View source for 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)|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" />.

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.

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>

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" />;

<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>

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).

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.

====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>