Homology Cobordism Invariants and the Cochran--Orr

Geometry & Topology 12 (2008) 387–430 387

Homology cobordism invariants and the Cochran–Orr–Teichner ﬁltration of the link concordance group

SHELLY LHARVEY

For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger– Gromov –invariant, we obtain new real-valued homology cobordism invariants n for closed .4k 1/–dimensional manifolds. For 3–dimensional manifolds, we show that n n N is a linearly independent set and for each n 0, the image of n is f j 2 g an infinitely generated and dense subset of R. In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define m a filtration F.n/ of the m–component (string) link concordance group, called the m .n/–solvable filtration. They also define a grope filtration Gn . We show that n vanishes for .n 1/–solvable links. Using this, and the nontriviality of n , we show C that for each m 2, the successive quotients of the .n/–solvable filtration of the link concordance group contain an infinitely generated subgroup. We also establish a similar result for the grope filtration. We remark that for knots (m 1), the successive D quotients of the .n/–solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when n 3. 57M27; 20F14

1 Introduction

The main objective of this paper is to investigate the set of 3–dimensional manifolds up to homology cobordism. To do this, we deﬁne, for each n N , an invariant of 2 .4k 1/–dimensional manifolds (k 1) that we call n . Loosely speaking, n.M / is deﬁned as a signature defect closely associated to a term of the torsion-free derived series of 1.M / and (for smooth manifolds) can be interpreted as the Cheeger–Gromov invariant of M associated to the n–th torsion-free derived regular cover of M . Using a derived version of Stallings’ Theorem[9], we show that n is an invariant of homology cobordism whereas the Cheeger–Gromov –invariant associated to an arbitrary cover is only a priori a homeomorphism invariant.

Published: 12 March 2008 DOI: 10.2140/gt.2008.12.387 388 Shelly L Harvey

Theorem 4.2 If M 4k 1 is rationally homology cobordant to M 4k 1 .k 1/ then 1 2 n.M1/ n.M2/. D

.n/ To define the invariant n , we first define a new characteristic series G of a group f H g G (Section 2) closely related to the derived series, that we call the torsion-free derived series and establish its basic properties. One should view the torsion-free derived series .n/ GH of a group G as a series that is closely related to the derived series but whose f g .n/ .n 1/ .n/ successive quotients GH =GH C are torsion-free as ZŒG=GH –modules. This can .n/ be compared to the rational derived series Gr (which we studied in[19]) which is a series whose successive quotients are torsion-free as abelian groups.

Let M be a .4k 1/–dimensional manifold and let G 1.M /. The invariant n.M / D is defined as follows (see Section 3 for more details). Recall that for any group ƒ, the 2 .2/ L –signature ƒ is a real-valued homomorphism on the Witt group of hermitian forms on finitely generated projective Uƒ–modules, where Uƒ is the algebra of unbounded operators associated to the von Neumann algebra of ƒ. Following Hausmann[20], we show (see Lemma 3.4 and Corollary 3.5) that for every coefficient system G .n 1/ W ! G=GH C , there exists a positive integer r and a 4k –dimensional manifold W such that the pair .rM; r/ is stably nullbordant via .W; ˆ 1.W / ƒ/. Let hW be W ! the intersection form associated to the regular ƒ–covering space of W and let be 1 .2/ the ordinary signature function. We show in Lemma 3.6 that . .hW / .W // is r ƒ independent of the stable nullbordism .W; ˆ/ and define

1 .2/ n.M / .hW / .W / : D r ƒ

More generally, we define for any coefficient system 1.M / (note W ! that one can also define via the Cheeger–Gromov construction for smooth man- lcs ifolds). Hence one could also study .n/.M / (respectively n .M /), the canoni- .n 1/ cally defined –invariant associated to 1.M /=1.M / C (respectively lcs .n/ D lcs D 1.M /=1.M /n 1 ) where G (respectively Gn ) is the n–th term of the derived series (respectivelyC lower central series) of G . Despite the fact that the torsion-free lcs derived series may seem a bit unwieldy, we focus on n (rather than .n/ or n ) since it gives an invariant of rational homology cobordism and provides new information about the structure of the .n/–solvable and grope filtrations of the (string) link concordance group (see below and Section 6). By contrast, .n/ is only a homeomorphism invariant. One can use Stallings’ Theorem[31] and follow through the proof of Theorem 4.2 to lcs show that n is a homology cobordism invariant. However, we choose to use n in our work since it is more directly related to the .n/–solvable and grope filtrations of lcs the link concordance group than n .

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We show that the n are highly nontrivial and independent for 3–manifolds (Section 5). To accomplish this, we construct an inﬁnite family of examples of 3–manifolds M.; K/ that are constructed by a method known as genetic infection. More speciﬁ- f g cally, to construct M.; K/ we start with a 3–manifold M and infect M by a knot K along a curve in the n–th term of the derived series of the fundamental group of M . We prove that i.M.; K// depends only on n and 0.K/, where 0.K/ is the integral of the Levine–Tristram signatures of K.

Theorem 5.8 Let M be a compact, orientable manifold, an embedded curve in M , 3 .n/ .n 1/ K a knot in S , and P 1.M /. If P P C for some n 0 then D 2 H H 0 0 i n 1 i.M.; K// i.M / Ä Ä I D 0.K/ i n: 3 Let HQ be the set of Q–homology cobordism classes of closed, oriented 3–dimensional manifolds. Using the set of examples M.; K/ with varying K we establish the f g following theorem.

3 Theorem 5.11 The image of n H R is (1) dense in R and (2) an inﬁnitely W Q ! generated subgroup of R .

Moreover, in Theorem 5.13, we show that n is a linearly independent subset of the 3 f g vector space of functions on HQ . We remark that S Chang and S Weinberger[5] use a similar type of signature defect, one associated to the universal cover of a manifold, to define a homeomorphism invariant of a .4k 1/–dimensional manifold (k 1). Using they show that if M is a .2/ .2/ .4k 1/–dimensional (smooth) manifold with k 2 and 1.M / is not torsion-free then there are infinitely many (smooth) manifolds homotopy equivalent to M but not homeomorphic to M . For the rest of the paper, we turn our attention to the study of link concordance. Recall 3 that if two links L1 and L2 in S are concordant then ML1 and ML2 are homology cobordant where ML is the zero surgery on L. We define n.L/ n.ML/ for a link 3 D L in S . Hence, by Theorem 4.2, n is a link concordance invariant. In[11], T Cochran, K Orr and P Teichner defined the .n/–solvable (and grope) filtration of the knot concordance group C . In[11] and their two subsequent papers

[12; 13], they showed that the quotients F.n/=F.n 1/ of the .n/–solvable ﬁltration of C the knot concordance group are nontrivial. In particular, they showed that for n 1; 2, D F.n/=F.n 1/ has inﬁnite rank and for all n 3, the quotient has rank at least 1. It is C

Geometry & Topology, Volume 12 (2008) 390 Shelly L Harvey still unknown if any of the quotients is infinitely generated for n 3. In the current m paper, we investigate the .n/–solvable (and grope) filtration F.n/ of the string link concordance group C.m/ and the subgroup generated by boundary links B.m/ for links with m 2 components. Since connected sum is not a well-defined operation for links, it is necessary to use string links to obtain a group structure. Using n we show that, for m 2, each of the successive quotients of the .n/–solvable filtration of the m boundary string link concordance group BF .n/ is infinitely generated.

m m Theorem 6.8 For each n 0 and m 2, the abelianization of BF.n/=BF.n 1/ m m C has infinite rank. In particular, for m 2, BF.n/=BF.n 1/ is an infinitely generated m m C subgroup of F.n/=F.n 1/ . C We note the previous theorem holds “modulo local knotting” (see Corollary 6.9), hence this result cannot be obtained using the work of Cochran–Orr–Teichner on knots. We m m also prove a similar statement for the grope filtrations BGn and Gn of the boundary and string link concordance groups respectively.

m m Theorem 6.13 For each n 1 and m 2, the abelianization of BGn =BGn 2 has m m m Cm infinite rank. Hence BGn =BGn 2 is an infinitely generated subgroup of Gn =Gn 2 . C C We also prove that the abelianization of BGm=BGm has nonzero rank for n 2 and n n 1 m 2 in Proposition 6.14. We conjecture these quotientsC groups are in fact infinitely generated.

To prove Theorem 6.8, we ﬁrst show that n is additive when restricted to B.m/, the subgroup of C.m/ consisting of m component boundary string links. We note that n is not additive on C.m/ itself.

Corollary 6.7 For each n 0 and m 1, n B.m/ R is a homomorphism. W !

Next, we show that .n 1/–solvable links have vanishing n . Thus, for each n 0, C m m n is a homomorphism from BF.n/=BF.n 1/ to R. C Theorem 6.4 If a 3–manifold M is .n/–solvable then for each .n/–solution W and k n, the inclusion i M W induces monomorphisms Ä W ! .k/ .k/ i H1.M K.1.W /=1.W /H // , H1.W K.1.W /=1.W /H //; W I ! I 1.M / 1.W / i .k 1/ , .k 1/ W 1.M /HC ! 1.W /HC and k 1.M / 0: D Thus, if L F.n/ then k 1.L/ 0 for k n. 2 D Ä

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To complete the proof of Theorem 6.8, we construct a collection of links that are .n/–solvable and have independent n . To do this we perform genetic infection on the m–component trivial link using some knot K along some carefully chosen curve in F .n/ where F is the fundamental group of the trivial link. This produces a collection of boundary links L.; K/ that are .n/–solvable and such that the image of n f g restricted to L.; K/ is infinitely generated. f g We remark that the invariant n is related to certain “finite” concordance invariants of boundary links associated to p–groups. Suppose L is a boundary link with m components. Then there is a surjective map G F where G 1.ML/ and W ! D F is the free group on m generators. By Theorem 4.1 of[9] (respectively Stallings’ .n/ .n/ Theorem[31]), G=GH F=F (respectively G=Gn F=Fn ). Moreover, ML is Š Š .n/ the boundary of a 4–dimensional manifold W over F . Since F=F and F=Fn are lcs residually finite p–groups, by work of W Luck¨ and T Schick, both n and l can be approximated by signatures of finite covers of W where the covering groups are p–groups. These finite p–group signatures are closely related to the concordance invariants of boundary links studied by S Friedl[17] and J C Cha and K H Ko[2]. We finish this paper by mentioning some applications to boundary link concordance in Section 7. In particular, we show k gives a homomorphism from certain gamma groups (modulo automorphisms of the free group) to R, generalizing work of S Cappell and J Shaneson. Here, B.n; m/ is the group of concordance classes of m component, n 2 n–dimensional boundary disk links in D C .

Proposition 7.3 For each n 1 mod 4 with n > 1, and each k 0, there is an Á induced homomorphism

k n 3.ZF Z/= Aut F R z W z C ! ! that factors through B.n; m/.

Acknowledgements The author was partially supported by an NSF Postdoctoral Fellowship, NSF DMS-0539044, and a fellowship from the Alfred P Sloan Foundation.

2 The torsion-free derived series

To define n , we must first introduce a new characteristic series of a group called the torsion-free derived series. In this section, we will define the torsion-free derived series and establish some its basic properties. If G is a group then G=G.1/ is an abelian group but may have Z–torsion. If one would like to avoid Z–torsion then, in direct analogy to the rational lower-central series,

Geometry & Topology, Volume 12 (2008) 392 Shelly L Harvey

.1/ k one can define Gr x G x ŒG; G for some k 0 , which is slightly larger .1/ D f.1/ 2 j 2 ¤ g .n/ than G , so that G=Gr is Z–torsion-free. Proceeding in this way, defining Gr to .n 1/ .n 1/ be the radical of ŒGr ; Gr , leads to what has been called the rational derived series of G [7; 19]. This is the most rapidly descending series for which the quotients of successive terms are Z–torsion-free abelian groups. Note that if N is a normal subgroup of G then N=ŒN; N is not only an abelian group but it is also a right module over ZŒG=N , where the action is induced from conjugation in G (Œxg Œg 1xg). D To define the torsion-free derived series, we seek to eliminate torsion “in the module .n/ sense” from the successive quotients. We define the torsion-free derived series GH of .0/ .n/ G as follows. First, set GH G . For n 0, suppose inductively that GH has been D .n/ defined and is normal in G (we will show that GH is normal in G below). Let Tn .n/ .n/ .n/ .n/ be the subgroup of GH =ŒGH ; GH consisting of ZŒG=GH –torsion elements, ie the .n/ elements Œx for which there exists some nonzero ZŒG=GH , such that Œx 0. 2 .n/ D (In fact, since it will be (inductively) shown below that ZŒG=GH is an Ore Domain, Tn is a submodule). Now consider the epimorphism of groups

.n/ .n/ n GH GH .n/ .n/ ! ŒGH ; GH

.n 1/ .n 1/ and define GH C to be the inverse image of Tn under n . Then GH C is, by .n/ .n/ .n/ definition, a normal subgroup of GH that contains ŒGH ; GH . It follows inductively .n/ .n 1/ .n 1/ .n/ı .n 1/ that GH contains GH C (and Gr C ). Moreover, since GH GH C is the quotient .n/ı .n/ .n/ .n/ of the module GH ŒGH ; GH by its torsion submodule, it is a ZŒG=GH torsion- free module[32, Lemma 3.4]. Hence the successive quotients of the torsion-free derived subgroups are torsion-free modules over the appropriate rings. We define G.!/ T G.n/ as usual. H D n

.n/ .n/ Proposition 2.1 For each 0 n < ! , GH is a normal subgroup of G and G=GH is Ä .n/ a poly-(torsion-free abelian) group. Consequently, ZŒG=GH is an Ore domain.

