The homotopy groups of knots. I. How to compute the algebraic 2-type

Abstract

Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes ϋΓ _ and K such that (a) K—K-UK+ and (b) iΓ_, K =K-ΠK K are connected and aspherical. 0 +1 + 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 for computing the algebraic 2-type of K i.e., the triple consisting of the 9 fundamental group π K, the second homotopy group π K x 2 as a TΓilΓ-module, and the first /^-invariant kK. The effectiveness of this procedure is demonstrated by letting K denote the complement of a smooth 2-knot (S 4, JcS 2). Then the above mentioned methods provide a way for computing the algebraic 2-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 2-type of 3-manifolds from their Heegaard splittings. This approach can be applied to many other well known classes of spaces. Various examples of the computation are given.