Abstract
We define the concepts of homotopy and fundamental group for geometric spaces as a generalization of metric spaces, digital spaces, and graphs; then, we compare them with corresponding concepts in these spaces. Also, we state some properties of the fundamental group of geometric spaces and some theorems to calculate them.
1. Introduction and Preliminaries
“Homotopy equivalence relation” and “fundamental group” are two famous and helpful concepts in mathematics; these concepts and related concepts are initially used as a tool to describe topological spaces via the properties of the fundamental groups of them in the category of groups. Many benefits of this tool have motivated researchers to adapt it to other spaces, such as metric spaces [1], digital spaces [2–5], and graphs [6–9].
Since in modeling, a composition of some different spaces is needed, it will be helpful to adapt these tools for a space that includes graphs, digital spaces, and metric spaces. In digital images, there are two different sets of points, black points and wight points, and each of them can be considered by its own adjacency relation (see [10]). However, studying a subject in real life requires more than two sets and relations for classifications. A good example is a medical image of the body, which usually includes images of various components, such as bone, muscle, cartilage, and blood vessels. A string of colored light bulbs is a simple example that consists of colored bubbles, LEDs inside the bubbles, wires, LEDs inside the rope, and so on (Figure 1). A simple model of a string of colored light bubbles in two dimension by the geometric space is similar to in Figure 2.
Another thing that motivates us to study geometric spaces is that defining a geometric space to model a subject in real life is more accessible than other spaces. For example, in the modeling of protein as the most important of cells in [11], the definition of the relevant topology seems to be not accurate, and in the continuation of the work, the modeling was completed with approximate numerical methods.
Freni [12] introduced the concept of “geometric space” when calculating the alpha relation in hypergroups (which was first called the “gamma relation” and is the smallest equivalence relation that makes an abelian group from a hypergroup). Afterwards, Torabi Ardakani and Pourhaghani [13], while extending the concept of the “topological hypergroup,” added the concept of “good morphism” as a map between geometric spaces. In this paper, we adapt the tool of fundamental groups and related concepts for geometric spaces as a generalization of those of metric spaces, digital spaces, and graphs.
Recall from [12] that a geometric space is a pair such that is a nonempty set whose elements are called points and is a nonempty family of subsets of whose elements are called blocks. If is a subset of , then it is called a -part of if implies for every . For a subset , the intersection of all -parts of containing is denoted by . There are five examples of geometric spaces shown in Figure 2.
In Figure 2, the set of points of geometric space is the black circle, and its blocks are four arcs of the black circle which are featured by colored closed curves. The set of points of the geometric space is the brown circle, and its blocks are five arcs of the brown circle, which are featured by colored closed curves. The set of points of the geometric space is the black points as a subset of , and its blocks are featured by colored closed curves. The graph is a geometric space, where the set of its points is the set of vertices of graph , and each block of it is the vertices of both sides of the corresponding edge. The shape is a composition of shapes ; its blocks are similar to the blocks of the corresponding shape in geometric spaces .
The -tuple of blocks of a geometric space is called a polygonal if for each . By the concept of polygonal, Freni [12] defined the relation as follows.
or there exists a polygonal such that and .
The relation is an equivalence and coincides with the transitive closure of the following relation:
Hence, is equal to , where [12]. for means that there exist some such that .
Recall from [13] that a map between the geometric spaces is called a good morphism if implies that for all .
Proposition 1 (see [13]). Let be a good morphism between geometric spaces; then, yields for all .
In this paper, we first define the concept of “homotopy” between good morphisms in geometric spaces and compare it with the concepts of “discrete homotopy” in metric spaces, “digital homotopy” in digital spaces, and “graph homotopy” in graphs; then, we present the concept of the “fundamental group” of geometric spaces and some properties of it; then, we state some theorems and propositions to find the fundamental groups of some geometric spaces up to isomorphism (Theorems 40 and 57, Proposition 47, and Corollaries 41, 51, and 62). In the final section, we present the relation between the discrete fundamental group, digital fundamental group, and graph homotopy group with the fundamental group of the corresponding geometric space.
2. Homotopic Geometric Spaces
In this section, we define the concept of homotopy between some maps in geometric spaces and state some of its properties. First, we introduce some concepts and properties, which are necessary to define the homotopy. To have some examples, we must recall some information from other objects in mathematics, such as metric spaces, digital spaces, and graph theory.
For every in a geometric space , the -class of is called the connected component of in . The geometric space is called connected if it has one unique connected component (see [12]). In other words, a geometric space is called connected if and only if for each pair of different points , there exists a polygonal of such that and . A component of the geometric space is the maximal connected subset of . Clearly, the connected component containing is equal to (see Proposition 2.1 in [9]).
In the following, we recall some concepts of digital spaces and some properties of them. For more information, see [14, 15].
Let be the set of integers, and let . Two distinct points and in are called -adjacent for if there exist at most distinct indices such that and for other indices . A(n) (-dimensional) digital image, denoted by , is a subset , where is an adjacency relationship on . A digital image with adjacency relation is called -connected if and only if for every pair of different points , there is a set of points in such that , , and and are -adjacent for . A -component of a digital image is a maximal -connected subset of . The function between digital images is called -continuous if it preserves the adjacency; see [15].
