Homology cobordism group of homology cylinders and ...k denotes the kth term of lower central series...

Homology cobordism group of homology cylindersand invariants related to lower central series

MINKYOUNG SONG

Abstract. The homology cobordism group of homology cylinders is a generalization of boththe mapping class group of surfaces and the string link concordance group. We considerextensions of Johnson homomorphisms of a mapping class group, Milnor invariants and Orrinvariants of links to homology cylinders, all of which are related to free nilpotent groups. Weestablish a combined filtration via kernels of extended Johnson homomorphisms and extendedMilnor invariants. We determine its image under the three invariants, and investigate relationsamong the invariants, and relations of the filtration to automorphism groups of free nilpo-tent groups and to graded free Lie algebras. We obtain the number of linearly independentinvariants by examining the successive quotients of the filtration.

1. Introduction

Let Σg,n be a compact, oriented surface of genus g with n boundary components.Roughly speaking, a homology cylinder over Σg,n is a homology cobordism betweentwo copies of Σg,n endowed with two embeddings i+ and i− of Σg,n (see Definition 2.1).Goussarov [Gou99] and Habiro [Habi00] independently introduced homology cylindersas important model objects for their theory of finite type invariants of 3-manifoldswhich play the role of string links in the theory of finite type invariants of links.Garoufalidis and Levine introduced in [GL05, Lev01] the homology cobordism groupHg,n of homology cylinders over Σg,n (see Definition 2.2 for the description of thegroup operation). We can regard Hg,n as an enlargement of the mapping class groupMg,n of Σg,n since there is a natural embedding Mg,n → Hg,n (see [Lev01, p. 247],[CFK11, Proposition 2.3]). We can identify Hg,n with more familiar groups. BothH0,0 and H0,1 are isomorphic to the group of homology cobordism classes of integralhomology 3-spheres. The group H0,2 is isomorphic to the concordance group of framedknots in homology 3-spheres. For n ≥ 3, H0,n is isomorphic to the concordance group offramed (n−1)-component string links in homology 3-balls, or equivalently, in homologycylinders over D2 = Σ0,1. Similarly, Hg,n can be considered to be the concordancegroup of framed (n− 1)-component string links in homology cylinders over Σg,1.

Hereafter, we assume that Σg,n has nonempty boundary, that is, n > 0. HenceF := π1(Σg,n) and H := H1(Σg,n;Z) are the free group and the free abelian groupof rank 2g + n − 1, respectively. We denote by Dk(H) the kernel of the bracket mapH ⊗ Lk(H) → Lk+1(H) where Lk(H) is the degree k part of the free Lie algebra overH. For a group G, Gk denotes the kth term of lower central series given by G1 = G,Gk+1 = [G,Gk].

To understand structures of Hg,n, several invariants and filtrations are introduced.First, Garoufalidis and Levine extended Johnson homomorphisms and Johnson fil-tration to Hg,1 [GL05, Lev01]. Based on them, Morita, Sakasai, Cochran-Harvey-Horn [Mor08, Sak12, CHH12] constructed their new invariants on Hg,1, respectively.

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HOMOLOGY CYLINDERS AND INVARIANTS RELATED TO LOWER CENTRAL SERIES 2

Cha-Friedl-Kim [CFK11] introduced a torsion invariant on Hg,n. The author nat-urally extended Garoufalidis-Levine’s Johnson homomorphisms and the filtration toHg,n in [So16]. Also, she defined extended Milnor invariants, extended Milnor filtra-tion, and Hirzebruch-type invariants. In this paper, we newly defined extended Orrinvariants, and address the three invariants related to lower central series:

• extended Johnson homomorphisms η̃k : Hg,n → Aut(F/Fk) (see Section 2.2)• extended Milnor invariants µ̃k : Hg,n → (F/Fk)2g+n−1 (see Section 2.3)• extended Orr invariants θ̃k : Hg,n(k)→ H3(F/Fk) (see Section 2.4),

with the following filtrations of Hg,n:• extended Johnson filtration Hg,n[k] := Ker η̃k• extended Milnor filtration Hg,n(k) := Ker µ̃k ∩H0g,n(= Ker µ̃k for k ≥ 2)• modified extended Johnson filtration H0g,n[k] := Hg,n[k] ∩H0g,n.

Here, H0g,n is the subgroup consisting of 0-framed homology cylinders (see Section 3.1).By modifying Hg,n[k] to H0g,n[k], and we have a filtration combined with Hg,n(k)

HBH0[1] = H(1)BH0[2]BH(2)B · · ·BH0[k]BH(k)BH0[k + 1]BH(k + 1)B · · · .

We determine images and successive quotients of this filtration under the three invari-ants.

1.1. Extended Johnson homomorphims

Johnson homomorphisms ηk : Mg,1 → Aut(F/Fk) and the Johnson filtration (alsocalled the relative weight filtration) on the mapping class group have connections tothe geometry of the moduli space of curves, 3-manifold topology, and number theory(e.g., see [Hai95, GL98, Put12, Mat13]). However, many basic questions remain open.In particular, the image of the Johnson homomorphisms are unknown. Morita [Mor93]

gave the induced injective homomorphismsMg,1[k]Mg,1[k+1] ↪→ Dk(H), but the precise images

are also unknown. Rationally, there are results for k ≤ 7 in [Joh80, Hai97, AN95,Mor99, MSS15]. On the other hand, Garoufalidis and Levine considered extendedJohnson homomorphisms η̃k : Hg,1 → Aut(F/Fk) and the filtration Hg,1[k] := Ker η̃k,and determine

Hg,1[k]Hg,1[k+1] as the whole Dk(H) in [GL05]. The homomorphisms have

been used to develop new invariants and to find conditions for the invariants to be(quasi-)additive in [Mor08, Sak12, CHH12]. The author naturally extended η̃k and thefiltration to Hg,n in [So16], and suggested a candidate of the image η̃k(Hg,n). Now weformulate the images and quotients of the combined filtration.

Theorem 1.1. The image of the extended Johnson homomorphism η̃k : Hg,n → Aut(F/Fk)is

Aut∗(F/Fk) :=

{φ ∈ Aut(F/Fk)

∣∣∣∣ there is a lift φ̃ ∈ Aut(F/Fk+1) such thatφ̃(xi) = α−1i xiαi for some αi ∈ F/Fk and φ̃ fixes [∂n]}

where x1, . . . , xn−1 ∈ F are homotopic to each boundaries. Moreover,

Hg,n[k]Hg,n[k + 1]

∼=H0g,n[k]H0g,n[k + 1]

∼= Ker{Aut∗(F/Fk+1)→ Aut∗(F/Fk)},

Hg,n(k)H0g,n[k + 1]

∼= Ker{Aut∗(F/Fk+1)→ Aut∗(F/Fk)} ∩ {φ | φ(xi) = xi}.


We construct a descending filtration of Aut∗(F/Fk) which corresponds to images ofthe combined filtration under η̃k. More details are discussed in Section 4.1.

1.2. Extended Milnor invariants

Milnor’s µ̄-invariants of links are actually invariants on string links without indeter-minacy [HL90]. Orr [Orr89] determined the precise number of linearly independentMilnor invariants of length k as the rank of Dk(H). The author defined extendedMilnor invariants µ̃k : Hg,n → (F/Fk)2g+n−1 and a filtration Hg,n(k) := Ker µ̃k onHg,n in [So16]. The invariants are not only an extension of the Milnor invariants for(string) links, but also a generalization of extended Johnson homomorphisms of Hg,1.If n = 1, they are equivalent to the extended Johnson homomorphisms on Hg,1, andif g = 0, the extended Milnor invariants are equivalent to the Milnor invariants ofstring links in homology 3-balls. Note that the Orr’s result above can be restated thatH0,n(k)H0,n(k+1)

∼= Dk(H). In [So16], it is revealed that for general g ≥ 0 and n > 0, thesuccessive quotients

Hg,n(k)Hg,n(k+1) are finitely generated free abelian if k ≥ 2, and upper

and lower bounds of their ranks are provided. Now we complete our previous work asfollows.

Theorem 1.2. For g ≥ 0, n ≥ 1 and k ≥ 2, the extended Milnor invariants induceisomorphisms

Hg,n(k)Hg,n(k + 1)

∼= Dk(H),H0g,n[k + 1]Hg,n(k + 1)

∼= Dk(H ′),H0g,n[2]Hg,n(2)

× Zn−1 ∼= D1(H ′)

where H ′ = H1(Σ0,n). Consequently, for k ≥ 2, there are (2g+n−1)Nk−Nk+1 linearlyindependent extended Milnor invariants of length k+1 distinguishing homology clinderswith vanishing extended Milnor invariants of lenght ≤ k. Here, Nk = 1k

∑d|k ϕ(d) (2g+

n− 1)k/d where ϕ is the Möbius function.

