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2. Preliminaries

In this section, in order to obtain an easier explanation of the results pre- sented in later sections, some tools from algebraic topology are shown. These tools are standard in a classical course on this topic. Our main references are the book in by J. Munkres [4] Chapters 9-12, and Chapter 1 in [5]. Let X be a -connected topological space. We consider π1(X, x0) the set of loops in X based on a fixed point x0 ∈ X, up to homotopy. This set is actually a group, called the fundamental group of the space X, with the composing operation defined for [α] and [β] two classes in π1(X, x0), via [α][β] := [α ∗ β], where ∗ denotes concatenation ([4], Chapter 9, §51) of loops based on x0. It is an easy exercise to prove that for a path- X the group π1(X, x0) does not depend on the point x0, so we can write π1(X). A topological space is called simply connected if its fundamental group is the trivial group.

2 2.1. Group actions Let X be a topological space, and let G be group. An action from the group G onto the space X is a map:

τ : G × X → X for each g ∈ G (1) such that for each g ∈ G, the map τ|{g}×X := τg : X → X is a and satisfies:

i. τe = idX for e the identity element in G.

ii. τg(τh(x)) = τgh(x) for all g,h ∈ G and any x ∈ X.

For short, we set τg(x)= gx. The orbit space X/G is defined to be the quotient space obtained from X by means of the equivalence relation x ∼ gx for all x ∈ X and all g ∈ G. The equivalence class of x is called the orbit of x. Example 2.1. Let X be a set. Suppose that we have an action from a group G on X. We can define an action from the group G on the set ℘(X) (parts of X) via the formula

g(A)= {ga ∈ X | a ∈ A} for g ∈ G, A ⊂ X. (2)

It is a straightforward verification that the action defined in Equation 2 satisfies the conditions to be a well defined action on ℘(X). Definition 2.2. If G is a group of of X, the action of G on X is said to be properly discontinuous if for every x ∈ X there is a neighborhood U of x such that g(U) is disjoint from U whenever g 6= e. (e is the identity element of G.) A topological space X is defined to be locally path-connected if for each point x ∈ X there exists a neighbourhood Ux of x such that Ux is path-connected. Theorem 2.3. [[5], Ch.1 Proposition 1.40 Part (c)] Let X be a simply con- nected and locally path-connected topological space endowed with an action from a group G. If the action is provided to be properly discontinuous, the group G is isomorphic to the group π1(X/G).

3. Construction for a finitely presentable group G

Let G be a finitely presentable group with presentation hS | Ri where S is a non empty finite set and R a finite subset of FS, the free group generated by S. By definition, every r ∈ R is a word of FS with length(r) letters in the alphabet S. Let us denote r[i] the i−th letter of r, r[0] = e be the identity element of FS and r[i, j] = r[i]r[i + 1] ··· r[j] for 0 ≤ i ≤ j ≤ length(r). Note that there is a homomorphism ϕ : FS → G such that Ker(ϕ) = hRi and G =∼ FS/hRi; ′ therefore, two elements w, w ∈ FS will represent the same element in G if ′ ′ ϕ(w)= ϕ(w ). This will be denoted w =G w .

3 Theorem 3.1. Let G be a finitely presentable group. There exists a reduced presentation hS | Ri in the sense that for every s ∈ S and every r ∈ R, s 6=G e and r[i, j]=G e only for i =0, 1 and j = length(r).

Proof. The case when there is a s ∈ S such that s =G e is trivial, and if there exists r ∈ R and n

(R −{r}) ∪{r[n,m], r[1] ··· r[n − 1]r[m + 1] ··· r[length(r)]}

For each presentation hS | Ri we can construct its Cayley complex XG. To do this we will perform an inductive construction. Let the vertices or 0- cells be the elements of G; these will form the 0-skeleton of the complex XG,0. Inductively, we will attach (n+1)-cells to the n-skeleton XG,n with attachment n maps γn : S → XG,n. For each element (g,s) ∈ G×S we put a 1-cell connecting the vertices g and gs. Now, for every element (g, r) ∈ G × R we attach a 2-cell to the path given by the edges denoted by

(gr[0, 0], r[1])(gr[0, 1], r[2]) ··· (gr[0,length(r) − 1], r[length(r)]) or equivalently, the path that passes through the vertices

gr[0, 0], gr[0, 1],...,gr[0,length(r)].