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Proof We prove this by induction on n. The statement is clear for n 0. Assume .n/ .n/ .n 1/ D GH is a normal subgroup of G and G=GH is PTFA. Let x GH C and g G . Then 1 .n/ 2 2 x and g xg lie in GH by assumption. By deﬁnition of the right module structure on .n/ .n/ .n/ 1 .n 1/ GH =ŒGH ; GH , n.g xg/ n.x/g. Since x GH C , n.x/ is torsion. To show 1 .n 1/ D 2 that g xg G C , it sufﬁces to show that n.x/g is torsion. Recall that the set of 2 H torsion elements of any module over an Ore domain is known to be a submodule[32, .n/ p 57]. Since ZŒG=GH is an Ore domain, it follows that the set of torsion elements in .n/ .n/ .n/ .n 1/ GH =ŒGH ; GH is a submodule. Thus, n.x/g is torsion and hence GH C is normal in G .

.n 1/ Consider the normal series for G=GH C :

.n 1/ .n/ .1/ GH C GH GH G 1 .n 1/ .n 1/ .n 1/ .n 1/ : D GH C G GH C G G GH C G GH C

Since the successive quotients of the above series are torsion-free abelian groups, .n 1/ G=GH C is PTFA.

.n/ For convenience, we will often write G=GH as Gn for any group G (not to be confused lcs with the terms of the lower central series of G which we will denote by Gn in this paper).

We remark that the torsion-free derived subgroups are characteristic subgroups but they are not totally invariant. That is, an arbitrary homomorphism A B need not .n/ .n/ W ! send AH to BH . To see this, let A x; y; z Œz;Œx; y , B x; y and A B D h j i D h i W.2/ ! be deﬁned by .x/ x, .y/ y and .z/ 1. Then we have Œx; y AH since D .1/D .1/ .1/ D .1/ 2 Œx; y is .z 1/–torsion in AH =ŒAH ; AH where z Œz A=AH but .Œx; y/ D 2 D Œx; y B.2/ B.2/ (see Proposition 2.3). 62 D H

.n/ .n/ Proposition 2.2 If A B induces a monomorphism A=AH , B=BH , then .n 1/ .n 1/ W ! W .n !1/ .n 1/ .A C / B C and hence induces a homomorphism A=A C B=B C . H H W H ! H

Proof Note that the hypothesis implies that induces a ring monomorphism

ZŒA=A.n/ ZŒB=B.n/: zW H ! H

.n 1/ Suppose that x A C . Consider the diagram below. 2 H

Geometry & Topology, Volume 12 (2008) 394 Shelly L Harvey

.n/ .n/ A- AH AH .n 1/ AHC

?x ? .n/ .n/ B- BH BH .n 1/ BH C

.n/ By deﬁnition, A.x/ is torsion. That is, there is some nonzero ZŒA=AH such 2 .n/ that .x/ 0. It is easy to check that is a homomorphism of right ZŒA=A – A x H D .n/ .n 1/ 1 modules using the module structure induced on B =B C by (since .a xa/ H H z 1 D .a/ .x/.a/). Thus .A.x//. / 0. Since is injective, .A.x// is a .n/ x z D z x .n 1/ ZŒB=BH –torsion element. But .A.x// B..x//, showing that .x/ BH C . .n 1/ .n 1/ x D 2 Hence .B C / B C . H H For some groups, such as free groups and free-solvable groups F=F .n/ , the derived series and the torsion-free derived series coincide.

.n/ .n 1/ Proposition 2.3 If G is a group such that, for each n, G =G C is torsion-free as a ZŒG=G.n/–module, then the torsion-free derived series of G agrees with the derived series of G . Hence for a free group F , F .n/ F .n/ for each n. H D

.n/ .0/ .n/ .n/ Proof By definition, GH G G . Suppose GH G . Then, under the .n/ .n/ .n/D D D .n 1/ hypotheses, GH =ŒGH ; GH is a torsion-free module and hence GH C ker n .n/ .n/ .n/ .n/ .n 1/ D D ŒG ; G ŒG ; G G C . H H D D .n/ .n 1/ .n/ It is well known that F =F C is a ZŒF=F –torsion-free module. This can be seen by examining the free ZŒF=F .n/ cellular chain complex for the covering space .n/ .n/ .n 1/ of a wedge of circles corresponding to the subgroup F . The module F =F C is merely the first homology of this chain complex. Since the chain complex can be chosen to have no 2–cells, its first homology is a submodule of a free module and thus is torsion-free. Hence, by the first part of this proposition, the derived series and the torsion-free derived series of a free group agree.

In[9], T Cochran and the author prove a version of Stallings’ Theorem[31] for the derived series. Speciﬁcally, it was there shown that if a map of ﬁnitely presented groups A B is rationally 2–connected then it induces a monomorphism W ! .n/ .n/ A=AH , B=BH W !

Geometry & Topology, Volume 12 (2008) Homology cobordism invariants 395 for all n 0 [9, Theorem 4.1] (see Theorem 2.4 and Proposition 2.5 below). For this paper, we will need the following “generalization” of that theorem. The following theorem is in fact a consequence of the proof of Theorem 4.1 of[9]. For the convenience of the reader, we will sketch the proof of Theorem 2.4 after the statement of Proposition 2.5.

Theorem 2.4 (Scholium to Theorem 4.1 of[9]) Let n be a nonnegative integer or n ! . If A B is a homomorphism that for each k n induces a monomorphism D W ! .k/ .k/ Ä H1.A K.B=BH // H1.B K.B=BH //, then for each k n, induces a W I .k !1/ I .k 1/ Ä monomorphism A=AHC , B=BH C . Moreover, if A B induces an isomor- .k!/ .k/ W ! phism H1.A K.B=BH // H1.B K.B=BH // for each k n, then for each W I ! .k/ I .k 1/ .k/ .k 1/ Ä k n, induces a monomorphism AH =AHC , BH =BH C between modules of Ä .k/ .k/ ! the same rank (over ZŒA=AH and ZŒB=BH respectively). In addition, if is onto .k 1/ .k 1/ then A=AHC B=BH C is an isomorphism. W ! The following proposition guarantees that one of the hypotheses of Theorem 2.4 is satisfied whenever is a rationally 2–connected map. Note that Proposition 2.5 and Theorem 2.4 together imply Theorem 4.1 of[9]. The justification for calling Theorem 2.4 a generalization of Theorem 4.1 of[9], is that, in the subsequent sections we will describe several conditions (see Proposition 3.15, Theorem 5.8, Theorem 6.4, Proposition 6.6, and Lemma 6.5) under which the hypothesis of Theorem 2.4 is satisfied but where the 2–connected hypothesis of Theorem 4.1 of[9] (and Proposition 2.5 and Proposition 4.3 of[9]) fails.

Proposition 2.5 (Proposition 4.3 of[9]) Let A be a finitely generated group and B a finitely related group. Suppose A B induces a monomorphism (respectively W ! isomorphism) on H1. Q/ and an epimorphism on H2. Q/. Then for each k 0, I I .k/ induces a monomorphism (respectively isomorphism) H1.A K.B=BH // .k/ W I ! H1.B K.B=BH //. I Proof of Theorem 2.4 We sketch the inductive proof of the first claim of the the- .1/ orem, referring the reader to[9] for more details. For n 0, A=AH is merely .0/ .0/ D H1.A Z/= Z-Torsion . But A=AH e so K.A=AH / Q. Thus our hypothesis, I f g D f g D that induces a monomorphism on H1. Q/, implies that induces a monomor- I phism on H1. Z/ modulo torsion. Now assume that induces a monomorphism .n/ I.n/ A=A B=B . We will prove that this holds for n 1. H H C .n 1/ .n 1/ It follows from Proposition 2.2 that .A C / B C . Thus from the commuta- H H tive diagram below we see that it suffices to show that induces a monomorphism

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.n/ .n 1/ .n/ .n 1/ A =A C B =B C . H H ! H H - .n/ .n 1/ - .n 1/ - .n/ - 1 AH =AHC A=AHC A=AH 1

n 1 n C ? ? ? - .n/ .n 1/ - .n 1/ - .n/ - 1 BH =BH C B=BH C B=BH 1

.n/ .n 1/ .n/ .n 1/ Now suppose that AH =AHC BH =BH C were not injective. From our discussions ! .n/ above in the proof of Proposition 2.2 we see that there would exist an a AH .n/ .n/ .n/ 2 representing a nontorsion class Œa in AH =ŒAH ; AH such that .a/ represents a .n/ .n/ .n/ torsion class in BH =ŒBH ; BH . But

.n/ .n/ .n/ .n/ A =ŒA ; A H1.A ZŒA=A /: H H H Š I H The torsion submodule is characterized precisely as the kernel of the canonical map

.n/ .n/ .n/ .n/ H1.A ZŒA=AH / H1.A ZŒA=AH / ZŒA=A.n/ K.A=AH / H1.A K.A=AH //: I ! I ˝ H Š I A similar statement holds for B . But the inductive hypothesis that A=A.n/ B=B.n/ H H guarantees that .n/ .n/ H1.A K.A=A // H1.A K.B=B // I H ! I H is injective. Moreover, the hypothesis of the theorem guarantees that

.n/ .n/ H1.A K.B=B // H1.B K.B=B // I H ! I H is injective, leading to a contradiction.

3 Deﬁnition of n

On the class of closed, oriented .4k 1/–dimensional manifolds, we will define a Q– homology cobordism invariant n for each n N . This will be defined as a signature 2 defect associated to the n–th term of the torsion-free derived series of the fundamental group of the manifold. We begin by recalling the definition of the L2 –signature of a 4k –dimensional manifold. For more information on L2 –signature and –invariants see Cochran and Teichner[13, Section 2], Cochran, Orr and Teichner[11, Section 5] and Luck¨ and Schick[26]. Let ƒ be a countable group and Uƒ be the algebra of unbounded operators affiliated .2/ to N ƒ, the von Neumann algebra of ƒ. Then Hermn.Uƒ/ R is defined by ƒ W ! .2/ ƒ .h/ trƒ.p .h// trƒ.p .h// D C

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for any h Hermn.Uƒ/ where trƒ is the von Neumann trace and p are the character- 2 .˙2/ istic functions on the positive and negative reals. It is known that ƒ can be extended to the Witt group of Hermitian forms on ﬁnitely generated projective Uƒ–modules.

Lemma 3.1 (see for example Corollary 5.7 of[11] and surrounding discussion) The 2 .2/ L –signature, ƒ , is a well-deﬁned real-valued homomorphism on the Witt group of hermitian forms on ﬁnitely generated projective Uƒ–modules. Restricting this homomorphism to nonsingular forms on free modules gives

.2/ L0.Uƒ/ R: ƒ W !

In particular, if h is a nonsingular pairing with a metabolizer then .2/.h/ 0. ƒ D Let W be 4k –dimensional manifold and ˆ 1.W / ƒ be a coefﬁcient system for W ! W . Let hW;ƒ be the composition of the following homomorphisms

PD (1) H .W Uƒ/ H .W;@W Uƒ/ H 2k .W Uƒ/ H .W Uƒ/ 2k I ! 2k I ! I ! 2k I where H .W Uƒ/ HomUƒ.H .W Uƒ/; Uƒ/. Since Uƒ is a von Neumann 2k I D 2k I regular ring, the modules H2k .W Uƒ/ are finitely generated projective right Uƒ– I .2/ .2/ modules. Then hW;ƒ Hermn.Uƒ/ and we define .W; ƒ/ ƒ .hW;ƒ/. We will .2/ 2 .2/ .2/ D sometimes write .W; ƒ/ as ƒ .W / or .W; ˆ/ when we want to emphasize the map ˆ. Suppose that ƒ is PTFA. Let U be a (possibly empty) union of components of the boundary of W . Then Zƒ embeds in its right ring of quotients Kƒ. Moreover, the map from Zƒ to Uƒ factors as Zƒ Kƒ Uƒ making Uƒ into a Kƒ Uƒ–bi- ! ! module. Since any module over a skew field is free, Uƒ is a flat Kƒ–module. Hence, H .W; U Uƒ/ H2.W; U Kƒ/ Kƒ Uƒ. In particular, H .W; U Kƒ/ 0 if 2k I Š I ˝ 2k I D and only if H2.W; U Uƒ/=0. I We will use the following facts about L2 –signatures throughout this paper. The first two remarks follow directly from the definition of .2/.W; ƒ/.

