The cobordism group of homology cylinders
Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis-Levine and Goda-Sakasai. Furthermore we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results hold for the homology cylinder analogue of the Torelli group as well.
Key words and phrases:Torsion Invariant, Homology Cylinder, Homology Cobordism
1991 Mathematics Subject Classification:Primary 57M27; Secondary 57N10
Given and , let be a fixed oriented, connected and compact surface of genus with boundary components. We denote by the group of orientation preserving diffeomorphisms of which restrict to the identity on the boundary. The mapping class group is defined to be the set of isotopy classes of elements in , where the isotopies are understood to restrict to the identity on the boundary as well. We refer to [FM09, Section 2.1] for details. It is well known that the mapping class group is perfect provided that [Po78] (e.g., see [FM09, Theorem 5.1]) and that mapping class groups are finitely presented [BH71, Mc75] (e.g., see [FM09, Section 5.2]).
In this paper we intend to study an enlargement of the mapping class group, namely the group of homology cobordism classes of homology cylinders. A homology cylinder over is roughly speaking a cobordism between surfaces equipped with a diffeomorphism to such that the cobordism is homologically a product. Juxtaposing homology cylinders gives rise to a monoid structure. The notion of homology cylinder was first introduced by Goussarov [Go99] and Habiro [Ha00] (where it was referred to as a ‘homology cobordism’).
By considering smooth (respectively topological) homology cobordism classes of homology cylinders we obtain a group (respectively ). These groups were introduced by Garoufalidis and Levine [GL05], [Le01]. We refer to Section 2 for the precise definitions of homology cylinder and homology cobordism. Henceforth, when a statement holds in both smooth and topological cases, we will drop the decoration in the notation and simply write instead of and .
It follows immediately from the definition that there exists a canonical epimorphism . A consequence of work of Fintushel-Stern [FS90], Furuta [Fu90] and Freedman [Fr82] on smooth homology cobordism of homology –spheres is that this map is not an isomorphism. In fact, using their results we can see the following:
Let . Then the kernel of the epimorphism contains an abelian group of infinite rank. If , then there exists in fact a homomorphism onto an abelian group of infinite rank such that the restriction of to the kernel of the projection map is also surjective.
An argument of Garoufalidis and Levine shows that the canonical map is injective. (See also Proposition 2.4.) It is a natural question which properties of mapping class groups are carried over to . In particular in [GS09] Goda and Sakasai ask whether is a perfect group and Garoufalidis and Levine [GL05, Section 5, Question 9] ask whether is infinitely generated (see also [Mo06, Problem 11.4]).
The following theorem answers both questions:
If , then there exists an epimorphism
which splits (i.e., there is a right inverse). In particular, the abelianization of contains a direct summand isomorphic to .
Note that Theorem 1.2 also implies that is not finitely related, since for a finitely related group its abelianization is also finitely related. Also refer to Remark LABEL:remark:proof_of_mainthm for a slightly more refined statement.
In many cases we can actually strengthen the result:
If , then there exists an epimorphism
Furthermore, the abelianization of contains a direct summand isomorphic to .
We remark that for the special case of , this is a consequence of Levine’s work on knot concordance [Le69a, Le69b] since one can easily see that maps onto Levine’s algebraic knot concordance group. The general cases of Theorem 1.3 for and for and are new.
In order to prove Theorems 1.2 and 1.3 we will employ the torsion invariant of a homology cylinder, first introduced by Sakasai (e.g., see [Sa06, Section 11.1.2], [Sa08, Definition 6.5] and [GS08, Definition 4.4]). In Section LABEL:section:torsion-invariant we recall the definition of the torsion of a homology cylinder and we study the behavior of torsion under stacking and homology cobordism. The result can be summarized as a group homomorphism
Here and is the multiplicative group of nonzero elements in the quotient field of the group ring . Loosely speaking, reflects the action of surface automorphisms on , and is the subgroup of “norms” in . (For details, see Section LABEL:section:torsion-invariant.)
An interesting point is that torsion invariants of homology cylinders may be asymmetric, in contrast to the symmetry of the Alexander polynomial of knots. Indeed, in Section LABEL:section:homomorphisms, we extract infinitely many –valued and –valued homomorphisms of from symmetric and asymmetric irreducible factors, respectively. Theorems 1.2 and 1.3 now follow from the explicit construction of examples in Section LABEL:section:construction-computation.
We remark that Theorem 1.3 covers all the possibilities of the asymmetric case, since it can be seen that for either or the torsion of a homology cylinder over is always symmetric in an appropriate sense (see Section LABEL:subsection:symmetry-asymmetry for details.)
We remark that our main results hold even modulo the mapping class group—the essential reason is that our torsion invariant is trivial for homology cylinders associated to mapping cylinders. More precisely, denoting by the normal subgroup of generated by , the torsion homomorphism actually factors as
In Section LABEL:section:pretzel, we study examples of homology cylinders which naturally arise from Seifert surfaces of pretzel links. We compute the torsion invariant and prove that these homology cylinders span a summand in the abelianization of .
