It is well known that the common proof of invariance of domain (so dimension of a Manifold is a topological invariant).
I have heard that there is a theory of infinite dimensional manifolds, i.e topological spaces locally homeomorphic to $\mathbb{R}^\mathbb{N}$ or perhaps an even bigger space like $\mathbb{R}^\mathbb{R}$. Note that I don't know anything about the theory of infinite dimensional manifolds, just that they exist.
Now the classic proof of invariance of domain fails, because $S^\infty$ is contractible. So my question, is there any finer method, using tools from algebraic topology, where given a Manifold of arbitrary dimension, I can determine its dimension?
Thanks