Remark 3.2 Suppose W is a compact, oriented 4k –dimensional manifold and ‰ 1.W / ƒ is a coefﬁcient system for W . W ! (1) If H .W; U Uƒ/ 0 (or H .W; U Kƒ/ 0 if ƒ is PTFA), where U is a 2k I D 2k I D (possibly empty) union of components of the boundary of W then .2/.W;ƒ/ 0. D (2) If ƒ 1 is the trivial group then .2/.W; ƒ/ .W / where is the ordinary D f g D signature function.

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(3) If ƒ ƒ then .2/.W; ƒ/ .2/.W; ƒ / (see for example, Proposition 5.13 0 D 0 of[11]).

(4) Suppose V is a compact, oriented 4k –dimensional manifold, ‰ 1.V / ƒ 0W ! is a coefﬁcient system for V and .V;‰0/ has the same oriented boundary as .W;‰/ (meaning the maps to ƒ agree on the boundary) then

.2/.W V ;‰ ‰ / .2/.W;‰/ .2/.V;‰ / [@W x [ 0 D 0 (see for example, Lemma 5.9 of[11]). (5) If W is closed then .2/.W; ƒ/ .W / (see for example, Lemma 5.9 of[11]). D

We now define .M / for a .4k 1/–dimensional manifold and coefficient system 1.M / . Let M be a closed, orientable, l –dimensional manifold with l 0 ! 6Á mod 4. It is well known that rM is the boundary of some compact, orientable manifold for r 1; 2 . J-C Hausmann showed further[20, Theorem 5.1] that rM is the boundary 2f g of a compact, orientable manifold W for which the inclusion map of M into W induces a monomorphism on 1 . That is, rM is stably nullbordant over 1.M / for some r 1; 2 in the language of Definition 3.3 below. In[5], S Chang and S Weinberger use 2 f g this fact to define a new “Hirzebruch type” invariant .2/ for a .4k 1/ dimensional 1 .2/ manifold M by setting .2/.M / . .W; 1.W // .W //. To define n we D r proceed in a similar manner, stably bounding over 1.M /n instead of 1.M /. To do this, we will show that rM is stably nullbordant over for any coefficient system 1.M / . !

Definition 3.3 Let M M1 Mm be a disjoint union of m connected, closed, D [ [ m oriented l –dimensional manifolds and S i 1.Mi/ i be a collection of D f W ! gi 1 coefficient systems for M . We say that .M; S/ is stably nullbordantD (or s-nullbordant) if there exists a triple .W; ˆ; T / where W is a compact, connected, oriented .l 1/– C dimensional manifold with @W M , ˆ 1.W / ƒ is a coefficient system for m D W ! W , and T i i , ƒ is a collection of monomorphisms such that for each D f W ! gi 1 1 i m, the following diagramD commutes (after modifying ˆ by a change of Ä Ä basepoint isomorphism)

i - 1.Mi/ i

(2) .i / i ? ? ˆ - 1.W / ƒ

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where i Mi W is the inclusion map. We call the triple .W; ˆ; T / a stable W ! (or s-)nullbordism for .M; S/. We say that .M1; S1/ is stably (or s-)bordant to .M2; S2/ if .M1 M2; S1 S2/ is s-nullbordant. If, in addition, i for each [ S [ Š i 1;:::; m, we say that M is stably (or s-)nullbordant over or that .M; / is D stably (or s-)nullbordant.

We remark that stable bordism is an equivalence relation since if .Wi; ˆi 1.Wi/ W ! ƒ ; i; i / is an s-nullbordism for .M M ; ; / (i 1; 2) then i i i 1 i Si 1 i i 1 f C g [ C f C g D .W W1 M W2; ˆ 1.W / ƒ1 ƒ2; ; / is an s-nullbordism for .M1 D [ 2 W ! 2 f 10 30 g [ M3; 1; 3 / when ˆ ˆ1 ˆ2 , and , are the obvious compositions. S f g D 10 30 The proof of the next lemma is similar to the proof of Hausmann’s Theorem 5.1[20].

Lemma 3.4 Let M M1 Mm be a disjoint union of closed, connected, D [ [ m oriented l –dimensional manifolds and S i 1.Mi/ i be a collection of D f W ! gi 1 coefﬁcient systems. If M is nullbordant then .M; S/ is s-nullbordant.D

Proof If 1 m is the free product of the collection i then there D f g is a natural inclusion i , for each i . By the homological coning construction ! of W Thurston and W Kan, is subgroup of an acyclic group ƒ [21, Section 3]. For each i , let i i , ƒ be the inclusion of i into the acyclic group ƒ . The W ! collection i i gives us a map f M K.ƒ ; 1/ such that .f Mi / i i f ı g W ! j D ı for each i and the following diagram commutes.

f - M K.ƒ ; 1/

? ? W - pt

Since K.ƒ ; 1/ is acyclic, the map K.ƒ ; 1/ pt induces an isomorphism on integral ! homology, hence an isomorphism on oriented bordism theory. Since M @W , this D implies that there is a compact manifold W and map g W K.ƒ ; 1/ such that 0 W 0 ! g @W f . Hence .W 0; g ; i / is an s-nullbordism for M . j 0 D f g

Since the l –dimensional oriented bordism group or.pt/ is 2–torsion when l 0 l 6Á mod 4, there is always an .l 1/–dimensional manifold W such that @W 2M . C D Hence we see that .2M; / is always s-nullbordant. f g

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Corollary 3.5 If M is a closed, connected, oriented l –dimensional manifold with l 0 mod 4 and 1.M / is a coefficient system for M then there is an integer 6Á W ! r 1; 2 such rM is s-nullbordant over . 2 f g In this paper, we will often assume that M is a 3–dimensional manifold. Since every closed, oriented 3–dimensional manifold is the boundary of a compact, oriented 4–dimensional manifold, .M; / is s-nullbordant for any 1.M / . W ! For a .4k 1/–dimensional closed, oriented manifold M along with homomorphism 1.M / , we define W ! 1 .M; / . .2/.W; ƒ/ .W // WD r for .W; ˆ/ any s-nullbordism for .rM; / where .W / is the ordinary signature of W . By the following lemma, this definition only depends on M and .

Lemma 3.6 .M; / is independent of the choice of .W; ˆ; T /.

Proof Let .W; ˆ; T / and .W ; ˆ ; T / be two s-nullbordisms for .r1M; / and 0 0 0 f g .r2M; / respectively. Assume that r1 r2 1. Let C W W be the closed, f g D D D [ 0 oriented 4k –manifold obtained by gluing W and W 0 along M . Then there is a coefﬁcient system for C,

ˆ ˆ 1.C / 1.W / 1.W / ƒ ƒ ƒ ƒ 0W D 1.M / 0 ! 1.M / 0 D 0 such that ˆ ˆ .˛/ iƒ.ˆ.˛// for all ˛ 1.W / where iƒ ƒ ƒ ƒ sends 0 D 2 W ! 0 ƒ to the word ƒ ƒ (similarly for ˛ 1.W /). Since ƒ and 2 2 0 2 0 W ! ƒ are monomorphisms, the maps i and i are monomorphisms. Hence by 0 0 ƒ ƒ0 W ! .2/ .2/ .2/ .2/ Remark 3.2 (3), .W; ƒ/ .W; ƒ ƒ / and .W ; ƒ / .W; ƒ ƒ /: D 0 0 0 D 0 Moreover, by Remark 3.2 (4)

(3) .2/ .C / .C / . .2/ .W / .W // . .2/ .W / .W //: ƒ ƒ ƒ ƒ ƒ ƒ 0 0 0 D 0 0 Since C is closed, by Remark 3.2 (5), the left hand side of (3) is 0. Therefore, .2/ .2/ ƒ .W / .W / ƒ .W 0/ .W 0/. The proofs when ri 1 are similar and are D 0 ¤ not included since most of our applications focus on 3–manifolds where r can be assumed to be 1.

Note that we may occasionally write .M; / as .M / or .M; / when the map is clear. As a result of Remark 3.2 (3), .M / only depends on the image of 1.M / in .

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Lemma 3.7 ( –induction) Suppose M is a closed, oriented .4k 1/–dimensional manifold and 1.M / is a coefficient system for M . If , is a W ! W ! 0 monomorphism then .M; / .M; /. D ı Definition 3.8 For each 0 n ! and .4k 1/–dimensional closed, oriented manifold Ä Ä M , we define the n–th order –invariant of M by

.n 1/ n.M / .M; n G G=G C / R WD W H 2 where G 1.M /. D We will now deﬁne the n–th order –invariant of a link in S 3 . First, suppose L S 3 is an m–component link in S 3 with linking numbers 0. Let N.L/ be a neighborhood of L in S 3 .

Proposition 3.9 Let L S 3 be a link for which all the pairwise linking numbers are 3 .!/ zero and G 1.S N.L//. The longitudes of L lie in GH . D

Proof Let i be the longitude of the i –th component of L. We will show that for each .n/ .1/ n 1, i GH . Since the linking numbers of L are zero, i GH . Suppose that .n/ 2 2 i G , for some n 1. Since the longitudes lie on the boundary tori, they commute 2 H with the meridians xi . Hence for each i , Œxi; i 1 is a relation in G . The relation D .n/ .n/ .n/ Œxi; i 1 in G creates the relation i.1 xi/ 0 in the module GH =ŒGH ; GH , D .n 1/ .nD/ showing that i G C since xi 1 in G=G . 2 H ¤ H

.!/ By contrast, the longitudes rarely lie in G! , much less lie in G , the former being true if and only if all of Milnor’s –invariants are zero. The Borromean Rings and the x .!/ Whitehead links provide examples where the longitudes lie in GH but not in G! . As a corollary, performing 0–framed surgery on a link with linking numbers 0 does not change the quotient of the link group by a term of its torsion-free derived series.

Corollary 3.10 (see Proposition 2.5 of[9]) Suppose L S 3 is a link with linking numbers 0. Let ML be the closed 3–manifold obtained by performing 0–framed 3 surgery on the components of L and i S N.L/ ML be the inclusion map. Then W ! for each n 1, 3 1.S N.L// 1.ML/ Š : 3 .n/ .n/ 1.S N.L// ! 1.ML/ H H Proof The kernel of i is the normal subgroup generated by the longitudes. But by 3 .n/ Proposition 3.9 above, the longitudes lie in 1.S N.L//H for all n 0.

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Hence it makes sense to make the following deﬁnition of n for a link. Note that the following deﬁnition does not require that the linking numbers be 0.

Deﬁnition 3.11 Let L be a link in S 3 . For each 0 n ! we deﬁne Ä Ä ! .M / 1 L n.L/ ML; n 1.ML/ .n 1/ : D W 1.ML/HC

m 1 2 As an easy example, we show that n.# S S / n.trivial link/ 0 for all i 1 D D 0 n ! . D Ä Ä Example 3.12 Let W be the boundary connected sum of m copies of S 1 D3 . Then m 1 2 @W #i 1S S . Moreover, the inclusion i @W W induces an isomorphism D D .n 1/ W ! on 1 . Let n F=FH C where F 1.W / is the free group with m generators. D .2/ D By deﬁnition, .@W / .W / .W /. Since W is homotopy equivalent to a n n D .2/ 1–complex, H .W Z / 0. Hence, .W / .W / 0. In particular, we have 2 n n m 1 2 I D D D n.#i 1S S / n.trivial link/ 0. D D D 3 The most important and easiest example to understand is 0 for a knot in S . In this case, 0 is determined by the Levine–Tristam signatures of the knot.

Example 3.13 Let K be a knot in S 3 . By Lemma 5.4 of[11] and Lemma 5.3 of[12] , Z 0.K/ !.K/d! D S 1 1 where !.K/ is the Levine–Tristram signature of K at ! S and the circle is 2 normalized to have length 1. Since ˇ1.MK / 1, the Alexander module of MK is D .n 1/ .1/ torsion[11, Proposition 2.11]. Therefore, for n 0, 1.MK /HC 1.MK / ; D hence n.K/ 0.K/. D A nice property of n is that it is additive under the connected sum of manifolds. To prove this, we show that the torsion free derived series behaves well under the inclusion A A B for suitable C . We start with a lemma. ! C

Lemma 3.14 Let C G be a homomorphism. If ˇ1.C / 0 then the image of .!/ W ! D is contained in GH .

