For manifolds of the same dimension, are submersions equivalent to immersions?

Solution 1:

You are correct on all three points.

The differential is a map between tangent spaces. If both tangent spaces have the same (finite) dimension, then an injective map is also a surjective map and is thus an isomorphism.

A local diffeomorphism between manifolds of the same dimension is indeed just an immersion or a submersion, as injectivity, surjectivity, and being an isomorphism on the level of tangent spaces are all equivalent.

If we have a smooth homeomorphism, your linked answer shows that it is a diffeomorphism if and only if it is an immersion. We know that a homeomorphism must be a map between manifolds of the same dimension, so here immersion is equivalent to submersion.