Hidden torsion, 3-manifolds, and homology cobordism
aa r X i v : . [ m a t h . G T ] D ec Hidden torsion, 3-manifolds, and homology cobordism
Jae Choon Cha and Kent E. Orr
Department of Mathematics, POSTECH, Pohang 790–784, Republic of Korea, andSchool of Mathematics, Korea Institute for Advanced Study, Seoul 130–722, Republic of KoreaE-mail address: [email protected]
Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USAE-mail address: [email protected]
Abstract.
This paper continues our exploration of homology cobordism of 3-manifolds usingour recent results on Cheeger-Gromov ρ -invariants associated to amenable representations. Weintroduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraicapproximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifoldswhich have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pair-wise homology cobordant, yet remain indistinguishable by any prior known homology cobordisminvariants. Additionally we give an answer to a question about transfinite lower central seriesof homology cobordant 3-manifold groups, asked by T. D. Cochran and M. H. Freedman.
1. Introduction
We investigate low dimensional manifolds via homology cobordism. Recall that twoclosed 3-manifolds M and M ′ are (topologically) homology cobordant if there is a 4-dimensional topological cobordism W between M and M ′ satisfying H ∗ ( W, M ) = 0 = H ∗ ( W, M ′ ). One can consider homology with twisted coefficients as well. We applyour recent results concerning the homology cobordism invariance of Cheeger-Gromov ρ -invariants [CO12] and utilizing a new form of torsion in manifolds groups we call localhidden torsion. Numerous central problems in low dimensional topology are special cases of the prob-lem of classifying homology cobordism types. Concordant knots in a three manifold, M, have homology cobordant exteriors, with coefficients in the group ring Z π M [CS74].Hence, to classify knots up to concordance we can understand homology cobordism ofthree manifolds. Casson and Freedman showed the knot slice problem contains “univer-sal” four dimensional topological surgery problems whose positive solution would yield aclassification theory for topological four manifolds [CF84, FQ90]. Levine embedded themapping class group, that is the group of isotopy classes of surface diffeomorphisms, inthe monoid of homology cobordism classes of the mapping cylinders associated to surfacediffeomorphisms (see [Lev01] for one boundary component case, [CFK11] for the general-ization to the multi-boundary component case). Hence, computing homology cobordism Mathematics Subject Classification.
Key words and phrases.
Homology cobordism, 3-manifold, Local hidden torsion, L -signature. classes of three manifolds potentially detects surface homeomorphisms and the mappingclass group. (See also [GS, GS11, CFK11].) And new results distinguishing smoothand topological knot concordance give a laboratory for experiments in four dimensionalsmoothing theory. Given a 4-dimensional homology cobordism W of M , the fundamental groups of thesemanifolds relate in subtle and poorly understood ways. In this paper, we exploit a newtype of torsion we call hidden torsion in a manifold, and use that torsion, along withrecent results from [CO12], to compute Cheeger-Gromov ρ -invariants as obstructions tohomology cobordism. Roughly speaking, we say that a curve in a manifold M represents hidden torsion if the curve has infinite order in π ( M ) and is essential in any homologycobordism of M , but some power is null-homotopic in some homology cobordism of M (Definition 2.3). Note that a “generic” 3-manifold has torsion-free fundamental group(e.g., all closed irreducible non-spherical 3-manifolds, by the Geometrization Conjecture).We give an algebraic version of this notion using Vogel’s homology localization ofgroups (equivalently, Levine’s algebraic closure of groups) which we call local hiddentorsion. We discuss the Vogel localization of groups in Section 2.1.
Definition 2.2.
Let G be a group, and b G the Vogel group localization of G . An element g ∈ G is local hidden torsion of G if g has infinite order in G , and nontrivial finite orderin its image under G → b G .We remark that local hidden torsion and hidden torsion , as defined above, agree for highdimensional manifolds. (See Theorem 2.5).We exploit local hidden torsion in 3-manifold groups to realize Cheeger-Gromov in-variants, which first appeared as homology cobordism obstructions in [CO12]. We provethe following theorem, and use this theorem to construct the examples in our main result,Theorem 1.2. Theorem 1.1.