Finally in Section LABEL:section:torelli, we consider the “Torelli subgroup” of which is the homology cylinder analogue of the Torelli subgroup of mapping cylinders. We prove that the conclusions of Theorems 1.2 and 1.3 hold for the Torelli subgroup , but for a larger group of surfaces. (See Theorem LABEL:thm:cyltorelli for details.) This extends work of Morita’s [Mo08, Corollary 5.2] to a larger class of surfaces.
In this paper, manifolds are assumed to be compact, connected, and oriented. All homology groups are with respect to integral coefficients unless it says explicitly otherwise.
This project was initiated while the three authors visited the Mathematics Department of Indiana University at Bloomington in July 2009. We wish to express our gratitude for the hospitality and we are especially grateful to Chuck Livingston and Kent Orr. The authors would like to thank Irida Altman, Jake Rasmussen, Dan Silver and Susan Williams for helpful conversations. We are very grateful to Takuya Sakasai for suggesting a stronger version of Theorem 1.1 to us and for informing us of Morita’s work in [Mo08]. The authors thank an anonymous referee for helpful comments, in particular the comment about Proposition 2.4. The first author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2007-0054656 and 2009-0094069). The last author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2009-0068877 and 2009-0086441).
2. Homology cylinders and their cobordism
In this section we recall basic definitions and preliminaries on homology cylinders, the notion of which goes back to Goussarov [Go99], Habiro [Ha00], and Garoufalidis and Levine [GL05], [Le01].
2.1. Cobordism classes of homology cylinders
Given and we fix, once and for all, a surface of genus with boundary components. When and are obvious from the context, we often denote by . A homology cylinder over is defined to be a –manifold together with injections , satisfying the following:
is orientation preserving and is orientation reversing.
Let . Then gives rise to a homology cylinder
where is given by for and . If is the identity, then we will refer to the resulting homology cylinder as the product homology cylinder.
Let be a knot of genus such that the Alexander polynomial is monic and of degree . Let be a minimal genus Seifert surface in the exterior of . Then cut along is a homology cylinder over in a natural way (e.g., see [Ni07, Proposition 3.1] or [GS08] for details).
Two homology cylinders and over are called isomorphic if there exists an orientation-preserving diffeomorphism satisfying . We denote by the set of all isomorphism classes of homology cylinders over . A product operation on is given by stacking:
This turns into a monoid. The unit element is given by the product homology cylinder.
Two homology cylinders and over are called smoothly homology cobordant if there exists a compact oriented smooth –manifold such that
and such that the inclusion induced maps and are isomorphisms. We denote by the set of smooth homology cobordism classes of elements in . The monoid structure on descends to a group structure on , where the inverse of a mapping cylinder is given by . (We refer to [Le01, p. 246] for details).
If there is a topological –manifold satisfying the above conditions, then we say that and are topologically homology cobordant. We denote the resulting group of homology cobordism classes by . Note that there exists a canonical surjection . We will see in the following sections that this map is in general not an isomorphism.
In the original papers of Garoufalidis and Levine [GL05, Le01] and in [GS09] the authors focus on the smooth case and denote the group by .
In this section we will discuss three types of examples:
homology cobordism classes of integral homology spheres, and
concordance classes of (framed) knots in integral homology spheres.
Surface automorphisms and homology cylinders
First recall that gives rise to a homology cylinder . Note that if , then is isomorphic to . In particular the map descends to a morphism of monoids. Proposition 2.4 below says that we can view the mapping class group as a subgroup of . To prove the proposition, we need the following folklore theorem:
Let be a surface with one boundary component, possibly with punctures. Let be a base point. Suppose is a homeomorphism with the following properties:
is the identity on ,
is the identity, and
fixes each puncture.
Then is isotopic to the identity where the isotopy restricts to the identity on .
The theorem is well-known, and we only give an outline of the proof. We choose disjoint circles in based at such that cut along the is a punctured disk. Since is the identity, we may assume (after applying an isotopy whose restriction to is the identity) that fixes each . Now the map on the punctured disk induced from is isotopic to the identity with fixed pointwise. This can be shown, for example, using the fact that the map from the pure braid group to is injective. ∎
The map is injective.
The proposition was proved by Garoufalidis and Levine in the case that and in the smooth category. (See [GL05, Section 2.4] and [Le01, Section 2.1].) Using their arguments partially, we will show that the proposition holds for any (and in the topological category as well).
First note that if , i.e., if is a sphere or a disk, then it is well-known that any orientation preserving diffeomorphism is isotopic to the identity. If , i.e., if is an annulus, then it is known that and it injects into (for example, this can be shown using the arguments in the subsection below entitled Concordance of (framed) knots in homology –spheres). Therefore we henceforth assume that .
Suppose is an orientation-preserving diffeomorphism such that
restricts to the identity on , and
Now fix and fix a base point lying in the -th component, say , of . We write . We denote by the lower central series of defined inductively by and , . The argument of Garoufalidis and Levine (which builds on Stallings’ theorem [St65]) shows that if is homology cobordant to the product homology cylinder, then is the identity map for any . Since is trivial, this implies that is the identity.
