Exotic singular structures on spheres

Levitt, Norman

doi:10.1090/S0002-9947-1975-0377909-0

Exotic singular structures on spheres
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Trans. Amer. Math. Soc. 205 (1975), 371-388 Request permission

Abstract:

It is shown how the category of PL-manifolds may be obtained from the smooth category by an iterative procedure, viz., first form singular smooth manifolds where smooth seven-spheres are allowed as links. Then, in the new category one has obtained, kill all eight-spheres in similar fashion. Repeating this process ad infinitum (but requiring only finitely many stages in each dimension), one obtains the category of PL-manifolds. By taking care that the set of “singular” points is always given enough structure, it is seen that this iterative process corresponds to a skeletal filtration of $BPL \bmod BO$. Also, a geometric interpretation of the Hurewicz map ${\pi _ \ast }(BPL,BO) \to {H_ \ast }(BPL,BO)$ is inferred.

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