= Euler characteristic and holes
{c}
{title2=$\chi=b_0-b_1+b_2-\cdots$}
For a finite <CW complex>, $b_0$ counts <connected components>: its <connected components> are its <path components>, as recorded by the <zeroth homology group>. For a finite <graph> regarded as a one-dimensional <CW complex>, no higher <homology groups> occur, so $b_1=E-V+b_0$ counts independent <graph cycles>, and $\chi=b_0-b_1$. For a connected planar <surface> obtained by removing $h$ disjoint open <disks> from the interior of a closed <disk>, $(b_0,b_1,b_2)=(1,h,0)$ and $\chi=1-h$; the <annulus> is $h=1$.
The word "hole" needs its <dimension>. A <two-sphere> encloses a cavity but has $b_1=0$ and $b_2=1$. A closed <orientable> <surface> of <genus> $g$ instead has $(b_0,b_1,b_2)=(1,2g,1)$ and $\chi=2-2g$: increasing the <genus> by one adds two independent one-dimensional <homology classes>. In higher <dimensions>, further <homology groups> enter with alternating signs; there is no universal "components minus handles" rule.
Back to article page
