If and lie completely in then we consider instead of the given the new -cube .
Starting with the case we can see, that there exists a -subcube of that contains a maximal number of points of , otherwise, would contain only one -component.
Using the separation property for this we find
This shows that is -connected. This contradicts our assumption on the existence of two -components in .
3 Digital Manifolds
An -connected set , for , is a (digital) -manifold under the pair , if the following properties hold:
In any -cube the set is -connected.
For every the set has exactly two -components and .
For every and every the point is -adjacent to and .
has the separation property.
Let , , be a -manifold under the pair . Then the set contains exactly two -components.
Let be any point in . The set consists per definitionem of the two -components and . Every point is -adjacent to both of these components. Furthermore, is -adjacent to both of them. The set also consists of two -components and , whereby the naming can be made consistent by saying is the component that contains a common point with and analogue for .
and are nonempty and contain a common point, therefore the union of the both sets is -connected.
Let and be any two points of . By definition there exist two points and with and . Since is -connected, there exist an -path of the form in . This path induces the following sets that are connected by the observation above:
If and lie in the same set or respectively, then the two points are -connected. Otherwise no -path between the two sets can exist.
Let be a -manifold. The following two definitions base upon the last theorem:
We recapitulate the notion of the elementary equivalence of paths given by G.T. Herman  Let and be paths in under the adjacency of the form
that satisfy , for , , for . Then and are called elementary -equivalent, if .
Two paths , in are called -equivalent, if there is a sequence of -paths with and , such that and are elementary -equivalent for . Spaces, in which every cycle is -equivalent to a point, are called -simple connected.
A -manifold , , under the pair is a -sphere, if it is -simple connected for some positive integer .
A pair of adjacency relations on is a separating pair if for all the set is a -sphere.
3.1 Double Points
A point is a double point under the pair , if there exist points and and a simple111A translation is called simple if no other translation exists with , . translation with , and .
This concept is the key to a local characterization of the good pairs . Without it, one could not consistently define topological invariants like the Euler-characteristic. It means that an edge between points in a set can be crossed by an edge between points of its complement and these four points lie in a square defined by the corresponding adjacencies. This crossing can be seen as a double point, belonging both to the foreground and to the background. Also, mention the close relationship to the separation property, which is a more general concept of similar interpretation.
An adjacency relation on is regular, if at least one of the following holds:
For every translation on is , iff .
For every rotation on is , iff .
Let be a separating pair of regular adjacencies on . If contains a double point for any , then exists a set, for which the Euler-characteristic cannot be defined.
Our argumentation follows section 7.5.3 in Klette and Rosenfeld . Choose any such that has a double point . There is a (discrete) line with generator through and with . Let . So are three consecutive points of this line. Because of the regularity of the point lies in .
Furthermore is a double point. In the case of the rotational invariance of this is immediately clear by rotation of in and in the case of translational invariance the proposition follows by exchanging of the roles of and and the translation of the points and to and in the definition 3.1 of double point.
The set is -simply connected and can be reduced to the single point by repeated deletion of (simple) points. Therefore its Euler-characteristic is
The set is topologically equivalent to a circle. Thereby
Let be an embedding of in , generated by and the translation between and . contains the points . The set is simple connected and after deletion of simple points a single point, such that . Let the sets and be the intersections of with the positive respectively negative closed (discrete) half-space of . By symmetry holds and by the sum-formula of the Euler-characteristic is
A separating pair of adjacencies in is a good pair, if for every the set contains no double points.
4 Theorem of Jordan-Brouwer
We want to prove a discrete variant of the theorem of Jordan-Brouwer. But we haven’t seen anything of it yet. This will change right now.
Theorem 2 (The Discrete Theorem of Jordan-Brouwer)
Let the set be a -dimensional manifold under the pair with . Then has exactly two path-connected components and is their common boundary.
This formulation is slightly more general than the original one, which only deals with spheres. It follows immediately from number 3.42 of chapter XV of Alexandrovs book , which reads as
Theorem 3 (Alexandrovs Theorem)
Any -dimensional pseudomanifold in is orientable, disconnects in exactly two path-components and is the common boundary of both of them.
We will see, how to construct a -pseudomanifold in from a discrete -manifold, thereby proving:
Theorem 4 (Main Theorem)
For every good pair in with . The theorem of Jordan-Brouwer is true.
4.1 The construction of a Simplicial Complex
We start with the following construction:
Given an adjacency and arbitrary points , we define:
These are the line segments in between the points and . The first equation resembles an edge of the adjacency in a geometric way for any .
At first we construct the complex inside an arbitrary -cube . Let be a -connected set, and be a good pair. Then we define a simplicial complex by the following process:
For any -cube with vertices let
be the barycenter of C. Observe, that the barycenter need not to lie in the discrete space containing the cube C. But since every such space can be embedded in a natural way in some Euclidean space, the barycenter is defined.
Given some cube of dimension , we evaluate the test as true, if and only if
The test is true, if is a subset of , if the line segment between two -neighbors in meet the barycenter of , or if the line segment between two points of , that are not -connected meets the barycenter of .
