Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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II.2 Abstract <strong>Simplicial</strong> Complexes 63<br />
0 1 2 0<br />
3 5 6 3<br />
4 7 8 4<br />
0 1 2 0<br />
Fig. II.10 A triangulation<br />
of the torus with oriented<br />
simplexes<br />
groups of boundaries and cycles). We notice that C2 ∼ = Z 18 , C1 ∼ = Z 27 , C0 ∼ = Z 9 .We<br />
represent the boundary homomorphisms <strong>in</strong> the next diagram<br />
0<br />
��<br />
C2<br />
∂2 ��<br />
C1<br />
∂1 ��<br />
C0<br />
Clearly, each vertex (and hence, each 0-cha<strong>in</strong>) is a cycle; hence, Z0 = C0. The<br />
elements {0}, {1}−{0}, ...{8}−{0} form a basis of Z0. Any two vertices can<br />
be connected by a sequence of 1-simplexes and so the 0-cycles {1}−{0}, ...,<br />
{8}−{0} are 0-boundaries. S<strong>in</strong>ce the boundary of a generic 1-simplex {i, j} can<br />
be written as<br />
∂1 ({i, j})={ j}−{i} = { j}−{0}−({i}−{0}),<br />
��<br />
0.<br />
we have that B0 ⊂ Z0 is generated by {1}−{0},...,{8}−{0} and thus,<br />
H0(T 2 ;Z) ∼ = Z .<br />
The homology class of any vertex is a generator of this group.<br />
Next, we compute H1(T 2 ;Z). The two 1-cha<strong>in</strong>s<br />
z 1 1 = {0,3} + {3,4} + {4,0} and z 2 1 = {0,1} + {1,2} + {2,0}<br />
are cycles and generate (<strong>in</strong> Z1) a free Abelian group of rank 2 which we denote by<br />
S ∼ = Z ⊕ Z. Let z ∈ Z1 be a 1-cycle z = ∑i kiσ i 1 ,<strong>in</strong>whichσi 1 are the 1-simplexes<br />
and ki ∈ Z. By add<strong>in</strong>g suitable multiples of 2-simplexes, it is possible to f<strong>in</strong>d a<br />
1-boundary b such that the 1-cycle z − b does not conta<strong>in</strong> the terms, which correspond<br />
to the diagonal 1-simplexes {0,5}, {1,6}, ..., {7,2}, {8,0}. Similarly,<br />
add<strong>in</strong>g suitable pairs of adjacent 2-simplexes (those form<strong>in</strong>g squares with a common<br />
diagonal) it is possible to f<strong>in</strong>d a 1-boundary b ′ such that the cycle z−b−b ′ conta<strong>in</strong>s<br />
only the terms correspond<strong>in</strong>g to the 1-simplexes {0,3}, {3,4}, {4,0},{0,1}, {1,2},<br />
and {2,0} (we leave the details to the reader, as an exercise). Because z−b−b ′ is a<br />
1-cycle, it follows that z − b − b ′ ∈ S. This argument shows that B1 + S = Z1. Letus<br />
now suppose that B1 ∩ S �= 0; then there exists a l<strong>in</strong>ear comb<strong>in</strong>ation of 2-simplexes<br />
have a common<br />
∑ j h jσ j<br />
2 such that ∑ j h j∂σ j<br />
2 ∈ S. If two 2-simplexes σ i 2<br />
and σ j<br />
2