Also, we clarify the image of µ̃k+1 : Hg,n(k) → (Fk/Fk+1)2g+n−1 in Theorem 4.5.Based on Theorem 1.2, the extended Milnor invariants can be regarded as a propergeneralization of the extended Johnson homormorphism on Hg,1 and the Milnor in-variants on H0,n. Habegger showed that the extended Johnson homomorphisms onHg,1 coincides with Milnor’s invariants on the string link concordance group, which iscalled a Milnor-Johnson correspondence [Habe00]. It also can be integrated with theextended Milnor invariants. We remark that vanishing of the extended Milnor invari-ants is the exact criterion that Hirzebruch-type intersection form defect invariants fromiterated p-covers are defined [So16].

1.3. Extended Orr invariants

Orr [Orr89] introduced homotopy invariants of based links to find the number of lin-early independent Milnor invariants. Igusa and Orr [IO01] saw the relation of theinvariants to k-slice and to Milnor invariants to prove the k-slice conjecture, whichgives a geometric characterization for the vanishing of the Milnor invariants. Theyreduced the invariants to invariants with values in H3(F/Fk). We extend (Igusa-)Orrinvariants to homology cylinders in Section 2.4, and obtain the following.


Theorem 1.3. For k ≥ 2, the extended Orr invariant θ̃k : Hg,n(k) → H3(F/Fk) issurjective. It induces isomorphisms

Hg,n(k)Hg,n(2k − 1)

∼= H3(F/Fk),

Hg,n(k)Hg,n(k + 1)

∼= Coker{H3(F/Fk+1)→ H3(F/Fk)}.

Cha and Orr defined transfinite Milnor invariants for closed 3-manifolds in [CO20].

Our θ̃k(M) is the same as the Cha-Orr invariant θk(M̂) of finite length in [CO20], with

the closure M̂ of M .The above theorems imply relations of the filtration ofHg,n to automorphism groups

of free nilpotent groups, graded free Lie algebras, and the third homology of freenilpotent groups, respectively. We also establish relations among the three invariants

in Section 4.4. Further, we look intoHg,n(1)Hg,n(2) , which is the only non-abelian case in

Section 4.6.

1.4. Organization

In Section 2, we recall definitions of homology cylinders, their homology cobordismgroup, extended Johnson homomorphisms, and extended Milnor invariants. Also weprovide previously known results about the two invariants. We define extended Orrinvariants for homology cylinders. In Section 3, we define 0-framed homology cylindersand refine the two filtrations to be totally ordered. In Section 4, we determine theirimages under the three invariants, and build a connection between the invariants. Wefind the rank of the successive quotients of the filtration. Finally in Section 5, we giveproofs of the theorems stated in Section 4.

In this paper, manifolds are assumed to be compact and oriented. Our results holdin both topological and smooth categories.

Acknowledgements

The author thanks Jae Choon Cha for helpful discussions. This work was supportedby IBS-R003-D1.

2. Definitions and known results

2.1. Homology cylinder cobordism group

We recall precise definitions about homology cylinders. Let Σg,n, or simply Σ, be asurface of g genus with n boundary components. We only consider the case n > 0.

Definition 2.1. A homology cylinder over Σ consists of a 3-manifold M with twoembeddings i+, i− : Σ ↪→ ∂M , called markings, such that

(1) i+|∂Σ = i−|∂Σ,(2) i+ ∪ i− : Σ ∪∂ (−Σ)→ ∂M is an orientation-preserving homeomorphism, and(3) i+, i− induce isomorphisms Hk(Σ;Z)→ Hk(M ;Z) for all k ≥ 0.

We denote a homology cylinder by (M, i+, i−) or simply by M .

Two homology cylinders (M, i+, i−) and (N, j+, j−) over Σg,n are said to be isomor-phic if there exists an orientation-preserving homeomorphism f : M → N satisfying


j+ = f ◦ i+ and j− = f ◦ i−. Denote by Cg,n the set of all isomorphism classes ofhomology cylinders over Σg,n. We define a product operation on Cg,n by

(M, i+, i−) · (N, j+, j−) := (M ∪i−◦(j+)−1 N, i+, j−)

for (M, i+, i−), (N, j+, j−) ∈ Cg,n, which endows Cg,n with a monoid structure. Theidentity is ((Σg,n× I)/(z, 0) = (z, t), id×1, id×0) where z ∈ ∂Σ, t ∈ I. Here and after,I denote the interval [0, 1].

Definition 2.2. Two homology cylinders (M, i+, i−) and (N, j+, j−) over Σg,n aresaid to be homology cobordant if there exists a 4-manifold W such that

(1) ∂W = M ∪ (−N)/ ∼, where ∼ identifies i+(x) with j+(x) and i−(x) withj−(x) for all x ∈ Σg,n, and

(2) the inclusions M ↪→ W , N ↪→ W induce isomorphisms on the integral homol-ogy.

We denote by Hg,n the set of homology cobordism classes of elements of Cg,n. Weusually omit the subscripts and simply write H for general g, n. By abuse of notation,we also write (M, i+, i−), or simplyM for the class of (M, i+, i−). The monoid structureon Cg,n descends to a group structure on Hg,n, with (M, i+, i−)−1 = (−M, i−, i+).We call this group the homology cobordism group of homology cylinders, or simplyhomology cylinder cobordism group. In fact, there are two kinds of groups Hsmoothg,nand Htopg,n depending on whether the homology cobordism is smooth or topological,and there exists a canonical epimorphism Hsmoothg,n � Htopg,n whose kernel contains anabelian group of infinite rank [CFK11]. In this paper, however, the author does notdistinguish the two cases since everything holds in both cases. The fact that themapping class group over Σg,n is a subgroup of Hg,n implies Hg,n is non-abelian except(g, n) = (0, 0), (0, 1) and (0, 2). For any pairs (g, n) and (g′, n′) satisfying n, n′ > 0,g ≤ g′, and g+ n ≤ g′ + n′, there is an injective homomorphism Hg,n ↪→ Hg′,n′ [So16].

xn-1x1

m1

l1 lg

mg

Figure 1. A generating set for π1(Σg,n)

Let ∂1, ∂2, . . . , ∂n be the boundary components of Σ. Choose a basepoint ∗ of Σ on∂n and fix a generating set {x1, . . . , xn−1,m1, . . . ,mg, l1, . . . , lg} for π1(Σ, ∗) such thatxi is homotopic to the ith boundary component ∂i and mj , lj correspond to a meridianand a longitude of the jth handle. Since our n is nonzero, the group is free on theabove 2g + n− 1 generators. Let F = π1(Σ, ∗) and H = H1(Σ). We may assume thatthe element [∂n] ∈ F is represented by

∏i xi∏j [mj , lj ], see Figure 1.


2.2. Extended Johnson homomorphisms and filtration

In [GL05, Lev01], Garoufalidis and Levine defined homomorphisms η̃k : Hg,1 → Aut(F/Fk)and a filtration Hg,1[k] := Ker η̃k as extensions of the Johnson homomorphisms andthe Johnson filtration of the mapping class groupMg,1 [Joh80]. It is a straightforwardconsequence of Stallings’ theorem in [Sta65]. The η̃k is defined to be (i+)

−1k ◦ (i−)k

where (i±)k : F/Fk → π1(M)/π1(M)k is the isomorphisms induced from i±. Theycan be considered on Hg,n in the same way. We call the maps η̃k on Hg,n extendedJohnson homomorphisms (they were referred to as ‘Garoufalidis-Levine homomor-phisms’ in [So16]). They proved that η̃k : Hg,1 → Aut0(F/Fk) are surjective whereF = π1(Σg,1) = 〈m1, . . . ,mg, l1, . . . , lg〉 and

Aut0(F/Fk) := {φ ∈ Aut(F/Fk) | there is a lift φ̃ : F/Fk+1 −→ F/Fk+1such that φ̃([∂n]) = [∂n]}.

and the image ofHg,1[k] under η̃k+1 is isomorphic to Ker{Aut0(F/Fk+1)→ Aut0(F/Fk)}.Recall that Lk(H) is the degree k part of the free Lie algebra over H, and Dk(H) :=Ker{H ⊗ Lk(H)→ Lk+1(H)}. For k ≥ 2,

Dk(H) ∼= Ker{Aut0(F/Fk+1)→ Aut0(F/Fk)}.

