How do the strong/weak Morse inequalities depend on the ring we are working over?

Question

The Morse inequalities relate the number of critial points to global invariants of the manifold $M$. The weak version states the following: $$ \# \operatorname{Crit}_k f \ge \dim HM_k(M, \mathbb Z_2) = \dim H_k(M, \mathbb Z_2) ,$$ where $f: M \to \mathbb R$ is a Morse function, $\operatorname{Crit}_k$ is the set of critical points of index $k$, and $HM$ is the Morse homology over $\mathbb Z_2 := \mathbb Z / 2 \mathbb Z$, and $H$ is singular homology. The last equality follows from the fact that Morse homology is isomorphic to singular homology.

We often also consider Morse/singular homology over $\mathbb Z$ (this involves choosing orientations, but does not require $M$ itself being oriented), where we similarly have $$ \# \operatorname{Crit}_k f \ge \operatorname{rank} HM_k(M, \mathbb Z) = \operatorname{rank} H_k(M, \mathbb Z) .$$ A natural question is to ask if these inequalities are equivalent. The klein bottle shows that this is not the case. The homology groups over $\mathbb Z_2$ are $HM(M, \mathbb Z_2) = (\mathbb Z_2, \mathbb Z_2^2, \mathbb Z_2)$, so $\dim HM(M, \mathbb Z_2) = (1, 2, 1)$, but over $ \mathbb Z$, we have $H(M, \mathbb Z) = (\mathbb Z, \mathbb Z \oplus \mathbb Z_2, 0)$, so $\operatorname{rank} H(M, \mathbb Z) = (1, 1, 0)$. In this case, the inequalities over $\mathbb Z_2$ are stronger, as they for example state $\# \operatorname{Crit}_1 f \ge 2$, while the inequalities over $\mathbb Z$ state $\# \operatorname{Crit}_1 f \ge 1$.

This observation leads to related questions based on different versions of the Morse inequalities.

Main Question 1: How do the (weak) Morse inequalities depend on the chosen ring ($\mathbb Z$ vs $\mathbb Z_2$)? Are those over $\mathbb Z_2$ always stronger than those over $\mathbb Z$?

Main Question 2: There exists versions of the Morse inequalities over $\mathbb Z$ taking torsion rank into account. In the example of the klein bottle, these inequalities result in the same thing either way: it seems to be independent on whether we are working over $\mathbb Z$ or $\mathbb Z_2$. Is this is general true?

Related Question: How do the strong Morse inequalities, stating $$ \sum_{k=0}^{m} (-1)^{k+m} \# \operatorname{Crit}_k f\ge \sum_{k=0}^{m} (-1)^{k+m} \dim HM_k(M, \mathbb Z_2) $$ depend on the chosen ring? (Checking the case of the klein bottle, it is easy to see that they are not equivalent)

Finally, two more minor questions:

Minor question 1: Can we say something like $\# \operatorname{Crit}_k f \ge \operatorname{rank} H_k(M, R)$ for other PIDs $R$, e.g. $\mathbb Z_p$ for some prime $p$? (This could possibly give stronger inequalities depending on $R$? E.g. $R=\mathbb Z_5$, and $M$ is the lens space $M(5,3)$)

Minor Question 2: Smale's theorem states that Morse inequalities are sharp in high enough dimensions on closed simply-connected manifolds. Does this refer to inequalities over $\mathbb Z$, $\mathbb Z_2$ or do they become the same under these assumptions?

Answers to any of these questions are welcome.

Conjecture The inequalities over $\mathbb Z$, with the torsion rank taken into account are always the strongest (and sometimes stronger than what most people call the "strong Morse inequalities")

Answer 1

You have the following result for a ring $R$ which is a principal ideal domain. Let me begin by introducing some notation.

Let us denote by $b_k$ the Beti number $\operatorname{rank}H_k(M)$, by $\mu_k$ the minimum number of generators of the torsion submodule $T_k$ of $H_k(M)$, and by $m_k=\operatorname{Crit}_k f$.

Then:

• For every $k\geq 0$: $$m_k\geq b_k+\mu_k+\mu_{k-1}.$$

• For every $k\geq 0$: $$m_k-m_{k-1}+\cdots +(-1)^k m_0\geq b_k-b_{k-1}+\cdots +(-1)^k b_0 + \mu_k.$$

It was recently stated for Discrete Morse Theory in this preprint. But the original result for smooth Morse Theory dates back to Pitcher*.

*: E. Pitcher. Inequalities of critical point theory.Bull. Am. Math. Soc.,64(1):1–30, 1958.