Let DIFF denote the category of smooth manifolds, TOP the category of topological manifolds and PL the category of piecewise linear manifolds.
In Kervaire 1960 it is shown for the first time that there exists a topological manifold that does not admit a differentiable (and therefore certainly not a smooth) structure.
I am interested in knowing the motivation for Kervaire to find the invariant and the manifold discussed in the paper.
My thoughts on it: At the beginning of topology it was not known whether DIFF = TOP. One possible reason could therefore have been to show that they are not equal. Another reason I could think of might have been his intent to work towards a resolution of the differentiable generalized Poincaré conjecture. Maybe there are more possible motivations that I failed to think of. If anyone knows the true motivation I'd be very grateful to know it.