Since Dk(H) is free abelian and the rank is well-known, the successive quotient groupsHg,1[k]Hg,1[k+1] , which is isomorphic to the image η̃k+1(Hg,1[k]), are clarified. We remarkthat the image ηk+1(Mg,1[k]) of the Johnson subgroup Mg,1[k] := Hg,1[k] ∩Mg,1 ofthe mapping class group is unknown.

Remark 2.3. From the result about the Artin representation for the concordancegroup of string links in D2 × I, Habegger and Lin [HL98, Theorem 1.1] proved thatη̃k : H0,n → Aut1(F/Fk) is surjective where F = π1(Σ0,n) = 〈x1, . . . , xn−1〉 and

Aut1(F/Fk) := {φ ∈ Aut(F/Fk) | φ(xi) = λ−1i xiλi for some λi ∈ F/Fk−1and φ(x1 · · ·xn−1) = x1 · · ·xn−1}.

Note that

Dk(H) ∼= Ker{Aut1(F/Fk+2)→ Aut1(F/Fk+1)}for k ≥ 2, and the Artin representation corresponds to the Milnor invariant on stringlinks. Two invariants give rise to the same filtration whose the graded quotients areisomorphic to Dk(H). See [HL98, Orr89].

2.3. Extended Milnor invariants and filtration

Let (M, i+, i−) be a homology cylinder over Σg,n. The chosen xi is of the form [α−1i ·βi ·

αi] for a closed path βi : I → I/∂I'−→ ∂i such that the latter map is a homeomorphism

and a path αi from βi(0) to ∗. Consider the loop (i+ ◦ α−1i ) · (i− ◦ αi) in M . If Mwere a framed string link exterior, this loop would be its ith longitude. Recall that(i±)k are the induced isomorphisms F/Fk → π1(M)/π1(M)k. We define µk(M)i :=(i+)

−1k ([(i+◦α

−1i )·(i−◦αi)]) for the class of the loop (i+◦α

−1i )·(i−◦αi) in π1(M, i+(∗)).

It is independent of the choice of αi, and it depends only on the choice of xi in π1(Σ, ∗).Also we define µ′k(M)j := (i+)

−1k ([(i+ ◦m

−1j ) · (i− ◦mj)]) and µ′′k(M)j := (i+)

−1k ([(i+ ◦

l−1j ) · (i− ◦ lj)]). We denote the (2g + n − 1)-tuple of µk(M)i, µ′k(M)j , and µ′′k(M)of F/Fk as µ̃k(M).


In [So16], the author proved that µ̃k is a crossed homomorphism from Hg,n to(F/Fk)

2g+n−1 with

µ̃k(MN)• = µ̃k(M)• · η̃k(M)(µ̃k(N)•),

and is a homomorphism on Ker µ̃k−1 and Ker η̃k for k > 2. We denote by Hg,n(k)the kernel of µ̃k. All the Hg,n(k) are normal subgroups of Hg,n, and {Hg,n(k)}kis a descending filtration. Since there is an injection H(k)H(k+1) ↪→ Dk(H) for k ≥ 2,it turned out that the successive quotient groups H(k)H(k+1) are finitely generated free

abelian if k ≥ 2.There is a relation between extended Johnson homomorphisms and extended Milnor

invariants that

(η̃k(M))(xi) = µk(M)−1i xi µk(M)i

(η̃k(M))(mj) = mj µ′k(M)j

(η̃k(M))(lj) = lj µ′′k(M)j in F/Fk.

(1)

Remark that the definitions of µ′k and µ′′k are slightly different from those in [So16].

2.4. Extended Orr invariants

To find the number of linearly independent Milnor invariants of a length, Orr de-fined concordance invariants θk of based links in S

3 with vanishing length ≤ k Milnorinvariants, whose value is in π3(Kk) where Kk is a mapping cone of the inclusionK(F, 1) → K(F/Fk, 1) in [Orr89]. Here, K(G, 1) is an Eilenberg-MacLane space, LetL be a link in S3 for which Milnor invariants of length ≤ k vanish, and τ be a basingF → π1(EL) for the exterior EL of L. Then, 0-framed longitudes of L are in π1(EL)k, soF/Fk ∼= π1(EL)/π1(EL)k. We have a map EL → K(π1(EL), 1) → K(F/Fk, 1) → Kkwhich sends meridians to null-homotopic loops, and so the map extends to S3 → Kk.The homotopy class is defined as θk(L, τ) ∈ π3(Kk). He found the precise numberby verifying that the Milnor invariants of length k + 1 vanish if and only if θk van-ishes in Coker{π3(Kk+1) → π3(Kk)}, which is isomorphic to Coker{H3(F/Fk+1) →H3(F/Fk)}. Igusa and Orr considered invariants valued in H3(F/Fk) by composingthe Hurewicz homomororphism with θk, and denoted them by θ̄k in [IO01]. This in-variant vanishes for a based link if and only if the Milnor invariants of length ≤ 2k− 1vanish for that link. The invariant θ̄k(L, τ) ∈ H3(F/Fk) is equal to the image of thefundamental class [ML] ∈ H3(ML) of the zero-surgery manifold ML of L under a mapML → K(F/Fk, 1) induced from the basing.

We can define an analogous invariant θ̃k : H(k) → H3(F/Fk). Recall that for ahomology cylinder M , there is an associated closed manifold M̂ obtained from Mby identifying i+(z) and ii(z) for each z ∈ Σ, which is called the closure of M . For ahomology cylinder M with trivial µ̃k, we have an isomorphism F/Fk ∼= π1(M̂)/π1(M̂)k,see [So16]. Define θ̃k(M) ∈ H3(F/Fk) as the image of of [M̂ ] under M̂ → K(F/Fk, 1).It is a homology cobordism invariant, and is a homomorphism. We note that θ̃k(M)

is the same as the Cha-Orr invariant θk(M̂) ∈ H3(F/Fk) in [CO20]. It can be alsothought as a generalization of the (Igusa-)Orr invariant θk(L, τ) ∈ H3(F/Fk) to basedlinks in a homology 3-sphere corresponding to M along the map from H to the set ofbased links in homology 3-spheres. The map will be appear in a future paper.


3. Refinement and combination of the two filtrations

We consider two descending filtrations of Hg,n: extended Johnson filtration Hg,n[k] =Ker η̃k and extended Milnor filtration Hg,n(k) = Ker µ̃k

H = H[1] BH[2] BH[3] B · · ·B H[k − 1] BH[k] BH[k + 1] B · · ·

H = H(1)BH(2)BH(3)B · · ·BH(k − 1)BH(k)BH(k + 1)B · · ·Though H[k] ⊇ H(k), the two filtrations are not linearly ordered since H(k) + H[k+1]unless n = 1. Only if n = 1, Hg,1[k] = Hg,1(k). We will refine and combine them toderive a linearly ordered filtration in Section 3.2.

As mentioned in Section 2.2, it is revealed that in the case of n = 1,Hg,1[k]Hg,1[k+1] is

isomorphic to Ker{Aut0(F/Fk+1)→ Aut0(F/Fk)} ∼= Dk(H) for k ≥ 2, and in the caseof g = 0,

H0,n(k)H0,n(k+1) is isomorphic to Ker{Aut1(F/Fk+2) → Aut1(F/Fk+1)}

∼= Dk(H)for k ≥ 2. We will merge and extend the theories on Hg,1 and H0,n to Hg,n by usingHg,n(k), that is, we will show that Hg,n(k)Hg,n(k+1) is isomorphic to Dk(H), but

Hg,n[k]Hg,n[k+1]

is not. We remind that F , H and Dk(H) depend on g and n since F = π1(Σg,n),H = H1(Σg,n).

Remark 3.1. The weight filtration of π1(Σg,n) and the induced filtration of the map-ping class group are studied in mixed Hodge theory. They are the same as the lowercentral series and the Johnson filtration only when n ≤ 1. But, the graded quotientsare also not isomorphic with those in this paper for n > 1.

3.1. 0-framed homology cylinders

We consider a natural surjection H1(F ) → H1(F/〈〈xi′ ,mj , lj | i′ 6= i〉〉) ∼= Z for eachi = 1, . . . , n− 1. Denote the image of µ2(M)i by µ̄2(M)i. When a homology cylinderM over Σg,n is thought as a framed string link in a homology-(Σg,1 × I), µ̄2(M)iindicates “self-linking number” of the i-th strand. We say that a homology cylinderM is 0-framed if µ̄2(M)i = 0 for each i. We denote by H0g,n the subgroup of Hg,nconsisting of the 0-framed homology cylinders.