We denote by the set of equivalence classes of orientation-preserving diffeomorphisms such that
is the identity on , and
fixes each boundary component setwise,
where we say that two such maps are equivalent if they are related by an isotopy which fixes pointwise. Then in by Theorem 2.3.
Now consider the map . It is known that the Dehn twists along boundary components of generate a central subgroup isomorphic to in , and is the subgroup generated by the Dehn twists along all, but the -th boundary component. (For example, see [FM09, Section 4.2].) Therefore, in . ∎
Note that this shows that is non-abelian provided that or . It is straightforward to see that in the remaining cases is abelian.
In the following we will see that the cobordism groups of homology cylinders over the surfaces with have been studied under different names for many years.
Homology cobordism of integral homology –spheres
We first consider the case . Recall that oriented integral homology –spheres form a monoid under the connected sum operation. Two oriented integral homology –spheres and are called smoothly (respectively topologically) cobordant if there exists a smooth (respectively topological) –manifold cobounding and . We denote by (respectively ) the group of smooth (respectively topological) cobordism classes of integral homology –spheres.
For the group (respectively ) is naturally isomorphic to the group (respectively ) (e.g., see [Sa06, p. 59]). Furuta [Fu90] and Fintushel-Stern [FS90] showed that has infinite rank. (See also [Sav02, Section 7.2].) On the other hand it follows from the work of Freedman [Fr82] that is the trivial group. (See also [FQ90, Corollary 9.3C].) This shows in particular that the homomorphism is not an isomorphism for .
Concordance of (framed) knots in homology –spheres
We now turn to the case , i.e., homology cylinders over the surface which we henceforth identify with the annulus . Let be an oriented knot in an integral homology –sphere. Let be the exterior of . It is not difficult to see there are pairs of maps satisfying (1)–(4) in the definition of a homology cylinder and satisfying the condition that is a meridian of . Furthermore, the isotopy types of such are in 1–1 correspondence with framings on . Indeed, the linking number of and the closed curve gives rise to a canonical 1–1 correspondence between the set of framings and . Conversely, a homology cylinder over determines an oriented knot endowed with a framing in the integral homology sphere, which is given by attaching a –handle along and then attaching a –handle.
We say that and are smoothly concordant if there exists a smooth cobordism of and such that is an integral homology and contains a smoothly embedded annulus cobounding and . The set of smooth concordance classes of knots in integral homology –spheres form a group under connected sum.
We will now see that we can also think similarly of the concordance group of framed knots in integral homology –spheres. Note that a concordance as above determines a 1–1 correspondence between framings on and . We say two framed knots in integral homology spheres are smoothly concordant if there is a concordance via which the given framings correspond to each other. Smooth concordance class of framed knots form a group as well, and it is easily seen that this framed analogue of is isomorphic to . Similarly, if we allow topologically locally flat annuli in topological cobordisms we obtain a group and its framed analogue .
As usual we adopt the convention that the group can mean either or . It follows easily from the definitions that we have an isomorphism . It follows from the work of Levine [Le69a, Le69b] that maps onto the algebraic knot concordance group which is isomorphic to , and furthermore the epimorphism from to splits. This discussion in particular proves the following special case of Theorem 1.3:
There exists a split surjection of onto .
Note that the subgroup in is exactly the factor, so that . In particular, Theorem 2.5 holds for as well as .
2.3. Proof of Theorem 1.1
Before we turn to the proof of Theorem 1.1 we introduce a gluing operation on homology cylinders. Let be a homology cylinder over a surface and let be a homology cylinder over a surface . Assume that and fix a boundary component of and fix a boundary component of . We can now glue and along and using an orientation reversing homeomorphism. Similarly we can glue and along neighborhoods of and . This gives a homology cylinder over , which we denote by . We refer to as the union of and along and .
Now let be the product homology cylinder over . The association gives rise to a monoid homomorphism . We refer to it as an expansion by along . Note that the expansion map descends to a group homomorphism .
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
We adopt the convention that stands either for or . Recall that in Section 2.2 we saw that we can identify with . Also recall that by Section 2.2 the group has infinite rank and that is the trivial group.
As we saw above, an expansion by a surface of genus with punctures gives rise to a homomorphism . We also consider the composition of expansions by a disk along all boundary components of . Loosely speaking, is the homomorphism given by filling in the holes.
There exists a map such that the composition
is the identity.
Let . Let be a fixed handlebody of genus . Given we write . Since has a Heegaard decomposition of genus there exist diffeomorphisms such that .
We write . We will see in Section LABEL:section:actionh1 that the action of a homology cylinder over on gives rise to a homomorphism . We will write . We now pick a splitting map , i.e. a map such that is the identity on . We pick such that .
Note that we can not arrange that is a homomorphism. Now consider the following map
Note that this map is indeed well-defined, i.e. the right hand side is an integral homology 3–sphere. Also note that this map descends to a map . It is easy to verify that
is indeed the identity map. This concludes the proof of the claim.
Before we continue we point out that the map is in general not a monoid morphism. We now obtain the following commutative diagram: