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The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey's framework of higher-order (non-commutative) Alexander invariants to them, we extract several pieces of information about the monoid and related objects.


INTRODUCTION
Let Σ g,1 be a compact connected oriented surface of genus g ≥ 1 with one boundary component. A homology cylinder (over Σ g,1 ) consists of a homology cobordism from Σ g,1 to itself with markings of its boundary. We denote by C g,1 the set of isomorphisms classes of homology cylinders. Stacking two homology cylinders gives a new one, and by this, we can endow C g,1 with a monoid structure (see Section 2 for the precise definition). The origin of homology cylinders goes back to Habiro [7], Garoufalidis-Levine [6] and Levine [14], where the clasper (or clover) surgery theory is effectively used to investigate the structure of C g,1 .
By a standard method, we can assign a homology cylinder to each homology 3-sphere or pure string link. Also, for a given homology cylinder, we can use an element of the mapping class group of Σ g,1 to construct another one by changing its markings. Since these operations preserve each monoid structure, C g,1 can be regarded as a simultaneous generalization of the monoid of homology 3-spheres, that of string links and the mapping class group, any of which plays an important role in the theory of 3-manifolds. On the other hand, there exists a natural way (called closing) to construct a closed 3-manifold from each homology cylinder. Therefore, through its monoid structure, C g,1 serves as an effective tool for classifying closed 3-manifolds.
The aim of this paper is to study the structure of C g,1 from rather an algebraic point of view. We mainly use non-commutative rings arising from group rings to define some invariants such as the Magnus representation for C g,1 and Reidemeister torsion invariants. Note that our Magnus representation extends that for the mapping class group defined by Morita [16], as the Gassner representation for string links due to Le Dimet [11] and Kirk-Livingston-Wang [10] does that for the pure braid group. See Birman's book [1] for generalities of the ordinary (pre-extended) Magnus representation, including free differentials.
After defining invariants using non-commutative rings, we shall need some devices to extract information from them. For that, we use the framework of higher-order Alexander invariants due to Cochran [2] and Harvey [8,9]. Higher-order Alexander invariants are those for finitely presentable groups interpreted as degrees of "non-commutative Alexander polynomials", which have some unclear ambiguity except their degrees. Historically, they are first defined for knot groups by Cochran, and then generalized for arbitrary finitely presentable groups by Harvey. Using them, Cochran and Harvey obtained various sharper results than those brought by the ordinary Alexander invariants -lower bounds on the knot genus or the Thurston norm, necessary conditions for realizing a given group as the fundamental group of some compact oriented 3-manifold, and so on. In the process of applying higher-order Alexander invariants to our case, we shall give its slight generalization (called torsion-degree functions) because of the difference of localizations of non-commutative rings used in the Magnus representation and higher-order Alexander invariants. Then we use it to study several properties of our invariants and relationships between them, from which we will obtain some information about the structure of C g,1 and related 3-manifolds.
The outline of this paper is as follows. In Section 2, we review the definition of homology cylinders as well as setting up our notation and terminology. Sections 3 and 4, which are the first main part of this paper, are devoted to define the Magnus representation and study its fundamentals, including some examples. In Section 5, we review the theory of higher-order Alexander invariants, following Harvey's papers [8,9], and then define its generalization. In Section 6, which is the second main part, we observe several applications of our invariants.

HOMOLOGY CYLINDERS
Throughout the paper, we work in PL or smooth category. Let Σ g,1 be a compact connected oriented surface of genus g ≥ 1 with one boundary component. We take a base point p on the boundary of Σ g,1 , and take 2g loops γ 1 , . . . , γ 2g of Σ g,1 as shown in Figure 1. We consider them to be an embedded bouquet R 2g of 2g-circles tied at the base point p ∈ ∂Σ g,1 . Then R 2g and the boundary loop ζ of Σ g,1 together with one 2-cell make up a standard cell decomposition of Σ g,1 .
We denote i + (p) = i − (p) by p ∈ ∂M again and consider it to be the base point of M. We write a homology cylinder by (M, i + , i − ) or simply by M. Two homology cylinders are said to be isomorphic if there exists an orientation-preserving diffeomorphism between the underlying 3-manifolds which is compatible with the embeddings of Σ g,1 . We denote the set of isomorphism classes of homology cylinders by C g,1 . Given two homology cylinders M = (M, i + , i − ) and N = (N, j + , j − ), we can construct a new homology cylinder M · N by Then C g,1 becomes a monoid with the unit 1 C g,1 := (Σ g,1 × I, id ×1, id ×0).
From the monoid C g,1 , we can construct the homology cobordism group H g,1 of homology cylinders as in the following way. Two homology cylinders M = (M, i + , i − ) and N = (N, j + , j − ) are homology cobordant if there exists a compact oriented 4-manifold W such that the inclusions M ֒→ W , N ֒→ W induce isomorphisms on the homology, where −N is N with opposite orientation. We denote by H g,1 the quotient set of C g,1 with respect to the equivalence relation of homology cobordism. The monoid structure of C g,1 induces a group structure of H g,1 . In the group H g,1 , the inverse of (M, i + , i − ) is given by (−M, i − , i + ). Example 2.1. For each element ϕ of the mapping class group M g,1 of Σ g,1 , we can construct a homology cylinder M ϕ ∈ C g,1 by setting where collars of i + (Σ g,1 ) and i − (Σ g,1 ) are stretched half-way along ∂Σ g,1 × I. This gives injective homomorphisms M g,1 ֒→ C g,1 and M g,1 ֒→ H g,1 . From this, we can regard C g,1 and H g,1 as enlargements of M g,1 .
Let N k (G) := G/(Γ k G) be the k-th nilpotent quotient of a group G, where we define Γ 1 G = G and Γ l G = [Γ l−1 G, G] for l ≥ 2. For simplicity, we write N k (X) for N k (π 1 X) where X is a connected topological space, and write N k for N k (F 2g ) = N k (Σ g,1 ).
Let (M, i + , i − ) be a homology cylinder. By definition, i + , i − : π 1 Σ g,1 → π 1 M are both 2connected, namely they induce isomorphisms on the first homology groups and epimorphisms on the second homology groups. Then, by Stallings' theorem [20] Using them, we obtain a monoid homomorphism It can be easily checked that σ k induces a group homomorphism σ k : H g,1 → Aut N k . We define filtrations of C g,1 and H g,1 by

THE MAGNUS REPRESENTATION FOR HOMOLOGY CYLINDERS
We first summarize our notation. For a matrix A with entries in a ring R, and a ring homomorphism ϕ : R → R ′ , we denote by ϕ A the matrix obtained from A by applying ϕ to each entry. A T denotes the transpose of A. When R = ZG for a group G or its right field of fractions (if exists), we denote by A the matrix obtained from A by applying the involution induced from (x → x −1 , x ∈ G) to each entry.
For a module M, we write M n and M n for the modules of column and row vectors with n entries respectively.
For a finite cell complex X and its regular covering X Γ with respect to a homomorphism π 1 X → Γ, Γ acts on X Γ from the right through its deck transformation group. Therefore we regard the ZΓ-cellular chain complex C * (X Γ ) of X Γ as a collection of free right ZΓ-modules consisting of column vectors together with differentials given by left multiplications of matrices. For each ZΓ-bimodule A, the twisted chain complex C * (X; A) is given by the tensor product of the right ZΓ-module C * (X Γ ) and the left ZΓ-module A, so that C * (X; A) and H * (X; A) are right ZΓ-modules.

Definition of the Magnus representation for homology cylinders.
In what follows, we fix an integer k ≥ 2, which corresponds to the class of the nilpotent quotient. The following construction is based on Kirk-Livingston-Wang's work of the Gassner representation for string links in [10].
Let (M, i + , i − ) ∈ C g,1 be a homology cylinder. By Stallings' theorem, N k and N k (M) are isomorphic. Since N k is a finitely generated torsion-free nilpotent group for each k ≥ 2, we can embed ZN k into the right field of fractions We consider the fields K N k and K N k (M ) to be local coefficient systems on Σ g,1 and M respectively.
Remark 3.2. The same conclusion as in Lemma 3.1 can be drawn for the homology with coefficients in any Zπ 1 (M)-algebra A satisfying the following property: Every matrix with entries in Zπ 1 (M) sent to an invertible one by the augmentation map Zπ 1 (M) → Z is also invertible in A. Note that K N k (M ) satisfies this property.
Since R 2g ⊂ Σ g,1 is a deformation retract, we have (2) The Magnus representation for C g,1 is the map r k : While we call r k (M) the Magnus "representation", it is actually a crossed homomorphism, namely we have the following.
Proof. We write M = M 1 · M 2 for simplicity. Let i : M 1 → M and j : ) is given as the composite of ) and This completes the proof.
Proof. Suppose M 1 = (M 1 , i + , i − ) and M 2 = (M 2 , j + , j − ) ∈ C g,1 are homology cobordant by a homology cobordism W . Let i : M 1 → W , j : M 2 → W be the natural inclusions. We may assume that M 1 ∪ M 2 is a subcomplex of W and that W has only one 0-cell p. Since Then we have the following commutative diagram: The left vertical maps give i r ′ k (M 1 ) and the right ones give j r ′ k (M 2 ). Applying I −1 + , we obtain r k (M 1 ) = r k (M 2 ).
Consequently, we obtain the Magnus representation r k : H g,1 → GL(2g, K N k ), which is a crossed homomorphism. If we restrict r k to C g,1 [k] (resp. H g,1 [k]), it becomes a monoid (resp. group) homomorphism. Example 3.6. For ϕ ∈ M g,1 ֒→ Aut F 2g , we can obtain where ρ k : ZF 2g → ZN k ⊂ K N k is the natural map and ∂/∂γ i are free differentials. From this, we see that r k generalizes the original Magnus representation for M g,1 in [16].

Computation of the Magnus matrix.
In [10], the Gassner matrix of a string link is computed from the Wirtinger presentation of the fundamental group of its exterior, which gives a finite presentation whose deficiency coincides with the number of strings. Recall that the deficiency of a finite presentation P = {x 1 , . . . , x n | r 1 , . . . , r m } of a finitely presentable group G is n−m, and the deficiency of G is the maximum of all over the deficiencies of finite presentations of G. In our context, we do not have such a useful method in general.
Note that there does exist an admissible presentation for each homology cylinder (M, i + , i − ). Indeed, take a Morse function with no critical points of indices 0 and 3. Then M can be seen as Σ g,1 × I with some 1-and 2-handles. Since the Euler characteristics of Σ g,1 × I and M are the same, the numbers of the attached 1-and 2-handles are the same. Therefore the presentation of π 1 M obtained from a presentation of π 1 (Σ g,1 × I) = F 2g with deficiency 2g by adding new generators and relations corresponding to the 1-and 2-handles has deficiency 2g again. Our claim follows from this. (See also Section 6.2.) Given an admissible presentation of π 1 M as in Definition 3.7, we define 2g × (2g + l), l × (2g + l) and 2g × (2g + l) matrices A, B, C by at ZN k (M), namely we apply the natural map to each entry of the matrices obtained by free differentials. (2) As matrices with entries in K N k (M ) , we have gives a presentation matrix of H 1 (M)/Φ + , where Φ + is the subgroup of H 1 (M) generated by i + (γ 1 ), . . . , i + (γ 2g ). (See [4] for this fact through the concept of presentations of a pair of groups.) By definition, H 1 (M)/Φ + = 0, and we have an exact sequence By the Hopfian property of Z 2g+l , we see that (2) Through a standard argument using Eilenberg-MacLane spaces, we can assume that a given admissible presentation is obtained from a cell decomposition of M. Then for some matrix X, where we write 0 k and 0 (k,l) for the zero matrices of sizes k × k and k × l respectively. (2) follows from this.
Note that from (2), we have r ′ , namely the Magnus matrix r ′ k (M) can be computed from any admissible presentation of π 1 (M).
Proposition 3.9. Let r ϕ k (M) be the Magnus matrix corresponding to the new generating system. Then Proof. We have the following admissible presentation of π 1 M with respect to γ ′ 1 , . . . , γ ′ 2g : The matrices A ′ , B ′ , C ′ corresponding to this presentation are given by , we obtain the formula.

EXAMPLE: RELATIONSHIP TO THE GASSNER REPRESENTATION FOR STRING LINKS
In [14], Levine gave a method for constructing homology cylinders from pure string links. By this, we can obtain many homology cylinders not belonging to the subgroup M g,1 . Also, we can see a relationship between the Gassner representation for string links and our representation.
For a g-component pure string link L ⊂ D 2 × I, we now construct a homology cylinder M L ∈ C g,1 as follows. Consider a closed tubular neighborhood of the loops γ g+1 , γ g+2 , . . . , γ 2g in Figure 1 to be the image of an embedding ι : D g ֒→ Σ g,1 of a g-holed disk D g as in Figure 2.
Then the manifold M L obtained from Σ g,1 × I by removing ι(D g ) × I and regluing C by h becomes a homology cylinder. This construction gives an injective monoid homomorphism L g → C g,1 from the monoid L g of (framed) pure string links to C g,1 . Moreover it also induces an injective homomorphism S g → H g,1 from the concordance group S g of (framed) pure string links to H g,1 . In particular, the (smooth) knot concordance group, which coincides with S 1 , is embedded in H g,1 . If we restrict these embeddings to the pure braid group, which is a subgroup of L g and S g , their images are contained in M g,1 .
By the Gassner representation, we mean the crossed homomorphism r G,k : L g → GL(g, K N k (Dg ) ) or r G,k : S g → GL(g, K N k (Dg ) ) given by a construction similar to that in the previous section. (In [11] and [10], only r G,2 is treated.) Comparing methods for calculating the Gassner and Magnus representations, we obtain the following. .
We mention two remarks about this theorem. First we identify F g = π 1 D g with the subgroup of F 2g = π 1 Σ g,1 generated by γ g+1 , . . . , γ 2g . Then the maps Second, the embeddings L g ֒→ C g,1 and S g ֒→ H g,1 have ambiguity with respect to framings. However we can check that the lower right part of r k (M L ) is independent of the framings.
Proof of Theorem 4.1. All we have to do is to give a suitable presentation of π 1 M L . We divide M L into two parts M and C as follows.
We take g points q 1 , . . . , q g and g paths l j from the base point p to q j as in Figure 3.
Let M be the union of Σ g,1 × I − D g × I and 2g paths i + (l j ) and i − (l j ) (j = 1, . . . , g). Then We denote by C the complement of an open tubular neighborhood of L in D g × I as before.
is given by the Wirtinger presentation of D × I − L. We glue C to M by using some fixed framing. Then it is easily seen that π 1 (M ∩C) is the free group generated by Using the above decomposition, we obtain where δ i are words in i − (γ g+1 ), . . . , i − (γ 2g ), z 1 , . . . , z l , i + (γ g+1 ), . . . , i + (γ 2g ) which depend on the framing. Rewrite the above presentation by using i + (γ j )'s and i − (γ j )'s instead of i + ( γ j )'s and i − ( γ j )'s. This process does not affect generators i − (γ g+j ), z j , i + (γ g+j ) and relations r j .
From the resulting admissible presentation, we can compute the Magnus matrix of M L . Then our claim follows by comparing it with the method for calculating the Gassner matrix of L from the Wirtinger presentation of π 1 C, which is given in [10].
Proof. In [10], they gave 3-component pure string links denoted by L 5 and L 6 having the condition that L 5 is a pure braid, while the conjugate L 6 L 5 L −1 6 is not. To show that L 6 L 5 L −1 6 is not a pure braid, they use the fact that r G,2 (L 6 L 5 L −1 6 ) has an entry not belonging to ZN 2 (D 3 ). Then our claim follows from Theorem 4.1 with respect to this example.
Then the presentation of π 1 M L is given by where we use the blackboard framing. We now compute r 2 (M L ). We identify N 2 and N 2 (M L ) by using i + . Using the presentation, we have Hence

HIGHER-ORDER ALEXANDER INVARIANTS AND TORSION-DEGREE FUNCTIONS
Here we summarize the theory of higher-order Alexander invariants along the lines of Harvey's papers [8,9]. For our use, we generalize them to functions of matrices called torsiondegree functions.
A group Γ is poly-torsion-free-abelian (PTFA, for short) if Γ has a normal series of finite length whose successive quotients are all torsion-free abelian. In particular, free nilpotent quotients N k are PTFA for all k ≥ 2. Note that every subgroup of a PTFA group is also PTFA. For each PTFA group Γ, the group ring ZΓ is known to be an Ore domain, so that it can be embedded in the right field of fractions K Γ := ZΓ(ZΓ − {0}) −1 , which is a skew field. We refer to [3], [17] for localizations of non-commutative rings.
We will also use the following localizations of ZΓ placed between ZΓ and K Γ . Let ψ : Γ ։ Z be an epimorphism. Then we have an exact sequence We take a splitting ξ : Z → Γ and put t := ξ(1) ∈ Γ. Since Γ ψ is again a PTFA group, ZΓ ψ can be embedded in its right field of fractions K Γ ψ = ZΓ ψ (ZΓ ψ − {0}) −1 . Moreover, we can also construct ZΓ(ZΓ ψ − {0}) −1 . Then the splitting ξ gives an isomorphism between ZΓ(ZΓ ψ − {0}) −1 and the skew Laurent polynomial ring is known to be a non-commutative right and left principal ideal domain. By definition, we have inclusions Harvey's higher-order Alexander invariants [9] are defined as follows. Let G be a finitely presentable group and let ϕ : G ։ Z be an epimorphism. For a PTFA group Γ and an epimorphism ϕ Γ : G ։ Γ, (ϕ Γ , ϕ) is called an admissible pair for G if there exists an epimorphism ψ : Γ ։ Z satisfying ϕ = ψ • ϕ Γ . For each admissible pair (ϕ Γ , ϕ) for G, we regard K Γ ψ [t ± ] = ZΓ(ZΓ ψ − {0}) −1 as a ZG-module, and we define the higher-order Alexander are called the Γ-degree and the refined Γ-degree respectively. (Our definition is slightly different from that of [9].) Note that the right For a connected space X and an admissible pair (ϕ Γ , ϕ) for π 1 X, we define δ ψ Γ (X) := δ ψ Γ (π 1 X) and δ ψ Γ (X) := δ ψ Γ (π 1 X). For a finitely presentable group G and an admissible pair (ϕ Γ , ϕ) for G, the (refined) Γ-degree can be computed from any presentation matrix of the right Therefore we can consider it to be a function on the set M(K Γ ψ [t ± ]) of all matrices with entries in K Γ ψ [t ± ]. Here we extend this function to M(K Γ ) as follows.
First, we extend deg ψ : It induces an isomorphism between The following lemma will be used in our generalization of Harvey's invariants. We denote by M(m, n, K Γ ) the set of all m × n matrices with entries in K Γ .