There are closed hyperbolic 3-manifolds M whose fundamental group haslocal hidden torsion lying in π ( M ) ω . Here { π α } denotes the lower central series of a group π , and ω is the first infinite ordinal.Recall that { π α } is defined inductively by π = π , π α = [ π, π α − ] for a successor ordinal α , and π α = T β<α π β for a limit ordinal α .The 3-manifolds we construct have non-trivial transfinite lower central series, a phe-nomenon first observed and studied in [CO98].We use this local hidden torsion and a special case of our main theorem in [CO12] (seeTheorem 3.8) to construct the examples satisfying parts (1) and (2) of the following the-orem. This torsion hides within the intersection of the lower central series and underliesour proof of parts (3) and (4) below. Theorem 1.2.
There are infinitely many closed hyperbolic 3-manifolds M = M , M , . . . with the following properties: (1) For each i , there is a homology equivalence f i : M i → M . That is, f i induces anisomorphism on H ∗ ( − ; Z ) . (2) Whenever i = j , M i and M i are not homology cobordant. IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 3
Furthermore, all prior known homology cobordism obstructions fail to distinguish theseexamples. In particular, (3)
For any homomorphism φ : π ( M ) → G with G torsion-free, the L -signaturedefects (= von Neumann-Cheeger-Gromov invariants) ρ (2) ( M, φ ) and ρ (2) ( M i , φ ◦ f i ∗ ) are equal for each i . In particular Harvey’s ρ n -invariants [Har08] of the M i arethe same. (4) Similarly, the following homology cobordism invariants for the M i are equal: (a) Multi-signatures (= Casson-Gordon invariants) for prime power order char-acters in [Gil81, GL83, Rub84, CR88](b)
Atiyah-Patodi-Singer ρ -invariants for representations factoring through p -groups in [Lev94, Fri05](c) Twisted torsion invariants for representations factoring through p -groupsin [CF](d) Hirzebruch-type Witt-class-valued invariants from iterated p -covers in [Cha10]Details and proofs are given as Propositions 5.4, 5.10, 5.7, and Theorem 5.6.We now outline the argument. We begin with a surface bundle over a circle withfundamental group G = ( Z t ) ⋊ Z . Here the quotient group Z acts by negation oneach factor of the free abelian subgroup. We compute the group localization of thisgroup, and show that this group localization has non-trivial transfinite lower centralseries, and that the ω term in this series contains torsion. We construct an homologycobordism from our surface bundle to a new three manifold with the property that theimage of the fundamental group of the new three manifold in its group localizationcontains some of the above torsion. Like the group localization, this new three manifoldhas non-trivial transfinite lower central series. We alter this three manifold, preservingit’s homology type by replacing a neighborhood of a curve representing hidden torsionwith a new homology circle, the complement of a knot, chosen to alter an appropriatelychosen Cheeger-Gromov ρ -invariant of the three manifold associated to this torsion groupelement.The resulting homology cobordism classes of the M i are indistinguishable via any priorknown invariant, as stated in (3) and (4). This follows, with some work, from observingthat the relevant local hidden torsion vanishes in the group nilpotent completion, andthrerefore cannot be detected in any nilpotent quotient group, and in particular, in any p -group. Additionally, the effect of this construction along the local hidden torsion isinvisible via any L -signature associated to a representation to a torsion-free group.By contrast, the Cheeger-Gromov invariants employed to detect these examples arisefrom representations to infinite, non-nilpotent amenable groups with torsion, in whichthe local hidden torsion is not eliminated.Additional applications, by the first author, of the main results of [CO12] exploitingtorsion can be found in [Cha].We obtain one additional result from our construction and answer a question posedby Cochran and Freedman (see Kirby’s problem list [Kir95, Problem 3.78]): Is the lowercentral series length of a 3-manifold group invariant under homology cobordism?
Recall that the lower central series length of a group π is defined to be the small-est ordinal α satisfying π α = π α +1 . Lower central series quotients π/π q , q finite, are JAE CHOON CHA AND KENT E. ORR preserved under homology cobordism by Stallings’s Theorem [Sta65]. Therefore, onemust consider 3-manifolds with transfinite lower central series length. In [CO98, Propo-sition 9.1], Cochran and the second author considered an algebraic analogue formulatedfor finitely presented groups and gave a negative answer. We give a 3-manifold groupcounterexample.
Theorem 1.3.
There are two homology cobordant closed 3-manifolds with fundamentalgroups of differing lengths, one having length ω , and the other length > ω .Convention. In this paper all manifolds are assumed to be oriented and compact.
