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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Exotic singular structures on spheres


Author: Norman Levitt
Journal: Trans. Amer. Math. Soc. 205 (1975), 371-388
MSC: Primary 57C25
DOI: https://doi.org/10.1090/S0002-9947-1975-0377909-0
MathSciNet review: 0377909
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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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DOI: https://doi.org/10.1090/S0002-9947-1975-0377909-0
Keywords: Manifold-like category, singular manifold, exotic structures on spheres, thickenings
Article copyright: © Copyright 1975 American Mathematical Society