For each I ⊂ {1, 2, . . . , m}, J ⊂ {1, 2, . . . , n} with #I = m − k, #J = n − k, let U I denote the square matrix defined by taking i-th columns from U for all i ∈ I, and V J denote the one defined by taking j-th rows from V for all j ∈ J. We also denote by A I c J c the one defined by taking i-th rows from A for all i ∈ I c := {1, 2, . . . , m} − I and then taking j-th columns for all j ∈ J c := {1, 2, . . . , n} − J.
is independent of the choice of I and J such that U I , V J are invertible, where sgn(II c ) ∈ {±1} (resp. sgn(JJ c )) is the signature of the juxtaposition of I and I c (resp. J and J c ), and we put det ∅ := 1.
Proof. (1) and (2) are deduced from easy observation using non-commutative linear algebra. To prove (3), it suffices to show in the cases where P 1 , P 2 , Q 1 , Q 2 are matrices of elementary transformations, and it can be easily checked.

Remark 5.2.
In the above situation, the sequence is exact. By taking the standard basis for each vector space, we regard the sequence as a based acyclic chain complex. Then we can take its torsion (see [15], [21] for generalities of torsions). This torsion coincides with ∆(A; U, V ) up to sign.
As seen in Lemma 5.1 (3), ∆(A; U, V ) does depend on U and V . The following definition and lemma give our rule to take U and V .
Proof. We prove only for V .
We take a basis v 1 , . . . , v n−k and put V = (v 1 , . . . , v n−k ). Then V satisfies the desired property.
The existence of P 2 follows immediately from (2). That of Q 2 is also clear since V and V ′ are bases of the same right K Γ ψ [t ± ]-module.
Definition 5.5. Let Γ be a PTFA group and let ψ : Γ ։ Z be an epimorphism. (3) Let G be a finitely presentable group. We take a presentation x 1 , . . . , x l | r 1 , . . . , r m of G.
For an admissible pair (ϕ Γ , ϕ), the matrix A : ). Then Harvey's invariants are given by where the second equality of each case follows from the direct sum decomposition shown by Harvey in [8].
Remark 5.7. Friedl [5] gave an interpretation of Harvey's invariants by Reidemeister torsions. The definition of our truncated torsion-degree functions has some overlaps with his description.

APPLICATIONS OF TORSION-DEGREE FUNCTIONS TO HOMOLOGY CYLINDERS
In this section, we study some invariants of homology cylinders arising from the Magnus representation and Reidemeister torsions by using torsion-degree functions associated to nilpotent quotients N k of π 1 Σ g,1 . N k is known to be a finitely generated torsion-free nilpotent group. In particular, it is PTFA.
Note that we can take a primitive element of H 1 (Σ g,1 ) instead of an epimorphism N k ։ Z to define a torsion-degree function since Hom(N k , Z) = H 1 (N k ) = H 1 (N 2 ) = H 1 (Σ g,1 ). We denote by P H 1 (Σ g,1 ) the set of primitive elements of H 1 (Σ g,1 ).

The Magnus representation and torsion-degree functions.
First, we apply torsion-degree functions to the Magnus matrix. However, it turns out that the result is trivial. Theorem 6.1. Let M be a homology cylinder. For any ψ ∈ P H 1 (Σ g,1 ), the torsion-degree d ψ N k (r k (M)) is always zero.
Proof. By definition, d ψ N k is additive for products of invertible matrices, and invariant under taking the transpose and operating the involution. Moreover, it vanishes for matrices in GL(ZN k ).
(2) (I 2g − r k (M)) ∂ζ ∂γ 1 , . . . , ∂ζ ∂γ 2g Proof. We take an admissible presentation of π 1 M as in Definition 3.7. We also take the matrices A, B, C ∈ ZN k corresponding to it. For simplicity, we put Then, by Proposition 3.8, Our claim follows by taking their first 2g columns.
(2) Let τ ζ ∈ M g,1 ⊂ C g,1 be the Dehn twist along ζ. It belongs to the center of C g,1 and acts on N k by conjugation x → ζ −1 xζ. Then where the fourth equality follows from the fact that M acts on N k trivially. On the other hand, it is easily checked that by using free differentials. Then (2) follows from this. (2) and (3) of Definition 5.3 respectively.
greater than 0. When A ∈ GL(ZN k ), the same holds for each entry of ( − −− → 1 − γ)A which is non-zero. However, if we choose f ∈ Aut F 2g such that ψ(f (γ 1 )) = 0, which does exist, (2). By a similar argument, we can show that ∂ζ (3), where we use f ∈ Aut F 2g preserving ζ and the chain rule for free differentials (see for instance [1,Proposition 3.3]).
Note that d ψ N k (I 2g −r k (M)) does not depend on the choice of the generating system of π 1 Σ g,1 . This follows from the formulas in Proposition 3.9 and Lemma 5.1 (3). In Section 6.3.3, we will see some examples showing that our invariants are non-trivial at least in the case of k = 2.
