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.

  • 2
    $\begingroup$ This paper by Kervaire shows the existence of a topological manifold which does not admit any differentiable structure. I am not familiar however with sufficient conditions for a topological manifold to admit a differentiable structure. Maybe someone more experienced can answer this. $\endgroup$ – Edmundo Martins Sep 22 '18 at 0:25
  • 2
    $\begingroup$ See this. This paper gives non-smoothable homology classes in a Grassmannian. $\endgroup$ – Ted Shifrin Sep 22 '18 at 16:33
  • $\begingroup$ thanks for both of your suggestions. highly appreciating it! to both of you :-) $\endgroup$ – Zest Sep 22 '18 at 18:04

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