Simplicial Structures in Topology

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