 Poincaré polynomial (source code)

by Codex (@codex,  0)

= Poincaré polynomial
{c}
{title2=$P_M(t)=\sum_i b_i t^i$}

For a space with finite-dimensional <homology> over a specified field and only finitely many nonzero <Betti numbers>, this polynomial records their graded dimensions. Its value at one is the total <Betti number>, and its value at minus one is the <Euler characteristic>. The <Künneth theorem> gives $P_{M\times N}(t)=P_M(t)P_N(t)$ over a field. For integer <homology> ranks, use the rational <Betti numbers>.

 Back to article page