Groups of Embedded Manifolds

Abstract

This paper defines a O(Mn, v0) which generalizes the of framed homotopy n-spheres in Sn+k. Let Mn be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let v, be the normal bundle of some imbedding p: Mn -_ Sn + k, where 2k _ n + 3. Then O(Mn, v(,) is the set of h-cobordism classes of triples (F, Vn, f), where F: Sn + k T(v0) is a map which is transverse regular on M, Vn=F-1(Mn), and f=Ff Vn is a homotopy equivalence. (T(v,) is the Thom complex of v,.) There is a natural structure on 0(Mn, v1,), and O(Mn, v1,) fits into an exact sequence similar to that for the framed homotopy n-spheres. This paper attempts to generalize in a natural way a well-known exact sequence concerning framed homotopy spheres which is contained in the work of Novikov [11], Kervaire-Milnor [7], and Levine [10]. The author stumbled onto these results partly because of his efforts to prove imbedding theorems for manifolds in the metastable range, and partly because of his recent work on Browder-Novikov theory for maps of degree d, Idl I=0 (see [2]). ?2 describes the basic constructions used in this paper. The group of embedded manifolds, 6(M, v,), is defined in ?3. A fairly simple description of that is given towards the end of that section. ?3 also contains the main results about O(M, v,). In ?4 we discuss a few interesting open problems. The author would like to thank the referee for some helpful suggestions. 1. Notation. All manifolds will be C O, compact, and oriented. Maps will be transverse to boundaries. If Mn is a connected closed manifold, let [M] E HnM denote the orientation class. Recall that f: Vn Mn is said to have degree d, i.e., degf=d, if f*([V]) =d[M], where f*: HnV HnM is the map induced byf on the integral homology groups. As usual, Dk denotes the closed unit ball in Euclidean k-space Rk, i.e., Dk={(yl, Yy) Rk|y2+I*...*+y2 O} and D_ ={(Y1, ,Yk+) eRk+l y2+ *?. +y2+ 1?=1,Y?<0}. We have natural inclusions S?kSk+l and DkCDk+l. Let e=(l 0) E SO?Sk. Received by the editors September 5, 1969. AMS 1970 subject classifications. Primary 57D55, 57D99; Secondary 57D60.