Fibrator properties of manifolds (Q2577116)

Fibrators are closed manifolds \(N\) which provide instant recognition of approximate fibrations introduced by \textit{D. Coram} and \textit{P. Duvall} [Rocky Mt. J. Math. 7, 275--288 (1977; Zbl 0367.55019); Pac. J. Math. 72, 41--56 (1977; Zbl 0368.55016)]. A surjective map between ANRs has a chance of being an approximate fibration if each point preimage is a copy, up to shape, of the fixed manifold \(N\). Work in this area attacks various kinds of converses: more specifically, it seeks to identify those \(N\) such that any map \(p :M\to M\) defined on a manifold \(M\) and for which each \(p^{-1}(b)\) is homeomorphic (or homotopy equivalent, or shape equivalent, or\dots) to \(N\) must be an approximate fibration. Attention is restricted to maps having finite-dimensional images \(B\), and the question is parsed in several ways, most notably, by the codimension of \(N\) relative to \(M\). ``Propelling this effort is the belief in the benefits of quick and effortless detection of approximate fibrations''. The paper is a survey whose aim is to present a coherent overview of results describing fibrator properties. Fix a closed, connected \(n\)-manifold \(N\). A proper map \(p :M\to B\) defined on an \((n+k)\)-manifold \(M\) is said to be \(N\)-shaped if each fiber \(p^{-1}(b)\) has the shape of \(N\); \(N\) itself is called a codimension \(-k\) (orientable) fibrator if, for every \(N\)-shaped map \(p : M\to B\), where \(M\) is a (respectively, orientable) \((n+k)\)-manifold, and where \(B\) is finite-dimensional, \(p\) is an approximate fibration. When \(B\) is a polyhedron and \(p\) as above is PL, then \(p\) is \(N\)-like if each fiber collapses to an \(n\)-complex homotopy equivalent to \(N\), \(N\) is a codimension-\(k\)(orientable) PL fibrator if, for every \(N\)-like map \(p : M \to B\), where \(M\) is a (respectively, orientable) PL \((n + k)\)-manifold, \(p\) is an approximate fibration. \(N\) is a PL (orientable) fibrator if it is a codimensional-\(k\) (orientable) PL fibrator for all \(k\). Briefly, the paper is organized as follows: Section 1 spells out relevant properties of approximate fibrations; Section 2 defines the fibrator properties studied up to date; Section 3 reviews results about codimension-2 fibrator properties; Section 4 reviews the much shorter list of results about codimension-\(k\) fibrator properties, \(k > 2\); Section 5 does the same in a slightly special PL category, where the results are fairly extensive; Section 6 presents what is known about the \(N\) which are fibrators in all codimensions with respect to PL maps between PL triangulated manifolds; Section 7 details fibrator properties of low dimensional manifolds \(N\); Section 8 treats some issues about the structure of manifolds admitting \(N\)-shaped maps and approximate fibrations; and the last Section offers a collection of unsolved problems.