Proposition 3.2. There is an exact sequence which splits:

1 −→ H0g,n −→ Hg,n −→ Zn−1 −→ 1.

Therefore, Hg,n ∼= H0g,n × Zn−1.

If we consider Mg,n as a subgoup of Hg,n, ϕ ∈ Mg,n corresponds to Iϕ := (Σ ×I, id, ϕ) ∈ Hg,n; see [Lev01, page 247], [CFK11, Proposition 2.4]. Note that Iϕ =(Σ × I, ϕ−1, id) and (Iϕ)−1 = (Σ × I, ϕ, id) = (Σ × I, id, ϕ−1) = Iϕ−1 in H. For i =1, . . . , n−1, we denote by τi the element of the mapping class groupMg,n correspondingto a Dehn twist along the boundary ∂i. Let T be the subgroup of M, and also of H,generated by τ1, . . . , τn−1. In general, Iϕ and a homology cylinder M do not commute,but IτM = MIτ for Iτ ∈ T . Thus, T is normal in H.

The following lemma yields a proof of the above proposition.

Lemma 3.3. (1) There is an isomorphism T ∼= Zn−1.(2) The short exact sequence 1→ T → H → H/T → 1 splits.(3) There is an isomorphism H/T ∼= H0.

Proof. (1) The group T has a representation 〈τ1, . . . , τn−1 | τiτj = τjτi ∀i, j〉.


(2) The map H → Zn−1 sending M to (µ̄2(M)1, . . . , µ̄2(M)n−1) gives rise to leftexactness.

(3) There is a homomorphism H/T → H0 which maps [M ] to MIτ where τ =τtn−1n−1 ◦ · · · ◦ τ

t11 with ti = µ̄2(M)i. It is an isomorphism.

�

3.2. Combination of the two filtrations

Let H0g,n[k] := H0g,n ∩ Hg,n[k]. For each k, H[k] ∼= Zn−1 × H0[k]. In fact, H0[k] =H(2) ∩ H[k] = H(k − 1) ∩ H[k], so H(k) ⊇ H0[k + 1]. Also we can define H0g,n(k) :=H0g,n ∩Hg,n(k), but then H(k) = H0(k) for every k ≥ 2. Thus, for simplicity, we resetH(1) := H0. Then H(k) = H0(k) for all k, and we obtain a combined descendingfiltration:

HBH0 = H0[1] = H(1)BH0[2]BH(2)B · · ·BH(k−1)BH0[k]BH(k)BH0[k+1]B · · ·

Note that H(k − 1) = H0[k] only if g = 0, and H[k] = H(k) only if n = 1. All H0,H[k], H0[k], and H(k) are normal subgroups of H.

The invariant µ̃k can be divided as follows for k ≥ 2:

µk :H0[k]H(k)

↪−→ (Fk−1/Fk)n−1

(µ′k, µ′′k) :H(k − 1)H0[k]

↪−→ (Fk−1/Fk)2g

In the case of k = 2, the bottom map is not a homomorphism, but a crossed homo-morphism with the action of η̃2 on H. The others are all homomorphisms. The two

subquotients of H are also finitely generated free abelian except H(1)H0[2] , and hence thereare isomorphisms

H0[k]H0[k + 1]

∼=H0[k]H(k)

× H(k)H0[k + 1]

,H(k)H(k + 1)

∼=H(k)H0[k + 1]

× H0[k + 1]

H(k + 1)

for k ≥ 2, which are not canonical. While H(k) = Ker(µk, µ′k, µ′′k) in H0, H0[k] =Ker(µk−1, µ

′k, µ′′k) in H0. From now on, we will investigate the four quotient groups

H(k)H(k+1) ,

H0[k]H0[k+1] ,

H(k)H0[k+1] , and

H0[k+1]H(k+1) for k ≥ 1.

Remark 3.4. (1) In the author’s paper [So16, page 920], H0 defined as {M ∈H | µ2(M) = 1} =: H0old is different from H0 here; The old one is smallerthan our new H0 unless n = 1. Though H0old ∩ H[k] = H0[k] for k ≥ 3,H0old ∩ H[2] = H(2) ( H0[2] for n > 1. The new definition is more suitablefor the concept of “zero-framed” homology cylinder since it corresponds to0-framed string link in homology 3-balls in the case of g = 0.

(2) There are some errors related to µ̃2 in [So16]. On [So16, Corollary 3.8 (4)], µ̃2is not a homomorphism on H or Kerµ2 though it is a homomorphism on H[2].[So16, Theorem 4.3] holds only for q ≥ 3, that is, the map µ̃2 : H/H(2) ↪→(F/F2)

2g+n−1 is not a homomorphism. It is only a crossed homomorphism.


4. Images and quotients of the filtration under the invariants

4.1. Images under extended Johnson homomorphism and Aut(F/Fk)

Clarification of the images of the filtration under the extended Johnson homomorphismsprovide a relationship between the filtration and a filtration of an automorphism groupof a free nilpotent group. Before we proceed, we introduce new notations as follows:

Aut∗(F/Fl) ∼={φ ∈ Aut(F/Fl)

∣∣∣∣ there is a lift φ̃ ∈ Aut(F/Fl+1) such thatφ̃(xi) = α−1i xiαi for some αi ∈ F/Fl and φ̃ fixes [∂n]}

Kl,k = Ker{Aut∗(F/Fl) −→ Aut∗(F/Fk)} (l ≥ k)Al,k = {φ ∈ Aut∗(F/Fl) | φk(mj) = mj , φk(lj) = lj , φk+1(xi) = xi}

= {φ ∈ Kl,k | φk+1(xi) = xi} (l > k)

Here, for an automorphism φ of F/Fl and t ≤ l, φt denotes the automorphism of F/Ftinduced from φ. Then, they form a descending filtration of Aut∗(F/Fl)

Aut∗(F/Fl) = Kl,1 = Al,1 BKl,2 BAl,2 B · · ·BAl,k−1 BKl,k BAl,k BKl,k+1 B · · · .

The subgroups Al,k and Kl,k are normal in Aut∗(F/Fl). Remark that the lift φ̃ canbe chosen to be an endomorphism in the definition of Aut∗(F/Fl) by the followingproposition.

Proposition 4.1. Suppose φ ∈ Aut(F/Fk) with k ≥ 2. Then any lift of φ to anendomorphism of F/Fl is an automorphism for l > k.

Proof. The argument comes from the proof of [And65, Theorem 2.1]. Let φ̃ be a lift

of φ to an endomorphism of F/Fl. Then (Im φ̃) · (Fk/Fl) = F/Fl. Since F/Fl isnilpotent, Im φ̃ = F/Fl (see [Hal76, Corollary 10.3.3]). Therefore, φ̃ is onto. Since

F/Fl is Hopfian, φ̃ is an automorphism. �

Therefore, Aut∗(F/Fl) is a generalization of both Aut0(F/Fl) of the case n = 1 andAut1(F/Fl) of the case g = 0. Now we relate the combined filtration of H in Section 3.2to a filtration of the automorphism group of F/Fl by determining the images of thesubgroups in the filtration of H under the extended Johnson homomorphisms.

Theorem 4.2. Suppose l > k ≥ 1.

(1) The composition η̃0l : H0g,n ↪→ Hg,nη̃l−→ Aut∗(F/Fl) is surjective.

(2) The inverse image (η̃0l )−1(Kl,k) = H0g,n[k]. Thus, H0[k]→ Kl,k is surjective.

(3) The inverse image (η̃0l )−1(Al,k) = Hg,n(k). Thus, H(k)→ Al,k is surjective.

We postpone the proof to the next section. From this theorem, we obtain thefollowing directly.

Corollary 4.3. For l > k ≥ 1, there are isomorphisms

H(k)H(k + 1)

∼=Al+1,kAl+1,k+1

,H0[k]H0[k + 1]

∼=Kl,kKl,k+1

∼= Kk+1,k,

H0[k + 1]H(k + 1)

∼=Kl+1,k+1Al+1,k+1

,H(k)H0[k + 1]

∼=Al,kKl,k+1

∼= Ak+1,k.


