Morse inequalities and perfectness depend on the coefficient field

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. The weak and strong Morse inequalities, the Morse polynomial identity, and perfectness all depend on the coefficient field: only the Euler characteristic identity is coefficient independent (Morse Euler characteristic identity).

More precisely, for a fixed Morse function $f$ on a closed manifold the Morse numbers $m_k(f)$ do not depend on $F$ (Morse numbers and the Morse polynomial), while the Betti numbers $b_k(M;F)$ can change with $F$ when $H_{\ast }(M;\mathbb{Z})$ has torsion (Poincare polynomial of a space and of a pair over a field); hence a function may be perfect over one field and not over another (Perfect Morse function over a field). The correction polynomial $Q$ of the identity $M_f=P_{M,F}+(1+t)Q$ is the coefficientwise measure of the loss (Morse polynomial identity), and the Euler identity survives because ${\sum }_k(-1{)}^k{b}_k(M;F)=\chi (M)$ for every field.

• Why the inequalities depend on $F$. The left side $M_f$ is geometric, while the right side is built from the $F$-Betti numbers; the field enters only through the homology coefficients, and the correction polynomial absorbs exactly the difference. Changing the characteristic can alter boundary-matrix ranks and therefore change Betti numbers, so the numerical content of the inequalities is not an integral statement.
• Where the field does not enter. The alternating sum of the Betti numbers is field independent; this is the content of the Euler identity. The handle-side construction of the chain complex also works over any field, but the ranks of its boundary maps do depend on the field when the attaching data has torsion in its incidence numbers.
• An explicit instance is worked on the examples page for real projective space, where a Morse function is perfect over ${\mathbb{F}}_2$ and not perfect over fields of characteristic different from two.