Remark 3.2. We will denote by ΣG the set of cells of the Cayley complex XG. We can identify the 0-cells with G, the 1-cells with G × S and the 2- cells with G × R; therefore, we will denote the elements of ΣG with elements of G ∪ (G × S) ∪ (G × R). We want the of every cell to be homeomorphic to a closed ball in some Rn, or equivalently, for the attaching maps to be embeddings. By construction, it is sufficient that the attaching maps are injective. This means that each 1-cell has non overlapping endpoints and the closed path in the boundary of each 2-cell does not self-intersect. Without loss of generality, we can assume that we are working with a reduced presentation. With such presentation, the edge (g,s) has endpoints g,gs ∈ G and g =G gs would imply the contradiction s =G e. If the closed path in the boundary of a 2-cell (g, r) self-intersects, it has to self intersect at a vertex; this would mean that gr[0,n] =G gr[0,m] and therefore r[n +1,m]=G e for n 6=0, 1 or m 6= length(r) which would be a contradiction. The previous condition on the Cayley complex allows us to construct a sim- plicial complex performing a generalized barycentric subdivision ([6], pp. 80-82). Let Σ˜ G ⊆ ℘(ΣG) be the simplicial complex with n-simplexes corresponding to the non-empty chains with the order 4 given by the condition “is in the bound- ary of” in ΣG. This way our simpleces have n-cells of XG such that for every pair of them, one is in the boundary of the other; as an example, for every r ∈ R there is a 2-simplex {(e, r), (e, r[1]),e} ∈ Σ˜ G. We want Σ˜ G to be a topological space. With this in mind, we endow Σ˜ G with the induced

4 ′ ′ by inclusion; that is, the topology τ = {S ⊆ Σ˜ G : ∀σ, σ ∈ S (σ ∈ S ∧ σ ⊆ ′ σ) → σ ∈ S} with base {star(σ): σ ∈ Σ˜ G} and star(σ)= {λ ∈ Σ˜ G : σ ⊆ λ}, the usual definition for simplicial complexes.

Lemma 3.3. Σ˜ G is path connected, simply connected and locally path connected.

Proof. We know that there is a weak homotopy equivalence between Σ˜ G and its geometric realization |Σ˜ G| ([2], Theorem 2). Also, we know that the latter is homeomorphic to XG as Σ˜ G is the generalized barycentric subdivision of XG. Therefore, there is a weak homotopy equivalence between XG and Σ˜ G. In particular, 0= π0(XG, x0)= π0(Σ˜ G, σ0),

0= π1(XG, x0)= π1(Σ˜ G, σ0), which means that Σ˜ G is path connected and simply connected. We also have that a space with an Alexandrov topology is locally path connected ([2], Corol- lary of lemma 6).

Now, we want to define an action of G on Σ˜ G. For this, we will define an action of G over ΣG and for each σ ∈ Σ˜ G ⊂ ℘(ΣG), we define the action of G with gσ = {gx : x ∈ σ}. Note that this is well defined as an action on Σ˜ G if and only if the action of G over ΣG sends chains to chains, or equivalently, if the action of G over ΣG preserves the order induced by inclusion. Every action of G on a set A ⊆ ℘(ΣG) that can be defined this way will be called an induced action.

Lemma 3.4. Let G act on Σ˜ G ⊆ ℘(ΣG) as an induced action, σ ∈ Σ˜ G and g ∈ G different from the identity of G. If the action of g on ΣG does not have fixed points and the action of g on ΣG preserves the dimension of the n-cells, then star(σ) ∩ g(star(σ)) = φ. Proof. As σ is non-empty, let λ ∈ σ so that star(σ) ⊆ star({λ}) and therefore star(gσ) ⊆ star({gλ}). If we assume that there exists α ∈ star(σ) ∩ star(gσ), then α ∈ star({λ}) ∩ star({gλ}). Thus λ, gλ ∈ α but we have that λ 6= gλ as the action of g on ΣG does not have fixed points. Now, as α is a chain, λ is in the boundary of gλ or gλ is in the boundary of λ which would give us a contradiction as λ and gλ are n-cells of the same dimension.

