Simplicial Structures in Topology

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