.n/ Proof We will show by induction on n that .C / GH for all n 0. This is trivial .n/ when n 0. Suppose for some n 0, that .C / GH . Then induces a map D .n/ .n/ .n/ .n/ .n 1/ C =ŒC; C GH =ŒGH ; GH GH =GH C . Since ˇ1.C / 1, C =ŒC; C is W ! .n/ .n 1/ ! D Z–torsion. However, GH =GH C is Z–torsion free, hence is trivial which implies .n 1/ .C / G C . H

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Proposition 3.15 Let A B be the amalgamated product of A and B where C , A C ! and C , B are monomorphisms and ˇ1.C / 0. For each 0 n ! , the inclusion ! D Ä Ä i A A B induces a monomorphism W ! C A A C B (4) i .n/ , .n/ : W A ! .A B/ H C H Proof Let G A B . For each n 0, we have the following Mayer–Vietoris D C sequence for group homology with KGn –coefﬁcients

H1.C KGn/ H1.A KGn/ H1.B KGn/ H1.G KGn/ ! I ! I ˚ I ! I ! where the coefficients systems for A; B; C and G are the obvious ones. By Lemma 3.14, the image of C G Gn is trivial hence H1.C ZGn/ H1.C Z/ Z ZGn . ! I Š I ˝ Since Q is a flat Z–module, KGn is a flat ZGn –module, and ˇ1.C / 0, it follows D that H1.C KGn/ H1.C Q/ Q KGn 0. Here, the map Q KGn is induced by I Š I ˝ D ! 1 , Gn . Thus i H1.A KGn/ H1.G KGn/ is a monomorphism. By Theorem ! W I ! .n/ I .n/ .n/ 2.4, i induces a monomorphism A=AH , G=GH as desired. Since A=AH , .n/ ! .!/ .!/ ! G=GH for all n 0, it follows immediately from the definition of AH and GH that .!/ .!/ A=A , G=G . H ! H

Proposition 3.16 Let k 1 and let M1 and M2 be closed, oriented, connected .4k 1/–dimensional manifolds. For each 0 n ! , Ä Ä n.M1#M2/ n.M1/ n.M2/: D C

Proof Let W be the 4k –manifold obtained by adding a 1–handle to .M1 M2/ I 4k 1 4k 1 t along some D D .M1 M2/ 1 and G 1.W /. Then @W M1 t.2/ t f g D D t M2 M1#M2 so .W; Gn/ .W / .M1; Gn/ .M2; Gn/ .M1#M2; Gn/. t D C Since the inclusion i M1#M2 W induces an isomorphism on 1 , by Lemma 3.7, W ! n.M1#M2/ .M1#M2; Gn/. Moreover G 1.M1/ 1.M2/ and the inclusion D D map i1 M1 W induces the inclusion map .i1/ 1.M1/ 1.M1/ 1.M2/ on W ! W ! 1 . Therefore, by Proposition 3.15, .i1/ 1.M1/n Gn is a monomorphism for all W ! n ! . Thus, by Lemma 3.7, n.M1/ .M1; Gn/ (similarly for M2 ). Ä D .2/ To finish the proof, it suffices to show that .W; Gn/ .W / 0. To see this, note D that .W; M1#M2/ is homotopy equivalent to .W 0; M1#M2/ where W 0 is obtained by attaching a .4k 1/–dimensional cell to M1#M2 . Therefore, H .W; M1#M2 ƒ/ 2k I D 0 for any coefficient system 1.W / and left Z –module ƒ. Therefore, W !.2/ H .W; M1#M2 UGn/ 0 and hence .W; Gn/ .W / 0. 2k I D D D

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4 Homology cobordism

Our primary interest in this paper is the study of manifolds up to (rational) homology cobordism. We begin with a deﬁnition.

m m Deﬁnition 4.1 Let M1 and M2 be oriented, closed m–dimensional manifolds. We say that M1 is Q–homology cobordant (respectively homology cobordant) to M2 m 1 if there exists a oriented, .m 1/–manifold W such that @W M1 M2 , and C C D [ S the inclusion maps ij Mj W induce isomorphisms on H . Q/ (respectively W ! I H . Z/). In this case we write M1 QH M2 (respectively M1 ZH M2 ) and deﬁne I the set of rational (respectively integral) homology cobordism classes of m–dimensional m m m m manifolds to be H M = QH (respectively H M = ZH ). Q D f g Z D f g

We will show that n is an invariant of Q–homology cobordism. Since two manifolds that are homology cobordant are necessarily rationally homology cobordant, n is an invariant of homology cobordism.

Theorem 4.2 If M 4k 1 is Q–homology cobordant to M 4k 1 .k 1/ then 1 2 n.M1/ n.M2/: D

Proof Let W be a 4k –dimensional manifold such that @W M1 M2 , ij Mj D [ S W ! W be the inclusion maps, E 1.W /, and Gj 1.Mj / for j 1; 2. Since D D D .ij / Hk .Mj Q/ Hk .W Q/ is an isomorphism for k 1; 2, .ij / H1.Gj Q/ W I ! I D W I ! H1.E Q/ is an isomorphism and .ij / H2.Gj Q/ H2.E Q/ is surjective. Hence I W I ! I by Theorem 4.1 of[9], for each n 0, the inclusion maps induce monomorphisms Gj E .ij / .n 1/ , .n 1/ : W .Gj /HC ! EH C

.n 1/ Let n E=EH C then we have coefficient systems .ˇj /n Gj n defined by D .n 1/ W ! .ij / .j /n where .j /n Gj Gj =.Gj /HC is the quotient map. By Remark ı W 3.2 (3), we have n.Mj / .Mj ; Gj n/. Therefore D ! .2/ n.M1/ n.M2/ .@W; n/ .W; n/ .W /: D D .2/ To finish the proof, we show .W; n/ .W / 0. Since .i1/ H2.M1 Q/ D D W I H2.W Q/ is surjective, the second homology of W comes from the boundary. Thus I the intersection of any two classes is H2.W Q/ is zero. In particular .W / 0. I D Let Kn be the classical right ring of quotients of Zn . Since Hi.W; M1 Q/ 0 for I D i 0; 1; 2 then by Proposition 2.10 of[11], Hi.W; M1 Kn/ 0 for i 0; 1; 2. By D .2/ I D D Remark 3.2 (1), .W; n/ 0. D

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Hence, for each n 0 and m 4k 1 with k 1 we have a map D m n H R: W Q ! Note that if L1 and L2 are concordant links then their 0–surgeries are homology cobordant hence n.L1/ n.L2/. Moreover, n of the trivial link is 0 as in Example D 3.12.

Corollary 4.3 For each n 0, n is a concordance invariant of links and is 0 for slice links.

We will use n to further investigate the structure of concordance classes of links in Section 6.

5 Nontriviality of n

We will show that the n are highly nontrivial. To do this we will show that the image 3 of n H R in R is dense and is an inﬁnitely generated subset of R. Before we W Q ! can do this, we must deﬁne a family of examples of 3–manifolds on which we can calculate n .

5.1 Examples: genetic modiﬁcation

We describe a procedure wherein one starts with a 3–manifold M (respectively a link L in S 3 ) and “infects” M (respectively L) along a curve in M (respectively S 3 L) with a knot K in S 3 to obtain a new 3–manifold M.; K/ with the same homology as M (respectively link L.; K/ in S 3 ). This construction is a specific type of satellite constructions which has been dubbed genetic infection (see Section 3 of[12]). We first describe the construction for a general 3–manifold. Let M be a compact, connected, oriented 3–manifold, be a curve embedded in M , and K be a knot in S 3 . Denote by N./ and N.K/ a tubular neighborhood of in M and K in S 3 respectively. Let and K be the meridians of and K respectively, and let l and lK be the longitudes of and K respectively. Note that if is not nullhomologous then the longitude of is not well-defined. In this case, we choose l to be an embedded curve on N./ that is isotopic to in M and intersects geometrically once. Define

(5) M.; K/ .M N.// .S 3 N.K// D [f

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3 1 where f @.S N.K// @.M N.K// is defined by f .K / l and f .lK / W ! D D . If is not nullhomologous, then there is a choice of longitude for and the homeomorphism type of M.; K/ will depend on this choice. Since H .S 3 N.K// is independent of K, an easy Mayer–Vietoris argument shows that M and M.; K/ have isomorphic homology groups. Now, consider the case when M S 3 N.L/ where L is an m–component link in S 3 D and is a curve in S 3 N.L/ S 3 . Even in the case that is not nullhomologous in 3 H1.S N.L// there is still a well defined longitude l for since is nullhomologous in S 3 . By choosing this longitude, we have a well-defined manifold M.; K/. We now further assume that bounds an embedded disk D in S 3 . It is well known that in this case, M.; K/ is homeomorphic to S 3 N.L.; K// where L.; K/ is another m–component link in S 3 . Moreover, one can check that L.; K/ can be obtained by the following construction. Seize the collection of parallel strands of L that pass through the disk D in one hand, just as you might grab some hair in preparation for braiding. Then, treating the collection as a single fat strand, tie it into the knot K. Note that in the special case that is a meridian of the i –th component Li of L then L.; K/ is the link obtained adding a local knot K to Li .

Remark 5.1 If L is a boundary link in S 3 then L.; K/ is also a boundary link in S 3 . Hence T .; K/ is always a boundary link where T is the trivial link.

Remark 5.2 Let ML be the result of performing 0–framed surgery on a link L in 3 3 S with all linking numbers 0 and let be a curve in S N.L/ ML that bounds 3 an embedded disk in S . Then ML.; K/ M . D L.;K/ Example 5.3 (Iterated Bing doubles of K) Let T be the trivial link with 2 compo- 3 nents and let bing be the curve in Figure 1. Then bing bounds a disk in S . L.; K/ is the link in Figure 2 and is more commonly known as the (untwisted) Bing double of .1/ .2/ 3 K, BD.K/. We note that bing F F where F 1.S N.T //. Moreover, 2 D any (untwisted) iterated Bing double of K can be obtained as T .; K/ where T is a .n/ .n 1/ trivial link with m 2 components and F F C for some n 1. 2 We now construct a cobordism C C.; K; W / between M and M.; K/ for which D the inclusion maps will behave nicely modulo the torsion-free derived series of their 3 respective fundamental groups. Recall that MK , the 0–surgery on K in S , is deﬁned 3 1 2 3 as MK .S N.K// q ST where ST S D and q @.ST/ @.S N.K// is D [ 2 D W ! deﬁned so that q . pt @D / lK . By[12, p 118], MK bounds a compact, oriented f g D 4–manifold W such that .W / 0 and 1.W / Z, generated by .K / where D Š MK W is the inclusion map. We have an exact sequence 2.W / H2.W / W ! ! !

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bing

3 Figure 1: bing S trivial link 2 f g

Figure 2: Bing double of K

H2.1.W //. Since 1.W / Z, the last term is zero, and the first map is surjective. Š Hence H2.W / image.2.W / H2.W //. Using the 4–manifold W, we can D ! construct a cobordism C C.; K; W / between M and M.; K/ as follows. Glue W , D 2 1 as above, to M I by identifying ST MK @W to N./ D S M 1 so that 1 D D f g l is identified with K and is identified to lK . It follows that @C M M.; K/. D t Let i M C and j M.; K/ C be the inclusion maps. W ! W ! We use this cobordism to show that the difference between i.M / and i.M.; K// .n/ depends only on 0.K/ and max n .1.M //H (see Theorem 5.8 below). We f j 2 g begin with some algebraic lemmas that will be employed in the proof of Theorem 5.8.

Lemma 5.4 i 1.M / 1.C / is an isomorphism. W !

Proof 1.W / Z which is generated by .K /. Moreover, K generates the Š 2 group 1.W .M I// 1. D /. Hence, by the Seifert–Van Kampen Theorem, \ D

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1.M I/ 1.C /, induced by the inclusion map, is an isomorphism. Therefore i ! is an isomorphism.

.i/ In particular, the inclusion map induces isomorphisms between 1.M /=1.M /H and .i/ 1.C /=1.C /H for all i . In order to show that j induces an isomorphism between .i/ .i/ 1.M.; K//=1.M.; K//H and 1.C /=1.C /H for all i , we appeal to Theorem 2.4 and Proposition 2.5. The next two lemmas will guarantee that the hypotheses of Proposition 2.5 are satisﬁed.

Lemma 5.5 j 1.M.; K// 1.C / is an epimorphism. W !