Acknowledgments.
The authors thank the anonymous referee for detailed comments.The first author was supported by National Research Foundation of Korea (NRF) grantsfunded by the Ministry of Education, Science and Technology (No. 2010–0029638 and2010–0011629). The second author was supported by NSF grant DMS–0707078, andSimons Foundation grant 209082.
2. Homology localization and hidden torsion
First introduced by Bousfield [Bou75], Vogel modified the construction of homology lo-calization, and with keen insight, revealed its fundamental role in the study of manifoldembeddings [Vog78]. For groups, Levine’s theory of algebraic closures provides a veryinfluential algebraic approach to the homology localization [Lev89b, Lev89a]. (For re-lated works, see, e.g., [LD88, Vog78, Cha08, CH05, Sak06, Cha10, Cha09, CO12, CH10,Gut79, Hec12].)The homology localization we use in this paper is a specific case of Vogel’s morecomplicated general theory in [Vog78], and follows the combinatorial approach introducedin [Lev89a, Lev89b, Lev92], suitably modified to remove the normal closure conditionused therein. Explicit definitions that precisely fit our purpose first appear in [Cha08]and [CO12]. For the reader’s convenience, we recall these:
Definition 2.1.
Let R be a commutative ring with unity.(1) A group homomorphism α : π → G is called R -homology 2-connected if f inducesan isomorphism on H ( − ; R ) and a surjection on H ( − ; R ). We denote by Ω R the collection of α : π → G with π and G finitely presented and α R -homology2-connected. We simply say α is 2-connected when R = Z .(2) A group K is R -local if for any α : π → G in Ω R and for any homomorphism f : π → K , there is a unique homomorphism g : G → K satisfying g ◦ α = f .(3) The Vogel-Levine R -homology localization of a group G is a group b G endowedwith a homomorphism p G : G → b G such that b G is R -local and p G is universal(initial) among morphisms with this property. That is, for any f : G → K with K local, there is a unique g : b G → K satisfying g ◦ p G = f .One knows that for any group G there exists unique ( b G, p G ) satisfying the above.(For a proof, e.g., see [Cha08].) Useful consequences of Definition 2.1 are the followingproperties:(1) The association G b G is a functor. IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 5 (2) { p G } is a natural transformation.(3) Any α : π → G in Ω R gives rise to an isomorphism b α : b π → b G .In this paper homology localization always refers to the group localization in Defini-tion 2.1.A homology equivalence induces a two-connected homomorphism between funda-mental groups. Hence, and central to applications to manifolds, a homology cobor-dism W between manifolds M i , i = 0 ,
1, determines inclusion-induced isomorphisms \ π ( M i ) ∼ = \ π ( W ). Local hidden torsion is an often more accessible homological approximation to hiddentorsion, suitable for some applications, including the primary application of this paper.We prove in Theorem 2.5 below that hidden torsion and local hidden torsion agree inhigh dimensional manifolds. This intrinsically low dimensional distinction deserves moreextensive examination in a future paper.We recall the definitions from the introduction:
Definition 2.2.
Let G be a group, and b G the homology localization of G . An element g ∈ G is local hidden torsion of G if g has infinite order in G , and nontrivial finite orderin its image under G → b G . Definition 2.3.
For a manifold M , an element g ∈ π ( M ) is called hidden torsion of M if g has infinite order in π ( M ) and is essential in any homology cobordism W of M ,but for some homology cobordism W of M , the image of g in π ( W ) has finite order.We remark that if g ∈ π ( M ) is hidden torsion, then for any N homology cobordantto M , there is a homology cobordism W between M and N satisfying the following:the image of g in π ( W ) has finite order. This follows from the obvious fact that if V is a homology cobordism with ∂V = M ∪ − N and V ′ is a homology cobordism with ∂V ′ = M ∪ − M ′ via which g ∈ π ( M ) is hidden torsion, then W = V ′ ∪ M ′ − V ′ ∪ M V isa homology cobordism from M to N .The following hidden torsion example in a hyperbolic 3-manifold may be viewed as ageometric analogy to the fact that an element g in a group G may have infinite ordereven when its homology class [ g ] ∈ H ( G ) has nontrivial finite order. Example 2.4.
