Inverse mapping theorem and local forms of continuous mappings

We present a homological version of the Inverse Mapping Theorem for open and discrete continuous maps between oriented topological manifolds, with assumptions on the degree of the maps, but without any assumption on differentiability. We prove that this theorem is equivalent to the known homological version of the Implicit Mapping Theorem. Additionally, we study conditions for a map between oriented topological manifolds to be locally like an injection or a projection, via alternative notions of topological immersions and submersions.