Proof Let ˛ be a curve in 1.C /. By Lemma 5.4, i is an isomorphism. Hence ˛ can be represented by a curve in M . Moreover, by general position, we can assume ˛ misses N./. Push ˛ into M 1 to get a curve ˇ in .M 1 / .N./ 1 / M.; K/ f g f g f g such that j .ˇ/ ˛. D

Lemma 5.6 j 1.M.; K// 1.C / induces an isomorphism on H1. Z/ and W ! I an epimorphism on H2. Z/. I

Proof Let G 1.M.; K// and E 1.C /. Since j induces an epimorphism on D D 1 , j induces an epimorphism on H1. /. The inclusion l M N./ M.; K/ W ! induces a epimorphism on H1. Z/. Moreover, the inclusion l M N./ M I 0W ! induces an epimorphism on H1. Z/ and the kernel of l0 is the subgroup generated I by H1.M N.//. 2 Consider the following commutative diagram where all of the maps are induced from inclusion maps: l--0 H1.M N.// H1.M /

l iŠ ? ? j-- H1.M.; K// H1.C /: Suppose ˛ H1.M.; K// and j .˛/ 0. Since l is surjective, there exists 2 D 2 H1.M N.// such that l . / ˛. Hence i .l0 . // j .l . // 0. Since i is D D D injective, ker.l0 / so is a multiple of . However, l ./ 0 since in M.; K/, 2 3 D is identiﬁed with lK , which bounds a surface in S K M.; K/. Therefore j H1.M.; K// H1.C / is a monomorphism. Since H1.M.; K// H1.G/ W ! D (similarly for E), j H1.G/ H1.E/ is an isomorphism. W !

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Let ˛ H2.E/, then there exists ˇ H2.C / such that k.ˇ/ ˛ where k H2.C / 2 2 D W ! H2.E/ is the map in the exact sequence k 2.C / H2.C / H2.E/ 0: ! ! ! By a Mayer–Vietoris argument, we see that H2.C / H2.W / H2.M I/ (in the Š ˚ obvious way). Since H2.W / im.2.W / H2.W // and any element of 2.W / D ! goes to zero under k , we can assume that ˇ H2.M I/ 0 H2.M /. Let S be 2 ˚f g Š ˇ a surface in M representing ˇ. We will construct a surface S M.; K/ such that ˇ0 k.j .ŒSˇ0 // k.i .ŒSˇ// ˛. This will complete the proof since k j j k0 D D ı D ı where k H2.M.; K// H2.G/. 0W We can assume that Sˇ intersects N./ in finitely many disks Di . To construct Sˇ0 , remove each of these disks and replace them with a copy of a chosen seifert surface for K in S 3 K M.; K/ oriented according the signed intersection of with S . ˇ Let F be the surface in W C obtained by gluing a disk in .M 1 / W whose f g \ boundary is with a copy of the chosen seifert surface for K. Since H2.1.W // 0, D k.ŒF/ is trivial in H2.E/. Moreover ŒS mŒF ŒS in H2.C / for some m so ˇ0 D C ˇ k.j .ŒSˇ0 // k.i .ŒSˇ//. D .i 1/ Note that MK @W C . For each i 0, let i 1.MK / 1.C /=1.C / C be D W ! H the composition of the map induced by inclusion 1.MK / 1.C / and the quotient .i 1/ ! map 1.C / 1.C /=1.C / C . ! H .i 1/ Lemma 5.7 If i 1.MK / 1.C /=1.C / C is the homomorphism as described W ! H above then 1 0 i n 1 Im.i/ f g Ä Ä I Š Z i n: 3 Proof Let E 1.C /. Recall that 1.MK / 1.S K/= lK where lK is the lon- D Š h i Q 1 gitude of K. Hence every element of 1.MK / can be represented by ˛ i giK gi 3 D where gi 1.S K/ and K is a fixed meridian of K. Let 1.MK / E 2 W !1 be induced by the inclusion of MK into C . Since K is identified to l in Q 1 1 .n/ E, we see that .˛/ i .gi/i .l/ .gi/ . Moreover, since l PH and .i 1/ .i 1/D .i 1/ 2 P=P C E=E C for all i , .˛/ E C for 0 i n 1. Therefore the image H Š H 2 H Ä Ä of i is trivial for 0 i n 1. Ä Ä We will prove that the image of i is Z for i n by induction on i . First we prove this .n/ .n/ .n/ is true for i n. Since .1.MK // EH , .Œ1.MK /; 1.MK // ŒEH ; EH .n 1/ D EH C . Hence we have a well-defined map

.1/ .n 1/ Z 1.MK /=1.MK / E=EH C : xW Š !

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.n 1/ .n 1/ .n 1/ 1 .n 1/ Since l PH C and P=PH C E=EH C , we have .K / i .l / EH C . 62 Š .n/D 62 Therefore is nontrivial. Moreover, since .1.MK // EH , the image of is x .n/ .n 1/ x contained is EH =EH C which is Z–torsion free. It follows that the image of n is isomorphic to Z.

To ﬁnish the induction, assume that the image of i is isomorphic to Z for some .j 1/ i n. Let A 1.MK /; then by Example 3.13, A=AHC Z for all j 0. D .i 1/ Š .i 1/ By assumption, induces a monomorphism A=AHC , E=EHC . Thus, by .i 2/ W .i 2/ ! .i 1/ .i 2/ Proposition 2.2, induces a map A=AHC E=EHC . Since AHC AHC , .i 2/ .i 2/W ! D the map A=AHC E=EHC is a monomorphism. Therefore the image of i 1 W ! C is Z.

.n/ .n 1/ Theorem 5.8 Let M.; K/ be as deﬁned in .5/ and P 1.M /. If P P C D 2 H H for some n 0 then 0 0 i n 1 i.M.; K// i.M / Ä Ä I D 0.K/ i n: Before proving Theorem 5.8, we establish two easy corollaries.

m 1 2 Corollary 5.9 Let be a embedded curve in #i 1S S that is nontrivial in F m 1 2 D D 1.#i 1S S /. If K is a knot with 0.K/ 0 then M.; K/ is not Q–homology D m 1 2 ¤ cobordant to #i 1S S . D .k/ .k/ .!/ Proof Since F is a free group, FH F for all k 0 and F 1 . Therefore .n/ .n 1/ D D f g F F C for some n 0. By Theorem 5.8, n.M.; K// 0.K/ 2 m 1 2 D C n.#i 1S S / 0.K/. Thus, if K is a knot with 0.K/ 0 then M.; K/ is D D m 1 2 ¤ not Q–homology cobordant to #i 1S S . D 3 Corollary 5.10 Suppose K is a knot in S with 0.K/ 0. Then no iterated Bing ¤ double of K is slice.

Proof Any iterated Bing double of K can be obtained as T .; K/ where is a nontrivial commutator of length m 1 where m is the number of components of C .n/ .n 1/ T .; K/ [6]. Therefore, is a nontrivial element in F =F C for some n 1 where F 1.T /. By Theorem 5.8, n.T .; K// 0.K/ 0. Hence T .; K/ is D D ¤ not slice.

Subsequent to our work, it has been shown that if the Bing double of K is slice then K is algebraically slice[4]. We will now prove Theorem 5.8.

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Proof of Theorem 5.8 Let W be as deﬁned before, V be an open neighborhood of MK @W in W , WV W V , and CV C WV where C C.; K; W / is D D D D as described in the discussion preceding Lemma 5.4. Then V is homeomorphic to 2 MK Œ0; 1 and CV can be obtained by gluing M Œ0; 1 to MK Œ0; 1 where D 2 1 is identiﬁed to ST D S in MK . It follows that CV has 3 boundary components; D in particular, @CV M M.; K/ MK . D t t

M.; K/ MK

M 0 f g Figure 3: The 4k –manifold CV with @CV M M.; K/ MK D t t

.i 1/ Let G 1.M.; K//, E 1.C /, and i E=EHC for i 0. Since CV C , D D D there is an obvious coefficient system 1.CV / i (similarly for M , M.; K/, ! and MK ). By Lemmas 5.4, 5.5 and 5.6, Theorem 2.4, and Proposition 2.5, the maps .i 1/ .i 1/ i P=PHC i and j G=GHC i are isomorphisms for all i 0. Therefore, W ! W ! i.M / .M; i/ and i.M.; K// .M.; K/; i/. Hence we have D D .2/ (6) i.M / i.M.; K// .MK ; i/ .CV ; i/ .CV / C D for all i 0. Since the abelianization of 1.MK / is Z, there is a unique surjective homomorphism 1.MK / Z up to isomorphism. Therefore, by Lemma 5.7 and Lemma 3.7 we have .MK ; i/ 0.MK / for all i n, and .MK ; i/ 0 for 0 i n 1. D D Ä Ä .2/ To finish the proof, we will show that .CV ; i/ .CV / 0 for all i 0. Since D Ui is a flat Zi –module it suffices to show that the map induced by the inclusion H2.@CV Zi/ H2.CV Zi/ is surjective for 1 i (recall that 1 1 /. I ! I Ä D f g

Recall that CV MK I D2 M I . Consider the Mayer–Vietoris sequence D [ 2 H2.MK I Zi/ H2.M I Zi/ H2.CV Zi/ H1. D Zi/ ! I ˚ I ! I ! I H1.M I Zi/ H1.MK I Zi/: ! I ˚ I

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By Lemma 5.7, if i n 1 then MK lifts to the i –cover hence the homology group Ä H1.MK I Zi/ Zi is generated by K . Moreover, in this case, lifts to the i – I Š2 cover and H1. D Zi/ H1.MK Zi/ is an isomorphism. Therefore H2.MK I ! I I Zi/ H2.M I Zi/ H2.CV Zi/ is surjective for i n 1. If i n, then I ˚ I 2 ! I Ä 2 the image of 1. D / in i is Z by Lemma 5.7. Hence H1. D Zi/ 0. I D Thus for i n we have that H2.MK I Zi/ H2.M I Zi/ H2.CV Zi/ I ˚ I ! I is surjective. Thus H2.@CV Zi/ H2.CV Zi/ is surjective for all i 1. I ! I 5.2 Nontriviality of examples

3 3 4 For each knot K in S , there exists a degree one map fK S K S T (where W ! T is the unknot) that induces an isomorphism on homology with Z coefficients and fixes the boundary. Hence, there is a degree one map fK M.; K/ M.; T / M x W ! D that induces an isomorphism on H. Z/. Recall that Hm is the set of Q–homology I Q cobordism classes of closed, oriented m–dimensional manifolds. For a fixed closed, oriented m–dimensional manifold M , we define Hm.M / Hm as follows: ŒN Q Q 0 2 Hm.M / if there exists an N such that ŒN ŒN Hm and there exists a degree Q D 0 2 Q one map f N M that induces an isomorphism on H . Q/. By Proposition W ! I 3.16, n is additive under the connected sum of manifolds. Therefore the images of m m n H .M / R and n H R are subgroups of R. W Q ! W Q !

Theorem 5.11 Let M be a closed, oriented 3–manifold, G 1.M / and n 0. If .n/ .n 1/ 3 D GH =GH C 1 then the image of n HQ.M / R is (1) dense in R and (2) an ¤ f g W ! 3 inﬁnitely generated subgroup of R. In particular, the image of n H R is (1) W Q ! dense in R and (2) an inﬁnitely generated subgroup of R .

Proof Let 2 < r < 2. By[3, Section 2], for all > 0, there exists a knot Kr; in S 3 such that ˇZ ˇ ˇ ˇ ˇ !.Kr;/d! rˇ < : ˇ S 1 ˇ 1 Here, !.K/ is the Levine–Tristram signature of K at ! S and the circle is 2R normalized to have length 1. By[11, Lemma 5.4], 0.MK / S 1 !.K/d! for any 3 D .n/ .n 1/ knot K in S . Let be a curve in M representing an element in G G C . Then H H by Theorem 5.8, n.M.; K// 0.MK /. Moreover, by the above remarks, there is D 3 a degree one map from M.; K/ to M hence r n.H .M //. 2 Q For arbitrary r R there exists a positive integer m such that r=m . 2; 2/. Let > 0 2 2 and set 0 =m. As above, there exists a knot Kr=m; such that 0.MKr=m; / D 0 j 0 r=m < . We remark that K1#K2 can be obtained as MK .; K2/ where is a j 0 1 meridian of K1 in MK . Hence by Theorem 5.8, 0.MK K / 0.MK / 0.MK / 1 1# 2 D 1 C 2

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(this can also be computed directly). Hence 0.MmK / m0.MK / where mK is the D connected sum of K with itself m times. Therefore ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 0.MmKr=m; / r m 0.MKr=m; / r=m < m0 : ˇ 0 ˇ D ˇ 0 ˇ D 3 As before, it follows that r n.H .M //. This completes the veriﬁcation that the 2 Q image of n is dense in R.

