Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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60 II <strong>Simplicial</strong> Complexes<br />
order<strong>in</strong>gs differ by an odd permutation). Now let us move to K. Beg<strong>in</strong> by tak<strong>in</strong>g a<br />
partial order<strong>in</strong>g of the set X <strong>in</strong> such a way that the set of vertices of each simplex<br />
σ ∈ Φ is totally ordered; <strong>in</strong> this way, we obta<strong>in</strong> an order<strong>in</strong>g class – that is to say,<br />
an orientation – for each simplex. A simplicial complex whose simplexes are all<br />
oriented is said to be oriented.<br />
Let K =(X,Φ) be an oriented simplicial complex. For every n ∈ Z, with n ≥ 0,<br />
let Cn(K) be the free Abelian group def<strong>in</strong>ed by all l<strong>in</strong>ear comb<strong>in</strong>ations with coefficients<br />
<strong>in</strong> Z of the oriented n-simplexes of K; <strong>in</strong> other words, if {σ i n} is the f<strong>in</strong>ite<br />
set of all oriented n-simplexes of K, thenCn(K) is the set of all formal sums<br />
∑i miσ i n , mi ∈ Z (called n-cha<strong>in</strong>s), together with the addition law<br />
∑ i<br />
piσ i n +∑ i<br />
qiσ i n := ∑(pi + qi)σ<br />
i<br />
i n.<br />
If n < 0, we set Cn(K) =0. Now, for every n ∈ Z, we def<strong>in</strong>e a homomorphism<br />
∂n = ∂ K n : Cn(K) −→ Cn−1(K) as follows: if n ≤ 0, ∂n is the constant homomorphism<br />
0; if n ≥ 1, we first def<strong>in</strong>e ∂n over an oriented n-simplex {x0,x1,...,xn}<br />
(viewed as an n-cha<strong>in</strong>) as<br />
∂n({x0,x1,...,xn})=<br />
n<br />
∑<br />
i=0<br />
(−1) i {x0,...,�xi,...,xn} ;<br />
f<strong>in</strong>ally, we extend this def<strong>in</strong>ition by l<strong>in</strong>earity over an arbitrary n-cha<strong>in</strong> of oriented<br />
n-simplexes. The homomorphisms of degree −1, that we have just def<strong>in</strong>ed, are<br />
called boundary homomorphisms.<br />
(II.2.15) Lemma. For every n ∈ Z, the composition ∂n−1∂n = 0.<br />
Proof. The result is obvious if n = 1. Let {x0,x1,...,xn} be an arbitrary oriented<br />
n-simplex with n ≥ 2. Then<br />
∂n−1∂n({x0,x1,...,xn})=∂n−1<br />
n<br />
∑<br />
i=0<br />
(−1) i {x0,...,�xi,...,xn}<br />
= ∑(−1) ji<br />
i (−1) j−1 {x0,...,�xi,..., �x j,...,xn}.<br />
This summation is 0 because its addendum {x0,..., �x j,...,�xi,...,xn} appears twice,<br />
once with the sign (−1) i (−1) j and once with the sign (−1) i (−1) j−1 . �<br />
This important property of the boundary homomorphisms implies that, for every<br />
n ∈ Z,theimageof∂n+1 is conta<strong>in</strong>ed <strong>in</strong> the kernel of ∂n; us<strong>in</strong>g the notation Zn(K)=<br />
ker∂n and Bn(K)=im∂n+1, we conclude that Bn(K) ⊂ Zn(K) for every n ∈ Z. Thus,<br />
to each <strong>in</strong>teger n ≥ 0, we can associate the quotient group<br />
Hn(K;Z)=Zn(K)/Bn(K);<br />
to each n < 0, we associate Hn(K;Z)=0.