Incidence coefficient

A number characterizing the coherence of the orientations of incident elements of simplicial, polyhedral (CW-) and other complexes. The concept of the incidence coefficient and its properties necessarily enter into the definition of an arbitrary abstract complex (cf. Complex (in homological algebra)).

Let $ t ^ {n} = ( a _ {0} \dots a _ {n} ) $ be an oriented simplex in $ \mathbf R ^ {N} $, i.e. a simplex in which a definite order of its vertices $ a _ {i} $ has been chosen, and let $ t _ {i} ^ {n-1} = ( a _ {0} \dots a _ {i-1} , a _ {i+1} \dots a _ {n} ) $ be its oriented face opposite to $ a _ {i} $. If $ i $ is even, then $ t ^ {n} $ and $ t _ {i} ^ {n-1} $ are coherently oriented, and the orientation of $ t _ {i} ^ {n-1} $ is induced by the orientation of $ t ^ {n} $; in this case they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = + 1 $. If $ i $ is odd, then $ t ^ {n} $ and $ t _ {i} ^ {n-1} $ are non-coherently oriented, and they are assigned the incidence coefficient $ [ t ^ {n} : t _ {i} ^ {n-1} ] = - 1 $.

Suppose now that $ t ^ {n} $ and $ t ^ {n-1} $ are elements (simplices) of a simplicial complex in $ \mathbf R ^ {N} $. Then their incidence coefficient is defined as follows. If $ t ^ {n} $ and $ t ^ {n-1} $ are not incident, then $ [ t ^ {n} : t ^ {n-1} ] = 0 $; if $ t ^ {n} $ and $ t ^ {n-1} $ are incident, then $ [ t ^ {n} : t ^ {n-1} ] = 1 $ or $ - 1 $, depending on whether they are coherently oriented or not.

Properties of incidence coefficients.

$$ \tag{1 } [ - t ^ {n} : t ^ {n-1} ] = \ [ t ^ {n} : - t ^ {n-1} ] = - [ t ^ {n} : t ^ {n-1} ] , $$

where $ - t ^ {n} $ is the oppositely-oriented simplex, i.e. the simplex oriented by an odd permutation of the vertices of $ t ^ {n} $;

$$ \tag{2 } \sum _ { k } [ t ^ {n} : t _ {k} ^ {n-1} ] [ t _ {k} ^ {n-1} : t ^ {n-2} ] = 0 , $$

where the summation extends over all oriented simplices $ t _ {k} ^ {n-1} $ (for some definitions of a simplicial complex (2) holds only if completeness is required).

Analogously, for a suitable definition of coherence of orientation the incidence coefficient of two elements of a polyhedral complex can be defined. Let $ \mathbf R ^ {n-1} $ be a subspace in $ \mathbf R ^ {n} $, let $ \mathbf R _ {1} ^ {n} $ be one of the half-spaces bounded by $ \mathbf R ^ {n-1} $, and let in $ \mathbf R ^ {n} $ be chosen an oriented vector basis $ ( e _ {1} \dots e _ {n} ) $. Then $ \mathbf R _ {1} ^ {n} $ and $ \mathbf R ^ {n-1} $ are called coherently oriented if $ ( e _ {2} \dots e _ {n} ) $ is a basis in $ \mathbf R ^ {n-1} $ and $ e _ {1} $ is directed into $ \mathbf R _ {1} ^ {n} $. Two cells $ \sigma ^ {r} $ and $ \sigma ^ {r-1} $ are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively.

References

[1]	P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian)
[2]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)
[3]	A. Dold, "Lectures on algebraic topology" , Springer (1980)