Example 1. (1)Let be a metric space, let , and let be the family of maximal connected subsets such that for all ; then, is a geometric space. Clearly, the connected component containing is equal to for all .(2)Let be a(n) (-dimensional) digital image, and let be the family of subsets such that and are -adjacent for all ; then, is a geometric space, and the adjacency relation in the digital image coincides with the relation in the corresponding geometric space . Clearly, the -component of containing is equal to in the geometric space .(3)Let be a graph; then, is a geometric space such that the relation “” in geometric spaces coincides with adjacency in the graph. Clearly, the connected component containing is equal to for all .Clearly, a geometric space is a digital image if for some , and it is a graph if each has two elements.
Definition 2. Let be the set of integers. The (standard) geometric space, denoted by , is the geometric space , where .
Recall from metric spaces that a function is called -Lipschitz for if for all .
Recall from the graph theory that a map is called a graph map if is a map and yields or . Let and let ; then, a based graph map is a graph map such that .
According to Example 1, any metric space induces more than one geometric space and any digital image and any graph induce a geometric space. In the following proposition, we consider the relation between Lipschitz functions of metric spaces, continuous functions of digital images, and graph maps with good morphisms of corresponding geometric spaces.
Proposition 3. (1)Let and let be an -Lipschitz between metric spaces; then, for such that , the induced map defined by for all is a good morphism.(2)The map between digital images is a -continuous function if and only if the induced map defined by for all is a good morphism.(3)The map is a graph map if and only if it is a good morphism.
Proof. (1)Since for all , is well defined. Let , and let ; then, there exists a block such that , so . Since is an -Lipschitz function, ; therefore, ; thus, , which implies that is a good morphism.(2)Since , is well defined if and only if is well defined. By Example 1, preserves the adjacency relation if and only if preserves the relation in the corresponding geometric spaces, which completes the proof.(3)It is straightforward.
Proposition 4 (see [16]). Let and be good morphisms between geometric spaces; then, is a good morphism.
Let be a geometric space for ; then, is a geometric space, where and (see [13]).
Proposition 5. Let be a good morphism; then, by is a good morphism for all and by is a good morphism for all .
Proof. Let . To show that is a good morphism, let such that ; then, . Since is a good morphism, . Similarly, is a good morphism for all .
Proposition 6. Let and be good morphisms between geometric spaces; then, is a good morphism.
Proof. Let for , and let ; then, and . Therefore, and since and are good morphisms; thus, ; hence, .
Definition 7. (see [16]). Let be a geometric space with ; then, the geometric space is called a geometric subspace if . The induced geometric subspace of , denoted by , is the geometric subspace , where . If there is no ambiguity, we delete the word “geometric.” The subspace is the induced subspace of , and we call it the trivial subspace.
For example, is an induced subspace of the geometric space for each , where is the family of blocks of that are contained in .
Definition 8. Let such that . A countable interval is the induced geometric subspace of . For conveniently, the countable interval is denoted by .
Definition 9. Let be a geometric space, and let . A path from to in the geometric space is a good morphism for some , such that and ; moreover, and are, respectively, called initial and final points of . If , then is called a loop based at . If is a constant map, then it is called a trivial loop. Furthermore, is called a simple path of length when if and only if or . If is a loop, then is called a simple loop.
Proposition 10. Let be a geometric space, and let . There exists a path from to if and only if .
Proof. Let ; then, there exists a polygonal of such that and . Define by , , and for , where is arbitrary. Clearly, is a good morphism; thus, is a path from to . In this case, is called a path corresponded to polygonal.
Conversely, let be a path from to ; then, is a good morphism; thus, ; hence, .
Let be a path from to in a geometric space ; then, ; hence, there exists such that for ; thus, is a polygonal; this polygonal is called corresponded to path .
Lemma 11 (Gluing lemma). Let be a geometric space for , and let such that for each , if , then there exists such that for some ; then, the map defined by the following: is a good morphism if is a good morphism for and for all .
Proof. Let be a good morphism for . Since for all , is well defined. To show that is a good morphism, let and . By the hypotheses, there exists a block such that for some , so . Since is a good morphism, .
Definition 12. Let be a geometric space for , and let be good morphisms. If for some , there exists a good morphism such that and for all , then and are called homotopic and denoted by . The map is called a homotopy between and . A good morphism is called null-homotopic if is homotopic to a constant map.
Recall from [1] that -Lipschitz functions between metric spaces are called -discrete homotopic if there exist and an -Lipschitz function such that and for all , where is equipped with the metric for all and the Cartesian product is equipped with the -metric.
Proposition 13. Let , and let -Lipschitz functions between metric spaces be -discrete homotopic; then, the induced good morphisms are homotopic for some such that .
Proof. Let and be -discrete homotopic; then, there exist and an -Lipschitz function such that and for all . Clearly, a suitable induced geometric space of is . Define by . By Proposition 3, is a good morphism and the hypothesis implies that and that , which completes the proof.
Recall from [2] that for each with , a digital interval is the digital image . Two -continuous functions between digital images are called digital-homotopic if there exist and a function that satisfy the following properties:(1) and for all (2)The induced function by for all is -continuous for all (3)The induced function by for all is -continuous for all In this case, is called digital-homotopy.
