Helli , i have question i Morse inequality why $$\sum_{q\geq0} M_q(a,b) t^q =\sum_{q\geq 0}\beta_q(a,b)t^q+(1+t)Q(t),$$ where $Q(t)$ is a polynomial with nonnegative integer coefficients
implise that:
1) $\displaystyle\sum_{j=0}^q (-1)^{q-j}M_j(a,b)\geq \sum_{j=0}^q (-1)^{q-j} \beta_j(a,b),q=0,1,2,...$ and
2) $\displaystyle\sum_{q=0}^{\infty}(-1)^q M_q(a,b)=\sum_{q=0}^{\infty} (-1)^q \beta_q(a,b)$
Since all the coefficients of $(1+t)Q(t)$ are non-negative integers, the initial equation says that $M_k(a,b)\ge\beta_k(a,b)$ for all $k$, and for all but a finite set of $k$, $M_k(a,b)=\beta_k(a,b)$. This implies 1).
If we set $t=-1$, the initial equation is just 2).