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

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... gh structure, it is seen that this iterative process corresponds to a skeletal filtration of BPL mod BO. Also, a geometric interpretation of the Hurewicz map nt(BPL, BO) -► Ht(BPL, BO) is inferred. I. Introduction. The object of this paper is to re-examine the differences between smooth and PL-manifolds via the geometry of PL-manifolds whose failure to be smooth may be measured, in some sense, by singularities. The germinal idea is this: From the ideas of Sullivan [a] and Baas [b] we know how to talk about a manifold with singularities; that is, a space which is of the form M"=M"0yjdMQs where M% is an «-manifold with boundary, S is of the form P"~r x cßr_ ', with P"~r, Qr~l manifolds, dM% =P"~r x Qf~l and c denotes unreduced cone. Now if M^,P"~r are smooth manifolds and Qr_1 = 2r_1 is a smoothness structure on the PL(r -1) sphere, we see that M" is a PL-manifold where lack of smoothness resides in the "singular" structure in a neighborhood of P"~r. We can iterate this process, i. e., we can find all PL-manifolds constructed in this way, using a certain set of exotic spheres as "allowable" singularities, and then go looking for exotic structures on spheres in this "new" category. We then use some of these spheres as singularities to create yet another category of singular manifolds (and so on), noting the important fact that at every stage the singular manifolds which we introduce retain underlying PL-manifold structures.