By Proposition 2.6 of[12], there exists an infinite set Ji i Z of Arf invariant f j 2 Cg zero knots such that 0.MJi / is linearly independent over the integers. Therefore f g 3 n.M.; Ji// is an infinitely generated subgroup of R. Since n.H .M // is an f g Q abelian group that contains this subgroup, it is itself infinitely generated.

Let T be a trivial link. If L is a boundary link then, just as for a knot, there is a degree one map S 3 L to S 3 T that fixes the boundary and induces an isomorphism on homology. In particular, there is a degree one map from the 0–surgery on a boundary link with m components to the connected sum of m copies of S 1 S 2 that induces an isomorphism on homology. Thus, for each m 1, we can consider the subset 3;b 3;b H .m/ H3 .#m S 1 S 2/ defined by ŒN H .m/ if ŒN ŒN H3 where Q Q i 1 0 2 Q 0 D 2 Q N is 0–surgery onD an m component boundary link in S 3 . As before, for each n and 3;b m, the image of n H .m/ R is a subgroup of R. W Q ! 3;b Corollary 5.12 For each n 0 and m 2, the image of n H .m/ R is (1) W Q ! dense in R and (2) an infinitely generated subgroup of R.

Proof Let M be the manifold obtained by performing 0–surgery on the trivial link m 1 2 with m 2 components. Then M #i 1S S and 1.M / F.m/. Since .n/ .n 1/ .n/ .n 1/ D D Š .n/ .n 1/ FH =FH C F =F C is nontrivial for n 0, there exists an FH FH C . D 3 3 2 We can assume that 1.S T / where T is the trivial link in S since the inclusion 3 2 S T M induces an isomorphism on 1 . Moreover, we can alter by a homotopy 3 ! 3 in S T to obtain a curve 0 that bounds a disk in S . Thus, as we showed in the proof 3 3;b of Theorem 5.11, the image of n M. ; K/ K is a knot in S H .m/ R is W f 0 j g Q ! dense in R and is an inﬁnitely generated subgroup of R.

We will now show that the n are independent functions. For each m 2, let Vm 3;b 3;b D f H .m/ R , be the vector space of functions from the set H .m/ to R and f W Q ! g Q V f H3 R be the vector space of functions from H3 to R. D f W Q ! g Q

Theorem 5.13 For each m 2, n n1 0 is a linearly independent subset of Vm . In f g D particular, n n1 0 is a linearly independent subset of V . f g D

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3;b Proof Let ˛ r1i r i where ri are nonzero real numbers, i H .m/ D 1 C C k k j W Q ! R and i < i if j < j . Suppose ˛ 0. Let M be 0–surgery on the m–component j j 0 0 3 D trivial link T in S and F 1.M /. As in the proof of Corollary 5.12 above, for D 3 3 each n 0, there is a curve n in S T such that n bounds a disk in S and .n/ .n 1/ n FH FH C . For each n 0, let Mn M.n; K/ for some K with 0.MK / 0 2 D ¤ (for example, let K be the right handed trefoil). By the remarks above Corollary 5.12, 3;b Mn H .m/. By Theorem 5.8, i.Mn/ 0 for i n 1 and n.Mn/ 0. Hence Q D Ä ¤ 0 ˛.Mi / r i .Mi /. Since i .Mi / 0, r 0. This is a contradiction. D k D k k k k k ¤ k D 6 The grope and .n/–solvable ﬁltrations

We now investigate the grope and .n/–solvable filtrations of the string link concordance group (with m 2 strands) first defined for knots by T Cochran, K Orr and P Teichner in[11]. We will show that the function n is a homomorphism on the subgroup of boundary links and vanishes for .n 1/–solvable links. Using this, we will show C that each of the successive quotients of the .n/–solvable filtration of the string link concordance group contains an infinitely generated subgroup (even modulo local knotting). We will also show that a similar statement holds for the grope filtration of the string link concordance group. The reason that we study string links instead of ordinary links is that the connected sum operation for ordinary links is not well-defined. Thus, in order to have a group structure on the set of links up to concordance, we must use string links. We begin by recalling some definitions. Recall that an m–component string link (sometimes called an m–component disk link[25; 23]) is a locally flat embedding f F I D3 of m oriented, ordered copies of I in D3 that is transverse to the W m ! F 2 boundary and such that f m @I is the standard m–component trivial 0–link in S , F 2 j j0 @I S . Two m–component string links f; g are concordant if there exists W m ! a locally flat embedding F F I I D3 I that is transverse to the boundary W m ! and such that F F f; F F g and F F j0 idI . The j m I 0 D j m I 1 D j m @I I D concordance classes offmg–component stringf g links forms a group under stacking (see Figure 5) which we denote by C.m/ (see[23] for more details). In the literature this group is often denoted C.m; 1/ or CSL.m/. This group is known to be nonabelian when m 2 [23] and abelian when m 1. Let B.m/ be the subgroup of boundary D disk links in C.m/. If L is a string link then the closure of L, denoted by Lb, is the oriented, ordered m–component link in S 3 obtained by adjoining to its boundary the standard trivial string link (see Figure 4). A string link is equipped with a well-defined set of meridians that we denote by 1; : : : ; m .

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L L D D y

Figure 4: The string link L and its closure L y

6.1 .n/–Solvable ﬁltration of C.m/

In[11, Definition 8.5, p 500] and[11, Definition 8.7, p 503], Cochran, Orr, and Teichner 1 define an .n/–solvable knot and link where n N0 . We recall the definitions here. 2 2 We first recall the definition of an .n/–Lagrangian. Let M be a fixed closed, oriented 3–manifold. An H1 –bordism is a 4–dimensional spin manifold W with boundary M such that the inclusion map induces an isomorphism H1.M / H1.W /. For any Š 4–manifold W , let W .n/ denote the regular covering of W that corresponds to the n–th term of the derived series of 1.W /. For each n 0, one can define the quadratic .n/ forms n; n on H2.W / in terms of equivariant intersection and self-intersection numbers of surfaces in W that lift to the cover W .n/ . If F is a closed, oriented, immersed, and based surface in W that lifts to W .n/ then F is called an .n/–surface. See[11, Chapter 7, pp 493–496] for the precise definitions.

Deﬁnition 6.1 Let W be an H1 –bordism such that 0 is a hyperbolic form.

(1) A Lagrangian for 0 is a direct summand of H2.W / of half rank on which 0 vanishes.

.n/ (2) An .n/–Lagrangian is a submodule L H2.W / on which n and n vanish and which maps onto a Lagrangian of the hyperbolic form 0 on H2.W /. (3) Let k n. We say that an .n/–Lagrangian L admits .k/–duals if L is generated Ä by .n/–surfaces l1;:::; lg and there are .k/–surfaces d1;:::; dg such that H2.W / has rank 2g and

.li; dj / ıij : k D

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We now deﬁne .n/–solvability for a 3–manifold or link in S 3 .

Deﬁnition 6.2 Let M be a closed, oriented 3–manifold and n N . M is called 2 .0/–solvable if it bounds an H1 –bordism W such that .H2.W /; 0/ is hyperbolic. M is called .n/–solvable for n > 0 if there is an H1 –bordism W for M that contains an .n/–Lagrangian with .n/–duals. The 4–manifold W is called an .n/–solution for M . A link L in S 3 is said to be .n/–solvable if the zero surgery on L is .n/–solvable. A string link L C.m/ is said to be .n/–solvable if L is .n/–solvable. 2 y

Definition 6.3 Let M be a closed, oriented 3–manifold and n N0 . M is said to be 2 .n:5/–solvable if there is an H1 –bordism for M that contains an .n 1/–Lagrangian C with .n/–duals. The 4–manifold is called an .n:5/–solution. A link L in S 3 is said to be .n:5/–solvable if the zero surgery on L is .n:5/–solvable. A string link L C.m/ 2 is said to be .n:5/–solvable if L is .n:5/–solvable. y Hence, for each m 1, we can define a filtration of the string link concordance group F m F m F m F m C.m/ .n:5/ .n/ .0:5/ .0/ m 1 by setting F.n/ to be the set of .n/–solvable L C.m/ for n 2 N0 . It is easy to m 2 2 check that F.n/ is a subgroup of C.m/ and in fact is a normal subgroup of C.m/ for 1 each m N and n N0 . 2 2 2 When m 1, C.1/ is the concordance group of knots which is an abelian group. It was D 1 1 shown in[11] that C =F Z2 given by the Arf invariant and C =F is J P Levine’s .0/ Š .0:5/ algebraic concordance group which is isomorphic to Z Z Z [24]. It was 1 ˚ 21 ˚ 41 also shown in[11; 12] that F 1 =F 1 for n 1; 2 has infinite rank. Moreover, in .n/ .n:5/ D [13], Cochran and Teichner showed that for each n N , F 1 =F 1 is of infinite order. 2 .n/ .n:5/ However, it is still unknown whether F 1 =F 1 has infinite rank for n 3. .n/ .n:5/ m m For m 2, we will show that for all n N0 , F.n/=F.n 1/ contains an infinitely 2 C generated subgroup, the subgroup “generated by boundary links” defined as follows. Define the .n/–solvable filtration of B.m/, the subgroup of boundary string links, by (7) BF m B.m/ F m : .n/ WD \ .n/ m m m m Then for each n 0, BF.n/=BF.n 1/ is a subgroup of F.n/=F.n 1/ . We will show that m mC C the abelianization of BF.n/=BF.n 1/ has infinite rank. It would be very interesting to m m C know whether BF.n/=BF.n:5/ is infinitely generated. It would be even more interesting m m m m to exhibit nontrivial links in BF.n:5/=BF.n 1/ . To show that BF.n/=BF.n 1/ is infinitely C C generated, we show that n is a homomorphism on the subgroup B.m/ and that n vanishes on .n 1/–solvable links. We begin with the latter. C

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Theorem 6.4 If a 3–manifold M is .n/–solvable then for each .n/–solution W and k n, the inclusion i M W induces monomorphisms Ä W ! .k/ .k/ i H1.M K.1.W /=1.W /H // , H1.W K.1.W /=1.W /H //; W I ! I 1.M / 1.W / i .k 1/ , .k 1/ W 1.M /HC ! 1.W /HC and k 1.M / 0: D

Thus, if L F.n/ then k 1.L/ 0 for k n. 2 D Ä

Proof Let W be an .n/–solution for M , G 1.M /, and E 1.W /. We D D will prove the result by induction on k . The result is clearly true for k 0 (here D 1.M / 0 for any M ). Assume the result is true for some k n 1. D Ä .k 1/ .k 1/ Let WH C be the regular cover of W corresponding to EH C , the k –th term of the .n/ .n/ .k 1/ torsion-free derived series of E. Since E E E C , W admits a “torsion- H H free .k 1/–Lagrangian” with “torsion-free .k 1/–duals.” Speciﬁcally, there are C H C H .k 1/ intersection and self-intersection forms k 1 and k 1 on H2.WH C /. We project C .kC 1/ the .n/–Lagrangian L and .n/–duals for L to H2.WH C / to get “torsion free .k 1/– C surfaces” l1;:::; lg; d1;:::; dg such that

H H k 1.li; lj / 0 and k 1.li; dj / ıij C D C D for 1 i; j g. Ä Ä .k 1/ Recall that for a group E, Ek 1 E=EH C . We will show that C D

l1;:::; lg; d1;:::; dg f g .k 1/ is a ZEk 1 –linearly independent set in H2.WH C /. Suppose C g X 0 mili nidi D C i 0 D H for some mi; ni ZEk 1 . Then applying k 1.lj ; / we get 2 C C g X H H 0 mi .lj ; li/ ni .lj ; di/ nj D k 1 C k 1 D i 0 C C D H Pg for 1 j g. We now apply k 1. ; dj / to the new equality 0 i 0 mili to Ä Ä C D D get mj 0 for all 1 j g. Therefore li; di 1 i g is a linearly independent D Ä Ä f j Ä Ä g

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set. Thus l1;:::; lg; d1;:::; dg generates a rank 2g free ZEk 1 –submodule of .k 1/f g C H2.WH C /. Moreover, since

rankKEk 1 H2.W KEk 1/ ˇ2.W / 2g; C I C Ä D by Proposition 4.3 of[11], l1;:::; lg; d1;:::; dg generates H2.W KEk 1/ as a f g I C KEk 1 –module. In particular, H2.W KEk 1/ is a free KEk 1 –module of rank 2g. C I C C

Consider the homomorphism hW;Ek 1 H2.W KEk 1/ H2.W KEk 1/ defined x C W I C ! I C by replacing the coefficients Uƒ in Section 3 (1) with Kƒ where ƒ Ek 1 . Since D C hW;Ek 1 .di/.lj / k 1.li; dj / KEk 1 ıij and hW;Ek 1 .li/.lj / k 1.li; lj / x C D C ˝ C D x C D C ˝ KEk 1 0 for 1 i; j g, we can use the same argument as before to show that if C D g Ä Ä h .P m l n d / 0 then m n 0 for all 1 i g. Hence h xW;Ek 1 i 0 i i i i i i xW;Ek 1 C D C D D D Ä Ä C is a monomorphism. However, since hW;Ek 1 is a monomorphism between modules of x C the same rank, hW;Ek 1 is an isomorphism. In particular, the map H2.W KEk 1/ x C I C ! H2.W; M KEk 1/ is surjective which implies that the map H1.M KEk 1/ I C I C ! H1.W KEk 1/ is a monomorphism. Hence, by Theorem 2.4, I C 1.M / 1.W / (8) i .k 2/ , .k 2/ W 1.M /HC ! 1.W /HC is a monomorphism. Moreover, .W; k E Ek 1; i Gk 1 Ek 1/ is therefore W ! C W C ! C.2/ an s-nullbordism for .M; k G Gk 1/. By definition, k .M / .W; Ek 1/ W ! C D C .W /.