We define an action of an element g ∈ G on ΣG for each n-cell with n=0,1,2 using Remark 3.2.This is:

• g ∈ G acting on the 0-cell h ∈ G is the 0-cell gh ∈ G • g ∈ G acting on the 1-cell (h,s) ∈ G × S is the 1-cell g(h,s) = (gh,s) ∈ G × S • g ∈ G acting on the 2-cell (h, r) ∈ G × R is the 2-cell g(h, r) = (gh,r) ∈ G × R

5 With the induced action of G over Σ˜ G, we can form a space Σ˜ G/G, the quotient space of Σ˜ G in which each point σ is identified with all its images gσ as g ranges over G. The points of Σ˜ G/G are thus the orbits Gσ = gσ : g ∈ G in Σ˜ G.

Theorem 3.5. G is isomorphic to π1(Σ˜ G/G).

Proof. It is clear that the action of G on ΣG we defined satisfies the hypothesis of Theorem 3.4 for every g ∈ G different from the group identity element. Therefore, the fact that the action is properly discontinuous follows directly from Lemma 3.4 and that {star(σ): σ ∈ Σ˜ G} is a base for Σ˜ G. Now, using Lemma 3.3, we have all the hypothesis of Theorem 2.3 (also including the path- connectedness assumed in all of Section 2) and therefore, using Theorem 2.3, we conclude that G is isomorphic to π1(Σ˜ G/G).

Now, we want to prove that Σ˜ G/G is finite. First we will characterize every orbit of Σ˜ G/G. Let Gσ ∈ Σ˜ G/G. As σ is a non-empty finite chain, there exists α = max σ. For all g ∈ G, G(gσ) = Gσ; therefore, without loss of generality, we can assume that α ∈{e} ∪ ({e}× S) ∪ ({e}× R). If we fix α, we have that there is only a finite number of β ∈ ΣG such that β 4 α and therefore there is a finite number of σ ∈ Σ˜ G/G with α = max σ. Furthermore, as G is finitely presentable, there is only a finite amount of α ∈{e} ∪ ({e}× S) ∪ ({e}× R) and we can conclude that Σ˜ G/G is finite.

Theorem 3.6. Let G have a finite presentation hS | Ri. Then G =∼ π1(Σ˜ G/G) with |Σ˜ G/G| =1+3|S| +Σr∈R (4length(r)+1)

Proof. We will prove the last statement of the theorem. Let Gσ ∈ Σ˜ G/G and α = max σ. As we saw in the last paragraph, we can assume α ∈{e} ∪ ({e}× S) ∪ ({e}× R). Therefore, there are three possibilities:

1. α = e: As e is minimal in ΣG, σ = {e}. 2. α = (e,s) for s ∈ S: σ ∈ {{α}, {α, e}, {α, s}}. 3. α = (e, r) for r ∈ R: As we can assume S ⊂ G, we can interpret r[0,i] ∈ FG as an element of G , and this case can be split in five: (a) σ = {α, r[0,i]} for i =0, ··· ,length(r) − 1. (b) σ = {α, (r[0,i], r[i + 1])} for i =0, ··· ,length(r) − 1. (c) σ = {α, (r[0,i], r[i + 1]), r[0,i]} for i =0, ··· ,length(r) − 1. (d) σ = {α, (r[0,i], r[i + 1]), r[0,i + 1]} for i =0, ··· ,length(r) − 1 (e) σ = {α}. and the formula |Σ˜ G/G| =1+3|S| +Σr∈R (4length(r) + 1) should be clear.

