Simplicial Structures in Topology

VI.2 Fundamental Group and Homology 211
Proof. We define the homomorphism
¯φ(g +[G,G]) := φ(g)+[H,H]
for every g +[G,G] ∈ G/[G,G]. �
Let us go back to the surfaces. The abelianized group of G = π(nT 2 ,∗) is the
group generated by the elements
together with the set of relations
hence,
a1,b1,a2,b2,...,an,bn
R = {a 1 1b 1 1a −1
1 b−1
1 ...a1nb 1 na −1
n b −1
n ;xyx −1 y −1 |x,y ∈ G};
π(nT 2 ,∗)/[π(nT 2 ,∗),π(nT 2 ,∗)] ∼ = Z 2n .
On the other hand, the abelianized group of H = π(nRP 2 ,∗) is the group presented
as Gp(S;R) where
S = {a1,a2,...,an}
R = {a 1 1a 1 1 ...a 1 na 1 n}∪{xyx −1 y −1 |x,y ∈ S};
therefore, the abelianized group of H is an Abelian group generated by the elements
together with the relation
a1,a2,...,an
2(a1 + a2 + ...+ an)=1
and thus, by setting h = a1 + ...+ an,wehave
π(nRP 2 ,∗)/[π(nRP 2 ,∗),π(nRP 2 ,∗)] ∼ = Gp(a1,...,an−1,h;2h) ∼ = Z n−1 × Z2.
Since Z2n and Zn−1 ×Z2 are not isomorphic, we conclude that the two fundamental
groups π(nT 2 ,∗) and π(nRP2 ,∗) are not isomorphic. By the way, this conclusion
proves once more that nT 2 and nRP2 are not homeomorphic.
What is interesting to note is that the abelianized groups of the fundamental
groups of nT 2 and nRP2 coincide with the corresponding homology groups
H1(nT 2 ;Z) and H1(nRP2 ;Z). Hence, it is reasonable to ask whether this is merely
a coincidence or it is a fact that holds true for specific types of polyhedra; the answer
to this question is found in the following result:
(VI.2.2) Theorem. The abelianized group of the fundamental group of a connected
polyhedron |K| is isomorphic to H1(|K|;Z).