Proposition 14. Let be -continuous functions between digital images. If the induced good morphisms by and for all are homotopic, then and are digital -homotopic.
Proof. Let be the homotopy map between and . Define by for all and . By Proposition 5, the induced functions and preserve the adjacency relation for all and , which completes the proof.
The inverse of the above proposition is not necessarily true because for such that and , and are -adjacent, and are 2-adjacent, and and necessarily are not adjacent.
Definition 15. Let be good morphisms and let ; then, and are called relative homotopic with respect to , denoted by , if there exists a homotopy map such that for all and ; moreover, is called relative homotopy.
Recall from the graph theory that the Cartesian product is the graph with vertex set and edges if either and or and .
The Cartesian product of graphs necessarily does not induce the Cartesian product of the induced geometric spaces, because for such that , , , and , but necessarily is not an edge.
Recall from [7] that based graph maps are called G-homotopic if there exist and a graph map such that and for all and for all , where is the graph .
Proposition 16. Let be based graph maps. If the induced good morphisms defined by and for all are relative homotopic with respect to , then and are G-homotopic.
Proof. Let be the homotopy map between and . Define by for all and . Proposition 3 completes the proof.
Theorem 17. Let be a geometric space for , let good morphisms be homotopic under , and let good morphisms be homotopic under ; then, is homotopic to .
Proof. Define byBy Proposition 4, is a good morphism, and by Proposition 6, is a good morphism; moreover, for all , so by Lemma 11, is a well-defined good morphism. Indeed and for all , which completes the proof.
Corollary 18. Let be a geometric space for , let good morphisms be homotopic under , and let ; then, and are homotopic.
Proof. Let be the inclusion map; then, is a good morphism and is homotopic to itself; hence, by Theorem 17, are homotopic; thus, and are homotopic.
Proposition 19. The homotopy relation between good morphisms from a geometric space to a geometric space is an equivalence relation.
Proof. Let be good morphisms between geometric spaces.
Reflexivity: clearly, defined by is a good morphism; thus, .
2.1. Symmetry
Let be a homotopy map between and ; then, by Proposition 4, defined by is a good morphism; and for all ; thus, .
2.2. Transitivity
Let be homotopic to under , and let be homotopic to under . Define by
We know that , so by using Lemma 11, is a well-defined good morphism, but and for all ; thus, .
3. Fundamental Group
In this section, we define the concept of “fundamental group” of a geometric space and state some properties of it. First, we introduce some concepts and properties, which are necessary to define it.
Definition 20. Let and be two paths in geometric space with . The product is defined asLemma 11 immediately yields the following corollary.
Corollary 21. The product of two paths in a geometric space is a path.
Proposition 22. The product of paths in a geometric space is associative.
Proof. Let , , and be paths in geometric space , such that and ; then, by the definition of the product of two paths, and are equal to defined byTo define the homotopy between paths, we need the following definition.
Definition 23. Let and be two paths in a geometric space . The path is called a trivial extension of if and there exist sets of paths and in such that , , and , and there exists a sequence such that for and is the trivial loop for all .
Clearly, any path is a trivial extension of itself. Let be a path in a geometric space , and let ; then, byis a trivial extension of .
Definition 24. Two paths and in a geometric space are called path homotopic and denoted by if there exists a trivial extension of and a trivial extension of such that there exists a relative homotopy map between and with respect to . In this case, is called a path homotopy.
Proposition 25. Let be a path in a geometric space from to , and let be a polygonal corresponded to the path with (or , resp.) for some ; then, (or , resp.) is a polygonal corresponded to the path .
Proof. Since (or resp.), (or , resp.), so (or , resp.) is a polygonal corresponded to the path .
Definition 26. Let be a polygonal of geometric space . If and for all , then it is called the reduced polygonal. If all blocks of a polygonal with for or for are omitted, then its reduced polygonal is obtained.
Lemma 27. Let be a path in a geometric space from to , and let be a polygonal corresponded to the path . Let for some be the reduced polygonal of . If is a path corresponded to from to , then and are path homotopic.
Proof. If , then we are done. Let . Define as follows: , , and for :Clearly, is an extension of the path . Define by and . By the structure of and , for , so is a good morphism. Indeed and , so is a path homotopy between and ; hence, .
The above lemma immediately yields the following proposition.
Proposition 28. Let and be paths in a geometric space from to . If reduced polygonals corresponded to and are equal, then and are path homotopic.
By Proposition 19, immediately we have the following proposition.
Proposition 29. The path homotopy relation between paths in a geometric space is an equivalence relation.
Proposition 30. Let and be paths in a geometric space from to . Let and for some be the reduced polygonals corresponded to paths and , respectively, such that ; then, and are path homotopic.
Proof. Let and be paths from to corresponded to polygonals and , respectively; hence, it follows that for , that for , that , that , and that for . Define bySince , is a well-defined good morphism, so is an extension of . Define by and . By the structure of and , for , so is a good morphism, but and ; thus, . Lemma 27 and Proposition 29 complete the proof.
Lemma 31. Let be a geometric space, and let , , and be paths from to ; then, , and are path homotopic.
