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Archiv der Mathematik

Homotopy ribbon concordance and Alexander polynomials Stefan Friedl and Mark Powell Abstract. We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L, then the Alexander polynomial of L divides the Alexander polynomial of J. Mathematics Subject Classification. 57M25, 57M27, 57N70. Keywords. Ribbon concordance, Alexander polynomial, Knot theory, Link theory.

1. Introduction. Let I := [0, 1]. An oriented, ordered m-component link J in S 3 is homotopy ribbon concordant mto 1an oriented, ordered m-component3 link S × I, locally flatly embedded in S × I, L if there is a concordance C ∼ = restricting to J ⊂ S 3 × {0} and −L ⊂ S 3 × {1}, such that the induced map on fundamental groups of exteriors π1 (S 3 \νJ)  π1 ((S 3 × I)\νC) is surjective and the induced map π1 (S 3 \νL)  π1 ((S 3 × I)\νC) is injective. Here νJ, νL, and νC denote open tubular neighbourhoods. When J is homotopy ribbon concordant to L, we write J ≥top L. From now on, we write XJ := S 3 \νJ, XL := S 3 \νL, and XC := (S 3 × I)\νC. The notion of homotopy ribbon concordance is a natural homotopy group analogue of the notion of smooth ribbon concordance initially introduced by Gordon [7] for knots: we say the link J is smoothly ribbon concordant to the link L, written J ≥sm L, if there is a smooth concordance from J to L such that the restriction of the projection map S 3 × I → I to C yields a Morse function on C without minima. The exterior of such a concordance admits a handle decomposition relative to XJ with only 2- and 3-handles, from which it

S. Friedl and M. Powell

Arch. Math.

is easy to see that the induced map π1 (XJ ) → π1 (XC ) is surjective. Gordon’s argument [7, Lemma 3.1] shows that π1 (XL ) → π1 (XC ) is injective. Thus a smooth ribbon concordance is a homotopy ribbon concordance. ±1 We define the Alexander polynomial ΔJ (t1 , . . . , tm ) ∈ Z[t±1 1 , . . . , tm ] of an oriented, ordered m-component link J to be the order of the torsion submodule of the Alexander module H1 (XJ ; Z[Zm ]). Here the precise coefficient system ϕ : π1 (XJ ) → Zm is determined by the oriented meridians and the ordering of L. Theorem 1.1. Suppose that J ≥top L. Then ΔL | ΔJ . For knots and for ≥sm instead of ≥top , Theorem 1.1 is a consequence of a more general theorem of Gilmer [6]. However Gilmer’s proof does not extend to the topological category. Further classical work on smooth ribbon concordance includes [6,15,16], and [20]. We want to explain a fairly simple proof of Theorem 1.1, thus we will not prove the most general result possible. But we expect that our argument can be generalised to twisted Alexander polynomials [8,10,11] and higher order Alexander polynomials [1], provided one uses a unitary representation that extends over the ribbon concordance exterior. Our proof can also be generalised to concordances between links in homology spheres. Having not found a convincing application, we have not carried out either of these generalisations in this short note. A number of articles have recently appeared