Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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182 V Triangulable Manifolds<br />
may be represented respectively by 4n-agons (polygons with 4n sides) and 2n-agons<br />
whose boundaries are respectively 4n-poligonal l<strong>in</strong>es and 2n-poligonal l<strong>in</strong>es with<br />
sides identified pairwise and denoted by<br />
a 1 1 b1 1 a−1<br />
1 b−1<br />
1 ...a1 n b1 n a−1<br />
n b−1<br />
n ,<br />
a 1 1 a1 1 a1 2 a1 2 ...a1 n a1 n .<br />
(V.2.3) Remark. We po<strong>in</strong>t out to the reader that the <strong>in</strong>terpretations of T 2 and RP 2<br />
given <strong>in</strong> this section correspond to their block triangulations given <strong>in</strong> Sect. III.5.1:<br />
for the torus T 2 , we have only one 0-block e0 = {0}, two 1-blocks (e 1 1 correspond<strong>in</strong>g<br />
to side a and e 2 1 correspond<strong>in</strong>g to side b), and one 2-block e2 = T 2 � (e0 ∪ e 1 1 ∪ e2 1 );<br />
for RP 2 , we have one 0-block e0 = {0}, one 1-block e1 correspond<strong>in</strong>g to side a,<br />
and one 2-block e2 = RP 2 � (e0 ∪ e1). A similar situation occurs for the closed<br />
surfaces nT 2 and nRP 2 . In fact, such closed surfaces are triangulable, for they<br />
derive from triangulable 2-manifolds (the triangulability is preserved by connected<br />
sums). Moreover, one can easily see that nT 2 has a block triangulation with only<br />
one 0-block, 2n blocks <strong>in</strong> dimension 1,<br />
e(a) 1 1,e(a) 2 1,...,e(a) n 1 and e(b) 1 1,e(b) 2 1,...,e(b) n 1 ,<br />
represented respectively by the sides a1,...,an and b1,...,bn; f<strong>in</strong>ally, it has one<br />
2-block<br />
e2 = nT 2 � (e0 ∪ (∪ n i=1e(a)i1 ) ∪ (∪ni=1 e(b)i1 )) .<br />
As for nRP 2 , we have a block triangulation with only one 0-block e0, n 1-blocks<br />
e 1 1 ,...,en 1 correspond<strong>in</strong>g to the a1,...,an, and only one 2-block e2 = nRP 2 � (e0 ∪<br />
(∪ n i=1 ei 1 )).<br />
As a consequence, we have the follow<strong>in</strong>g<br />
(V.2.4) Theorem. The closed surfaces S 2 ,nT 2 , and nRP 2 (with n ≥ 1) are, pairwise,<br />
not homeomorphic.<br />
Proof. In order to prove this result, we compute the homology groups H1(S2 ;Z),<br />
H1(nT 2 ;Z), H1(nRP2 ;Z) and show that no two of them are isomorphic.<br />
We know that H1(S2 ;Z) ∼ = 0. We now compute the first homology group with<br />
coefficients <strong>in</strong> Z of the other two manifolds through the block triangulation that we<br />
have just described. Beg<strong>in</strong>n<strong>in</strong>g with nT 2 ,wegivee(a) i 1 and e(b)i 1 an orientation, by<br />
choos<strong>in</strong>g a generator β(a) i 1 and β(b)i 1 , respectively, for each<br />
•<br />
H1(e(a) i 1 , e(a) i 1 ;Z) and H1(e(b) i 1 ,<br />
•<br />
e(b) i 1 ;Z)