Orientation of a simplex

We have now described in detail how we may associate with a polyhedron a combinatorial structure called a simplicial complex. This can be done in many ways and thus the problem of deriving true topological invariants from the structure is non-trivial. We are concerned in this chapter to describe how certain important invariants may be obtained from a simplicial complex, but we shall not prove until the next chapter that they are, in fact, invariants of the underlying polyhedron. We stress (as we did in the Introduction) that these invariants are computable from the complex; they are algebraic in nature and attach to a polyhedron certain abelian groups, its homology groups.

We now begin the description of the way in which the homology groups are obtained from a simplicial complex. The first step is to orientthe simplexes.

We recall that, in Euclidean geometry, two different sets of axes may or may not determine the same orientation of the space. In considering the orientation of a simplex, the role of an ordering of a set of axes is played by an ordering of the vertices. Two orderings of the vertices are said to determine the same orientation of the simplex if and only if an even permutation transforms one ordering into the other; if the permutation is odd, the orientations are said to be opposite.

Suppose that the simplex sp has been oriented by selecting an ordering of the vertices; we call the pair consisting of sp and its orientation an oriented simplex and write it as σp or + σp.