Surgery and Geometric Topology

Ideas from the theory of topological stability of smooth maps are trans ported into the controlled topological category For example the controlled topologi cal equivalence of maps is discussed These notions are related to the classi cation of manifold approximate brations and manifold strati ed approximate brations In turn these maps form a bundle theory which can be used to describe neighborhoods of strata in topologically strati ed spaces Introduction We explore some connections among the theories of topological stability of maps controlled topology and strati ed spaces The notions of topological equivalence of maps and locally trivial families of maps play an important role in the theory of topological stability of smooth maps We formulate the analogues of these notions in the controlled topological category for two reasons First the notion of controlled topological equivalence of maps is a starting point for formulating a topological version of Mather s theory of the topological stability of smooth maps Recall that Mather proved that the topologically stable maps are generic for the space of all smooth maps with the C topology between closed smooth manifolds see Mather Gibson Wirthm( uller du Plessis and Looijenga The hope is to identify an analogous generic class for the space of all maps with the compact open topology between closed topological manifolds Controlled topology at least gives a place to begin speculations Second the controlled analogue of local triviality for families of maps is directly related to the classi cation of approximate brations between manifolds due to Hughes Taylor and Williams We elucidate that relation in x Another important topic in the theory of topological stability of smooth maps is that of smoothly strati ed spaces cf Mather Quinn initiated the study of topologically strati ed spaces and Hughes has shown that )manifold Mathematics Subject Classi cation Primary N N A Secondary R R N