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