A Simple Proof of the Algebraic Unknotting of Spheres in Codimension Two

Abstract

By a knot we mean a locally flat PL embedding f:Sn Sn+2 of the nsphere in the (n + 2)-sphere. By Alexander duality the complement of the knot C = Sn+2 f(Sn) has the homology groups of the circle S1, but C is not, in general, homotopy equivalent to S1. The purpose of this note is to give a simple proof of the following theorem. THEOREM 1. Let f:Sn _ Sn+2 be a knot with complement C. If ri(C) =ri (SDfor i c (n + 1)/2, then C is homotopy equivalent to S'. This theorem is known from previous work (for n = 1 by [1], for n = 2 by [2, 8], for n = 3 by [4], and for n 2 4 by [3]), but the available proofs are all rather long and technical. One can also combine our proof of Theorem 1 with the s-cobordism theorem (as done in [7, p. 92]) to show that for n 2 4 the knot is actually trivial, thus avoiding the ambient surgery techniques of [3]. That Theorem 1 is the best possible result can be seen by considering [5, p. 25]. Proof of Theorem 1. First we replace C by a finite CW-complex. Let X be the complement in Sn+2 of an open regular neighborhood of f(Sn). Thus X is a compact orientable (n + 2)-manifold with boundary S1 X sn, and is homotopy equivalent to C. We will show that X is homotopy equivalent to a CW-complex of dimension (n + 1)/2 or (n + 2)/2, depending on whether n is odd or even. If n is odd, this implies that the universal cover of X is contractible, and so X must be homotopy equivalent to S1. If n is even, we can then show (following the argument of Wall [9, Prop. 3.3] in the 2dimensional case) that X is homotopy equivalent to a wedge of S1 with some (n + 2)/2-spheres. Since X has the homology of S1, this implies that X is homotopy equivalent to S1. We use the following special case of a basic result of Wall which relates the cohomology with local coefficients in the group ring of a finite CW-complex to its homotopy dimension.