i recently started studying differential topology and smooth manifolds. I'm really new to this area but a frequent question that came to my mind quite often is the following
consider a topological manifold $M$ that is not smooth (maybe lots of sharp edges on its surface). Are there any ways to "smoothen" such manifolds such that they become differentiable at every point ? If the answer is yes, under what circumstances i.e. what are the requirements for $M$ and how can this be done?
I already figured out that if $X$ is a topological space, $Y$ a differentiable manifold and there is a homeomorphism $$ \phi:X \to Y$$ $X$ can be equipped with a differentiable structure.
However, is there more to discover? I really do not know what exactly to search for or what books i should look for.
I highly appreciate any suggestion about keywords/theorems or preferably books that cover similar topics.
Thank you very much.