.2/ Since .W / 0, to complete the proof, we show that .W; Ek 1/ 0. Re- D 2g C D call that H2.W; M KEk 1/ H2.M KEk 1/ .KEk 1/ where the isomor- I C Š I C Š C phism is given by the composition of the Poincare duality and Kronecker map. Thus H2.W KEk 1/ H2.W; M KEk 1/ is a surjective map between finitely generated I C ! I C KEk 1 –modules of the same rank; hence is an isomorphism. Since Ek 1 is an C C Ore domain, UEk 1 is flat as a right KEk 1 –module hence H2.W UEk 1/ C C I C ! H2.W; M UEk 1/ is an isomorphism. By naturality, the .k 1/–Lagrangian above I C C also becomes a metabolizer for hW;Ek 1 with UEk 1 coefficients. Thus, hW;Ek 1 C C .2/ C is a nonsingular pairing with metabolizer so by Lemma 3.1, .W; Ek 1/ 0. C D

We now show that n is additive on boundary links. We begin with an algebraic lemma that will be useful in the proof.

Lemma 6.5 Let A be a ﬁnitely related group, E be a ﬁnitely generated group and i A , E be a monomorphism that induces an isomorphism on H1. Q/. If there W ! I is a retract r E A then for each n 0, both i and r induce an isomorphism W !

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.n/ .n/ .n/ .n/ i A=AH , E=EH and r E=EH A=AH respectively. Moreover, for each W ! W ! n 0, .n/ .n/ .n/ K .n/ K rankK.E=E / H1.E .E=EH // rankK.A=A / H1.A .A=AH //: H I D H I

Proof Since r is a retract and i induces an isomorphism on H1. Q/, r induces an I isomorphism on H1. Q/ and a surjective map H2. Q/. Hence, by Theorem 4.1 of I I [9] (see also Theorem 2.4 and Proposition 2.5 of this paper), r induces isomorphisms .n/ .n/ r E=EH A=AH for all n 0. We will prove i induces monomorphisms W .n/ ! .n/ i A=AH , E=EH by induction. This is clear when n 0; 1. Assume that the W ! D result holds for some n 1, then by Proposition 2.2, i induces a homomorphism n 1 .n 1/ .n 1/ n 1 i C A=AHC E=EH C . Moreover, if we postcompose i C with W ! .n 1/ Š .n 1/ r E=EH C A=AHC ; W ! n 1 we get the identity. Hence i C is an isomorphism. The last statement follows from the last part of Theorem 2.4.

If L1 and L2 are m–component string links then L1L2 is the m–component string link obtained by stacking L1 on top of L2 as depicted in Figure 5. This stacking operation induces the multiplication in the group C.m/.

L1L2 D

Figure 5: The product of L1 and L2 , L1L2

Proposition 6.6 If L1; L2 C.m/ and L1; L2 are boundary links then for each n 0, 2 y y n.L1L2/ n.L1/ n.L2/: D C Proof If L and L are boundary links then L L is also a boundary link. Let y1 y2 1 2 M1 , M2 and M be the closed 3–manifolds obtained by performing 0–framed Dehn surgery on L , L , and L L respectively. Let M be the 3–manifold obtained y1 y2 1 2 0 Geometry & Topology, Volume 12 (2008) 1 1 420 Shelly L Harvey

0 0 0 0 0 0 L1 L1

0 0 0 H) ˛1 ˛m c

L2 0 0 0 L2 0 0 0

Figure 6: The 3–manifolds M1#M2 and M 0 by performing 0–framed Dehn surgery along the curves ˛1; : : : ; ˛n in M1#M2 as in Figure 6. We will show that M 0 is homeomorphic to M . To see this, we ﬁrst isotope the curve c in M 0 (by a “handle slide”) to obtain the 3–manifold on the left-hand side of Figure 7. Note that by Li , we mean the .m 1/–component string link obtained z C from Li by adding a copy of the “ﬁrst string” by doing a 0–framed pushoff. Then by doing a slam-dunk[8, The Slam-Dunk Theorem, p 15] move on the middle surgery diagram of Figure 7, we see that M 0 is homeomorphic to the 3–manifold on the right-hand side of Figure 7. We continue this process until we arrive at the surgery description of M 0 in Figure 8. Hence M 0 is homeomorphic to M .

Thus .M1#M2/ M @W where W is a 4–dimensional manifold that is obtained by t S D adding 0–framed 2–handles to .M1#M2/ I along the curves ˛1 1 ; : : : ; ˛m 1 . j j f g f g Let E 1.W /, A 1.M1/, B 1.M2/ and ; : : : m be the standard meridians D D D 1 of Lj included into Mj . Then E A B= 1 2; : : : ; 1 2 : Š h 1 D 1 m D mi Recall that M1 M2 M1#M2 @W where W is obtained by adding a 1–handle to t t D 0 0 .M1 M2/ I . Let V W M M W then @V M1 M2 M and the inclusion t D [ 1# 2 0 D t t S map of W into V induces an isomorphism 1.V / 1.W /. Let i; j and k be the Š .n 1/ inclusion maps of M1; M2 and M into V respectively and let n E=EH C . To D complete the proof we will show that (1) i A E (similarly for j ; k ) induces an .n 1/ W ! isomorphism A=A C n and (2) .V; n/ .V / 0 for each n 0. Thus, H ! D n.M1/ n.M2/ n.M /. C D

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0 0 0 0 0 0 L 0 L L1 z 1 z 1

0 0 0 0 0 0 slam-dunk 0 0 0 H) H) c Š Š

L2 0 0 0 L2 0 0 0 L2 0 0

Figure 7

0 0 0

Figure 8: The 3–manifold M that is homeomorphic to M 0

We ﬁrst prove (1). Let F x1;:::; xm be the free group on m generators and D h i j iB F B and iA F A be the inclusion maps of sending xi . Then W ! W ! 7! i E A F B . Since B is the fundamental group of 0–surgery on a boundary link, Š there is a retract rB B F giving a retract r E A where the inclusion is i as W ! .n 1/ W ! above. Therefore, by Lemma 6.5, A=A C n is a monomorphism for all n 0. H ! The proof for B is the same as for A.

.n/ .n/ Let G 1.M /. We will show that k induces an isomorphism G=GH E=EH D ! for each n 0. Since A has a retract to F and iA induces an isomorphism on H1 , by Lemma 6.5, rankKA H1.A KAn/ rankKF H1.F KFn/ m 1 for all n I D n I D n 0. Similarly, rankKA H1.A KAn/ rankKE H1.E KEn/. Hence, for all n 0, n I D n I

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rankK H1.E Kn/ m 1. Since G is also the fundamental group of 0–surgery n I D on a boundary link, rankKG H1.G KGn/ m 1. n I D We show by induction on n that k G E induces an isomorphism H1.G KEn/ W ! I ! H1.E KEn/ for all n 0. First, E G= Œi; gi where i are the given meridians I D h i for the string link L1L2 included into G . Hence H1.G KE0/ H1.E KE0/ is I ! I an isomorphism. Now, assume that H1.G KEn/ H1.E KEn/ is an isomorphism .n 1I/ ! .n 1/ I for all k n. By Theorem 2.4, G=G C E=E C is an isomorphism. Hence, Ä H ! H rankKEn 1 H1.G KEn 1/ rankKGn 1 H1.G KGn 1/ m 1. Since k G E C I C D C I C D W ! is surjective, k induces a surjective map H1.G KEn 1/ H1.E KEn 1/ between I C I C KEn 1 –modules of the same rank; hence is an isomorphism. Thus, for each n 0, C.n/ .n/ G=G E=E is an isomorphism by Theorem 2.4. H ! H To prove (2), note that there is an exact sequence coming from the long exact sequence of the pair .W; M / with coefﬁcients in K.n/ 0 Im.˛/ H2.W; M / H1.M / H1.W / H1.W; M / 0 ! ! ! ! ! ! where M MA MB and ˛ H2.W / H2.W; MA MB/. Since W is ob- D [ W ! [ tained by adding a 1–handle and m 2–handles to M I , rankK H2.W; M / rankK H1.W; M / m 1. Since rankK H1.W / m 1 and rankK H1.M / D D D 2m 2, it follows that rankK Im.˛/ 0. Hence H2.M / H2.W / is a surjective D ! homomorphism.

Corollary 6.7 For each n 0 and m 1, n B.m/ R is a homomorphism. W ! We now establish the main theorems of this paper.

m m Theorem 6.8 For each n 0 and m 2, the abelianization of BF.n/=BF.n 1/ m m C has inﬁnite rank. In particular, for m 2, BF.n/=BF.n 1/ is an inﬁnitely generated m m C subgroup of F.n/=F.n 1/ . C

Proof Let T be the m–component trivial link in S 3 with m 2. Then F 3 3 D .S T / 1.MT / is free group on m–enumerators. Let be a curve in S T D .n/ .n 1/ such that is a trivial knot and the homotopy class of is in F F C . Deﬁne 3 (9) S T .; K/ K is a knot in S with Arf invariant zero B.m/: D f j g Since the trivial link is .n/–solvable, by the proof Proposition 3.1 of[12] (this result holds if one replaces .n/–solvable knot with .n/–solvable link), T .; K/ is .n/– m solvable if the Arf invariant of K is 0. Hence S BF . By Theorem 6.4, n .n/

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m vanishes on F.n 1/ . Therefore C m BF.n/ n m R W BF.n 1/ ! C is a homomorphism.

By Theorem 5.8, n.T .; K// 0.K/. Moreover, by Proposition 2.6 of[12], there D is an inﬁnite set of Arf invariant zero knots Ji such that 0.Ji/ is Z–linearly f g f g m m independent. Since R is abelian, this implies that the abelianization of BF.n/=BF.n 1/ has inﬁnite rank. C

Since adding a local knot to L B.m/ doesn’t change n 0 , we can show that m m 2 BF.n/=BF.n 1/ is infinitely generated “modulo local knotting.” We make this precise starting withC the following definition. Let K.m/ be the subgroup of B.m/ of split string links. K.m/ is a normal subgroup of C.m/ and hence is a normal subgroup m m m of B.m/. For each n 0, define KF.n/ F.n/ K.m/. Then KF.n 1/ is a normal m m m D \ m C m subgroup of KF.n/ and KF.n/=KF.n/ is a normal subgroup of BF.n/=BF.n 1/ . Note that adding a local knot to L B.m/ is the same as multiplying L by an elementC of 2 K.m/. As a corollary of the proof of Theorem 6.8 we have the following result.

Corollary 6.9 (Theorem 6.8 remains true modulo local knotting) For each n 1 and m 2, the abelianization of m m BF.n/=BF.n 1/ m Cm KF.n/=KF.n/ m m m m has inﬁnite rank; hence .BF.n/=BF.n 1//=.KF.n/=KF.n// is an inﬁnitely generated m m m C m subgroup of .F.n/=F.n 1//=.KF.n/=KF.n//. C

Proof This follows from the proof of Theorem 6.8 once we show that n vanishes for .n/–solvable string links with n 1. To see this, let L KF m where n 1. Since 2 .n/ L is .n/–solvable for n 1, 0.L/ 0. Since L K.m/, L can be obtained from D 2 the trivial string link by tying local knots into the strings. That is, L Lm.m; Km/ y D where Li is deﬁned inductively by: L0 is the trivial link and Li 1 Li.i; Ki/ for 3 3 .1/ C D some i 1.S Li/ 1.S Li/r and knot Ki . Hence by applying Theorem 2 Pm 5.8 multiple times we have n.L/ i 1 0.Ki/ 0.L/ 0. D D D D

m m Question 6.10 Is BF.n/=BF.n:5/ (modulo local knotting for m 2 and n 1) inﬁnitely m m generated for each n 0 and m 1? Is BF.n:5/=BF.n 1/ (modulo local knotting for C m 2 and n 1) inﬁnitely generated for each n 0 and m 1?