Suppose K is a hyperbolic slice knot in S . Let M be the ( n/ K on S , so that H ( M ) = Z /n . By Thurston’s hyperbolic Dehn surgery the-orem [Thu78], M is hyperbolic for all but finitely many n , and consequently π ( M ) istorsion-free for those n . Since the meridian µ of K in M is a generator of H ( M ), itfollows that the order of µ in π ( W ) is at least n for any homology cobordism W of M .In particular, µ is not null-homotopic in W .Taking a concordance C ∼ = S × [0 ,
1] in S × [0 ,
1] between K and the unknot andby filling in the exterior of C with D × S × [0 ,
1] along the ( n/ W , between M and the lens space L = L ( n, µ isisotopic to the generator of π ( L ) = Z /n in W , µ has order n in π ( W ). This shows that µ represents hidden torsion of M .Since H ( Z /n ) = 0, the abelianization map π ( M ) → H ( M ) = Z /n induces anisomorphism on homology localization. Since the homology localization of any abelian JAE CHOON CHA AND KENT E. ORR group is the group itself, it follows that the localization of π ( M ) is the abelianizationmap π ( M ) → Z /n . Therefore µ represents local hidden torsion of π ( M ). Theorem 2.5.
Suppose M is a closed n -manifold with n > . Then an element g ∈ π ( M ) is hidden torsion of M if and only if g is local hidden torsion of π ( M ) .Proof. For convenience we write π = π ( M ). It is known (e.g., see [Cha08, Theorem 2.6])that the homology localization b π is the direct limit of a sequence of 2-connected homo-morphisms between finitely presented groups G i : π = G −→ G −→ · · · −→ G n −→ · · · First we prove the only if direction. Suppose g ∈ π is hidden torsion. If g were trivialin b π , then (the image of) g would be trivial in some G i . But, since n >
3, by appealingLemma 2.6 below, there is a homology cobordism W of M for which π ( W ) = G i and theinclusion-induced map π ( M ) → π ( W ) is equal to the above π → G i . This contradictsthe hypothesis that g is hidden torsion. It follows that g is nontrivial in b π . Also, thereis a homology cobordism W for which g k is trivial in π ( W ) for some k >
0, by thedefinition of hidden torsion. It follows that g k is trivial in b π ( W ) ∼ = b π .For the if part, suppose that g is local hidden torsion. For any homology cobordism W of M , \ π ( W ) ∼ = b π . Therefore g is nontrivial in \ π ( W ), and consequently in π ( W ).Since g has finite order in b π , so does g in some G i . Appealing to Lemma 2.6 below,choose a homology cobordism W of M satisfying π ( W ) ∼ = G i . Then g has finite orderin π ( W ). This shows that g is local torsion. (cid:3) The following fact used in the above proof of Theorem 2.5 is not due to us and seemsknown to experts. Whereas the key idea is used in various applications in the literature,we did not find an explicitly written proof elsewhere. We give a proof for the convenienceof the readers.
Lemma 2.6.
Suppose M is a closed n -manifold with n > and φ : π ( M ) → G is a 2-connected homomorphism with G finitely presented. Then there is a homology cobordism W of M such that π ( W ) = G and the inclusion M → W induces φ .Proof. Choose generators g , . . . , g k of G . For convenience we denote by F h g j i the freegroup on the g , . . . , g k viewing these as symbols, and often regard an element in F h g j i as its image in G under the obvious surjection F h g j i → G .Since φ gives rise to an isomorphism on H ( − ), for each i = 1 , . . . , k there exist γ i ∈ π ( M ) and a product of commutators h i ∈ F h g j i satisfying g i = φ ( γ i ) · h i in G .Now consider the ( n + 1)-manifold V obtained by attaching handles to M × [0 ,
1] asfollows: V = M × [0 , ∪ ( k ∪ ( k j -th 1-handle corresponds to the j -th generator g j so that the fundamentalgroup of V (1) = M × [0 , ∪ (1-handles) is identified with free product π ( M ) ∗ F h g j i , andthe i -th 2-handle is attached along an embedded circle representing g − i γ i h i ∈ π ( V (1) ).Obviously there is a homomorphism ψ : π ( V ) → G that the given φ : π ( M ) → G factorsthrough. Also, computing H ∗ ( V, M ) from the handle decomposition, it is easily seen thatthe V is a homology cobordism, since each h i is in the commutator subgroup. IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 7