Proof. By Proposition 29, it is enough to show that and . Clearly, defined by and is an extension of . Define by and . Since is a good morphism, , so is a good morphism; thus, , and hence .
Clearly, defined by and is an extension of . Define by and . Since is a good morphism, , so is a good morphism; thus, ; therefore, .
Proposition 32. Let be a geometric space, and let . Let for each with , there exists a polygonal in such that , , and ; then, each path in is path homotopic to a path in with the same initial and final points.
Proof. Let be a path from to ; then, there exists a reduced polygonal in corresponded to . Let be a path from to corresponded to polygonal ; then for . We can write the path , where defined by and is a path for . For each , there are two cases: Case 1: If in , then define by . Case 2: If in , then, by the hypothesis, there exists a polygonal in such that , and , so is defined as a path from to corresponded to ; by Lemma 31, for all ; thus, by Proposition 10, , where is a path in .
Immediately, the above proposition yields the following corollary.
Corollary 33. Let be a geometric space, and let such that preserves the connectedness of (i.e., for each , if in , then in ). If , then each path in is homotopic to a path in .
Definition 34. Let and be two paths in geometric space , such that . The operation between two path homotopy classes and is defined as , where is the product of and .
Lemma 35. Let be a homotopy map between two good morphisms and ; then byfor all is a homotopy map between and . If is a relative homotopy with respect to , then is too.
Proof. To show that is a homotopy between and , first, we show that is a well-defined good morphism. We can write asfor all , so by Lemma 11, is a well-defined good morphism. Indeed and ; thus, .
Let be a relative homotopy with respect to ; then, for all ,which completes the proof.
Proposition 36. The operation between path homotopy classes in a geometric space is well defined.
Proof. Let be path homotopic under , and let be path homotopic under , such that . Define byfor all , where , , and are, respectively, defined byBy Lemma 35, and are path homotopy between and , respectively. By Lemma 11, is a well-defined good morphism. Indeedwhich completes the proof.
Theorem 37. Let be a geometric space; then, the operation “” has the following properties.
Associativity. Let , and be paths in the geometric space . If is defined, then is defined, and they are equal.
Right and Left Identity. Let be a path from to ; then, and , where is a trivial loop at .
Inverse. Let be a path from to . Let be the path defined by ; then and .
Proof. Associativity. Clearly, by the definition of operation “” and Proposition 22, and are equal to .
Right and Left Identity. Since and is an extension of , ; thus, . Similarly, .
Inverse. Define byBy Lemma 11, is a good morphism. Since and , so . Indeed ; hence, . Similarly, .
Definition 38. Let be a geometric space, and let for . Let be a(n) (induced) geometric subspace of such that . The triple is called a pointed geometric space. The triple is called a(n) (induced) pointed geometric subspace. A map is called a pointed good morphism if is a good morphism and .
Immediately, by Theorem 37, we have the following corollary.
Corollary 39. Let be a pointed geometric space, and let (if there is no ambiguity ) be the family of all path classes of all loops in based at ; then, with operation is a group, which is called the fundamental group or first homotopy group of the geometric space .
Example 2. Let be a metric space, where is the unit circle and is the length of the shortest arc connecting to . We calculate the fundamental group of induced geometric space for multivalued radius . In general, there are two cases: Case I: the fundamental group is a trivial group and Case II: the fundamental group is isomorphic to . Case I. Let , let , and let , where is the arc clockwisely from to (which is presented in Figure 3 by colored close curve) for . By Corollary 33, it is enough to calculate the fundamental group of . Define by and , where for , so is a loop at and is the reduced polygonal corresponded to . Define by for all . Since , and , is a good morphism, but and for all ; thus, . Since every nonconstant simple loop at in has the reduced polygonal (or resp.), then, by Proposition 28, every nonconstant simple loop at in is in the path homotopy class (or ); hence, is a trivial group. By a similar argument, is a trivial group when . Case II. Let , let , and let , where is the arc clockwisely from to (which is presented in Figure 3 by colored close curve) for . For each , if , then , so there are two cases. Case 1: there exists such that . Case 2: there exists such that , where ; thus, by Proposition 32, it is enough to find the fundamental group of .Define by and , where for , so is a loop at and is the reduced polygonal corresponded to . Since every nonconstant simple loop at in has the reduced polygonal (or resp.), by Proposition 28, every nonconstant simple loop at in is in the path homotopy class (or resp.). Indeed there is not homotopy map between and , so ; thus, the fundamental group of is generated by ; hence, as groups.
By a similar argument, as groups when .
Let where , let , and let , where is the arc clockwisely from to (which is presented in above figure by colored close curve) for . Now, we rewrite the argument of the case for . Similarly, it is enough to calculate the fundamental group of . Let by and , where for , so is a loop at and is the reduced polygonal corresponded to . Since every nonconstant simple loop at in has the reduced polygonal (or resp.), by Proposition 28, every nonconstant simple loop at in is in the path homotopy class (or resp.). Indeed there is not homotopy map between and , so ; thus, the fundamental group of is generated by ; hence as groups.
Let be a geometric space, and let . If is a path in from to , then we identify by for all .
Theorem 40. Let be a geometric space, and let such that ; then as groups.
