= Subdivision invariance of Euler characteristic
{title2=$\chi(K')=\chi(K)$}
In the illustrated <simplicial subdivision>, inserting the three <edge> midpoints turns one filled <triangle> into four. The counts change from $(V,E,F)=(3,3,1)$ to $(6,9,4)$, but $6-9+4=1$. Splitting a single <edge> adds one <vertex> and one <edge>, so its net contribution is $1-1=0$. Adding a diagonal inside a polygonal face adds one <edge> and one face, giving $-1+1=0$.
These local accounts explain common refinements. For arbitrary finite <simplicial subdivisions>, including <barycentric subdivision> in any <dimension>, the underlying <geometric realization> is unchanged. The <topological invariance of Euler characteristic> therefore guarantees the same alternating count, even when the individual counts change substantially.
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