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Difference between revisions of "Incidence coefficient"

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be an oriented simplex in  $  \mathbf R  ^ {N} $,  
 
be an oriented simplex in  $  \mathbf R  ^ {N} $,  
 
i.e. a simplex in which a definite order of its vertices  $  a _ {i} $
 
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} ) $
+
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} $.  
 
be its oriented face opposite to  $  a _ {i} $.  
 
If  $  i $
 
If  $  i $
 
is even, then  $  t  ^ {n} $
 
is even, then  $  t  ^ {n} $
and  $  t _ {i}  ^ {n-} 1 $
+
and  $  t _ {i}  ^ {n-1} $
are coherently oriented, and the orientation of  $  t _ {i}  ^ {n-} 1 $
+
are coherently oriented, and the orientation of  $  t _ {i}  ^ {n-1} $
 
is induced by the orientation of  $  t  ^ {n} $;  
 
is induced by the orientation of  $  t  ^ {n} $;  
in this case they are assigned the incidence coefficient  $  [ t  ^ {n} :  t _ {i}  ^ {n-} 1 ] = + 1 $.  
+
in this case they are assigned the incidence coefficient  $  [ t  ^ {n} :  t _ {i}  ^ {n-1} ] = + 1 $.  
 
If  $  i $
 
If  $  i $
 
is odd, then  $  t  ^ {n} $
 
is odd, then  $  t  ^ {n} $
and  $  t _ {i}  ^ {n-} 1 $
+
and  $  t _ {i}  ^ {n-1} $
are non-coherently oriented, and they are assigned the incidence coefficient  $  [ t  ^ {n} :  t _ {i}  ^ {n-} 1 ] = - 1 $.
+
are non-coherently oriented, and they are assigned the incidence coefficient  $  [ t  ^ {n} :  t _ {i}  ^ {n-1} ] = - 1 $.
  
 
Suppose now that  $  t  ^ {n} $
 
Suppose now that  $  t  ^ {n} $
and  $  t  ^ {n-} 1 $
+
and  $  t  ^ {n-1} $
 
are elements (simplices) of a [[Simplicial complex|simplicial complex]] in  $  \mathbf R  ^ {N} $.  
 
are elements (simplices) of a [[Simplicial complex|simplicial complex]] in  $  \mathbf R  ^ {N} $.  
 
Then their incidence coefficient is defined as follows. If  $  t  ^ {n} $
 
Then their incidence coefficient is defined as follows. If  $  t  ^ {n} $
and  $  t  ^ {n-} 1 $
+
and  $  t  ^ {n-1} $
are not incident, then  $  [ t  ^ {n} :  t  ^ {n-} 1 ] = 0 $;  
+
are not incident, then  $  [ t  ^ {n} :  t  ^ {n-1} ] = 0 $;  
 
if  $  t  ^ {n} $
 
if  $  t  ^ {n} $
and  $  t  ^ {n-} 1 $
+
and  $  t  ^ {n-1} $
are incident, then  $  [ t  ^ {n} :  t  ^ {n-} 1 ] = 1 $
+
are incident, then  $  [ t  ^ {n} :  t  ^ {n-1} ] = 1 $
 
or  $  - 1 $,  
 
or  $  - 1 $,  
 
depending on whether they are coherently oriented or not.
 
depending on whether they are coherently oriented or not.
Line 44: Line 44:
  
 
$$ \tag{1 }
 
$$ \tag{1 }
[ - t  ^ {n} :  t  ^ {n-} 1 ]  = \  
+
[ - t  ^ {n} :  t  ^ {n-1} ]  = \  
[ t  ^ {n} :  - t  ^ {n-} 1 ]  =  -
+
[ t  ^ {n} :  - t  ^ {n-1} ]  =  -
[ t  ^ {n} :  t  ^ {n-} 1 ] ,
+
[ t  ^ {n} :  t  ^ {n-1} ] ,
 
$$
 
$$
  
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$$ \tag{2 }
 
$$ \tag{2 }
 
\sum _ { k }
 
\sum _ { k }
[ t  ^ {n} :  t _ {k}  ^ {n-} 1 ]
+
[ t  ^ {n} :  t _ {k}  ^ {n-1} ]
[ t _ {k}  ^ {n-} 1 :  t  ^ {n-} 2 ]
+
[ t _ {k}  ^ {n-1} :  t  ^ {n-2} ]
 
  =  0 ,
 
  =  0 ,
 
$$
 
$$
  
where the summation extends over all oriented simplices  $  t _ {k}  ^ {n-} 1 $(
+
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).
+
(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 $
+
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} $,  
 
be a subspace in  $  \mathbf R  ^ {n} $,  
 
let  $  \mathbf R _ {1}  ^ {n} $
 
let  $  \mathbf R _ {1}  ^ {n} $
be one of the half-spaces bounded by  $  \mathbf R  ^ {n-} 1 $,  
+
be one of the half-spaces bounded by  $  \mathbf R  ^ {n-1} $,  
 
and let in  $  \mathbf R  ^ {n} $
 
and let in  $  \mathbf R  ^ {n} $
 
be chosen an oriented vector basis  $  ( e _ {1} \dots e _ {n} ) $.  
 
be chosen an oriented vector basis  $  ( e _ {1} \dots e _ {n} ) $.  
 
Then  $  \mathbf R _ {1}  ^ {n} $
 
Then  $  \mathbf R _ {1}  ^ {n} $
and  $  \mathbf R  ^ {n-} 1 $
+
and  $  \mathbf R  ^ {n-1} $
 
are called coherently oriented if  $  ( e _ {2} \dots e _ {n} ) $
 
are called coherently oriented if  $  ( e _ {2} \dots e _ {n} ) $
is a basis in  $  \mathbf R  ^ {n-} 1 $
+
is a basis in  $  \mathbf R  ^ {n-1} $
 
and  $  e _ {1} $
 
and  $  e _ {1} $
 
is directed into  $  \mathbf R _ {1}  ^ {n} $.  
 
is directed into  $  \mathbf R _ {1}  ^ {n} $.  
 
Two cells  $  \sigma  ^ {r} $
 
Two cells  $  \sigma  ^ {r} $
and  $  \sigma  ^ {r-} 1 $
+
and  $  \sigma  ^ {r-1} $
 
are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively.
 
are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 20:44, 22 December 2020


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)
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