6 4. The case of G a finite abelian group

Despite the generality and beauty of the presentation described in the previ- ous section, it is important to highlight the difficulty in finding explicit examples using the results therein. Nevertheless, we know that finite abelian groups are more handleable than non-abelian ones. We realize that if a group G is an abelian group, we can find an explicit finite topological space X associated to the group G in the sense that π1(X) =∼ G. This is great news, since explicit examples can be considered. In general, for a finite abelian group G it is possible to define explicitly the topology on a finite set X such that π1(X) =∼ G. For the groups Zn =∼ Z/nZ there is a simpler way to do it than for an arbitrary group. Furthermore, by the known property π1(X ×Y ) =∼ π1(X)×π1(Y ) of the fundamental group together with the Theorem of Classification of Finite Abelian Groups, it is enough to present the construction for the group Zn for n ∈ N.

4.1. The construction for the group Zn

Let G = Zn with n> 1 and n ∈ N. Let us consider the space X˜G consisting of n disks D1,...,Dn with their boundary circles identified. There is a generator g of G which acts on this by sending Di to Di+1 via a 2π/n rotation with i being taken mod n. Let a1,...,a2n be 2n equidistant points on the boundary of X˜G and ordered ′ ′ ˜ ′ in the clockwise direction. Also, let a1,...,an be points in XG such that ai is the center of the disk Di for 1 ≤ i ≤ n. We denote bi,1,...,bi,2n as the 2n ′ open edges between a1,...,a2n and ai respectively for each 1 ≤ i ≤ n. Notice that the edges bi,1,...,bi,2n divide the disk Di in 2n areas. Let us denote the open part of these 2n areas by ci,1,...,ci,2n maintaining the coherence between these and the notation of the segments. The collection of all these elements will be denoted by T˜G. We proceed now to endow the set T˜G with a topology that makes continuous the map F : X˜G → T˜G which sends an element in X˜G to the element in T˜G that represents the location of the point. For instance, if x ∈ X˜G is located on the open edge represented by b1,1 then F (x)= b1,1. Thus, we will endow T˜G with the final topology with respect to the F . Now we are going to define the following equivalence relation in X˜G: x ∼ y if both elements are located in a region represented by the same element in T˜G. Then, we endow the set T˜G with the topology that makes this space homeomorphic to the quotient space X˜G/ ∼. This means that the map F = H ◦ q : X˜G → T˜G is continuous, where q : X˜G → X˜G/ ∼ is the quotient map and H : X˜G/ ∼ → T˜G is a homeomorphism. Notice that the map

q∗ : π1(X˜G) → π1(X˜G/ ∼) is surjective and hence we have that π1(T˜G) = 0. Thus, we have that T˜G is simply-connected, path-connected, locally path-connected and G is a covering space action on it . Therefore, the map

p : T˜G → T˜G/G

7 is the universal and hence π1(T˜G/G) =∼ G = Zn. Lastly, if we take into account the Fundamental Theorem of finitely gener- ated abelian groups and the fact that the fundamental group of a finite product of topological spaces is isomorphic to the finite product of the fundamental groups of the topological spaces, then it is possible to construct a finite topo- logical space with fundamental group G for G any finitely generated abelian group. Now we are going to show explicitly what the open sets in the resulting topology are. For 1

′ ′ ′ ′ ′ ′ A = {a1,...,a2n}, A = {a1,...,an}, B = {b1,...,b2n}, Bi = {bi,1,...,bi,2n}, Ci = {ci,1,...,ci,2n},

′ ′ n for 1 ≤ i ≤ n. LetΣ = A ∪ A ∪ B ∪ B ∪ C, where B = ∪i=1Bi and n C = ∪i=1Ci. Endow the set Σ with the following topology: for ci,j ∈ C the basic will be Uci,j = {ci,j }. For bi,j ∈ B the basic open set will be

Ubi,j = {bi,j } ∪ Uci,j−1 ∪ Uci,j = {ci,j−1,bi,j ,ci,j } with j taken mod n and for ′ ′ n ′ ′ ′ 1 bi ∈ B it will be Ubi = j=1 Ucj,i ∪{bi} = {c ,i,...,cn,i,bi}. For ai ∈ A the basic open set will be US = B ∪ C ∪{a } and for a′ ∈ A′ it will be U ′ = ai i i i i ai n ′ ′ ′ ′ ′ ′ j=1 Ubj,i ∪{bi−1,ai,bi} = {bi−1,ai,bi,b1,i,c1,i−1,c1,i,...,bn,i,cn,i−1,cn,i} with Sj taken mod n and i taken mod 2n. The collection of all these open sets to- gether with the form a topological basis because every intersection of any two elements is empty or contains at least one element of the form ci,j , and the latter implies that the open set Uci,j (which belong to the collection) is contained in the intersection.

