= Rank cancellation in the Euler-Poincare formula
{title2=$\dim C_i=b_i+r_i+r_{i+1}$}
Use the <cellular chain complex> or <simplicial chain complex> over $\mathbb Q$, rather than the infinite <singular chain complex>. Put $r_i=\operatorname{rank}\partial_i$, with $r_0=r_{d+1}=0$. The <rank-nullity theorem> gives $\dim\ker\partial_i=\dim C_i-r_i$; quotienting the <chain cycles> by the <chain boundaries> then gives $b_i=\dim C_i-r_i-r_{i+1}$. Each <boundary operator>'s rank occurs in adjacent degrees with opposite signs, so
$$
\sum_{i=0}^{d}(-1)^i\dim C_i=\sum_{i=0}^{d}(-1)^ib_i.
$$
Finiteness makes this cancellation an ordinary finite sum. This is the algebra behind the <cell-count formula for Euler characteristic>.
Back to article page
