View source for Euler characteristic

''of a finite CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364001.png" />''

The integer

<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/e/e036/e036400/e0364002.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364003.png" /> is the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364004.png" />-dimensional cells in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364005.png" />. It was given this name in honour of L. Euler, who proved in 1758 that the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364006.png" /> of vertices, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364007.png" /> of edges and the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364008.png" /> of faces of a convex polyhedron are connected by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e0364009.png" />. This relation was known, in an implicit form, already to R. Descartes (1620). It turns out that

<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/e/e036/e036400/e03640010.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640011.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640012.png" />-dimensional [[Betti number|Betti number]] of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640013.png" /> (the Euler–Poincaré formula). The Euler characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640014.png" /> is a homology, homotopy and topological invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640015.png" />. In particular, it does not depend on the way in which the space is partitioned into cells. Consequently one can speak, for example, of the Euler characteristic of an arbitrary compact polyhedron, meaning by it the Euler characteristic of any of its triangulations. On the other hand, the Euler–Poincaré formula permits the extension of the concept of the Euler characteristic to a larger class of spaces and pairs of spaces for which the right-hand side of the formula remains meaningful. This formula has been generalized to the case of an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640016.png" />, where it expresses the Euler characteristic in terms of the dimensions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640017.png" /> of the homology groups with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640018.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/e/e036/e036400/e03640019.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640020.png" /> be a locally trivial fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640021.png" />. If the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640024.png" /> satisfy certain conditions, then their Euler characteristics are connected by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640025.png" />. In particular, the Euler characteristic of the direct product of two spaces is equal to the product of their Euler characteristics. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640026.png" />, which holds for any excisive triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640027.png" />, makes it possible to compute the Euler characteristics of all compact two-dimensional manifolds. The Euler characteristic of a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640028.png" /> handles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640029.png" /> deleted open discs is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640030.png" />, while that of a sphere with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640031.png" /> Möbius strips and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640032.png" /> deleted discs is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640033.png" />. The Euler characteristic of an arbitrary compact orientable manifold of odd dimension is equal to half that of its boundary. In particular, the Euler characteristic of a closed orientable manifold of odd dimension is zero, since its boundary is empty.



====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640034.png" /> is homotopic to the identity mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640035.png" />, then the Lefschetz fixed-point theorem (cf. [[Lefschetz theorem|Lefschetz theorem]], [[#References|[a1]]]) states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640036.png" /> is non-zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036400/e03640037.png" /> must have a [[Fixed point|fixed point]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)  pp. 156</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.R. Munkres,   "Elements of algebraic topology" , Addison-Wesley  (1984)</TD></TR></table>