Surgery on piecewise linear manifolds and applications

Abstract

SURGERY ON PIECEWISE LINEAR MANIFOLDS AND APPLICATIONS BY WILLIAM BROWDER AND MORRIS W. HIRSCH 1 Communicated by S. Smale, March 30, 1966 1. Introduction and statement of results. In this note we indicate a method of performing surgery on piecewise linear ( = PL) mani- folds, and show how to prove piecewise linear analogs of theorems on the homotopy type and classification of smooth manifolds 2 (Browder [ l ] , Novikov [lO], Wall [13]). The basic principles are two: to use normal microbundles instead of normal vector bundles, and to put a differential structure a on a neighborhood V of an embedded sphere S CM that represents a homotopy class we wish to kill. Then smooth ambient surgery can be performed on V„, and the resulting cobordism triangulated. Let Mi, M 2 be closed PL w-manifolds embedded in S n+k with nor- mal microbundles vi, vi. A normal equivalence b: (Mi, vi)~~K-^2» ^2) is a microbundle equivalence b: v\—*vi covering a homotopy equiva- lence Mi—>ikf 2 . Let T(vi) be the Thorn complex of Vi (see [12]), and let CiCE.T n +kT(vi) be the homotopy class of the collapsing map S n+k —>T(vi). We call Ci a normal invariant for Mi. If dM^O, a similar construction defines a normal invariant for M as an element in w n +k(T(vM), T(vu\dM)). T H E O R E M 1. Let X be a 1-connected polyhedron satisfying Poincare duality in a dimension n^5. Let £ be a PL k-microbundle over X, and let aCzir n +kT{t;) be such that h(a) =&(g), where h: Tr n +kT(£)— : >H n+ kT(l;) is the Hurewicz homomorphism y ?«(£)> &))> then M has a normal microbundle induced from £, and ais a normal invariant of M; (b) If n is even, M- {point} has a normal microbundle induced from £. Work partially supported by the National Science Foundation (USA) and De- partment of Scientific and Industrial Research (UK) at the Cambridge Topology Symposium, 1964. We are informed that some of our results have been obtained independently by R. Lashof and M. Rothenberg.