Piecewise linear fibrations
A PL \((=\) piecewise linear) fibration is a Hurewicz fibration p: \(E\to B\) such that E, B are compact polyhedra and p is a PL map. Very important examples are PL manifold bundles (i.e. PL locally trivial bundles whose fibers are compact PL manifolds with boundary). The present paper establishes a PL link between the category of PL fibrations and the category of PL manifold bundles, and the author expects that this will provide an adequate framework to apply algebraic K-theory to the study of PL manifold bundles [cf. \textit{A. E. Hatcher}, Ann. Math. 85, 218-245 (1975; Zbl 0305.57009)]. (1) The classifying space of PL fibrations is a semi-simplicial complex \({\mathcal S}\) where a typical n-simplex is a PL fibration \(E\to \Delta\) n \((=\) standard n-simplex). Let \({\mathcal B}_ m'\) be the subcomplex of \({\mathcal S}\) having as a typical n-simplex a PL manifold bundle \(E\to \Delta\) n of fiber dimension m and satisfying an additional niceness condition. There is a natural map \({\mathcal B}'_{\infty}=dir \lim \{{\mathcal B}_ 1'\to {\mathcal B}_ 2'\to...\}\to {\mathcal S}\) which is proved to be a homotopy equivalence. This implies in particular that each PL fibration \(E\to B\) occurs as the 0-level of a PL fibration E'\(\to B\times [0,1]\) whose 1-level is a PL manifold bundle. (2) Let K be a compact polyhedron. A PL fibration rel K is a PL fibration p: \(E\to B\) such that \(B\times K\) is fiber-preserving PL embedded in E and each inclusion \(\{b\}\times K\to p^{-1}(b)\) is a homotopy equivalence. Let \({\mathcal S}(K)\) be the subcomplex of \({\mathcal S}\) having as a typical n-simplex a PL fibration \(E\to \Delta\) n rel K. For any compact PL manifold N, a PL manifold bundle rel N is a PL manifold bundle p: \(E\to B\) which is a PL fibration rel N such that \(B\times N\) is fiber- preserving PL collared in E. Let \({\mathcal C}(N)\) be the subcomplex of \({\mathcal S}(N)\) where a typical n-simplex is a PL manifold bundle \(E\to \Delta\) nrel N. The final result asserts that the natural map \({\mathcal C}_{\infty}(N)=dir \lim \{{\mathcal C}(N)\to {\mathcal C}(N\times [0,1]\to...\}\to {\mathcal S}(N)\) is a homotopy equivalence.
scientific article; zbMATH DE number 4067882