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