The Homotopy Groups of Knots I. How to Compute the Algebraic 3-type


Samuel J. Lomonaco, Jr.(*)


Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes K- and K+ such that (a) K = K- UNION K+ and (b) K-, K0 = K- INTERSECT K+, K+ are connected and aspherical. The main theorem of this paper gives a practical procedure for computing the homology H*(K~) of the universal cover K~ of K. It also provides a practical method of computing the algebraic 3-type of K, i.e., the triple consisting of the fundamental group p1(K), the second homotopy group
p2(K) as a p1(K)-module, and the first k-invariant kK.

The effectiveness of this procedure is demonstrated by letting K denote the complement of a smooth 2-knot (S4, kS2). Then the above mentioned methods provide a way for computing the algebraic 3-type of 2-knots, thus solving problem 36 of R.H. Fox in his 1962 paper, "Some problems in knot theory." These methods can also be used to compute the algebraic 3-type of 3-manifolds from their Heegard splittings. This approach can be applied to many well known classes of spaces. Various examples of the computation are given.
(*) Partially supported by the L-O-O-P Fund.