= Euler characteristic
{c}
{title2=$\chi$}
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The <Euler characteristic> compresses a finite <CW complex>'s <homology> into one integer: count its <topological cells> with alternating signs, or count its <Betti numbers> with the same signs. A filled <triangle> has <Euler characteristic> one; its <cycle graph> boundary has <Euler characteristic> zero. The missing face changes the answer even though the <vertices> and <edges> are identical.
For a finite <CW complex> $X$, including the <geometric realization> of a finite <simplicial complex>,
$$
\chi(X)=\sum_i(-1)^i\dim_\mathbb QH_i(X;\mathbb Q).
$$
Here $b_i=\dim_{\mathbb Q}H_i(X;\mathbb Q)$ is the rational <Betti number>. More generally this ordinary <homology> definition makes sense whenever every rational <homology group> is finite-dimensional and only finitely many are nonzero. An arbitrary infinite <CW complex> need not satisfy these assumptions; its formal infinite alternating cell count is not a definition of this invariant.
The usefulness is that a finite description can reveal something about a space without computing every <homology group>. By the <cell-count formula for Euler characteristic>, count pieces rather than solve all the <boundary operators>. If two spaces give different answers, <topological invariance of Euler characteristic> rules out a <homeomorphism>, and even a <homotopy equivalence>. Equal answers are less decisive: the <Euler characteristic of an annulus> and the <Euler characteristic of a torus> are both zero, while their <homology groups> differ. This is a compact diagnostic, with a clearly defined limit on what it can tell us.
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