Simplicial Structures in Topology

36 I Fundamental Concepts
where 0 ≤ k ≤ n and a stands for any element of S. LetF(S)=WS/E be the set of
equivalence classes defined by E in WS; we denote the equivalence class of a word
w with [w]. We give here the definition of a product in F(S) by juxtaposition:
[s ε1 1 ...sεn n ][s εn+1
n+1 ...sεn+m n+m ]=[sε 1
1 ...sεn+m n+m ].
(I.2.3) Lemma. The set F(S) with the operation juxtaposition is a group.
Proof. Clearly, the juxtaposition is associative; the identity is the class of the empty
word and is denoted with 1; finally, the inverse of [ω] =[s ε1 1 sε 2
2 ...sεn n ] is [ω] −1 =
[s −ε1 n s −εn−1 n−1 ...s −ε1 1 ]. �
The group F(S) is the free group generated by S; the elements of S are the
generators of F(S). When S has a finite number of elements, we may also write
F({s1,s2,...,sl})=〈s1,s2,...,sl〉.
It is useful to write the elements of F(S) without the square brackets. In practice,
it is usual to simplify the notation by means of integral exponents, not necessarily
±1, such as in s1 1s11 s11 s11 = s41 or s−1
2 s−1
2 = s−2
2 . It is also customary to write s instead
of the word s1 defined by s ∈ S; all this allows us to write equalities such as s2s−1 = s,
ss−1 = 1, and so on.
(I.2.4) Remark. Awordω = s ε1 1 sε 2
2 ...sεn n is reduced if, for every element a of S,the
“subword” aεa−ε does not appear in ω (that is to say, if it is not possible to do any
“cancellation”). In each class [ω]=[s ε1 1 sε 2
2 ...sεn n ], there is one and only one reduced
word. We may therefore define F(S) by considering only the reduced words defined
by S.
Given a nonempty subset R ⊂ F(S), letR be the intersection of all normal subgroups
of F(S) that contain R; the quotient group
F(S)/R = F(S;R)
is the group generated by the set S with the relations R in F(S). The elements of
F(S;R) are the lateral classes (modulo R)ofelementsofF(S); for each element ω ∈
F(S), ωR denotes its class modulo R. In practice, to construct F(S;R) we only take
reduced words of S and free them from all the subwords of R. IfS = {s1,s2,...,sl}
and R = {w1,w2,...,wr}, we normally write
〈s1,s2,...,sl | w1 = w2 = ···= wr = 1〉 = F(S;R).
Examples: 1. S = {s}, F(S) =〈s〉 �Z; this group could also be described as
S = {s,t}, R = {t}, 〈s,t |t = 1〉 = F(S;R) � Z.
2. S = {s}, R = {s 2 }, 〈s | s 2 = 1〉�Z2.
3. S = {s,t}, R = {sts −1 t −1 }, 〈s,t | sts −1 t −1 = 1〉�Z × Z, thefree Abelian group
generated by the elements s and t.
4. Any group G may be viewed as a group generated by elements and relations: take
S = G and RG = {(st) 1 t −1 s −1 | s,t ∈ G}; then, G � F(G;RG).