6.2. N k -torsion and torsion-degree functions. For a homology cylinder M = (M, i + , i − ) ∈ C g,1 , we put Σ + := i + (Σ g,1 ). Since the relative complex C * (M, Σ + ; K N k (M ) ) obtained from any smooth triangulation of (M, Σ + ) is acyclic by Lemma 3.1, we can consider its Reidemeister torsion τ (C * (M, Σ + ; K N k (M ) )). Definition 6.5. The N k -torsion of a homology cylinder M = (M, i + , i − ) ∈ C g,1 is given by Recall that Reidemeister torsions are invariant under subdivision of the cell complex (M, Σ + ) and simple homotopy equivalence. Now we consider τ N k (M) more closely. First we give a cell decomposition of ∂M ∼ = Σ g,1 ∪ (−Σ g,1 ) by pasting two copies of that of Σ g,1 in Figure 1. We denote by R + 2g the subcomplex i + (R 2g ). Take a triangulation of ∂M by refining the cell decomposition, and extend it to one of M. Then by the multiplicativity of torsions and the fact that Σ + is simple homotopy equivalent to R + 2g . Starting from a 3-simplex of M facing the boundary, we can deform M onto a 2-dimensional subcomplex M ′ by a simple homotopy equivalence keeping the 1-skeleton of M invariant. Then τ (C * (M, R + 2g ; K N k (M ) )) = τ (C * (M ′ , R + 2g ; K N k (M ) )). Next, we take a maximal tree T of the 1skeleton of M ′ and collapse it to a point. This process also preserves the simple homotopy type of (M ′ , R + 2g ), so that τ (C * (M ′ , R + 2g ; K N k (M ) )) = τ (C * (M ′ /T, R + 2g /(T ∩ R + 2g ); K N k (M ) )).
Proof. We take an admissible presentation of π 1 M 1 and the matrices A M 1 , B M 1 , C M 1 corresponding to it. We denote this presentation by for short. Similarly, we take an admissible presentation Then we obtain an admissible presentation . The corresponding partial matrix where i : M 1 → M 1 · M 2 and j : M 2 → M 1 · M 2 are the natural inclusions. From this, we have This completes the proof.
We call it the closing of M. It is easily seen that if M ∈ C g,1 [k], we have the natural isomor- Here we identify these groups.
Proof. Take an admissible presentation of π 1 M as in Definition 3.7, and construct the corresponding matrices A, B, C ∈ M(ZN k ).
Adding 2g relations i + (γ j ) = i − (γ j ) (j = 1, . . . , 2g) and deleting the generators i + (γ j ) by using them, we obtain a presentation of π 1 C M . From this presentation, we have a presentation matrix J C M of H 1 (C M , p; ZN k ) given by where the second equality follows from Proposition 3.8.
Hence to show our claim, it suffices to prove the case where this value is just 2g + l − 1 (see Definition 5.5 (2)). By Fundamental formula of free calculus, we have On the other hand, we have by Lemma 6.3 (2). Then we can define ∆ (J C M ; ξ, µ), where we put T . Lemma 6.9. µ belongs to (ZN k ) 2g+l .
Proof. Recall that ( A B ) is a presentation matrix of H 1 (M, Σ + ; ZN k ), so that we have an exact sequence where the injectivity of the second map follows from the fact that H 1 (M, Σ + ; K N k ) = 0. Hence to prove the lemma, it suffices to show that ∂ζ ⊕(ZN k ) 2g . Then by observing the boundary corresponding to the relation T is homologous to 0 (1,2g) 0 (1,l) ∂ζ ∂ − → γ T , which comes from C 1 (Σ + , p; ZN k ). Our claim follows from this. Now we continue the proof of Theorem 6.8. We can show that (ξ, µ) is ψ-primitive for J C M as in the proof of Proposition 6.4. Then We have From the above argument, we obtain This completes the proof. First we attach a 2-handle I × D 2 along I × i ± (∂Σ g,1 ), so that we obtain a homology cylinder (M ′ , i ′ + , i ′ − ) over a closed surface Σ g , which corresponds to the embedding Σ g,1 ֒→ Σ g . Then we put ), x ∈ Σ g and call T M the mapping torus of M. Indeed, for M ϕ ∈ M g,1 ⊂ C g,1 , the resulting manifold T Mϕ is the usual mapping torus of ϕ extended naturally to the mapping class of Σ g . If we take an admissible presentation of π 1 M briefly denoted by i − ( − → γ ), − → z , i + ( − → γ ) | − → r , then a presentation of π 1 T M is given by where λ is the loop Note that these groups are torsion-free nilpotent. We consider N k (Σ g ) to be a subgroup of N k (T M ). For simplicity, we denote N k (Σ g ) by N k,0 and N k (T M ) by N k,T .
We can show that H * (M, i + (Σ g,1 ); K N k,T ) = 0 (see Remark 3.2). Hence by a similar argument, the Magnus representation r k,T : C g,1 → GL(2g, K N k,T ) and the N k,T -torsion are defined. Then we obtain the following factorization formula of the N k,T -degree for the mapping torus of a homology cylinder.
