The Second Homotopy Group of a Spun Knot

In this paper we take a totally different approach via Reidemeister homotopy chains, which gives a simpler and more natural proof enabling us to visualize geometrically the free derivatives. Briefly the proof is as follows: We collapse the complement C A of the arc A in C to a 2-dimensional CW-complex K, where C is the closure of the component of S3 S2 containing A. Then it follows from [S] that the matrix of the boundary homomorphism a2 : &(I?) -+ C,(d) is simply Fox’s Jacobian matrix (ari/axj) of free derivatives, where i? is the universal cover of K and C,(g) and C,(k) denote the respective chain groups of k. Moreover, it follows from the asphericity of knots [7] that there are no nontrivial 2-cycles. Thus the structure of the chain complex is completely determined. Spinning K about aC = S2, we obtain a 3-dimensional CW-complex K* which is a deformation retract of S 4 k(S ‘). The structure of the chain complex of the universal cover K * of K* is determined from the previous chain complex. This enables us to calculate H,(K *), and hence, by Hurewicz’s theorem, n2(S4 k(S2)).