4.2. Images under extended Milnor invariants and free Lie algebras

We consider the restricted invariant µ̃k+1 : H(k) → (Fk/Fk+1)2g+n−1, which is a ho-momorphism for each k ≥ 2. Recall that Lk(H) is the degree k part Lk(H) of thefree Lie algebra over H, and Dk(H) is the kernel of the bracket map H ⊗ Lk(H) →Lk+1(H). Note that Lk(H) is isomorphic to Fk/Fk+1. We consider the image of thehomomorphism invariant µ̃k+1 : H(k) → (Fk/Fk+1)2g+n−1. We identify the codomain(Fk/Fk+1)

2g+n−1 with H ⊗ Lk(H) along

(α1, . . . , αn−1, β1, . . . , βg, γ1, . . . , γg) 7→∑i

xi ⊗ αi +∑j

(mj ⊗ γj − lj ⊗ βj).

We define a surjective homomorphism pk as follows.

pk : (Fk/Fk+1)2g+n−1 −→ Fk+1/Fk+2

(α1, . . . , αn−1, β1, . . . , βg, γ1, . . . , γg) 7−→n−1∏i=1

[xi, αi]

g∏j=1

[mj , γj ][βj , lj ].

Then, Ker pk is identified with Dk(H) along the above identification.We can think Σg,n as a boundary connected sum of Σ0,n and Σg,1 with π1(Σ0,n) =

〈x1, . . . , xn−1〉 =: F ′ and π1(Σg,1) = 〈m1, . . . ,mg, l1, . . . , lg〉 =: F ′′. Let H ′ :=H1(Σ0,n) and H

′′ := H1(Σg,1). Then F = F′ ∗ F ′′ and H = H ′ ⊕H ′′.

Lemma 4.4. For every k ≥ 1,Ker pk ∩ {(Fk/Fk+1)n−1 × 02g} = Ker pk ∩ {(F ′k/F ′k+1)n−1 × 02g}

= Ker{p′k : (F ′k/F ′k+1)n−1 → F ′k+1/F ′k+2}

where p′k sends (α1, . . . , αn−1) to∏n−1i=1 [xi, αi]. In other words,

Dk(H′) ∼= Ker{H ′ ⊗ Lk(H)→ Lk+1(H)} ⊂ Dk(H).

We clarify the image of the extended Milnor invariants as follows.

Theorem 4.5. For k ≥ 2, the image of µ̃k+1 : H(k) → (Fk/Fk+1)2g+n−1 is Ker pk ∼=Dk(H). In addition, the inverse image of Ker p

′k∼= Dk(H ′) is H0[k + 1].

Now we relate the successive quotients of the filtration of Hg,n to Dk(H).

Theorem 4.6. For k ≥ 2, there are isomorphismsH(k)H(k + 1)

∼= Dk(H),H0[k + 1]H(k + 1)

∼= Dk(H ′), andH[2]H(2)

∼= D1(H ′).

The proof of the above theorems are achieved by associating the image Ak+2,k ofH(k) under η̃l with Ker pk ∼= Dk(H). We also postpone the precise proof to the nextsection.

The diagram in Figure 2 on page 12, summarizes results so far. The horizontalsurjections are (restrictions of) the extended Johnson homomorphisms. What writtenover the injection arrows are the cokernels of the maps.

In the above diagram, each subgroup of H on the left surjects to the subgroup ofAut∗(F/Fl) located at the same position on the right under η̃l. What written on theinjection arrows are the cokernels of the maps. The cokernels of the maps at the sameposition on both sides are isomorphic except Zn−1 which corresponds to the “framing”µ̄2(−)i.


H

H0 Kl,1= Aut∗(F/Fl)

H(1) Al,1

H[2]

H0[2] Kl,2

H(2) Al,2

H[3]

H0[3] Kl,3

H(3) Al,3

H[k]

H0[k] Kl,k

H(k) Al,k

H[k + 1]

H0[k + 1] Kl,k+1

H(k + 1) Al,k+1

Coke

r

=Zn−1

η̃l

Coker=

K2,1

η̃0l

Coke

r

=Zn−1

Coker=

K3,2

Coker =

A2,1

Coker

=

D1(H′ )

Zn−1

Coke

r

=Zn−1

Coker =

A3,2

Coker=

D2(H

)

Coker

=D2

(H′ )

Coke

r

=Zn−1

Coker=

Kk+

1,k

Coker

=Dk−1

(H′ )

Coke

r

=Zn−1

Coker =

Ak+

1,k

Coker=

Dk(H

)

Coker

=Dk

(H′ )

Figure 2.


4.3. Images under extended Orr invariants

We can verify the image of the extended Orr invariants and relation between thequotients of the filtration and H3(F/Fk) as follows. Note that the injection F

′ → Finduces H3(F

′/F ′k) → H3(F/Fk), which is also injective. By abuse of notation, wethink H3(F

′/F ′k) as a subgroup of H3(F/Fk).

Theorem 4.7. For k ≥ 2, θ̃k : H(k)→ H3(F/Fk) is surjective. The inverse image ofIm{H3(F/Fl)→ H3(F/Fk)} under θ̃k is H(l) for k ≤ l ≤ 2k − 1.

To prove the surjectivity, we use the fact that H3(G) is isomorphic to the k-dimensional oriented bordism group Ωk(G) of K(G, 1) for any group G. Any element of(X3 → K(F/Fk, 1)) ∈ Ω3(F/Fk) ∼= H3(F/Fk) can be realized as a homology cylinderwith trivial µ̃k, which is obtained by cutting X along an embedded Σg,n. The surfaceshould be chosen so that π1(Σg,n)→ π1(X)→ F/Fk sends the generetors xi,mj , lj to[xi], [mj ], [lj ] in F/Fk.

Corollary 4.8. For k ≥ 2, the invariant θ̃k induces an isomorphism

H(k)H(k + 1)

−→ Coker{H3(F/Fk+1)ψk+1,k−−−−→ H3(F/Fk)}.

We can obtain them in two ways. One approach is to apply the known fact that

H3(F/Fk) ∼=2k−2⊕i=k

Z(2g+n−1)Nk−Nk+1 and Dk(H) ∼= Coker{H3(F/Fk+1) → H3(F/Fk)}

for k ≥ 2 (see [IO01]). Alternatively, we can prove them directly. We leave the proofto the reader.

The corollary implies that the invariants θ̃k with values in the cokernel and µ̃k+1on H(k) are equivalent. The invariant θ̃k with values in H3(F/Fk) is equivalentto µ̃2k−1 on H(k). We have an isomorphism θ̃k : H(k)H(2k−1) → H3(F/Fk). Remarkthat µ̃2k−1 : H(k) → (Fk/F2k−1)2g+n−1 is also a homomorphism. It follows fromLemma 5.2(3).

4.4. Relation between the invariants

11 1

H(2k − 1) H(k + 1) H0[k + 1]

H(k)

Ak+1,k Ker pk ∼= Dk(H) H3(F/Fk)1 1

1 Cokerψk+1,k

Aut(F/Fk+1) (Fk/Fk+1)2g+n−1∼= H ⊗ Lk(H)

η̃k+1µ̃k+1

θ̃k

∼


The map ψk+1,k is the natural map H3(F/Fk+1) → H3(F/Fk) appeared in Corol-lay 4.8. The above diagram commutes, and the diagonal and vertical sequences are ex-act. The bottom left map (Fk/Fk+1)

2g+n−1 → Aut(F/Fk+1) sends (α1, . . . , αn−1, β1, . . . , βg, γ1, . . . , γg)to the automorphism xi 7→ α−1i xiαi, mj 7→ mjβj , and lj 7→ ljγj , and its image is inKer{Aut(F/Fk+1) → Aut(F/Fk)}. The map and its restriction on Ker pk are not ho-momorphisms. All the other arrows in the above diagram are homomorphisms. Thelower right homomorphism H3(F/Fk) → Dk(H) is defined using the Massey productof classes in H1(F/Fk) in [GL05].

4.5. Rank of quotients of the filtration

From Theorem 4.6, we get the exact ranks of quotient groups:

Corollary 4.9. Suppose k ≥ 2, then

H(k)H(k + 1)

∼= Z(2g+n−1)Nk−Nk+1

where Nk = rankFk/Fk+1 =1k

∑d|k ϕ(d) (2g+n−1)k/d, and ϕ is the Möbius function.

Also, we have

H0[k + 1]H(k + 1)

∼= Z(n−1)N′k−N

′k+1 and

H0[2]H(2)

∼= Z(n−12 ) = Z

12 (n−1)(n−2)

where N ′k = rankF′k/F

′k+1 =

1k

∑d|k ϕ(d) (n− 1)k/d.