Proof. The first assertion is a slight generalization of [8,Proposition 8.4], and we now follow the proof. Let ψ ∈ H 1 (T M ) be the Poincaré dual of the surface i ′ Let (T M ) N k,T be the N k,T -cover of T M , and let (T M ) ψ be the Z-cover of T M with respect to ψ. (T M ) ψ is the product · · · · M ′ · M ′ · M ′ · · · · of countably many copies of M ′ , and (T M ) N k,T can be regarded as the N k,0 -cover of (T M ) ψ . Then Here we remark that the image of the composite π 1 ((T M ) ψ ) → π 1 T M → N k,T is contained in N k,0 . The same holds for the composite π 1 M ′ → π 1 T M → N k,T . We also remark that is a finite dimensional K N ψ k,T -vector space, so that δ ψ N k,T (T M ) is finite. To show the second assertion, we take an admissible presentation of π 1 M, and construct the matrices A, B, C ∈ ZN k,T as before. From the presentation, we have a presentation matrix J T M of H 1 (T M , p; ZN k,T ) given by We remark that λ belongs to the center in N k,T . As presentation matrices of H 1 (T M , p; ZN k,T ), this matrix is equivalent to the square matrix By Proposition 3.8, Lemma 6.3 and Fundamental formula of free calculus, we have Then it is easily checked that As in Proposition 6.4 and Lemma 6.9, we can show that ( ξ, µ) is ψ-primitive for J ′ T M . Then This completes the proof.
6.3.3. The case of k = 2 (commutative case). Since ZN 2 = ZN 2 (Σ g ) and K N 2 = K N 2 (Σg) are commutative, we can use the ordinary determinant for computation. Moreover, we can obtain some invariants before taking degrees. For example, define where A (i,j) is the matrix obtained from a matrix A by removing its i-th row and j-th column. ∆(M) is well-defined by Lemma 5.1. Note that this invariant is based on that for string links given in [10], and we call it the Alexander rational function of M.
Theorem 6.12. Let M ∈ C g,1 [2], and let ∆ C M , ∆ T M be the Alexander polynomials of C M , T M , respectively. Then where .
= means that these equalities hold in K N 2 and K N 2 (T M ) up to ±N 2 and ±N 2 (T M ) respectively.
Proof. We prove only the first assertion. The proof is almost the same as that for Theorem 6.8 under the following remarks. We follow the notation used there. We may assume that By definition, ∆ C M is the greatest common divisor of {det J C M (i,j) T } 1≤i,j≤2g+l . We show that it is nothing other than As seen in Lemma 6.9, T is a vector in (ZN 2 ) 2g+l . If ∆ is in ZN 2 , it attains the greatest common divisor. To show it, suppose ∆ = h 1 /h 2 where h 1 ∈ ZN 2 and h 2 ∈ ZN 2 − {0} are relatively prime. From the definition of ∆, we have Hence h 2 is a common divisor of (1 − γ −1 i ) ∂ζ ∂γ j i,j 's, and it is 1. That is, h 2 is a unit in does not depend on the choice of an admissible presentation, and it gives τ N 2 (M). Indeed the matrix A B is a presentation matrix of H 1 (M, Σ + ; ZN 2 ), and its determinant gives a generator of the 0-th elementary ideal, which is principal and invariant under Tietze transformations. This completes the proof.
The formula in Theorem 6.8 holds as elements of Z ∪ {∞}, so that the additivity loses its meaning when the value is ∞. Note that δ ) has a non-trivial free part. The following are some examples of homology cylinders which have non-trivial Alexander rational functions. By using Theorem 6.15 in the next subsection, we obtain many situations where the formula sufficiently works. When k ≥ 3, the computation becomes quite difficult in general. Example 6.13. Assume that g = 1. The Dehn twist τ ζ ∈ M 1,1 belongs to C 1,1 [3]. Then, we have Then ∆(τ ζ ) = 1 ∈ ZN 2 , which is non-trivial.
Example 6.14. Assume that g ≥ 2. Let τ 1 , τ 2 and τ 3 be Dehn twists along simple closed curves c 1 , c 2 and c 3 as in Figure 5. Then τ 1 τ −1 2 , τ 3 ∈ C g,1 [2]. By a direct computation, we can check that 6.4. N k -torsions and Harvey's Realization Theorem. By Proposition 6.6, the degree of the N k -torsion gives a monoid homomorphism d ψ N k (τ N k (·)) : C g,1 [2] −→ Z ≥0 for each ψ ∈ P H 1 (Σ g,1 ) and an integer k ≥ 2. To see some properties of these homomorphisms, including their non-triviality, we use a variant of Harvey's Realization Theorem in [8,Theorem 11.2] which gives a method for performing surgery on a compact orientable 3manifold to obtain a homology cobordant one having distinct higher-order degrees. Theorem 6.15. Let M ∈ C g,1 be a homology cylinder. For each primitive element x of H 1 (Σ g,1 ) and any integers n ≥ 2 and k ≥ 1, there exists a homology cylinder M ′ such that Proof. The proof is based on Harvey's proof of Realization Theorem in [8,Theorem 11.2]. However, since we now use the lower central series instead of the rational derived series, we can shorten the argument.
