Every Group Is a Fundamental Group ∗ Abhishek Gupta Abstract We See That Every Group Can Be Realized As the Fundamental Group of a 2-Dimensional CW-Complex

Every Group is a Fundamental Group ∗ Abhishek Gupta Abstract We see that every group can be realized as the fundamental group of a 2-dimensional CW-complex.

1 Free Groups and Presentations

Free Groups Given a set S the free group on S, which we shall denote by F (S), is a group which has the following universal property: Given any group G and a set map f : S → G there is a unique group homomorphism fˆ : F (S) → G ˆ such that f|S = f. The free group on any set exists and it is the set of all words of finite length in elements of S. The verification that this is a group and satisfies the universal property is left as an exercise. Presentations A presentation is given by hS|Ri where S is the set of generators and R is the set of relators. The group described by the presentation is a free group on S quotiented by the smallest normal subgroup generated by R (or in other words the intersection of all normal subgroups of F (S) containing R). Intuitively we set the relators to identity in the free group. Consider for example,

1. ha, b, c|∅i is the free group on three generators as there are no relations.

2. ha, b|aba−1b−1i is the free abelian group on two generators(i.e. Z×Z) as we are setting ab = ba. More rigorously deﬁne the homomorphism f : F ({a, b}) → Z × Z and show that the kernel is precisely the smallest normal subgroup containing aba−1b−1. This proves that it is the free abelian group on two generators.

3. ha|ani is the cyclic group of order n.

m n −1 −1 4. ha, b|a , b , aba b i is the group Zm × Zn. Theorem 1. Every group has a presentation.

Proof. Take a set S of generators (which every group is assured to have) of G and f : S,→ G the inclusion, and take the relators as the kernel of the map fˆ : F (S) → G. Every finite group has a presentation in which S,R are both finite sets. For S take the whole group itself and for S take the multiplication table. Such groups are called finitely presented.

2 CW complexes

CW complexes are nice spaces that are extensively used in algebraic topology. They are defined inductively as follows. Let X0 be a discrete space (i.e. a space with discrete topology). Now attach 1-cells (closed 1 0 1 unit interval, lets denote them by e0), by forming the adjunction space X = X t Dα/ ∼ where ∼ is the 1 1 0 1 1 0 equivalence relation that identifies x ∈ Dα with ψα(x) ∈ X using maps ψα : ∂Dα → X . Repeat this n n−1 process to attach the n-cells (denoted as en) Dα, α ∈ S(n) to the (n − 1)-skelton X obtained in the 1 previous step where S(n) is some index set depending on n and the equivalence relation identifies x ∈ Dα ∗This can be found at http://home.iitk.ac.in/∼gabhi/

1 n n−1 with ψα(x) ∈ X . We could stop at some finite n or continue indefinitely. In the former case the complex so obtained is said to be a finite dimensional CW complex. In CW ‘C’ stands for closure-finite meaning the boundaries are attached to only finitely many lower simplices. ‘W’ refers to the weak topology i.e., after forming this space it is given the weak topology. Again we make this more clear by examples: 1. All simplicial complexes are CW complexes.

2. The Euclidean space of dimension n. For example say R3. Take the space X0 to be the discrete space of all points of R3 with integer coordinates. Now attach 1-cells to form the 1-skeleton X1 which is the usual good-looking obvious lattice in R3. Now attach 2-cells to form a 3-d array of boxes. Then attach 3-cells to ﬁll these boxes.

3. Sn. This is an example which shows that CW decomposition need not be unique. The numbers of cells of diﬀerent dimensions could change between two decompositions. Consider ﬁrst a 0-cell e0. 1 Attach to this, say, a e1 by joining both ends to e0 and you get S . If an e2 had been attached instead 2 of an e1 it would have resulted in S . Similarly attaching an en to e0 along the boundary gives an Sn. Another decomposition would be beginning with two 0-cells and attaching two 1-cells to get S1, then proceeding from this stage itself to attach two 2-cells as two hemispheres to get a S2. So in this n case the decomposition is S = e0 t e0 t e1 t e1 · · · t en t en/ ∼ where ∼ is a an equivalence relation that relates the boundaries of a k − cell to the k − 1 skeleton.

n 4. The projective RP . Well, the cell decomposition for this is e as this is obtained by identifying both n+1 the n-cells of S for all n. So the decomposition is just e0 t e0 t e1 t e1 · · · t en t en/ ∼ In fact most of the nice spaces that we deal with are homeomorphic to some CW complex. We will see more in the presentation complex examples.