We can also compute the ranks of H(k)H0[k+1] andH0[k]H0[k+1]

∼= H[k]H[k+1] for k ≥ 2 using theabove corollary.

4.6. First quotient of the extended Milnor filtration

In order to complete the investigation, it remains to characterize H(1)H(2) andH0[1]H0[2]

∼= H(1)H0[2] ,which are all the non-abelian cases except

H0,n(1)H0,n(2) . We found that

H(1)H0[2] is isomorphic

to A2,1 = Aut∗(F/F2). Recall that H/H[2] ∼= Aut∗(H) where

Aut∗(H) =

{φ ∈ Aut(H)

∣∣∣∣ φ fixes [∂i] for all i = 1, · · · , n andpreserves the intersection form of Σ}

(see [GS13, Proposition 2.3 and Remark 2.4]). Note that

Aut∗(H) ∼={[

In−1 A0 P

] ∣∣∣∣ P ∈ Sp(2g,Z), A ∈M(n−1)×2g(Z)}∼= Z(n−1)2g o Sp(2g,Z)

where Mn−1,2g(Z) is the additive group of (n− 1)× 2g matrices over Z, and Sp(2g,Z)is the symplectic group over Z. So, H(1)H0[2] ∼=

H0[1]H0[2]

∼= Z(n−1)2g o Sp(2g,Z). The factAut∗(H) = Aut∗(F/F2) can be also checked directly. In the remaining part of this

section, we focus on H(1)H(2) . Let us discuss whether the exact sequence

1 −→ H0[2]

H(2)−→ H(1)

H(2)−→ H(1)

H0[2]−→ 1


splits. We remark that if g = 0, i.e. the case of string links in homology 3-balls,H0,n(1) = H00,n[2], so

H0,n(1)H0,n(2)

=H00,n[2]H0,n(2)

∼= Z(n−12 ) = Z

12 (n−1)(n−2).

This corresponds to the mutual linking while H/H(1) ∼= Zn−1 corresponds to theself-linking. If n = 1, then Hg,1(1) = Hg,1 and Hg,1[k] = H0g,1[k] = Hg,1(k), so

Hg,1(1)Hg,1(2)

=Hg,1(1)H0g,1[2]

=Hg,1Hg,1[2]

∼= Aut∗(H) ∼= Sp(2g,Z).

For general g and n, we prove a partial result. We recallH ′′ = H1(〈m1, · · · ,mg, l1, · · · , lg〉) ⊂H (see page 11). Let H0[1.5] := {M ∈ H0 | η̃2(M)(H ′′) ⊂ H ′′}. Then, M ∈ H[1.5]

if and only if η̃2(M) corresponds to the matrix

[I 00 P

]. So, H

0[1.5]H0[2]

∼= Sp(2g,Z).

However, H0[1.5] is not normal in H0.

Proposition 4.10. There is a split exact sequence

1 −→ H0[2]

H(2)−→ H

0[1.5]

H(2)−→ H

0[1.5]

H0[2]−→ 1.

Therefore,

H0g,n[1.5]Hg,n(2)

∼= Z(n−12 ) × Sp(2g,Z) ∼=

H0,n(1)H0,n(2)

× Hg,1(1)Hg,1(2)

,

Hg,n[1.5]Hg,n(2)

∼= D1(H ′)× Sp(2g,Z) ∼=H0,nH0,n(2)

× Hg,1Hg,1(2)

.

Proof. We define a map f : H0[1.5]H(2) →

H0[2]H(2) so that

(1) µ′2(f(M)), µ′′2(f(M)) are trivial,

(2)∏i[xi, µ2(f(M))i] = 1 in F2/F3, and

(3) for each i, µ2(f(M))−1i µ2(M)i is represented by a product of xr,mj , lj with

r > i, 1 ≤ j ≤ g.Since

∏i[xi, µ2(f(M))

−1i µ2(M)i] =

∏i[xi, µ2(f(M))

−1i ][xi, µ2(M)i] =

∏i[xi, µ2(M)i]

is given and {[xi, xr], [xi,mj ], [xi, lj ] | r > i, 1 ≤ j ≤ g} is a subset of a Hall basis forF2/F3, which is a basis for a subgroup of F2/F3 generated by {[xi, z] | z ∈ F/F2},µ2(f(M))

−1i µ2(M)i is uniquely determined as a product of xr,mj , lj for r > i, 1 ≤ j ≤

g. Hence so is µ2(f(M))i. An element ofH0[2]H(2) is determined by µ2(−). Therefore, f

is well-defined.To see that the map is a homomorphism, we should check that µ2(f(M)f(N)) =

µ2(f(MN)). It is enough to show that µ2(f(M)f(N))−1i µ2(M)i is represented by a

product of xr,mj , lj for r > i, 1 ≤ j ≤ g. In H0[2], µ2 is a homomorphism.

µ2(f(M)f(N))−1i µ2(MN)i = µ2(f(N))

−1i µ2(f(M))

−1i µ2(M)i η̃2(M)(µ2(N)i)

= µ2(f(M))−1i µ2(M)i µ2(f(N))

−1i η̃2(M)(µ2(N)i)

is a product of xr,mj , lj for r > i, 1 ≤ j ≤ g since η̃2(M) preserves the powers of xifor M ∈ H0[1.5]. This homomorphism f is a left splitting, so the given exact sequencesplits. �


1 Z(n−12 ) Z(

n−12 ) × Sp(2g,Z) Sp(2g,Z) 1

1 1

1H0[2]H(2)

H0[1.5]H(2)

H0[1.5]H0[2]

1

H(1)H(2)

KH(1)H0[2]

Aut∗(H)

1 1

M(n−1)×2g(Z) Z(n−1)2g

1 1

∃ splitting

6

∃ splitting

∃ spli

tting

∃splitting

@ splitt

ing

@ splitting

@splittin

g

@ splitting

@splitting

@splittin

g

Figure 3.

To sum up, we have the diagram in Figure 3. Here, K denotes the kernel of H(1)H(2) →H0[1.5]H0[2] . All sequences in a straight line are part of short exact sequences that the trivial

terms are omitted. But, only those which end with the top right term H0[1.5]H0[2] are right

split, and the top horizontal sequence is the only one which splits.

5. Proofs

Proof. (Proof of Theorem 4.2)

(1) To prove the surjectivity of the map η̃l : H → Aut∗(F/Fl), we mainly follow theproof of [GL05, Theorem 3], and simplify a part of it. Suppose φ ∈ Aut∗(F/Fl).We can construct maps f± : Σ → K(F/Fl, 1) so that (f+)# : π1(Σ) = F →F/Fl is the natural surjection and (f−)# = φ◦(f+)#. We may assume f+|∂Σ =f−|∂Σ since f+|∂i and f−|∂i are freely homotopic for i = 1, . . . , n− 1 and f+|∂nand f−|∂n are homotopic. Let f = f+ ∪ f− : Σ ∪∂ Σ → K(F/Fl, 1). We claimthat there are a compact oriented 3-manifold M bounded by Σ ∪∂ Σ and amap Φ: M → K(F/Fl, 1) extending f . The obstruction is [f ] ∈ Ω2(F/Fl) ∼=H2(F/Fl). Using a lift φ̃ ∈ Aut(F/Fl+1) of φ, we can also consider f̃ : Σ ∪∂Σ → K(F/Fl+1, 1) as above which factors through f . Since H2(F/Fl+1) →H2(F/Fl) is the 0-map, [f ] = 0 ∈ H2(F/Fl). Thus, we have M and Φ.


We will do surgery M to be a homology cylinder over Σ by killing Ker{Φ∗ :H1(M) → H}. As [GL05, Lemma 4.6], for α ∈ Ker Φ∗, there exists ᾱ ∈π1(M) such that ᾱ ∈ Ker{Φ# : π1(M) → F/Fl} and ᾱ represents α. Thus,we can surger M along a curve representing α. Note that the surgery on anembedding ϕ : S1 ×D2 → (interior of M) induces H1(M)/〈α〉 ∼= H1(M ′)/〈β〉where α = [ϕ(S1 × 1)], β = [ϕ(1 × S1)], and M ′ is the resulting manifold[KM63, Lemma 5.6].

