Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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II.2 Abstract <strong>Simplicial</strong> Complexes 59<br />
II.2.3 The Homology Functor<br />
We now def<strong>in</strong>e the homology functor<br />
H∗(−;Z): Csim → Ab Z ,<br />
Fig. II.9<br />
another important functor with doma<strong>in</strong> Csim.<br />
Let K =(X,Φ) be an arbitrary simplicial complex. We beg<strong>in</strong> our work by giv<strong>in</strong>g<br />
an orientation to the simplexes of K. Let σn = {x0,x1,...,xn} be an n-simplex<br />
of K; the elements of σn can be ordered <strong>in</strong> (n + 1)! different ways. We say that<br />
two order<strong>in</strong>gs of the elements of σn are equivalent whenever they differ by an even<br />
permutation; an orientation of σn is an equivalence class of order<strong>in</strong>gs of the vertices<br />
of σn, provided that n > 0. An n-simplex σn = {x0,x1,...,xn} has two orientations.<br />
A 0-simplex has clearly only one order<strong>in</strong>g; its orientation is given by ±1.<br />
If σn = {x0,x1,...,xn} is oriented, the simplex {x1,x0,...,xn} for example, is<br />
denoted with −σ. Ifn ≥ 1, a given orientation of σn = {x0,x1,...,xn} automatically<br />
def<strong>in</strong>es an orientation <strong>in</strong> all of its (n − 1)-faces: For example, if σ2 = {x0,x1,x2} is<br />
oriented by the order<strong>in</strong>g x0 < x1 < x2, its oriented 1-faces are<br />
{x1,x2} , {x2,x0} = −{x0,x2} and {x0,x1}.<br />
More generally, if σn = {x0,x1,...,xn} is oriented by the natural order<strong>in</strong>g of the<br />
<strong>in</strong>dices of its vertices, its (n − 1)-face<br />
σn−1,i = {x0,x1,...,�xi,...,xn} = {x0,x1,...,xi−1,xi+1,...,xn}<br />
(opposite to the vertex xi with i = 0,...,n) has an orientation given by (−1) i σn−1,i;<br />
we say that σn−1,i is oriented coherently to σn if i is even, and is oriented coherently<br />
to −σn if i is odd. We observe explicitly that the symbol � over the vertex xi means<br />
that such vertex has been elim<strong>in</strong>ated.<br />
We are now ready to order a simplicial complex K =(X,Φ). We recall that the<br />
technique used to give an orientation to a simplex was first to order its vertices <strong>in</strong><br />
all possible ways, and then choose an order<strong>in</strong>g class (there are two possible classes:<br />
the class <strong>in</strong> which the order<strong>in</strong>gs differ by an even permutation, and that <strong>in</strong> which the