THE FUNDAMENTAL IDEAL AND tt2 OF HIGHER DIMENSIONAL KNOTS

Let (S*, k(S2)) be a knot formed by spinning a polyhedral arc a. about the standard 2-sphere S" in the 3-sphere S3. Then the second homotopy group of S"—k(S2) as a Z^-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by a—\, where a is the image in ■¡r1(Si—k(S2)) of a generator of (2) The first homology group of the kernel of ^(SP-hiS1))-* WtfF-kffl) 0. Introduction. A presentation of the second homotopy group of an arbitrary spun knot [3] was calculated as a ZTr^module in [4], [5], [6]. Professor John Milnor has conjectured that this is a presentation of the fundamental ideal modulo a suitably chosen element. In this paper, we show that this is actually the case. In particular, Theorem 2. Let k(S)2 be a 2-sphere formed by spinning an arc x about the standard 2-sphere S2 in the 3-sphere S3. Let a denote the image in TT1(St—k(S2)) of a generator of ttx(S2—x) and let g be the fundamental ideal in Zttx, i.e., the two sided ideal generated by all elements of the form g—l (ge-TTx). Then the second homotopy group TT2(S*—k(S2)) as a Zttxmodule is 5/(a —1), where (a—I) denotes the left ideal generated by a—I. The action ofZ-n-x on tt2 is that induced by left multiplication on $. Moreover, we have yet another characterization of 7r2, namely Theorem 1. Let (S*,k(S2)) be defined as in Theorem 2 above. Then TT2(Si—k(S2)) as a Zirx-module is the first homology group of the kernel of 7r1(S*-rc(S,2))^771(S4-/c(S'2)). 1. Definition of a spun knot. Let S2 he a standard 2-sphere in the 3sphere S3 and let a be a polyhedral arc with endpoints lying on S2 and with interior lying entirely within one of the two components of S3 — S2. Received by the editors June 5, 1972. AMS (MOS) subject classifications (1970). Primary 57C45; Secondary 55E99, 55E15.