Proof. Since , by Proposition 10, there exists a path in from to . Define by for all . By Corollary 21, is a path in . Clearly, the initial and final points of it are , so ; thus, is well defined.
To show that is a group homomorphism, let ; then, by Theorem 37 and the concept of trivial extension of a path, we haveTo show that is an isomorphism, let . Define by for all ; then, is a well-defined group homomorphism, by the similar argument of . By Theorem 37 and the concept of trivial extension of a path, we havefor all . Similarly, , for all ; thus, is an inverse of , which completes the proof.
Corollary 41. If is a connected geometric space, then as groups, for each .
Proposition 42. Let be a geometric space, and let . If and are two homotopic paths from to , then .
Proof. We must show that or for all . It is enough to show that if , then and .
Let under . Define by . Since is a good morphism, is too, butfor all andfor all ; thus, .
Since , .
Theorem 43. Let be a connected geometric space, and let . The group is a commutative group if and only if for all paths and in from to .
Proof. Let be a commutative group, let , and let and be paths in from to . Since , the commutativity of yields , so ; thus, ; then, by Theorem 37, ; hence, .
Conversely, consider in from to , and let ; then, is a path in from to , so by the hypothesis, ; thus, ; therefore, . Since , , so by Theorem 37, ; thus, ; hence, by Theorem 37, .
Theorem 44. Let be a pointed good morphism between pointed geometric spaces; then, defined by for all is a group homomorphism.
Proof. Let be a loop in at . By Proposition 4, the map is a good morphism, so it is a path in , but ; then, is a loop in at ; thus, by Theorem 17, ; hence, is well defined.
To show that is a good homomorphism, let ; then,where is the group operation of for .
Theorem 45. Let and be pointed good morphisms between geometric spaces; then, ; moreover, if is the identity map in the geometric space , then .
Proof. Let ; then,Let be the identity map; then,for all ; thus, .
Corollary 46. Let be a bijective pointed good morphism between geometric spaces such that the inverse map is a pointed good morphism; then, is a group isomorphism.
Proof. By Theorem 44, is a group homomorphism. By Theorem 45, and , so is a group isomorphism.
Proposition 47. Let be a geometric space, and let ; then, as groups.
Proof. Let be the inclusion map; then, is a good morphism; thus, by Theorem 44, is a group homomorphism. To show that is one-to-one, let such that ; then, , so there exists a homotopy map in the geometric space such that . Since is a good morphism and is a connected subset of , by Proposition 1, is a connected subset of , but ; thus, ; therefore, is a homotopy map between and in the subspace ; hence, is one-to-one.
To show that is onto, let ; then, is a loop in at . By Proposition 1, ; thus, ; therefore, .
3.1. Deformation Retracts
In this section, we introduce the concept of “deformation retract,” some of its properties, and some related concepts.
Definition 48. Let be a geometric space, let , and let be the inclusion map. A retraction of onto is a good morphism such that (i.e., ). In this case, is called a retract of . A retract is called a deformation retract if . In this case, is called a deformation retraction of . A retract is called a strong deformation retract if . In this case, is called a strong deformation retraction of .
Proposition 49. Let be a geometric space, and let be a retraction of with retract map ; then, the group homomorphism is onto and the group homomorphism is one-to-one, where is the inclusion map and .
Proof. Let be a retract of . Since , by Theorem 45, , but is a group isomorphism; thus, is onto, and is one-to-one.
Theorem 50. Let be two pointed good morphisms between pointed geometric spaces such that ; then, the induced group homomorphisms are equal.
Proof. Let . Since , by Theorem 17, , but is a loop at and ; thus, , and hence .
Corollary 51. Let be a strong deformation retract of geometric space ; then, as groups, for all .
Proof. Let be the inclusion map, and let be a strong retract map of ; then, and , so by Theorem 45, for all . Since is a group isomorphism, is one-to-one; thus, by Proposition 49, is a group isomorphism.
In Corollary 51, if is a singleton, then is a trivial group.
Theorem 52. Let be a geometric space for , and let be two homotopic good morphisms under homotopy map . Let , and let defined by be a path in from to ; then, the group homomorphisms are equal.
Proof. Let . We must show that , or identically . Define by for all ,. By Propositions 4 and 6, is a good morphism, but and , so . Define byClearly, is well defined, and by Lemma 11, is a good morphism, but and . Since is a trivial extension of , by Theorem 17, ; thus, , but , which completes the proof.
In Theorem 52, if , then is a loop in at and by for all is a group automorphism; moreover, if , then is the trivial loop at , so ; thus, , similar to Theorem 50.
Using Theorem 52, immediately, we have the following corollary.
Corollary 53. Let be homotopic good morphisms between geometric spaces, and let . If is a one-to-one (onto, or trivial, resp.) group homomorphism, then is a one-to-one (onto, or trivial resp.) group homomorphism.
Corollary 54. Let be a null-homotopic pointed good morphism between geometric spaces; then, is a trivial group homomorphism.
Proof. Since is null-homotopic, under homotopy map , where is the constant map at . Define by , so is a path from to . By Theorem 52, ; then, for all ,Hence, by Theorem 37, , which completes the proof.
3.2. Homotopy Type
In this section, we define the concept of “homotopy type” and present some of its properties.