′ ′ Now we define the following equivalence relation: ai ∼ aj iff i ≡ j mod 2. ′ ′ ai ∼ aj for all 1 ≤ i, j ≤ n. bi ∼ bj iff i ≡ j mod 2. bi,s ∼ bj,t iff 2(j − i) ≡ t − s mod 2n and ci,s ∼ cj,t iff 2(j − i) ≡ t − s mod 2n. Let Σ/ ∼ be the quotient space and

p :Σ → Σ/ ∼, the quotient map. Let g :Σ → Σ be the function defined by a′ if x = a′  i+2 i    ai+1 if x = ai    ′ ′ g(x)=  bi+2 if x = bi ,   bi+1,j+2 if x = bi,j     ci+1,j+2 if x = ci,j  with i taken mod n, j taken mod 2n. We define the group G = hgi with composition of functions as the binary operation. G is isomorphic to Zn because

8 gn(x)= x; furthermore, the action of the group on Σ is properly discontinuous:

g(U ′ ) ∩ U ′ = U ′ ∩ U ′ = ∅, g(U ) ∩ U = U ∩ U = ∅, ai ai ai+2 ai ci,j ci,j ci+1,j+2 ci,j g(U ′ ) ∩ U ′ = U ′ ∩ U ′ bi bi bi+2 bi = ∅, g(Ubi,j ) ∩ Ubi,j = Ubi+1,j+2 ∩ Ubi,j = ∅,

g(Uai ) ∩ Uai = Uai+1 ∩ Uai = ∅.

Next we are going to prove that Σ is path-connected and locally path- connected: assume there are U, V open sets with U 6= ∅ and V 6= ∅ such that U ∪ V = Σ and U ∩ V = ∅. Notice that (A ∪ A′) 6⊂ U because otherwise we would have V = ∅. Similarly, (A ∪ A′) 6⊂ V . Therefore, there exist x, y ∈ A ∪ A′ such that x ∈ U and y ∈ V . Thus, Ux ⊂ U and Uy ⊂ V , but Ux ∩ Uy 6= ∅ because x, y ∈ A ∪ A′. Hence, U ∩ V 6= ∅ which is a contradiction. Then, Σ is connected and also path-connected (these two invariants are equivalent in finite topological spaces). To see that Σ is locally path-connected the only thing remaining to be proved is that each one of the basic open sets is path-connected. For x, y ∈ Σ we de- note by fy,x : [0, 1] → Ux the function such that fy,x(a) = y for 0 ≤ a < 1, fy,x(1) = x. Notice that if Uy ⊂ Ux, then fy,x is continuous; in fact Uy ⊂ Ux for every x ∈ Σ and y ∈ Ux because Σ is finite. Hence, for any x ∈ Σ and y ∈ Ux the function fy,x is continuous. Thus, there exists a path connecting each element y in Ux with x and therefore Ux is path-connected. Finally, it will be proved that Σ is simply connected in order to apply Theo- rem 2.3 to show that Zn is the fundamental group of Σ/ ∼ and for this we need the following proposition. Proposition 4.1. Let f : I → T , g : I → T be two paths such that f(0) = g(0), f(1) = g(1) and for any x ∈ I, f(x) ∈ U ′ and g(x) ∈ U ′ . Then f and g are ai ai homotopic.

Proof. From [3] Remark 1.2.8 we have that Ua is simply connected and therefore f and g must be homotopic.

