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Links Not Concordant to the Hopf Link

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Links Not Concordant to the Hopf Link LINKS NOT CONCORDANT TO THE HOPF LINK STEFAN FRIEDL AND MARK POWELL Abstract. We give new Casson-Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link. 1. Introduction 1.1. Definitions and previous results. An m–component (oriented) link is an embedded locally flat ordered collection of m disjoint (ori- ented) circles in S3. We say that two m–component oriented links L = 0 0 0 L1 ∪···∪Lm and L = L1 ∪···∪Lm are concordant if there exists a col- lection of m disjoint, embedded, locally flat, oriented annuli A1,...,Am 3 0 in S × [0, 1] such that ∂Ai = Li ×{0} ∪ −Li ×{1}, i =1,...,m. Throughout the paper we denote by H the Hopf link equipped with the orientation such that the linking number is 1, i.e. H consists of two generic fibres of the Hopf map h: S3 → S2 (z1, z2) 7→ [z1, z2], 3 2 2 2 2 where S ⊂ C is the set of (z1, z2) ∈ C with |z1| + |z2| = 1 and S2 ∼= CP1. There is a rich literature on obstructions for two links to be concor- dant. Roughly speaking the obstructions fall into the following cate- gories: (1) obstructions to the components (viewed as knots) being concor- dant (the landmark papers being the work of Levine [Lev69], Casson and Gordon [CG86] and Cochran, Orr and Teichner [COT03]); (2) Milnor’s µ-invariants (see Milnor [Mil57]), in particular the link- ing numbers; (3) obstructions from the multivariable Alexander polynomial and the Levine-Tristram signatures (see Murasugi [Mur67], Kawauchi [Kaw77][Kaw78, Theorem B], Nakagawa [Nak78] and Tristram [Tri69]); (4) twisted Alexander polynomials, twisted signature invariants and L2–signature invariants, where the corresponding representa- tions factor through a nilpotent quotient (e.g. a p–group) or the algebraic closure of the link group (see Cha and Ko [CK99b, CK06], Cha and Friedl [CF10], Friedl [Fri05], Harvey [Har08], 1 2 STEFAN FRIEDL AND MARK POWELL Levine [Lev94] and Smolinsky [Smo89]); furthermore see Cha and Orr [CO09] for a closely related approach; (5) twisted Alexander polynomials and twisted signature invariants of boundary links, where the corresponding representations fac- tor through the canonical epimorphism onto the free group (see Levine [Lev07], Cha and Friedl [CF11]); (6) by considering a prime power branched cover over one compo- nent of a link, one can view the preimages of the other compo- nents as a link in a rational homology sphere; using this reduc- tion in the number of components one can reduce problems in link concordance to the study of ‘rational knot concordance’ (see Cochran-Orr [CO90, CO93], Cha-Kim [CK08], Cha-Livingston- Ruberman [CLR08] and Cha-Kim-Ruberman-Strle [CKRS10]); (7) by doing a Dehn filling on one component one can also reduce problems in link concordance to the study of knot concordance in integral or rational homology spheres. This approach, which for example appears in Cha-Ko [CK99a] and the aforementioned [CKRS10], is often referred to as ‘blowing down along one com- ponent’. Davis [Dav06] proved a partial converse to the above obstructions: if L is a 2-component link with linking number one and such that the ±1 ±1 multivariable Alexander polynomial ∆L ∈ Z[s , t ] equals one, then the link is concordant to the Hopf link. This result generalised ear- lier results by Freedman ([Fre84, FQ90]) on knot concordance and by Hillman [Hil02]. In this paper we will give new obstructions to a link being concor- dant to the Hopf link by using signature invariants corresponding to appropriately chosen metabelian representations. This approach is in- spired by the classic work of Casson and Gordon on knot concordance [CG86]. Generalising their work to the setting of link concordance re- quires overcoming several delicate technical issues. One particularly difficult step (Theorem 2.1 in the present paper) appeared in an earlier paper by the authors [FP10]. 