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 0 =K-ΠK +1 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 for computing the algebraic 2-type of K 9 i.e., the triple consisting of the fundamental group π x K, the second homotopy group π 2 K 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.

On a banded link presentation of knotted surfaces

“…Milnor was later broadly presented in [4]. Theorem 2.1 (Lomonaco [10], Kawauchi, Shibuya, and Suzuki [8]). For any knotted surface F, there exists a knotted surface F satisfying the following:…”

Applying Lipson's state models to marked graph diagrams of surface-links

“…[2]). It is known [4,7,8,14] that any surface-link L can be deformed into a surface-link L ′ , called a hyperbolic splitting of L, by an ambient isotopy of R 4 in such a way that the projection p : L ′ → R to the fourth coordinate satisfies the following:…”

“…S. J. Lomonaco, Jr. [14] and K. Yoshikawa [17] introduced a method of describing surface-links by marked graph diagrams.…”

The fundamental crossed module of the complement of a knotted surface

“…Therefore we have a crossed module π 1,2 (M ) for any based topological space. See for example [31,34] for calculations of π 1,2 (M ) when M is the complement of a knotted surface in S 4 . A first idea about how to employ the notion of a crossed module to define invariants of manifolds could be to consider the crossed module π 1,2 (M ).…”