In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. My question is: Which is the "easiest" non-smoothable manifold? Easiest in the sense that among all the non-smoothable manifolds, this manifold has the easiest construction process.
Thanks everyone for your help !!
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Best Answer
Here are explicit equations for nonsmoothable manifolds (all of which admit triangulations). I do not know if these are the "easiest" but they are surely much more explicit than a description of the E8-manifolds, which is constructed as a result of some infinite, and very implicit, process (Freedman's work).
Consider the homogeneous equation $$ z_1^5 + z_2^3 + z_3^2 +z_4^2 + z_5^2 +\sum_{j=1}^5 e^{j-1} z_j^6=0 $$ in the complex projective space $CP^5$. Here instead of $e$ one can take any transcendental number. Then the solution set of this equation is a piecewise-linear complex 4-dimensional (real 8-dimensional) manifold which is not homeomorphic to a smooth manifold.
See "Algebraic equations for nonsmoothable 8-dimensional manifolds" by N.Kuiper, Math. Publ. of IHES, 1967.