Geometry & Topology, Volume 12 (2008) 424 Shelly L Harvey

6.2 Grope ﬁltration of C.m/

m There is another filtration of the link concordance group called the grope filtration, Gn , that is more geometric than the .n/–solvable filtration.

Deﬁnition 6.11 A grope is a special pair (2–complex, base circles). A grope has a height n N .A grope of height 1 is precisely a compact, oriented surface † with 2 a nonempty boundary. A grope of height .n 1/ is deﬁned inductively as follows: C Let ˛i; i 1;:::; 2g be a standard symplectic basis of circles for †, the bottom f D g stage of the grope. Then a grope of height .n 1/ is formed by attaching gropes C of height n (with a single boundary component, called the base circle) to each ˛i along the base circle. A model of a grope can be constructed in R3 and thus has a regular neighborhood. Viewing R3 as R3 0 , R3 Œ 1; 1, this model grope has f g ! a 4–dimensional regular neighborhood. When we say that a grope is embedded in a 4–dimensional manifold, we always mean that there is an embedding of the entire 4–dimensional regular neighborhood.

m 3 4 We say that L C.m/ is in Gn if L S @.D / bounds a embedded grope of 4 2 y 2 D height n in D . Note that if L1; L2 C.m/ and Lci for i 1; 2 bounds an embedded 4 2 D 4 grope of height n in D then L1L2 bounds an embedded grope of height n in D . m Therefore, Gn is a subgroup of C.m/. Moreover, since 1 L1L2L Lc2; 1 D m Gn is a normal subgroup of C.m/. We call 0 1Gm Gm C.m/ n 1 the grope filtration of C.m/. We define the grope filtration of the concordance group of boundary string links by BGm 4Gm B.m/. For more about the grope filtration of n D n \ a knot, see Cochran, Orr and Teichner[11; 12] and Cochran and Teichner[13]. For more about gropes, see Freedman and Quinn[15] and Freedman and Teichner[16]. The .n/–solvable and grope filtrations are related by the following theorem of T Cochran, K Orr and P Teichner.

Theorem 6.12 (Cochran–Orr–Teichner[11, Theorem 8.11]) If a link L bounds a grope of height .n 2/ in D4 then L is .n/–solvable. C Hence for all n 0 and m 1, Gm F m (and hence BGm BF m ). Note that n 2 .n/ n 2 .n/ Cochran–Orr–Teichner only state theC above theorem for knots butC their proof holds for links in S 3 as well. We show that certain quotients of the grope ﬁltration are nontrivial.

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m m Theorem 6.13 For each n 1 and m 2, the abelianization of BGn =BGn 2 has m m m Cm inﬁnite rank; hence BGn =BGn 2 is an inﬁnitely generated subgroup of Gn =Gn 2 . C C

.n/ .n 1/ Proof Let be a curve such that the homotopy class of is in F F C where 3 F 1.S T / and n 0. By Lemma 3.9 of[13], after changing by a homotopy D in S 3 T , we can assume that bounds a disk in S 3 and that bounds an embedded 3 height n grope in S N.T /. Consider the set S as deﬁned in (9) (but now using these specially chosen isotopy classes for ). We showed in the proof of Theorem 6.8 that n.S/ is a Z–linearly independent subset of R. We will show that if L S then L m m m 2 m bounds a grope of height .n 1/. Since BGn 3 BF.n 1/ , n BGn 1=BGn 3 R C C W ! is a well deﬁned homomorphism. This will completeC the proof of theC theorem.C

Let L S then L T .; K/ where K is a knot. By Murakami and Nakanishi 2 D [28], K can be obtained from the unknot by doing a sequence of delta moves. Hence, in the language of Habiro[18], K is related to the unknot by a ﬁnite sequence of simple C2 –moves and ambient isotopies. Therefore, by Theorem 3.17 of[18], K is Fl the result of clasper surgery on the unknot along i 1 C.Ti ;ri / where .Ti; ri/ is a rooted symmetric tree of height 1 and the leaves areD copies of the meridian of the unknot. Therefore, L T .; K/ is the result of clasper surgery on the trivial link Fl D T along i 1 C.Ti ;ri / where .Ti; ri/ is a rooted symmetric tree of height 1 and the leaves are copiesD of . Since bounds an embedded height n grope in S 3 N.T /, by Corollary 3.14 of[13], L bounds a height .n 1/ grope in D4 . C

We remark that there are knots that are the result of a union of clasper surgeries on the unknot along rooted trees of height 2 and have nonzero 0 (see for example Figure 3.6 of[13]). By the same argument that was used in the proof of Theorem 6.13, we can show that if you choose such a K then L T .; K/ bounds a grope of height n 2 D m m C for n 0. As a result, we see that the groups BGn 2=BGn 3 are nontrivial for n 0. C C

Proposition 6.14 For each n 2 and m 2, the rank of the abelianization of m m m m BGn =BGn 1 is at least 1; hence Gn =Gn 1 contains an inﬁnite cyclic subgroup. C C

Just as in the case of the .n/–solvable ﬁltration, Theorem 6.13 and Proposition 6.14 are true “modulo local knotting.” We formalize this below. Thus these results cannot be m m obtained using the results in[11; 12; 13]. For each n 0, deﬁne KGn Gn K.m/. m m m m D \ Then KGn 1 is a normal subgroup of KGn and KGn =KGn is a normal subgroup of m mC BGn =BGn 1 . The proof of the following corollary is similar to the proof of Corollary 6.9 so we willC omit the proof.

Geometry & Topology, Volume 12 (2008) 426 Shelly L Harvey

Corollary 6.15 (1) For each n 3 and m 2, the abelianization of m m BGn =BGn 2 m mC KGn =KGn 2 C m m m m has inﬁnite rank; hence .BGn =BGn 2/=.KGn =KGn 2/ is an inﬁnitely generated m m m mC C subgroup of .Gn =Gn 2/.KGn =KGn 2/. C C

(2) For each n 3 and m 2, the rank of the abelianization of m m m m .BGn =BGn 1/.KGn =KGn 2/ C C m m m m is at least 1; hence .Gn =Gn 1/=.KGn =KGn 2/ contains an inﬁnite cyclic subgroup. C C 7 Applications to boundary link concordance

We point out some applications of our work to the study of boundary link concordance and to the abstract determination of certain –groups and relative L–groups that have been previously studied by Cappell–Shaneson and Le Dimet in the context of the classification of links up to concordance (sometimes called cobordism). Recall that boundary links are amenable to classification because each component bounds a disjoint Seifert surface (alternatively because the fundamental groups of their exteriors admit epimorphisms to the free group). In fact it was originally hoped that every odd-dimensional link was concordant to a boundary link, so the classification of link concordance would have been reduced to the case of boundary links. Despite the collapse of this hope[10], boundary links remain an important case for study. S Cappell and J Shaneson first considered pairs .L; / called F –links where L is an 3 m–component boundary link and is a fixed map 1.S L/ F that is a splitting n ! map for a meridional map. They defined a suitable concordance relation, called F – concordance , which entailed an ordinary link concordance between L0 and L1 but required that the fundamental group of the exterior of the concordance admit a map to F extending 0 and 1 . See Cappell and Shaneson[1] for details. Let the m– n 2 component F –concordance classes of (n–dimensional) F –links in S C be denoted by CF.n; m/. This is an abelian group if n > 1, or if n m 1 in which case it is D D equal to the classical knot concordance group. The boundary concordance group of boundary links, B.n; m/, is obtained by dividing out be the action of Aut0.F/, the group of generator-conjugating automorphisms of the free group, which eliminates the dependence on choice of . This classification (for n > 1) was later accomplished by K Ko[22] and W Mio[27] in terms of Seifert matrices and Seifert forms. More

Geometry & Topology, Volume 12 (2008) Homology cobordism invariants 427 recently, D Sheiham completed a more explicit classiﬁcation in terms of signatures associated to quivers[30].

n 2 For a link L in S C (with n > 1), define ML to be the .n 2/–dimensional manifold n 2 C obtained by doing surgery on S C along the components of L with the unique normal framing. For each link L in S n 2 where n 1 mod 4 with n > 1, and each k 0, C Á we define .L/ .ML/. It is then relatively straightforward (see Proposition k D k 7.1) to show that the –invariants considered herein give a rich source of invariants of CF.n; m/ (n 1 mod 4 and n > 1). One should compare Levine[25] where Á –invariants associated to representations into finite unitary groups are used in an analogous fashion.

Proposition 7.1 For any k 0, n 1 mod 4 with n > 1, the invariant induces a Á k homomorphism CF.n; m/ R that factors through B.n; m/. zk ! Proof The situation can be summarized in the following diagram where B.n; m/ is the group of concordance classes of m component, n–dimensional boundary disk links in Dn 2 (sometimes called boundary string links if n 1); where is the natural lift C D deﬁned by Levine[25, Proposition 2.1] (only for n > 1); and I is the natural forgetful map. B.n; m/ -

? -- --I Boundary Links k - CF.n; m/ B.n; m/ CF.n; m/= Aut0.F/ f g R D concordance

Since k is an invariant of concordance of links, the horizontal composition, denoted , exists. zk The group CF.n; m/, n > 1, has been classiﬁed by the aforementioned authors in terms of –groups, Seifert matrices and quiver signatures. The biggest question in this ﬁeld is whether or not I is injective. Our results offer further evidence that it is injective by showing that many powerful signature invariants (the k ) of boundary links, that a priori are only invariants of F –concordance, are actually ordinary concordance invariants.

Question 7.2 How many of Sheiham’s quiver-signatures are captured by information from k ?

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If the were strong enough to detect all of Sheiham’s signatures then it would f k g follow that the kernel of I is torsion. Since Cappell and Shaneson have essentially identiﬁed CF.n; m/, n > 1, with the quotient of a certain gamma group n 3.ZF Z/ (relative L–group) modulo the C ! image of Ln 3.ZF/ [1, Theorem 2, Theorem 4.1] we have the following result. C Proposition 7.3 For each n 1 mod 4 with n > 1, and each k 0, there is an Á induced homomorphism

k n 3.ZF Z/= Aut F R: z W z C ! ! Using the techniques of this paper we can also show that each extends to the zk corresponding –groups of the algebraic closure F of the free group, in terms of y which Le Dimet has successfully “classiﬁed” the higher-dimensional concordance group of disk links[23].

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[10] T D Cochran, K E Orr, Not all links are concordant to boundary links, Ann. of Math. .2/ 138 (1993) 519–554 MR1247992 [11] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and L2 - signatures, Ann. of Math. .2/ 157 (2003) 433–519 MR1973052 [12] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123 MR2031301 [13] T D Cochran, P Teichner, Knot concordance and von Neumann -invariants, Duke Math. J. 137 (2007) 337–379 MR2309149 [14] P M Cohn, Free rings and their relations, second edition, London Mathematical Society Monographs 19, Academic Press [Harcourt Brace Jovanovich Publishers], London (1985) MR800091 [15] M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press, Princeton, NJ (1990) MR1201584 [16] M H Freedman, P Teichner, 4-manifold topology. I. Subexponential groups, Invent. Math. 122 (1995) 509–529 MR1359602 [17] S Friedl, Link concordance, boundary link concordance and eta-invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005) 437–460 MR2138572 [18] K Habiro, Claspers and ﬁnite type invariants of links, Geom. Topol. 4 (2000) 1–83 MR1735632 [19] S L Harvey, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895–945 MR2153977 [20] J-C Hausmann, On the homotopy of nonnilpotent spaces, Math. Z. 178 (1981) 115–123 MR627098 [21] D M Kan, W P Thurston, Every connected space has the homology of a K.; 1/, Topology 15 (1976) 253–258 MR0413089 [22] K H Ko, Seifert matrices and boundary link cobordisms, Trans. Amer. Math. Soc. 299 (1987) 657–681 MR869227 [23] J-Y Le Dimet, Cobordisme d’enlacements de disques,Mem.´ Soc. Math. France (N.S.) (1988) ii+92 MR971415 [24] J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244 MR0246314 [25] J Levine, Concordance of Boundary Links, J. Knot Theory Ramiﬁcations 16 (2007) 1111–1119 [26] WLuck¨ , T Schick, Various L2 -signatures and a topological L2 -signature theorem, from: “High-dimensional manifold topology”, World Sci. Publ., River Edge, NJ (2003) 362–399 MR2048728

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Department of Mathematics, Rice University PO Box 1892, MS 136, Houston TX 77005-1892, USA [email protected] http://math.rice.edu/~shelly/

Proposed: Peter Teichner Received: 9 April 2007 Seconded: Peter Ozsvath, Ron Fintushel Revised: 2 September 2007

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