# Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $$D^2 \subset B^4$$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there has been at least one published incorrect proof of this result, namely, in this paper.

I was looking at "The homotopy groups of knots. I. How to compute the algebraic 2-type." by S. J. Lomonaco, Jr. and I noticed that in this paper the author seems to be using the fact that ribbon disk complements are aspherical in his calculations (the problem seems to start at Theorem 7.4).

Any 2-knot $$K \subset S^4$$ is the union of two ribbon disks, namely if we think of $$S^4 = B_1 \cup B_2$$ as the union of two 4-balls there are ribbon disks $$D_1,D_2 \subset B_1,B_2$$ with $$K$$ isotopic to $$D_1 \cup D_2$$ - namely, with respect to the standard height function restricted to $$K$$, isotope $$K$$ so that there are an equal number of minima and maxima, minima come before maxima, and rearrange the 1-handles so that all of the fusion bands come before all of the fission bands. If we assume that ribbon disk complements are aspherical, then the complement of $$K$$ is the union of two aspherical spaces along an aspherical space (since classical knot complements are aspherical).

The primary work in the aforementioned paper by Lomonaco goes towards proving Theorem 7.1, which gives a method for computing the homology of the universal cover of a space that can be decomposed as the union of two connected aspherical spaces along another connected aspherical space. Additionally, in this theorem, the author shows how to compute the homology of the cover as a $$\mathbb{Z}\pi_1$$-module and the $$k$$-invariant. This theorem is then applied to the above "bridge position" of a 2-knot in the previous paragraph as well as to Heegaard splittings of 3-manifolds.

I have no real reason to suspect that Theorem 7.1 is incorrect (although I have not checked through the homological algebra), but I am coming here for a little reassurance. Is this the case? Also, are there any generalizations of Theorem 7.1 for more complicated aspherical splittings (say with three connected aspherical components all with pairwise connected aspherical intersection and with a connected aspherical triple intersection)? I imagine the answer here would be some sort of spectral sequence.

Additionally, am I correct in saying that the calculations in say Figure 5, 6, 7, and 9 are incorrect? Or maybe they are just not correctly justified (relying on the unproven conjecture above) but still true? If you can show that the results to the calculations are incorrect then there is either another error in the paper that I am not aware of, or you've got a disproof of the above mentioned conjecture (and hence also the Whitehead asphericity conjecture). Therefore, I am imagining that the results are true but a different technique is needed to justify them.

• My original answer (now deleted) did not address the question posed, so a portion is now a comment. It is not the case that all 2-knots have aspherical complements. A paper of E. Dyer and A. T. Vasquez, Can. J. Math. 25 (1973), 1132–1136 shows that if a higher dimensional knots has aspherical complement then the knot group is $\mathbb{Z}$! In dimension 2 this implies topologically unknotted; Dyer-Vasquez show that asphericity implies unknotted in higher dimensions as well. For example, any nontrivial fibered 2-knot complement has nonzero $\pi_2$. – Danny Ruberman May 29 at 12:03