The classification of PL fibrations.
Copyright policy )The classification of PL fibrations.
F. Waldhausen has developed machinery to compute the rational homotopy of a space \(\Omega Wh^{PL}(K)\) which conjecturally classifies PL fibrations with fibers of the homotopy type of a PL space K and with fiber homotopy trivilizations [\textit{F. Waldhausen}, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math. 32, Part I, 35-60 (1978; Zbl 0414.18010)]. That this space did classify these PL fibrations was one of several important assertions claimed by \textit{A. E. Hatcher} [Ann. Math., II. Ser. 102, 101-137 (1975; Zbl 0305.57009)]. Hatcher's assertions, together with the work of several authors (e.g. Burghelea, Dwyer, Hsiang, and Staffeldt) that carry out the computations of Waldhausen, yield many important results concerning the homotopy groups of automorphism groups of manifolds. The concepts and ideas presented in Hatcher's paper, although insightful and influential, are now known to be insufficient. The paper under review gives a proof that \(\Omega Wh^{PL}(K)\) does indeed classify these PL fibrations. The proof builds upon Hatcher's idea of using iterated mapping cylinders to classify PL fibrations. Although these iterated mapping cylinders are well-defined and functorial in the topological category, it is not so in PL. The author generalizes \textit{M. M. Cohen}'s simplicial mapping cylinder [ibid. 85, 218-245 (1967; Zbl 0147.426)] to an iterated mapping cylinder and then generalizes arguments of J. H. C. Whitehead to show that this construction is fiberwise homeomorphic to the topological iterated mapping cylinder. By building on more highly structured versions of Hatcher's constructions, the author finally obtains the classification. The proofs of the other assertions claimed in Hatcher's paper have been announced but have not yet appeared in print.
PL fibrations, Flatness and tameness of topological manifolds, homotopy groups of automorphism groups of manifolds, Classification of fiber spaces or bundles in algebraic topology, Waldhausen K-theory, simplicial mapping cylinder, iterated mapping cylinders, 57Q60, 57Q10, Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), PL-topology, PL-Whitehead space of a PL-space, algebraic K-theory of topological spaces, 55R15
PL fibrations, Flatness and tameness of topological manifolds, homotopy groups of automorphism groups of manifolds, Classification of fiber spaces or bundles in algebraic topology, Waldhausen K-theory, simplicial mapping cylinder, iterated mapping cylinders, 57Q60, 57Q10, Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), PL-topology, PL-Whitehead space of a PL-space, algebraic K-theory of topological spaces, 55R15
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