On the fibre homotopy type of normal bundles.

Abstract

ON THE FIBRE HOMOTOPY TYPE OF NORMAL BUNDLES Morris W. Hirsch 1 . INTRODUCTION It was proved by Atiyah [1] that the fibre homotopy type of the stable normal sphere bundle of a manifold M is an invariant of the homotopy type of M. Theorem A below (discovered before I learned of Atiyah’s proof) gives an elementary proof of this fact, and also applies to nonstable cases. (See [5], for example. One purpose of the present work is to supply item 6 in the bibliography of [5].) The situation con- sidered is a homotopy—commutative diagram 1' M0 '—‘”M1 8o /31 U1 with f a homotopy equivalence, gi: Mi —g V embeddings (i = 0, 1), and U1 a closed tubular neighborhood of g1(M1). Theorem A implies that the normal sphere bundles of go and g1 are fibre-homotopically equivalent. Theorem B applies Theorem A to the problem of choosing go (given f and g) so that it will have as many independent normal vector fields as g1 . The proof of Theorem A in the case dim V 2 dim M + 3 depends on Lemma 2, due to Milnor, which states that if U0 is a closed tubular neighborhood of g0(M 0) inside int U1 , then U1 — int U0 is an h-cobordism between the boundaries bU1 and bU0 . This Lemma is no longer universally true if dim V = dim M + 2; Theorem C (which is independent from Theorems A and B) exhibits a special case where it is true. An immediate corollary is that if M0 X R1‘ is diffeomorphic to M1 gl Rk, then M0 gl Sk-1 is h— cobordant to M1 X Sk'1 . (The interesting case is k = 2.) All manifolds, immersions, and embeddings are smooth. Throughout the paper, Mo and M1 are compact unbounded manifolds of dimen- sion m, and V is a Riemannian manifold of dimension v. 2. FIBRE HOMOTOPY TYPE If at and B are bundles, then oz ~ 3 indicates that a and /3 are isomorphic, while 0: g 3 means that oz and B have the same fibre homotopy type. For this con- cept, the reader is referred to Dold Let f: M -g V be an immersion. If V is the normal vector space bundle of f, then 13‘ will denote the normal sphere bundle of f, and conversely. THEOREM A. Let gi: Mi —g V be embeddings (i = O, 1). Let U1 C V be a closed tubular neighborhood of g1(M1) such that g0(M0) C U1 . Let f: MO —g MI be a homotopy equivalence making a homotopy—commutative diagram Received July 31, 1964. 225