Vol. 95, No. 2, 1981

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

Samuel James Lomonaco, Jr.

Vol. 95 (1981), No. 2, 349–390

Let K be a CW complex with an aspherical splitting, i.e., with subcomplexes K and K+ such that (a) K = KK+ and (b) K,K0 = KK+,K+ are connected and aspherical. The main theorem of this paper gives a practical procedure for computing the homology HK 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 fundamental group π1K, the second homotopy group π2K as a π1K-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 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.

Mathematical Subject Classification 2000
Primary: 57Q45
Secondary: 55Q52
Received: 21 May 1979
Revised: 10 June 1980
Published: 1 August 1981
Samuel James Lomonaco, Jr.