Definition 55. A good morphism between two geometric spaces is called a homotopy equivalence if there exists a good morphism such that and , where is the identity map for ; moreover, is called a homotopy inverse of . If is a homotopy equivalence, then and are called homotopic equivalent and denoted by . Two spaces that are homotopy equivalent are called to have the same homotopy type.
Theorem 56. The homotopy equivalence relation between geometric spaces is an equivalence relation.
Proof. Reflexivity. Since is a good morphism for all geometric space and , .
3.2.1. Symmetry
If is a homotopy equivalence between two geometric spaces and , then there exists a good morphism such that and , so is a homotopy equivalence between and .
3.2.2. Transitivity
If is a geometric space for , , and , then there exist good morphisms and such that for , , and . By Proposition 4, and are good morphisms, andhence .
Theorem 57. Let be a good morphism, and let . If is a homotopy equivalence map, then is a group isomorphism.
Proof. By Theorem 44, is a group homomorphism. Since is a homotopy equivalence map, there exists a good morphism and homotopy maps and such that and . Define by for all ; then, is a path in from to , so by Theorems 45 and 52, , but by the proof of Theorem 40, is a group isomorphism; thus, is one-to-one. By a similar argument, is a group isomorphism, so is onto, which completes the proof.
According to Theorem 57, if two geometric spaces have the same homotopy type, then the fundamental groups of them are isomorphic in the group theory.
3.3. Contractible Space
In this section, we define the concept of “Contractible” and present some of its properties.
Definition 58. A geometric space is called contractible if the identity map is null-homotopic.
Theorem 59. Geometric space is contractible if and only if each good morphism is null-homotopic for all geometric space .
Proof. Let be contractible; then, there exists such that , so by Theorem 17, ; thus, is a null-homotopic map. Conversely, let each good morphism be null-homotopic for all geometric space ; then, the good morphism is null-homotopic; thus, is contractible.
Theorem 60. Each contractible geometric space is connected.
Proof. Let be a contractible geometric space; then, for some , under homotopy map . For each , clearly, defined by is a path from to ; thus, for each , is a path from to ; hence, is connected.
Theorem 61. Geometric space is contractible if and only if and have the same homotopy type for some .
Proof. Let be contractible; then, there exists such that . Define by and by for all . Clearly, and are good morphisms, but and for all ; thus, and , respectively, but by Theorem 56, , and since S is contractible, ; hence, and have the same homotopy type.
Conversely, and have the same homotopy type; then, there exist good morphisms and such that and , but clearly, and for all ; hence, ; thus, is contractible.
Corollary 62. Let be a contractible geometric space; then, is the trivial group for all .
Proof. By Theorem 61, and have the same homotopy type for some ; then, by Theorem 57, . Theorems 40 and 60 and Proposition 10 complete the proof.
The converse of Theorem 60 necessarily is not true; see Example 3.
Example 3. Consider the geometric space . Since , this geometric space is connected. Define by , and ; then, is a loop in the geometric space at . Clearly, the fundamental group is generated by ; thus, as groups; therefore, by Corollary 62, this geometric space is not contractible.
Definition 63. A connected geometric space is called simply connected if is the trivial group for all .
By Corollary 62, each contractible geometric space is simply connected. To examine the simply connection of a geometric space, according to Theorem 40, it is enough to examine it for some .
Theorem 64. A geometric space is simply connected if and only if any two paths having the same initial and final points are path homotopic.
Proof. Let , let , and let and be two paths in from to ; then, is a loop in at ; therefore, ; thus, ; hence, by Theorem 37, .
Conversely, let and ; then, by the hypothesis, , so ; thus, .
4. The Relation between the Fundamental Group of Geometric Space with the Fundamental Group of Other Spaces
In this section, we try to find the relation between the discrete fundamental group, digital fundamental group, and graph homotopy group with the fundamental group of corresponding geometric space. First, we prove a helpful lemma in general.
Lemma 65. Let be a nonempty set with an operation on . Let be an equivalence relation on . Assume that is an equivalence class of relation for all and . Let be a group, for , where , for all and . If for all , then there exists a surjective homomorphism from to .
Proof. Define by for all . Since , for all , is well defined. To prove that is a group homomorphism, let , then . Since is an equivalence relation on , the set is a partition for set , for . So, for each , we have such that because is a partition of ; thus, is surjective.
Clearly, the homomorphism in the above lemma is one-to-one if and only if .
4.1. Graph Homotopy Groups
In this section, we discuss about the relation between graph homotopy groups with the fundamental group of geometric space. First, we recall the concept of graph homotopy groups and some related concepts and properties in graph. For more information, see [7].
Let and be the graph . The distinguished base point and the boundary of the 1-cube of graph , respectively, are and . The extension of to a graph map for is for and for [7]. Clearly, the extension of a graph map is a special case of trivial extension of a path in the induced geometric space.
The family of homotopy classes of graph maps is denoted by where . The multiplication on is defined as the -homotopy class of the map by
Obviously, the multiplication on coincides with the product of two homotopy classes of induced paths in the corresponding geometric space.
Proposition 66. Let be a graph with ; then, there is an onto group homomorphism from to .
