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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)]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504001.png" /> be an oriented simplex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504002.png" />, i.e. a simplex in which a definite order of its vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504003.png" /> has been chosen, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504004.png" /> be its oriented face opposite to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504005.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504006.png" /> is even, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504008.png" /> are coherently oriented, and the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i0504009.png" /> is induced by the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040010.png" />; in this case they are assigned the incidence coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040012.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040014.png" /> are non-coherently oriented, and they are assigned the incidence coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040015.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040017.png" /> are elements (simplices) of a [[Simplicial complex|simplicial complex]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040018.png" />. Then their incidence coefficient is defined as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040020.png" /> are not incident, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040021.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040023.png" /> are incident, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040025.png" />, depending on whether they are coherently oriented or not.
+
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.
 
Properties of incidence coefficients.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
[ - t  ^ {n} : t  ^ {n-1} ]  = \
 +
[ t  ^ {n} : - t  ^ {n-1} ]  = -
 +
[ t  ^ {n} : t  ^ {n-1} ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040027.png" /> is the oppositely-oriented simplex, i.e. the simplex oriented by an [[odd permutation]] of the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040028.png" />;
+
where $  - t  ^ {n} $
 +
is the oppositely-oriented simplex, i.e. the simplex oriented by an [[odd permutation]] of the vertices of $  t  ^ {n} $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\sum _ { k }
 +
[ t  ^ {n} : t _ {k}  ^ {n-1} ]
 +
[ t _ {k}  ^ {n-1} : t  ^ {n-2} ]
 +
= 0 ,
 +
$$
  
where the summation extends over all oriented simplices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040030.png" /> (for some definitions of a simplicial complex (2) holds only if completeness is required).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040031.png" /> be a subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040032.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040033.png" /> be one of the half-spaces bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040034.png" />, and let in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040035.png" /> be chosen an oriented vector basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040036.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040038.png" /> are called coherently oriented if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040039.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040041.png" /> is directed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040042.png" />. Two cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050400/i05040044.png" /> are coherently oriented if they are contained in a certain coherently-oriented half-space and subspace, respectively.
+
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====
 
====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=39856
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article