We take a loop representing x ∈ H 1 (Σ g,1 ), and denote it by x again. We also take a loop γ whose homology class in H 1 (Σ g,1 ) is independent of x.
In fact, we can show it by constructing homology cylinders that are homology cobordant to the unit 1 C g,1 . From this we see that C g,1 [2], C g,1 [3], . . . , Ker(C g,1 → H g,1 ) are not finitely generated monoids. Note that d ψ N k (τ N k (M)) = 0 if M ∈ M g,1 , since Σ g,1 × I is simple homotopy equivalent to Σ g,1 and hence τ N k (M) is trivial. 6.5. Appendix:Application of torsion-degree functions to Aut F acy n . In [18], we defined the Magnus representation r k : Aut F acy n → GL(n, K N k (Fn) ) for Aut F acy n , where F acy n is a completion of F n in a certain sense and is called the acyclic closure of F n . The natural map F n → F acy n is known to be injective and 2-connected. In particular, N k (F n ) = N k (F acy n ), and we denote it briefly by N k in this subsection. Aut F acy n can be regarded as an enlargement of Aut F n . Indeed we have the enlarged Dehn-Nielsen homomorphism σ acy : H g,1 → Aut F acy 2g extending the classical one σ : M g,1 ֒→ Aut F 2g . That is, we have the commutative diagram Aut F 2g ֒→ Aut F acy 2g ֒→ σ ↑σ acy M g,1 ֒→ H g,1 .
Note that σ acy is not injective. The Magnus representation for homology cylinders is nothing other than the composite H g,1 σ acy −→ Aut F acy 2g r k − → GL(2g, K N k ). We now consider the map d ψ N k • r k : Aut F acy n → Z for ψ ∈ P H 1 (F n ), where P H 1 (F n ) denotes the set of primitive elements of H 1 (F n ). Since d ψ N k (A) = 0 for A ∈ GL(ZN k ), it follows that d ψ N k • r k Aut Fn is trivial. When n = 2g, d ψ N k • r k Im σ acy is also trivial as seen in Theorem 6.1. On the other hand, d ψ N k • r k is actually non-trivial on Aut F acy n as we will see below. Since r k is a crossed homomorphism, we have the following. Proposition 6.17. For f, g ∈ Aut F acy n and ψ ∈ P H 1 (F n ), we have d ψ N k (r k (f g)) = d ψ N k (r k (f )) + d ψ•f N k (r k (g)).
In particular, if we restrict d ψ N k • r k to IAut F acy n := Ker(Aut F acy n → Aut N 2 = GL(n, Z)), it becomes a homomorphism. Remark 6.18. Aut F acy n acts on P H 1 (F n ) from the right, and hence acts on Map(P H 1 (F n ), Z) from the left. We regard d · N k (r k (·)) as a map Aut F acy n → Map(P H 1 (F n ), Z). Then Proposition 6.17 shows that d · N k (r k (·)) is a 1-cocycle in C 1 (Aut F acy n , Map(P H 1 (F n ), Z)). We can see that it is non-trivial in H 1 (Aut F acy n , Map(P H 1 (F n ), Z)) from the proof of Theorem 6.19 below.
Theorem 6.19. For every n ≥ 2, IAut F acy n is not finitely generated. In fact, H 1 (IAut F acy n ) has infinite rank.
Proof. Let F n = x 1 , x 2 , . . . , x n . We take ψ := x * 1 ∈ P H 1 (F n ). Consider the endomorphism f k of F n given by where we define Y 1 = x 1 and Y l = [x 2 , Y l−1 ] for l ≥ 2. Since f k is 2-connected, it induces an automorphism of F acy n (see [18,Section 4]). We denote it by f k again. It belongs to IAut F acy n . For such an automorphism, the Magnus matrix r l (f k ) can be computed by using free differentials. That is, we have at ZN k . Then d ψ N l (r l (f k )) = deg ψ (det(r l (f k ))) = deg ψ ∂f (x 1 ) . By a direct computation, we have ∂f (x 1 ) When l = k, we have Y k−1 = 1 ∈ N k , so that ∂f (x 1 ) and d ψ N k (r k (f k )) = deg ψ ∂f (x 1 ) This shows that {d ψ N k (r k (·))} k≥2 are all non-trivial, and independent of each other. Our claim follows from this.

ACKNOWLEDGEMENT
The author would like to express his gratitude to Professor Shigeyuki Morita for his encouragement and helpful suggestions. He also would like to thank Professor Masaaki Suzuki for valuable discussions and advice. This research was supported by JSPS Research Fellowships for Young Scientists.