Proof. Let be a loop at in geometric space , so is a graph map by . Let , then, by Proposition 16, , where . Since the multiplication “” on coincides with the product “” on , by Lemma 65, by , for all , is an onto group homomorphism.
The following examples show that and sometimes are isomorphic and sometimes are not isomorphic.
Example 4. (1)Let be a graph; then, the induced geometric space is equal to (Figure 4). With respect to Example (1) in page 120 and Proposition 5.10 in [7], is a trivial group. By Example 3, as groups.(2)Let be a graph; then, the induced geometric space is equal to (Figure 4). With respect to Example (3) in page 120 and Proposition 5.10 in [7], .By a similar argument about the fundamental group of geometric space in Example 3, we have as groups.
Let be a graph with , and let be G-homotopic under G-homotopy map . If , for every and every , then the induced map of in corresponding geometric space is a homotopy map; thus, the homomorphism of Proposition 66 is an isomorphism.
4.2. Digital Fundamental Group
In this section, we consider the relation between the digital fundamental group and the fundamental group of geometric space. First, we recall the concept of digital fundamental group and some related concepts and properties in digital space. For more information, see [2, 10, 17, 18].
Recall from [2] that a pointed digital image is where is a digital image and . A pointed digital continuous function is a -continuous function such that . Let be pointed -continuous function. A -homotopy function , between and , is called a pointed digital homotopy if for [2].
A digital-path in a digital image is a -continuous function . It is called a digital-loop if ; in this case, is called the base point of the loop. If is a constant function, it is called a trivial loop [2].
Recall from [2, 18] that and be two paths such that , then the product of and is the path with
Let and be -loops in . is called trivial extension of if there exist sets of -paths and in such that , , and and there are indices such that for and is a trivial loop if . Two -loops and are pointed digital homotopic if there exist trivial extensions and of and , respectively, with the same cardinality in their domains and a pointed digital homotopy between and . The homotopy class of -loop is denoted by , and the family of all homotopy classes of -loops in is denoted by . The set is a group with product operation is a group which is called the digital fundamental group [2].
Let be a pointed digital image, then, by Example 1, the corresponding geometric space of it is . By Proposition 3, the digital -path in coincides with a path in the corresponding geometric space . Clearly, the product of two digital -paths coincides with the product of two induced paths in corresponding geometric space. Obviously, the trivial extension of a digital -loop coincides with the trivial extension of a loop in geometric space. Now, we can identify the relation between the digital fundamental group and the fundamental group of corresponding geometric space.
Proposition 67. Let be a pointed digital image; then, there is an onto group homomorphism from to .
Proof. Let be a loop at in geometric space , so is a digital -loop by . Let , then, by Proposition 14, , where . Since the multiplication “” on coincides with the product “” on , by Lemma 65, by , for all , is an onto group homomorphism.
The following examples show that and sometimes are isomorphic and sometimes are not isomorphic.
Example 5. (1)Consider the pointed digital image , then its corresponding geometric space is (Figure 5). By Example 5.4 in [10] and Theorem 4.16 in [2], is a trivial group. Consider with and , for . Clearly, is a good morphism, so is a homotopy map between and ; thus, is contractible; therefore, by Corollary 62, is a trivial group.(2)Let be , respectively. Let be a pointed digital image; then, the corresponding geometric space is , where (Figure 5). With respect to Theorem 4.16 in page 6 in [2], is a trivial group. By Example 3, .Let be a pointed digital image, and let be digital homotopic -loops at under digital homotopy map . If and , and and are -adjacent, for each and each , then the induced map of in corresponding geometric space is a homotopy map; thus, the homomorphism of Proposition 67 is an isomorphism.
4.3. Discrete Fundamental Group
In this section, we discuss about the relation between the discrete fundamental group of metric space with the fundamental group of geometric space. First, we recall the concept of discrete fundamental group (fundamental group at scale ) and some related concepts and properties in metric space. For more information, see [1].
Let be a metric space and , and let , recall from [1] that an -path from to is a finite sequence of points such that , , and , for . An -path from to is called an -loop based on if . The family of -loops based on is denoted by . The concatenation of two -paths and is defined as is a monoid under concatenation of -loops [1].
Recall from [1] that the -homotopy equivalence on is an equivalence relation which is defined as follows:(1)Each -loop is -homotopy equivalent to , where . It allows to increase the length of the sequence by adding to the end of the sequence.(2)Two -loops and are -homotopy equivalent if there is an -homotopy grid:where(a)each row is an -loop based on (b)each column is an -path
Let be a family of -homotopy equivalence class of ; then, with the operation of concatenation is a group which is called the fundamental group at scale. A metric space is called connected at scale if for all , there are such that , , and , for . If is connected at scale , then the fundamental group does not depend on . So, it is denoted by [1].
Proposition 68. Let be a metric space and . is an -path if and only if by , for , is a path in the geometric space .
Proof. With respect to the definition of the geometric space in Example 1, if and only if in geometric space , for all , which completes the proof.
Obviously, the concatenation of two -paths in a metric space coincides with the product of two induced paths in the corresponding geometric space . Increasing the length of an -loop in by repeating the base point at the end of the sequence of the -loop, in the definition of -homotopy equivalence, is a particular case of trivial extension of the induced loop in the corresponding geometric space.
