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

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")

• I think we can do morse homology over an arbitrary ring. You need to chose orientations as for the case of integer coefficients, but no problems arise I believe. Mar 29, 2021 at 15:33
• Oh, yes of course. Choosing orientations fixes the issue I was worried about. I'll update the post. Mar 31, 2021 at 17:14

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.