Mapping cylinder neighborhoods of some ANR’s
R. T. Miller · Bulletin of the American Mathematical Society · 1975
Let X be a closed subset of the interior of a manifold Q.A submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: dM -• X such that M is homeomorphic to the mapping cylinder of r, fixing dM and X in the natural way.Since mapping cylinders strong deformation retract to their targets, a set that possesses a mapping cylinder neighborhood in some manifold is a finite dimensional ANR.Moreover, if the set is compact, its mapping cylinder neighborhood is a compact manifold.It follows from Kirby and Siebenmann [K-S] that the neighborhood, hence the set, has finite homotopy type.Regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods.More generally, R. D. Edwards showed [E] that stably, locally finite cell complexes in manifolds have mapping cylinder neighborhoods.We prove the following THEOREM.Let M' be a manifold.Suppose M = M' U (open outside collar of dM'), and that M supports a fixed point free flow whose flow lines give an oriented foliation on M that is transverse and outward pointing on dM'.Suppose X is a locally compact ANR embedded as a closed, 1 -LC, codimension 4 subset of the interior of a manifold Q.Then X x M' has a mapping cylinder neighborhood in Q x M. AMS (MOS) subject classifications (1970). Primary 54C55, 57A40.