As a first step, kill the torsion-free part of Ker Φ∗. Choose α ∈ Ker Φ∗ whichmaps to a primitive element of H1(M,∂M). It is possible since H1(M) ∼=H ⊕Ker Φ∗ and Ker Φ∗ → H1(M,∂M) is surjective. Let C be a simple closedcurve in the interior of M representing α. Surgery on C results in a newmanifold M ′ with H1(M

′) ∼= H1(M)/〈α〉. After a sequence of such surgeries,H1(M,∂M) becomes a torsion group. By calculating ranks of the groups inthe long exact sequence of homology groups of the pair (M,∂M), we concludethat rankH1(M) = rankH and rank Ker Φ∗ = 0.

Now Ker Φ∗ = TorH1(M) ∼= TorH1(M,∂M) = H1(M,∂M), so the linkingpairing l : Ker Φ∗ ⊗Ker Φ∗ → Q/Z is non-singular.

Lemma 5.1. Suppose M is a 3-manifold whose linking pairing TorH1(M)⊗TorH1(M)→ Q/Z is non-singular. If α is a torsion element in H1(M), thenthere is a surgery along a simple closed curve C representing α which reduces|TorH1(M)|, that is, |TorH1(M ′)| < |TorH1(M)| where M ′ is the resultingmanifold.

More precisely, if l(α, α) 6= 0, then there is a normal framing on C such thatthe order of β in H1(M

′) is less than that of α in H1(M), and if l(α, α) = 0,then there is a normal framing of C such that the order of β in H1(M

′) isinfinite. In the latter case, the surgery generates a torsion-free summand. Byapplying the first step above, we can kill it. Hence, after a sequence of surgeries,Φ∗ : H1(M)→ H becomes an isomorphism.

Let i± : Σ → M be the restrictions of Σ ∪∂ Σ → M so that f ◦ i± = f±.We check that (M, i+, i−) is a homology cylinder over Σ; Clearly, i± : Σ→Msatisfies i+|∂ = i−|∂ . Since Φ∗ and f±∗ are isomorphisms on H1(−), so arei±∗ on H1(−). It follows that i± also induce isomorphisms on H2(−) fromH1(M,∂M) = 0 and H2(M) = 0.

To prove the surjectivity of H0 → Aut∗(F/Fl), we will find a 0-framedhomology cylinder from the above M which also maps to φ under η̃l. Let

ti := µ̄2(M)i ∈ Z and τ = τ tn−1n−1 ◦· · ·◦τt11 where τi is a Dehn twist along ∂i. We

look at M ′ := (M, i+, i−◦τ). For each i = 1, . . . , n−1, µ2(M ′)i = x−tii µ2(M)i,so µ̄2(M

′)i = 0 and hence M′ ∈ H0. We have η̃0l (M ′) = η̃l(M) = φ.

(2) (η̃0l )−1(Kl,k) = H0 ∩H[k] = H0[k].

(3) SupposeM is inH0[k]. If η̃0l (M) ∈ Al,k, then η̃k+1(M)(xi) = µk(M)−1i xiµk(M)i =

xi, hence [xi, µk(M)i] = 1 in F/Fk+1. It implies µk(M)i = xti ∈ F/Fk for some

t ∈ Z. Since M is in H0, µk(M)i is trivial. The converse is also true. Hence,(η̃0l )

−1(Al,k) = H0[k] ∩ {M ∈ H0 | µk(M) = 1} = H(k).�

Before proving Theorem 4.6, we present the proof of Lemma 4.4 and some ingredi-ents.


Proof. (Proof of Lemma 4.4) Naturally, Ker p′k → Ker pk ∩ {(Fk/Fk+1)n−1 × 02g} isan injective homomorphism. We check that it is an isomorphism. Let αi ∈ Fk/Fk+1for i = 1, . . . , n − 1 satisfying

∏i[xi, αi] = 1 ∈ Fk+1/Fk+2. We will show that αi is in

F ′k/F′k+1 for all i.

Consider the Magnus expansion of F into the algebra of formal power series innoncommutative variables

M : F ↪−→ Z[[X1, . . . , Xn−1,M1, . . . ,Mg, L1, . . . , Lg]]

which sends xi,mj , lj of F to 1 + Xi, 1 + Mj , 1 + Lj respectively. It is well knownthat a ∈ Fk if and only if M(a) − 1 is a sum of monomials of degree at least k; see[MKS76, Section 5]. Denote by Pk the set of homogeneous polynomials of degree k inZ[[X1, . . . , Xn−1,M1, . . . ,Mg, L1, . . . , Lg]]. We have an isomorphism Mk : Fk/Fk+1 →Pk which sends aFk+1 ∈ Fk/Fk+1 to the homogeneous part of degree k in M(a) −1 (see [MKS76, Corollary 5.12]). Let Mk(αi) = hi. The inverse image of Pk ∩Z[[X1, . . . , Xn−1]] under the above map is F ′k/F ′k+1. Hence it is enough to show thathi is in Z[[X1, . . . , Xn−1]].

By calculation, Mk+1([xi, ai]) = Xihi−hiXi, and then, Mk+1(∏i[xi, ai]) =

∑i(Xihi−

hiXi) = 0 since∏i[xi, αi] = 1. For a monomial z in the Xi,Mj , Lj , define

l(z) =

t if z = uyv where v is a monomial in the Xi of degree t,

y ∈ {M1, . . . ,Mg, L1, . . . , Lg},u is a monomial in the Xi,Mj , Lj ,

∞ if z is a monomial in the Xi.

i.e. l(z) is the length of the rightmost submonomial written in the Xi only. Writehi =

∑α ciαziα, where ciα ∈ Z, ziα is a monomial. To show that l(ziα) = ∞ for

all i, α, suppose l(ziα) < ∞ for some i, α. Let l := mini,α{l(ziα)}. Then l is finite.Consider a term ci0α0Xi0zi0α0 with l(zi0α0) = l in the expression∑

i,α

ciαXiziα −∑i,α

ciαziαXi = 0.

The term ci0α0Xi0zi0α0 is not eliminated with any other term of the form ciαziαXi sincel(Xi0zi0α0) < l(ziαXi). Also, ci0α0Xi0zi0α0 is not eliminated with any other term of theform ciαXiziα if either i 6= i0 or l(ziα) 6= l. It follows that

∑{α|l(zi0α)=l}

ci0αXi0zi0α =

0, and so∑{α|l(zi0α)=l}

ci0αzi0α = 0. Thus, all the terms in hi with l(−) = l areeliminated in hi. This contradicts the choice of l. �

Lemma 5.2. (1) For each i, the map Fk/Fk+1 → Fk+1/Fk+2 which sends α to[xi, α] is an injective homomorphism if k ≥ 2. If k = 1, then the restrictionmap H1(〈xi′ ,mj , lj | i′ 6= i〉)→ F2/F3 is injective.

(2) Let φ ∈ Ak+2,k. Then φ(xi) = (αφi )−1xiαφi , φk+1(mj) = mjβ

φj , φk+1(lj) =

ljγφj for some α

φi , β

φj , γ

φj ∈ Fk/Fk+1. The formula φ(

∏xi∏

[mj , lj ]) =∏xi∏

[mj , lj ]implies ∏

[xi, αφi ]∏

[mj , γφj ][β

φj , lj ][β

φj , γ

φj ] = 1 in F/Fk+2.

If k ≥ 2, [βφj , γφj ] = 1 in F/Fk+2.

(3) For a group G, if an automorphism of G/Gk induces the identity on G/Gl forsome l < k, then it is also identity on G1+α/Gl+α for any α ≤ k − l.


Proof. (1) We use the Magnus expansion as in the proof of Lemma 4.4. For α ∈Fk/Fk+1, let Mk(α) = h. Then Mk+1([xi, α]) = Xih−hXi. Suppose [xi, α] = 1in Fk+1/Fk+2, then Xih−hXi = 0. A similar argument to the proof gives thath is a polynomial in only Xi with the given i. It implies α ∈ 〈xi〉. If k ≥ 2,α = 0. If k = 1, α = xti for some t ∈ Z.

(2) We omit the calculation.(3) Use induction on α. From the hypothesis, it is true for α = 0. Assume that

φl+α−1|Gα/Gl+α−1 = id. It is enough to check that φl+α([g, h]) = [g, h] forg ∈ G, h ∈ Gα. Let φl+α(g) = ga with a ∈ G/Gl+α. Then a is in Gl/Gl+α.Also, φl+α(h) = hb for some b ∈ Gl+α−1/Gl+α. Applying commutators iden-tityes, we see that φl+α([g, h]) = [ga, hb] = [ga, b][ga, h][[ga, h], b] = [ga, h] =[g, h][[g, h], a][a, h] = [g, h] in G/Gl+α.