′ Let α : I → Σ be any loop with α(0) = α(1) = a1. For a fixed 1 ≤ i ≤ n, −1 the open set α (Uai ) can be viewed as the countable union of disjoint open intervals. Let (d, d′) be one of those intervals. Notice that α(d) ∈ A′ ∪ B′ and ′ ′ ′ ′ ′ α(d ) ∈ A ∪ B as well. Moreover, we have that A ∪ B ∩ Uai = ∅, and therefore ′ ′ there is ǫ> 0 such that α(x) 6= ai for x ∈ (d, d+ǫ)∪(d −ǫ, d ). If we restrict the function α to the interval [d + ǫ/2, d′ − ǫ/2] and consider it as a path, then by Proposition 4.1 we have that this path is homotopic to a path γ with ai 6∈ Im(γ). −1 Applying this process for every open interval in α (Uai ) and every i, we can construct a loop α1 homotopic to α such that A ∩ im(α1) = ∅. Let α2, α3 be the paths defined by

′ ′ aj if α1(x)= bi,j b1,j if α2(x)= aj  ′  α2(x)=  bj if α1(x)= ci,j , α3(x)=  . ′ ′ ′ α1(x) if α1(x) ∈ A ∪ B c1,j if α2(x)= bj  

9 −1 ′ ′ Notice that α2 is indeed a path because α2 (Ux) = ∅ for x 6∈ A ∪ B , −1 −1 −1 −1 ′ ′ ′ ′ 3 α2 (Uai )= α1 (Uai ) and α2 (Ubi )= α1 (Ubi ) for 1 ≤ i ≤ 2n. Similarly, α is −1 −1 −1 −1 a path because α (U )= α (U ′ ) and α (U )= α (U ′ ). 3 b1,j 2 aj 3 c1,j 2 bj The map H : [0, 1] × [0, 1] → T defined by

α1(s) if t ∈ [0, 1/4)  α2(s) if t ∈ [1/4, 1/2] H(s,t)=   α3(s) if t ∈ (1/2, 3/4)  a1 if t ∈ [3/4, 1]  1 −1 is continuous: for 1 ≤ i ≤ 2n we have H− (U ′ ) = α (U ′ ) × [0, 3/4) and bi 1 bi 1 −1 1 −1 − ′ ′ ) × [0, 3/4). For 1 < i ≤ n, H− (U ) = α H (Uai ) = α1 (Uai bi,j 1 (Ubi,j ) × −1 −1 [0, 1/4) and H (Uci,j )= α1 (Uci,j ) × [0, 1/4); in the case i = 1 we have

−1 −1 −1 H (Ub1,j )= α1 (Ub1,j ) × [0, 1/4) ∪ α3 (Ub1,j ) × (1/2, 3/4)   and

−1 −1 −1 H (Uc1,j )= α1 (Uc1,j ) × [0, 1/4) ∪ α3 (Uc1,j ) × (1/2, 3/4) .   And lastly, we have

−1 −1 H (Ua )= α (Ua ) × [0, 1/4) ∪ ((1/2, 1] × (1/2, 1]) . 1 1 1 

Then α1 is null-homotopic, and therefore α is null-homotopic as well because these two loops are homotopic.

4.2. Examples We present some examples by constructing finite topological spaces associ- ated to the finite groups Z2, Z3 and Z4 respectively. A useful way to describe finite spaces is by using Hasse diagrams (cf. [3] Example 1.1.1). The Hasse diagram of a partially ordered space X is a graph whose vertices are the points of X and whose edges are the ordered pairs (x, y) such that x

′ ′ ′ ′ I. For Z2 we have the set Σ2 = {a1,a2,a,b1,b2,b1,...,b4,c1,...,c4} with

basic open sets Uci = {ci} for 1 ≤ i ≤ 4, Ubi = {bi,ci−1,ci} for 1 ≤ ′ ′ 2 ′ i ≤ 4 with i taken mod 4, Ub1 = {b1}∪{ci ∈ Σ | i is odd}, Ub2 = ′ {b2}∪{ci ∈ Σ2 | i is even}, Ua = C ∪ B ∪{a} with C = {c1,...,c4} ′ ′ ′ ′ and B = {b1,...,b4}, Ua1 = {a1,b1,b2} ∪ C ∪{bi ∈ Σ2 | i is odd} and ′ ′ ′ ′ 2 Ua2 = {a2,b1,b2} ∪ C ∪{bi ∈ Σ | i is even}. Below is the Hasse diagram for Σ2:

