PL approximations of fiber preserving homeomorphisms of vector bundles (Q1375781)

The group of fiber-preserving (briefly, f.p.) homeomorphisms of an \(n\)-dimensional vector bundle \(\xi\equiv (\pi:E\to X)\) over a countable-dimensional space \(X\) is investigated. The main theorem asserts that in case \(n\geq 5\), every stable f.p. homeomorphism \(h:E\to E\) can be approximated by f.p. PL homeomorphisms in the sense that for every open cover \({\mathcal U}\) of \(E\) there is a f.p. PL homeomorphism \(g:E\to E\), \({\mathcal U}\)-near to \(h\). (A f.p. homeomorphism \(h:E\to E\) is stable if \(h= h_1\cdots h_n\), where each f.p. homeomorphism \(h_i\) is PL on an open neighborhood \(U_i\) of the image \(s_i(X)\) of some section \(s_i:X\to E\)). This theorem is a fibrewise counterpart of a known result of \textit{E. H. Connell} [Ann. Math., II. Ser. 78, 326-338 (1963; Zbl 0116.14802)]. As an application it is shown that if \(X\) is compact and \(n\geq 5\), then the subgroup \({\mathcal H}^{\text{PL}} (\xi)\) of f.p. PL homeomorphisms of \(\xi\) has locally homotopy negligible complement in the group \({\mathcal H}^{\text{PL}_1} (\xi)\) of f.p. homeomorphisms of \(\xi\) which are PL on the open unit ball.