Simplicial Structures in Topology

210 VI Homotopy Groups
Exercises
1. Let X be the space defined by the union of the unit circle S 1 and the closed
segment [(4,0),(5,0)]; take the vertices a0 =(1,0) and a1 =(4,0). Prove that the
fundamental groups of X based at a0 and a1 are not isomorphic.
2. Let f : (Y,y0) → (X,x0) be a homotopy equivalence. Prove that π(Y,y0) ∼ =
π(X,x0).
3. Let (X,x0) and (Y,y0) be two topological-based spaces. Prove that π(X ×Y,x0 ×
y0) ∼ = π(X,x0) × π(Y,y0).
VI.2 Fundamental Group and Homology
Let ∗ be a generic base point of either nT 2 or nRP 2 ; by Theorem (VI.1.15),
π(nT 2 ,∗) ∼ = Gp(a1,b1,a2,b2,...,an,bn;a 1 1b 1 1a −1
1 b−1
1 ...a 1 nb 1 na −1
n b −1
n )
π(nRP 2 ,∗) ∼ = Gp(a1,a2,...,an;a 1 1a 1 1 ...a 1 na 1 n).
For n ≥ 2, these groups are not Abelian; to better understand whether they are
isomorphic, we resort to a little algebraic trick: we abelianize them. An element of
the form ghg −1 h −1 of a given group G is called a commutator of G; the subset of G
[G,G]={x ∈ G|x = ghg −1 h −1 , g,h ∈ G}
is a normal subgroup of G known as the commutator subgroup of G and is the
smallest subgroup H of G for which G/H is Abelian. The Abelian group G/[G,G]
is called the abelianized group of G.
(VI.2.1) Lemma. A group homomorphism φ : G → H induces a homomorphism
¯φ : G/[G,G] → H/[H,H]
such that the following diagram commutes:
G
��
G/[G,G]
φ
¯φ
��
H
��
��
H/[H,H]
Moreover, if φ is an isomorphism then also ¯φ is an isomorphism.