# 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.

How to Cite This Entry:
Incidence coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incidence_coefficient&oldid=51041
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article