One could imagine the following situation: Two points define a barycenter of some -cube and at the same time two points are -connected and the segment meets this barycenter. This is a case we do not want but we will see in Lemma 9, that the concept of double points is the key to avoid this.
We construct the complex inductively. First, consider the vertices:
If the simplices of dimension are defined, we are able to construct the -simplices for all . We thereby consider only the simplexes that are contained in the convex hull of the cube :
This means, we connect the barycenters of the subcubes of that satisfy the test to the already constructed -simplices, thus forming -simplices. Remember, that if the test is true, then the barycenter of is a vertex of our complex .
The union of all will be called .
The set is a simplicial complex for every fixed pair and a set .
We have to check the definition 2 of a simplicial complex.
First, we show that every face of a simplex in is an element of . Observe, that is in for some . If , so has no proper faces and the proposition is true.
Consider . For every let
be the simplex formed by deletion of the ’th vertex.
For the simplex is by construction of a member of and also lies in .
For , is a simplex in and a face of . By construction the simplex is in . Using induction, we see that the simplex is in .
Second, given different simplices of the form and , we need to show their disjointness. Suppose the vertex sets and are disjoint. Then, has to be empty, too. We prove this by induction over .
If , so the proposition is true. So let and let the claim be true for all proper subsets of . arises from a -simplex by attachment of a barycenter of a subcube of . By assumption, is no vertex of . By the inductive hypothesis is for . This is the boundary of . Therefore, cannot be true.
If holds, then is a common proper face of and .
None of the vertices of lies in the inside of and vice versa, since the simplices arise by successively attaching barycenters to a vertex of . No face of can meet the inside of , because of the construction of the simplices.
Let be a finite -connected subset of under the pair . can be covered by a finite number of -cubes . We define:
is a simplicial complex for any finite set under the pair .
The result follows immediately from the preceding lemma.
If is not -connected but finite then we can construct component wise.
The next lemma is crucial in our theory, since it establishes the connection between the topology of the discrete set and the topology of the geometric realization in .
Let be a set under the pair and the simplicial complex from the construction above. Then the following properties hold:
Two points are in the same -component of iff they are in the same component of .
Two points are in the same -component of iff they are in the same component of and does not contain double points.
The boundary of a component meets the boundary of a component iff there exists a point in that is -adjacent to some point in .
1. To prove the first equation, we observe that a point is in , iff it is a vertex of and also a point of . This is the case, iff is in . The second equation follows by an analog argument.
2. () Let be points from an -component of . There exists an -path such that is in for all . This is a piecewise linear path.
() Let be two points of the same component of . There exists a real path such that and . Let be the simplices met by . Every point of the path in a simplex can be mapped to the boundary by a linear homotopy. After a finite number of such homotopies the path is mapped to a path, which only meets edges and vertices of . This path can the be transformed into an -path by the points on it, that are in .
3. The proof is analog to 2. But the two directions do not hold in general if the pair has double points. Let be as in definition 3.1 such that and are in different components of , and . The point is in by construction and therefore is in , too.
4. Let and be two components.
() Suppose the intersection of the boundaries of and contains a point . This point is in a -simplex of that has a vertex in a -cube , which meets and . There are points , such that and . In exists a -path from to . Since this path need to contain two -adjacent points such that one of them is in and the other is in .
() Suppose there exist a -path from to . This path can be extended to a -path by insertion of certain points from . There exist two -neighbors and in . By construction of the point has to be in the simplicial boundary of and therefore in the intersection of the boundaries of and .
4.2 The Construction of a Pseudomanifold
We now apply the construction of to digital -manifolds with the goal to define a combinatorial -pseudomanifold (see definition 3).
Let be a digital -manifold in the rest of the section. The first issue to be solved, is that needs not to be homogenous -dimensional by construction, since -simplices are easily introduced.
We solve this issue by finding a deformation retract of , that has no -simplices in . The new complex is constructed as follows: At first we remove all vertices from that are barycenters of the cubes with only one -component in . Then we remove all simplices, that contain such a barycenter as vertex.
The next lemma shows that this reduction preserves all the important topological properties of the initial complex .
The complex is a triangulation of a strong deformation retract of .
We begin by proving the next lemma:
Let be a -simplex and be any vertex of . Then is a strong deformation retract of .
We give a linear homotopy from to .
Given let be the point on the ray that lies in , that is the boundary of . The function
is a linear homotopy between the given sets.
is a simplicial complex and by existence of , the set is a strong deformation retract of .
Let now be the homotopy defined above for every containing as vertex. The removal of from induces the following linear homotopy
Since only a finite number of points are removed from , we can consider the construction of a finite sequence of linear homotopies of the form given above. Therefore, it follows that is a strong deformation retract of the complex .
4.3 Properties of a Digital Manifold
Let , , be a digital -manifold under the pair . For every -cube , the set contains at most one -component in and , respectively.
Since is a digital -manifold, for all the set