On normal microbundles

Abstract

Topology Vol. 5, pp. 229-240. Pergamon Press, 1966. Printed in Great Britain ON NORMAL MICROBUNDLES MORRIS W. HIRSCH? (Received 6 January 1966) $1. INTRODUCTION JOHN MILNOR invented the microbundle [6, 71, and proved the following basic theorems: (A). (EXISTENCE OF INVERSE MICROBUNDLES). For any microbundle 5 over a polyhedron, there exists a microbundle q such that 5 @ II is triGal. (B). (STABLE EXISTENCE OF NORMAL MICROBUNDLES). Let M be a submanifold of V. Then M x 0 has a normal microbundle in V x R4 for sonze q. Lashof and Rothenberg [5] proved : (C) (STABLE ISOTOPY OF NORMAL MICROBUNDLES). If to and (I are normal nzicrobundles on M in. V, then for some k, to @ k its use requires addition of a trivial line bundle. Since every manifold is covered by a finite number of open sets that are homeomorphic to open sets in Euclidean space, (C) and (B) follow. It is well known that t The author is a Sloan Fellow.