�

Remark 5.3. For M ∈ H0, the boundary condition η̃2(M)([∂n]) = [∂n] implies∏j [mj , µ

′′2(M)j ][µ

′2(M)j , µ

′′2(M)j ][µ

′2(M)j , lj ] = 1 ∈ F2/F3. The term [µ′k(M)j , µ′′k(M)j ]

does not vanishes in Fk/Fk+1 for the case k = 2, differently from the other k. It isthe cause of failure for (µ′2, µ

′′2) to be a homomorphism on H(1)/H0[2] and that for

H(1)/H(2)→ D1(H) to be defined when g 6= 0.

Proof. (Proof of Theorem 4.5 and Theorem 4.6) We will prove that there is a surjectionΘ: Ak+2,k → Ker pk such that Θ ◦ η̃k+2 = µ̃k+1 on H(k).

Suppose k ≥ 2 and φ ∈ Ak+2,k. The βφj , γφj ∈ Fk/Fk+1 in Lemma 5.2(2) are

uniquely determined. If we consider x−1i φ(xi) = [x−1i , (α

φi )−1] = [xi, α

φi ] ∈ Fk+1/Fk+2,

Lemma 5.2(1) implies that αφi is uniquely determined in Fk/Fk+1 for k ≥ 2. ByLemma 5.2(2), (αφ1 , . . . , α

φn−1, β

φ1 , . . . , β

φg , γ

φ1 , . . . , γ

φg ) is in Ker pk. Hence, we can define

a map

Ak+2,kΘ−−→ Ker pk ⊂ (Fk/Fk+1)2g+n−1

φ 7−→ (αφ1 , . . . , αφn−1, β

φ1 , . . . , β

φg , γ

φ1 , . . . , γ

φg ).

The relation (1) in Section 2.3 implies Θ ◦ η̃k+2 = µ̃k+1 on H(k).To see the surjectivity of Θ, let θ = (α1, . . . , αn−1, β1, . . . , βg, γ1, . . . , γg) ∈ Ker pk

with αi, βj , γj ∈ Fk/Fk+1. We show that there are lifts α̃i, β̃j , γ̃j ∈ Fk/Fk+2 of αi, βj , γjsuch that the automorphism φ of F/Fk+2 sending xi,mj , lj to α

−1i xiαi,mj β̃j , lj γ̃j

respectively is in Aut∗(F/Fk+2) with a lift φ̃ ∈ Aut(F/Fk+3) sending xi to α̃−1i xiα̃i.In detail, let α̃i = αiai, β̃j = βjbj , γ̃j = γjcj with ai, bj , cj ∈ Fk+1/Fk+2. Thenφ̃(xi) = a

−1i φ(xi)ai, φ(mj) = φk+1(mj)bj , φ(lj) = φk+1(lj)cj . The φ̃([∂n]) is computed


as

φ̃

(∏xi∏

[mj , lj ]

)=∏

a−1i φ(xi)ai∏

[φk+1(mj)bj , φk+1(lj)cj ]

=∏

φ(xi)[φ(xi)−1, a−1i ]

∏[φk+1(mj), φk+1(lj)][φk+1(mj), cj ][bj , φk+1(lj)][bj , cj ]

=∏

φ(xi)[α−1i x

−1i αi, a

−1i ]∏

[φk+1(mj), φk+1(lj)][mjβj , cj ][bj , ljγj ][bj , cj ]

=∏i

φ(xi)∏j

[φk+1(mj), φk+1(lj)]∏i

[x−1i , a−1i ]∏j

[mj , cj ][βj , cj ][bj , lj ][bj , γj ][bj , cj ]

=∏i

φ(xi)∏j

[φk+1(mj), φk+1(lj)]∏i

[xi, ai]∏j

[mj , cj ][bj , lj ] if k ≥ 2

=∏i

xi∏j

[mj , lj ] · ξ ·∏i

[xi, ai]∏j

[mj , cj ][bj , lj ] for some ξ ∈ Fk+2/Fk+3.

Since any element in Fk+2/Fk+3 can be represented by∏

[xi, ai]∏

[mj , cj ][bj , lj ] forai, bj , cj ∈ Fk+1/Fk+2, we can choose ai, bj , cj so that ξ−1 =

∏[xi, ai]

∏[mj , cj ][bj , lj ].

Thus φ̃ fixes [∂n], and hence φ ∈ Aut∗(F/Fk+2). The elements αi, βj , γj ≡ 1 mod Fk,so φk+1(xi) = xi, φk(mj) = mj , φk(lj) = lj , and φ ∈ Ak+2,k. Therefore, Θ(φ) = θ.

Lastly, we show that Θ is a homomorphism. For φ, ψ ∈ Ak+2,k,

Θ(φ) + Θ(ψ) = (αφ1αψ1 , . . . , α

φn−1α

ψn−1, β

φ1 β

ψ1 , . . . , β

φg β

ψg , γ

φ1 γ

ψ1 , . . . , γ

φg γ

ψg ).

By Lemma 5.2 (3), ψk+1|F2/Fk+1 = id follows from ψk = id. Hence,

(ψ ◦ φ)(xi) = ψ((αφi )−1xiα

φi )

= ψk+1((αφi )−1)(αψi )

−1xiαψi ψk+1(α

φi )

= (αφi )−1(αψi )

−1xiαψi α

φi

We obtain αψ◦φi = αψi α

φi . Also,

(ψ ◦ φ)k+1(mj) = ψk+1(mjβφj )

= mjβψj ψk+1(β

φj )

= mjβψj β

φj

since ψk+1|F2/Fk+1 = id. So, βψ◦φi = β

ψi β

φi . Similarly for lj . Since Fk/Fk+1 is abelian,

Θ(φ) + Θ(ψ) = Θ(ψ ◦ φ).To show the surjectivity of H0[k + 1]→ Ker p′k for k ≥ 2, we observe Θ(Kk+2,k+1).

For φ ∈ Kk+2,k+1, αφi is uniquely determined for each i if k ≥ 2 and βφi , γ

φi are

trivial in Fk/Fk+1, hence the Θ(φ) is in (Fk/Fk+1)n−1 × 02g ⊂ (Fk/Fk+1)2g+n−1. By

Lemma 4.4, Θ(Kk+2,k+1) can be considered as a subset of Ker p′k. Let θ ∈ Ker p′k be(α1, . . . , αn−1) satisfying

∏[xi, αi] = 1. It is the case that βj = γj = 1 for all j in the

proof of the surjectivity of Θ. Similar to the argument, we obtain φ ∈ Ak+2,k such thatφ(xi) = (αiai)

−1xiαiai, φ(mj) = mjbj , φ(lj) = ljcj for some ai, bj , cj ∈ Fk+1/Fk+2and Θ(φ) = θ. The φ is in Kk+2,k+1.


To prove the last isomorphism H[2]H(2)∼= Ker p′k in Theorem 4.6, we construct an

isomorphism

Θ̄S : Zn−1 × K3,2A3,2

−→ Ker{pk| : (F/F2)n−1 × 02g −→ F2/F3}.

For φ ∈ K3,2, αφi is uniquely determined in H1(〈xi′ ,mj , lj | i′ 6= i〉) by Lemma 5.2(1).For an element ((ti), φ) ∈ Zn−1 × K3,2A3,2 , Θ̄

S maps it to (xt11 αφ1 , . . . , x

tn−1n−1 α

φn−1, 1, . . . , 1).

Then we can check similarly that the map is an isomorphism. �

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Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673,

KoreaEmail address: [email protected]

1. Introduction1.1. Extended Johnson homomorphims1.2. Extended Milnor invariants1.3. Extended Orr invariants1.4. OrganizationAcknowledgements

2. Definitions and known results2.1. Homology cylinder cobordism group2.2. Extended Johnson homomorphisms and filtration2.3. Extended Milnor invariants and filtration2.4. Extended Orr invariants

3. Refinement and combination of the two filtrations3.1. 0-framed homology cylinders3.2. Combination of the two filtrations

4. Images and quotients of the filtration under the invariants4.1. Images under extended Johnson homomorphism and Aut(F/Fk)4.2. Images under extended Milnor invariants and free Lie algebras4.3. Images under extended Orr invariants4.4. Relation between the invariants4.5. Rank of quotients of the filtration4.6. First quotient of the extended Milnor filtration

5. ProofsReferences

Homology cobordism group of homology cylinders and ...k denotes the kth term of lower central series...

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