Arbitrary Smooth structure

Question

Is it possible to give a smooth structure to any objects? Say two lines intersecting at a point. It seems there is a smooth structure though at the intersecting point it is not locally euclidean if one views it in $R^2$. Is there a limit at which one cannot assign smooth structure?

Answer 1

It is not always possible to assign a smooth structure to a geometric object, even if it is locally Euclidean. There exist 4-manifolds with no possible smooth structure.