3 Presentation Complex

For any group G we now deﬁne the presentation complex of G and show that it has G as its fundamental group. Let hS|Ri be a presentation of G. The complex P (G) has one 0-cell, which we shall refer to as x0. One 1-cell for each generator of hS|Ri attached to the zero cell in the obvious manner which results in a one dimensional CW complex, which is in fact a wedge of circles indexed by generators. We now attach one 2-cell along each word in the set of relators. For attaching a 2-cell along a word make a polygon of circumferences of the circles representing the letters of the word, if an inverse appears go in the other direction and ﬁnally attach the 2-cell to the boundary of the polygon. An example will make this clear. Consider the following wedge. Say, we were to attach a 2-cell along the word bc−1a2bd then we do it as shown:

a b x0 x0

d c−1 x0 d b x0 x0

b a

x x c 0 a 0

1. Wedge sum of circles. Consider the presentation ha, b|∅i. The presentation complex is a wedge of two circles (no 2-cells attached) as shown. Recall that the fundamental group of ﬁgure 8 is a free

2 group on two generators. This tells us that the presentation complex of G has π1 = G in this case. In fact this is always true and the space that we are looking for is nothing but this presentation complex as we shall see. In general, for a free group F (I) = hI|∅i the presentation complex is a wedge sum 1 1 of circles ∨iS , i ∈ I for some index set I, thus the fundamental group of ∨iS is F (I).

2. The presentation ha|∅i has the circle as its presentation complex. It is a special case of the previous part.

3. Consider the group G = ha, b|aba−1b−1i then the presentation complex is

b x0 x0

a−1 a

x0 x0 b−1

Its hard not to see that this is a torus and that the group G we began with was Z × Z. This tells that again π1(P (G), x0) = G. 4. The projective plane. The presentation ha|a2i has the complex:

a a

This is clearly the projective plane. Thus we have found out the fundamental group of projective plane.

5. The presentation ha, b|abab−1i. It has the following presentation complex. The Klein bottle group is

b x0 x0

a a

x0 x0 b−1

3 the Baumslag-Solitar group BS(1, −1). General BS(m, n) = ha, b|bamb−1a−ni, these are important for counterexamples in group theory(BS(2, 3) is non-Hopfian, so is the free group on any infinte set). Recall that the first homology group of Klien bottle was Z × Z/2. As an exercise one can show that abelianization of the above fundamental group agrees with the first homology group Z × Z/2 (map a to (0, 1) and b to (1, 0)).

6. Dunce caps (beware: different books define dunce caps differently). The presentation is ha|ani, i.e. the cyclic group. The presentation complex is easy to draw. The projective plane is a special case as it is a 2-cap.

Note: We have thus seen how we can calculate a lot of fundamental groups using this approach. We have basically worked backwards and started with a group, constructed its presentation complex and to our good luck we obtained a familiar space. More strongly we could use in most of the cases the van Kampen theorem on which the following theorem relies.

Theorem 2. Let hS|Ri be a presentation of G and P (G) be deﬁned as above. Then π1(P (G), x0) = G. To prove this theorem we shall use the van Kampen theorem which we will assume.

Theorem 3 (van Kampen). Let X be the union ∪αAα with Aα path connected all containing the basepoint. Then the theorem says that the homomorphism φ : ∗απ1(Aα, x0) → π1(X, x0) obtained by mapping a word to the corresponding loops is surjective if Aα ∩ Aβ is path connected. Moreover if Aα ∩ Aβ ∩ Aγ is path connected then the kernel of the above map is the smallest normal subgroup containing the elements of the −1 type iαβ(ω)iβα(ω) where iαβ is the natural map iαβ : π1(Aα ∩ Aβ) → π1(Aα). We see some quick applications to understand the above theorem. Using this theorem we prove the lemma that will ﬁnish it.

Theorem 4. Let (X, x0) be a pointed CW complex. Let (Y, x0) be the space obtained from X by attaching 2-cells. Then the natural map φ : X → Y is surjective and has the kernel N which is the normal subgroup generated by the elements of the type fα ∗ ψα ∗ fα. In other words adding 2-cells kills the boundary of the 2-cell in the fundamental group.