Perfect Morse function over a field

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $M$ be a closed smooth $n$-manifold, let $f:M\to \mathbb{R}$ be a Morse function and let $F$ be a field (Field). Write $m_k(f)$ for the Morse numbers and $M_f(t)$ for the Morse polynomial of $f$ (Morse numbers and the Morse polynomial), and $b_k(M;F)={\dim }_F{H}_k(M;F)$ for the $F$-Betti numbers with Poincare polynomial $P_{M,F}(t)={\sum }_k{b}_k(M;F)t^k$ (Poincare polynomial of a space and of a pair over a field).

Then $f$ is $F$-perfect, or perfect over $F$, when $m_k(f)=b_k(M;F)\text{for every }k,$ equivalently $M_f(t)=P_{M,F}(t)$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

Perfectness is always understood with respect to a field: it may hold over one field and fail over another when $H_{\ast }(M;\mathbb{Z})$ has torsion. No orientation of $M$ and no Morse-Smale condition is required by the definition.

• Equivalent forms. Since a polynomial over $\mathbb{Z}$ is determined by its coefficient sequence, the identity $M_f(t)=P_{M,F}(t)$ holds exactly when $m_k(f)=b_k(M;F)$ for every $k$; both polynomials have nonnegative integer coefficients and are zero in degrees outside $[0,n]$, so no degree-range correction is hidden.
• What perfectness asserts. It is equality in every weak Morse inequality at once; equivalently, it is the vanishing of the correction polynomial of the Morse polynomial identity proved later on this page.
• Existential status. The definition names a property of a pair $(f,F)$; it asserts nothing about existence, and it does not require $f$ to be excellent.