Relative knots and tangential equivalences

Abstract

Levine has studied the groups, 6m,n, of knotted homotopy n-spheres in Sm. These groups can be interpreted as n-cobordism classes of certain ball pairs (Dm, Dn). In this paper we define a relativized version, 6m-n(Vk), of these groups, where Vk is a /¿-manifold. dm,n(V) is the group of n-cobordism classes of pairs (VxDm, VxDn), although in practice we find it convenient to give the formal definition in terms of maps rather than manifold pairs. Of course, 0m,n(point)= &m,n as a special case. We shall show that the groups 6m,n(V) and their framed analogues ff$-n(V) fit into exact sequences (see §3) generalizing those of [11] and [4]. Finally, we use these groups to obtain necessary and sufficient conditions under which a "partial" stable tangential homotopy equivalence /: M*-> Wn (i.e., /|[n/2]-skeleton is tangential) is actually a stable tangential homotopy equivalence (see §9). We rely heavily on the techniques of Rn will denote n-dimensional real vector space with unit cube /" and unit discFn-all with their natural orientations. Let Sn~1 = 8Dn, Jn-1 = dln-ln-1, where int X or X denotes the interior of X. Let Cl (X) be the closure of X. Z and Z, will mean the integers and cyclic group of order /, respectively.

A classification of immersed knots

“…(It should be clear that 8m-n and 8f'n can be so interpreted. See also [1].) Define where we identify the normal bundle va with the element in 7rn_iSOi that it determines by the classification theorem of bundles over spheres.…”