10 ′ ′ a1 a a2

′ ′ b1 b1 b2 b3 b4 b2

c1 c2 c3 c4

Figure 1: Hasse diagram for Z2

′ ′ ′ ′ II. For Z3 we have the set Σ2 = {a1,a2,a,b1,b2,b1,...,b6,c1,...,c6} with

basic open sets Uci = {ci} for 1 ≤ i ≤ 6, Ubi = {bi,ci−1,ci} for 1 ≤ ′ ′ ′ i ≤ 6 with i taken mod 6, Ub1 = {b1}∪{ci ∈ Σ2 | i is odd}, Ub2 = ′ {b2}∪{ci ∈ Σ2 | i is even}, Ua = C ∪ B ∪{a} with C = {c1,...,c6} ′ ′ ′ 1 6 ′ 2 and B = {b ,...,b }, Ua1 = {a1,b1,b2} ∪ C ∪{bi ∈ Σ | i is odd} and ′ ′ ′ ′ Ua2 = {a2,b1,b2} ∪ C ∪{bi ∈ Σ2 | i is even}. The corresponding Hasse diagram is:

′ ′ a1 a a2

′ ′ b1 b1 b2 b3 b4 b5 b6 b2

c1 c2 c3 c4 c5 c6

Figure 2: Hasse diagram for Z3

′ ′ ′ ′ III. Finally, for Z4 we have the set Σ2 = {a1,a2,a,b1,b2,b1,...,b8,c1,...,c8}

with basic open sets Uci = {ci} for 1 ≤ i ≤ 8, Ubi = {bi,ci−1,ci} for ′ ′ 2 ′ 1 ≤ i ≤ 8 with i taken mod 8, Ub1 = {b1}∪{ci ∈ Σ | i is odd}, Ub2 = ′ {b2}∪{ci ∈ Σ2 | i is even}, Ua = C ∪ B ∪{a} with C = {c1,...,c8} ′ ′ ′ 1 8 ′ 2 and B = {b ,...,b }, Ua1 = {a1,b1,b2} ∪ C ∪{bi ∈ Σ | i is odd} and ′ ′ ′ ′ Ua2 = {a2,b1,b2} ∪ C ∪{bi ∈ Σ2 | i is even}. The corresponding Hasse diagram is:

Remark 4.2. The cardinality of the finite topological space XZn constructed

11 ′ ′ a1 a a2

′ ′ b1 b1 b2 b3 b4 b5 b6 b7 b8 b2

c1 c2 c3 c4 c5 c6 c7 c8

Figure 3: Hasse diagram for Z4

in this section for the group Zn is the same as in Theorem 3.6 of the previous section applied to the group Zn. However, from the discussion at the beginning of this section, the finite topological space associated to the group Zn ×Zm by the method above turns out to be XZn ×XZm . The cardinality of the latter is greater than the cardinality provided by Theorem 3.6 applied to the group Zn × Zm. Actually, neither of the two constructions presented in this paper provides us with a minimal finite topological space associated to a finitely presented group G, in the sense of having minimal cardinality in the class of spaces XG such that π(XG) =∼ G. The Hasse diagrams presented above show that the construction described throughout Section 3 provides reduced spaces in the sense that the topology described in XZn can not be reduced by using beat points [3].

References

[1] P. Alexandroff, Mat. sbornik, Diskrete Raume 2 (1937) 501–519.

[2] M. C. McCord, et al., Singular homology groups and homotopy groups of finite topological spaces, Duke Mathematical Journal 33 (3) (1966) 465–474. [3] J. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2011. URL https://books.google.com.co/books?id=KR3zCAAAQBAJ [4] J. Munkres, Topology, Featured Titles for Topology Series, Prentice Hall, Incorporated, 2000. [5] A. Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge, 2000. URL https://cds.cern.ch/record/478079 [6] A. T. Lundell, S. Weingram, The topology of CW complexes, Springer Sci- ence & Business Media, 2012.

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