Proposition 69. Let be a metric space and . Let and ; these two -loops are -homotopy equivalent if and only if the induced loops in the corresponding geometric space are path homotopic.
Proof. Let by and , for , be the induced loops of -loops and , respectively.
Let -loops and be -homotopy equivalent, then, by definition of -homotopy equivalence, there is a gridsuch that each row is an -loop based at , and each column is an -path; moreover, and for . Define by , and for and . By Proposition 68, each row and each column of the above grid are a path in geometric space , so is a good morphism, and hence is a path homotopy map between the loops and .
Conversely, let the induced loops and be path homotopic; then, there exists a homotopy map with , , for and , for . Define gridBy Proposition 5, and are good morphisms for and , so by Proposition 68, each row of grid is an -loop at and each column of is an -path, which completes the proof.
Lemma 70. Let be a metric space and ; then, -loops and are -homotopic.
Proof. By definition, is -homotopy equivalent to . ConsiderClearly, each row is an -loop, and each column is an -path; thus, -loops and are -homotopy equivalent. Since -homotopy is an equivalence relation, it is transitive, which completes the proof.
Lemma 71. Let be a metric space and ; then, -loop is -homotopy equivalent to -loop .
Proof. By a similar argument of the proof of Lemma 70, the -loop is -homotopy equivalent to -loop . Repeating this argument completes the proof.
In the following, we compare the -homotopy equivalence class of an -loop in metric space with the homotopy equivalence class of the induced loop in the corresponding geometric space.
Proposition 72. Let be a metric space and . Let and be -loops, where . Let and be induced loops of and , respectively, in geometric space . If is a trivial extension of , then and are -homotopy equivalent.
Proof. By definition of trivial extension, there exist sets and of paths in geometric space and there exists a sequence such that for and is the trivial loop for all . Since and are defined by and respectively, for and , for and is equal to or for all ; thus, the proof is completed by Lemma 70 and 71.
By Proposition 72 and Proposition 69, we have the following corollary.
Corollary 73. Let be a metric space and . Let and ; then, the -homotopy equivalence class of is equal to the path homotopy class of the induced loop in geometric space .
Now, we can identify the relation between the discrete fundamental group of metric space and the fundamental group of corresponding geometric space.
Proposition 74. Let be a metric space with and ; then, there is a group isomorphism from to .
Proof. If , then by , for , is the induced loop of -loop . Since the concatenation of two -paths in a metric space coincides with the product of two induced paths in the corresponding geometric space , by Corollary 73 and Lemma 65, by , for all , is an onto group homomorphism. However, , for all , so the kernel of is empty; thus, is an isomorphism.
In the following examples, we compare the discrete fundamental group of Hawaiian earring and the fundamental group of geometric space of Hawaiian earring.
Example 6. Conciser to the well-known Hawaiian earring in Euclidean metric space , whereBy Proposition 74, the discrete fundamental group of is isomorphic to fundamental group of corresponding geometric space of . In the following, we calculate the fundamental group of induced geometric space at for multivalued radius and the fundamental group of an arbitrary geometric space of .
(a)
(b)
(c)
(d)
(e)
Case 1. Let where ; then and .(1)Let , then , ( in Figure 6); thus, by Proposition 28, each loop at on is homotopic to , where ; thus, is a trivial group.(2)Let , then ; thus, by a similar argument of case , each loop at on is homotopic to , where ( in Figure 6). Let be a regular inscribed hexagonal in the circle (1). Since , , for , where is the center of ; thus, the loop is homotopic to which is a trivial extension of loop . However, is a loop on ; thus, is homotopic to ; therefore, is a trivial group.(3)Let , then ; thus, by a similar argument of case , each loop at on is homotopic to , where ( in Figure 6).Let for . If for some , then . In the following, we show that a loop at on is homotopic to a loop at on when ; then, by transitivity of homotopy relation, all the loops at on and on are homotopic.
Let be a regular inscribed hexagonal in the circle , and let be in the intersection of and line , for , where is the center of (Figure 6). Clearly, , for . Now, we define a path from to . Select , for and , such that . Let be in the intersection of and line , for and ( in Figure 6). Clearly, for and for and . Let be the disc with radius and center and be the disc with radius and center , for and . Clearly, , for and .
Since , the discs cover the arc of from to , for . Since , for each , there exists a block such that , for . Similarly, there exists a block such that , for and ; thus, is a polygonal corresponded to paths and , for ; therefore, by Proposition 28, the paths and are path homotopic, for . By a similar argument, paths and are homotopic and and are homotopic; hence, by transitivity of the homotopy equivalence relation, loops and are homotopic.
We can continue some way like above to find other homotopic loops at on and , for , but it is not our issue. For some enough little , there exists such that the loops at on and are not homotopic, for , and the loops at on are not homotopic to , for . By a similar argument of fundamental group of in algebraic topology and Example 2 Case II, is isomorphic to free product of -copies of .
Case 2. Now, we define a geometric space on which is different from . Let be a geometric space where , and contains four arcs of circle like that in Figure 6, for all . By Case II in Example 2 and a similar argument of fundamental group of in algebraic topology, is isomorphic to free product of infinite copies of .
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.