Euler Characteristic in Algebraic Topology
Definition 1
Let us assume a simplex $\Delta$ is given. When the number of vertices of $\Delta$ is $n$, the number of edges is $m$, and the number of faces is $f$, the Euler Characteristic $\chi$ of $\Delta$ is defined as follows. $$ \chi := n - m + f $$
Theorem
Given the simplicial homology group formed from a simplicial complex $K$ for a topological space $X$, the Euler Characteristic can be generalized and calculated as follows.
Euler-Poincaré Theorem
The alternating sum of the $p$th Betti number $\beta_{p}$ of the homology group is called the Euler Characteristic of $X$. $$ \chi = \sum_{p \ge 0} \left( -1 \right)^{p} \beta_{p} $$ Thus, the Euler Characteristic is a Topological Invariant.
Euler-Poincaré Formula
Given boundary matrices $\partial_{p}$ and $\partial_{p-1}$ in Smith normal form, the following can be obtained for the number $z_{p}$ of Zero Columns in $\partial_{p}$ and the number $b_{p-1}$ of occurrences of $1$ in the diagonal elements of $\partial_{p-1}$. $$ \begin{align*} \chi =& \sum_{p \ge 0} \left( -1 \right)^{p} \left( z_{p} + b_{p-1} \right) \\ =& \sum_{p \ge 0} \left( -1 \right)^{p} \left( z_{p} - b_{p} \right) \end{align*} $$
Derivation
The detailed proof of the Euler-Poincaré theorem is omitted. Refer to Munkres2.
Efficient computation of homology groups: The $\mathcal{C}$ of Betti numbers for $H_{p} \left( \mathcal{C} \right)$ is called the $p$th Betti Number. The $\beta_{p}$ for a finite complex $K$ is as follows. $$ \beta_{p} = \rank Z_{p} - \rank B_{p} $$
The Betti number of a homology group can be efficiently calculated through the boundary matrix in Smith normal form, thus deriving the Euler-Poincaré formula from the Euler-Poincaré theorem.
Explanation
The Euler Characteristic of a topological space is an Invariant, touching the essence of the topological space, as the term Characteristic suggests. This provides a plausible and intuitive explanation in the context of topology known to the public. For example, if a cube and a sphere can be molded and shaped into each other like clay dough—being homeomorphic—then $\chi$ remains the same.
- Since a cube has $8$ vertices, $12$ edges, and $6$ faces, $$\chi = 8 - 12 + 6 = 2$$
- And a sphere has $1$ vertices and $1$ edges for creating an equator and $2$ faces for the hemispheres, $$\chi = 1 - 1 + 2 = 2$$
Relation to Betti Numbers
According to the Euler-Poincaré theorem, the concept of the Euler Characteristic defined in simplices is extended to general topological spaces, and it can be concretely calculated according to the Euler-Poincaré formula.
The fact that the Euler Characteristic is represented as the alternating sum of Betti numbers is an interesting statement itself because Betti numbers explain characteristics of topological spaces that cannot be grasped by the Euler Characteristic alone, thereby covering the intuitive understanding of the Euler Characteristic. It’s as if the Euler Characteristic has been decomposed into Betti numbers for each $p$ dimension, and scholars naturally have an interest in phenomena where Euler Characteristics are identical while Betti numbers differ.
Euler Characteristic in Graph Theory
Originally, the Euler characteristic was most famous in graph theory, where Euler’s polyhedron theorem or Euler formula is a theorem in Graph Theory for a connected planar graph that states $\chi = 2$. If the polyhedron becomes as simple as the simplex or not just any convex polyhedron we think of easily, then the generalized definition introduced in this post becomes necessary.
Euler Characteristic in Geometry
Defined as an integer satisfying the equation of Gauss-Bonnet theorem.
Euler Characteristic in Algebraic Topology
Defined as the alternating sum of Betti numbers for each dimension.