My book is An Introduction to Manifolds by Loring W. Tu.
Let $N$ and $M$ be smooth manifolds with dimensions. Let $p \in N$. Let $F: N \to M$ be a smooth map.
Question 1. Are these correct?
A. If $F$ is injective, then $\dim N \le \dim M$, by this (from Momentum Maps and Hamiltonian Reduction by Juan-Pablo Ortega, Tudor Ratiu)
B. If $F$ is open, then $\dim N \ge \dim M$, by this (from Momentum Maps and Hamiltonian Reduction by Juan-Pablo Ortega, Tudor Ratiu).
C. If $F$ is a submersion, then $F$ is open and so $\dim N \ge \dim M$. Alternatively, we may use this.
D. If $F$ is an immersion, then $\dim N \le \dim M$, by this.
Question 2. Given that injections, immersions and submersions imply either $\dim N \le \dim M$ or $\ge$, I guess surjections imply one of those too. Which if any does surjection imply, and why?
I think it's the same as submersion (and open): $\dim N \ge \dim M$.
An example would be retraction $r(x) = \frac{x}{||x||}, r: \mathbb R^2 \setminus 0 \to S^1$ (I recall this is smooth. Not sure). At the very least, I think the example (if correct) proves that surjections definitely do not imply $\dim N \le \dim M$.