1.2. The linking pairing for 2–component links with linking number 1. In the following we let L be a 2–component link of linking 3 number 1. Throughout this paper we write X = XL = S \ νL, where νL is a tubular neighbourhood of L. The boundary ∂XL can be written 1 2 as the union of two tori ∂XL and ∂XL which correspond to the two i ∼ link components. Moreover, for each i, we can decompose ∂XL = 1 1 1 1 1 S ×D ∪S1×S0 S ×D , where each cross section S ×{pt} is a meridian of the link component Li. We denote this splitting by i ∼ i 1 0 i ∂XL = Ya ∪S ×S Yb , 1 2 1 2 for i =1, 2. Define Ya = Ya t Ya and Yb = Yb t Yb . LINKSNOTCONCORDANTTOTHEHOPFLINK 3 ' 2 Note that we have a canonical isomorphism H1(XL; Z) −→ Z which 2 we will use to identify the groups. Now let ϕ: H1(XL; Z) = Z → A be an epimorphism onto a finite abelian group. In the following we ϕ denote by p: XL → X the corresponding covering space and we write ϕ −1 ϕ −1 Ya = p (Ya) and Yb = p (Yb). In Section 2.2 we will see that there exists a non–singular linking form ϕ ϕ ϕ ϕ λ: TH1(X ,Ya ; Z) × TH1(X ,Ya ; Z) → Q/Z, where, given an abelian group G, we write TG := TorZ(G). Now suppose that L is concordant to the Hopf link via a concordance C ∼= S1 × I ⊂ S3 × I. Denote the exterior of the concordance by EC . The inclusion maps XL → EC and XH → EC are Z-homology equivalences. Note that ϕ canonically extends to a homomorphism H1(EC ; Z) → A which we also denote by ϕ. In the following we denote ϕ by p: EC → EC the corresponding covering space. Before we state our first result we need to introduce one more def- inition. We say that a homomorphism ϕ: Z2 → A to a finite abelian group is admissible if ϕ is given by the direct sum of two projections on 2 2 each summand of Z , i.e. ϕ has to be of the form Z = Z⊕Z → Zk ⊕Zl. We now have the following philosophy. Proposition 1.1. Let L be an oriented 2–component link with linking number 1 which is concordant to the Hopf link via a concordance C. Then for any admissible epimorphism ϕ: Z2 → A to a group of prime power order, ϕ ϕ ϕ ϕ P := ker(TH1(X ,Ya ; Z) → TH1(EC ,Ya ; Z)), is a metaboliser of λ, i.e. P = P ⊥. Following the philosophy of [CG86], we will now use this proposi- tion to show that certain metabelian representations will extend over a concordance exterior. 1.3. Definition of the invariant τ(L, χ). We define the closed man- ifold ML := XL ∪∂XL XH : an ordering and an orientation of the link components of L and H de- termines a canonical identification of ∂XL with ∂XH which identifies the corresponding meridians and longitudes. We say that a homomor- phism ϕ: H1(ML; Z) → A is admissible if it restricts to an admissible homomorphism on ∼ H1(XL; Z) ⊆ H1(ML; Z) = H1(XL; Z) ⊕ Z. ϕ ϕ We denote the corresponding finite cover by ML . Note that MH is a 3-torus. We denote the 3-torus S1 × S1 × S1 by T . Let 0 0 ∼ 3 φ : π1(ML) → H1(ML; Z) := H = Z 4 STEFAN FRIEDL AND MARK POWELL be the Hurewicz map. Let ϕ 0 φ: π1(ML ) ⊆ π1(ML) → H be the restriction of φ0, and define H := im φ. Note that also H ∼= Z3. We pick an isomorphism ' ' ψ : π1(T ) −→ H1(T ; Z) −→H. Finally let ϕ 1 χ: H1(ML ; Z) → S ⊂ C be a character which factors through Zql for some prime q, and denote the trivial character by tr. Following Casson-Gordon, in Section 3.2, we define a Witt group obstruction 0 τ(L, χ) ∈ L (C(H)) ⊗Z Z[1/q], in terms of the twisted intersection form of a 4-manifold whose bound- ϕ ary is a finite number of copies of (ML t −T, χ × φ t tr ×ψ). 1.4. Statement of the main theorem and examples. The follow- ing is our main theorem. Theorem 1.2. Suppose that L is concordant to the Hopf link. Then for any admissible homomorphism ϕ: H1(ML) → A to a finite abelian p-group A, there exists a metaboliser P = P ⊥ for the linking form ϕ ϕ ϕ ϕ TH1(XL ,Ya ) × TH1(XL ,Ya ) → Q/Z with the following property: for any character of prime power order ϕ 1 χ: H (M ) → Z k → S , which satisfies that χ| ϕ factors through 1 L q H1(XL ) ϕ ϕ ϕ δ 1 χ| ϕ : H (X ) → H (X ,Y ) −→ Z k ,→ S H1(XL ) 1 L 1 L a q and that δ vanishes on P , we have that 0 τ(L, χ)=0 ∈ L (C(H)) ⊗Z Z[1/q]. This theorem is reminiscent of ‘Casson-Gordon type’ obstruction theorems in the context of knot concordance, especially of the main theorem of [CG86]. The proof of Theorem 1.2 is indeed guided by the knot theoretic case coupled with the main result of [FP10]. In particular we will show that if a link L is concordant to H and if χ satisfies the conditions of the theorem, then there exists a 4-manifold ϕ ϕ ϕ ϕ ϕ ϕ WC = EC ∪ (XH × I) whose boundary is ML t −MH = ML t −T , and which has trivial (twisted) intersection form. In Section 4 we will discuss an example of a link for which our main theorem detects that it is not concordant to the Hopf link.
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