Let N = Ker ψ . Note that N is contained in the commutator subgroup [ π ( V ) , π ( V )]since ψ induces an isomorphism on H ( − ). Since ψ is surjective and G is finitely pre-sented, N is normally finitely generated in π ( V ), namely for some finitely many elements n , . . . , n r ∈ N , the normal subgroup in π ( V ) generated by the n i is equal to π ( V ).Choosing disjoint embedded circles in the interior of V representing the n i and perform-ing surgery on V along these circles, we obtain an ( n + 1)-manifold, say V ′ . By standardarguments, using the fact that each n i lies in the commutator subgroup of π ( V ), oneshows that π ( V ′ ) = G , H ( V ′ ) = H ( V ) ⊕ Z r , and H i ( V ′ ) = H i ( V ) for i >
2, wherethe Z r factor of H is introduced by surgery.We will do surgery to eliminate the Z r factor of H ( V ′ ). For this purpose, firstwe obtain appropriate spherical elements in H as follows. Since φ : π ( M ) → G is2-connected, the map H ( V ) ∼ = H ( M ) −→ H ( G ) ∼ = H ( π ( V ′ ))is surjective. It follows that we can choose elements c i = ( x i , y i ) ∈ H ( V ′ ) = H ( V ) ⊕ Z r = H ( M ) ⊕ Z r ( i = 1 , . . . , r ) satisfying the following: the y i form a basis of Z r andthe c i lie in the kernel of the map H ( V ′ ) → H ( π ( V ′ )). By the exact sequence π ( V ′ ) −→ H ( V ′ ) −→ H ( π ( V ′ )) −→ c i are spherical. We may assume that the c i are given asdisjoint embedded 2-spheres, and can perform surgery along the c i on V ′ , since this isbelow middle dimension ( n + 1 ≥ c i ; the result ofsurgery, say W , is an ( n + 1)-manifold with H ∗ ( W ) ∼ = H ∗ ( V ) ∼ = H ∗ ( M ) as desired. (cid:3) In the remainder of this paper, we construct three manifolds whose groups contain localhidden torsion that cannot be seen in the nilpotent completion, and then use this hiddentorsion to construct new manifolds homology equivalent, but not homology cobordant toeach other.
3. Torus bundle group and homology cobordism invariance
We construct homology cobordism invariants in three steps. In Section 3.1, we computethe integral homology localization b Γ of a torus bundle group Γ. In Section 3.2, weconstruct a representation of b Γ into an amenable D ( Z /p )-group, where D ( Z /p ) denotesStrebel’s class in [Str74]. Lemma 6.8 in [CO12] clarifies how to achieve this using the“mixed type commutator series” defined in [Cha]. (Here successive terms of the derivedseries of a group arise from distinct coefficient subrings of the rational numbers.) InSection 3.3, by applying the Z /p -coefficient version of a key result in [CO12] (stated asTheorem 3.7 in this section), we obtain our homology cobordism invariant.We remark that we need the localization of Z at p , Z ( p ) =, and Z /p , as coefficients inthe latter two steps to apply the main result of [CO12]. While it suffices to use integralgroup homology localization in the first step, we present our computation of group local-ization for any subring R of Z (2) . This small generalization may prove useful elsewhere,and omitting it does not simplify the computation in any way. (See Theorem 3.1.) JAE CHOON CHA AND KENT E. ORR
We consider the group Γ defined below, and compute its group localization. As statedpreviously, this group localization has a non-trivial transfinite lower central series whichcontains torsion. We remark that Levine investigated earlier a different version of alge-braic closure for this group [Lev91, Proposition 2].Let Γ be the groupΓ = ( Z t ) ⋊ Z = h x, y, t | [ x, y ] = 1 , txt − = x − , tyt − = y − i . In the first expression the Z factor is the subgroup generated by t , and Z t is the Z [ t ± ]-module with underlying abelian group Z on which t acts by negation t · a = − a . Theelements x and y generate the two Z t factors. This group Γ is the fundamental groupof an oriented torus bundle over S whose monodromy on S × S is given by ( z, w ) ( z − , w − ).We construct the homology localization of our group Γ as the limit of groups Γ k − , k positive.Γ k − = D x, y, t (cid:12)(cid:12)(cid:12) [ x, y ] (2 k − , [[ x, y ] , t ] , [[ x, y ] , x ] , [[ x, y ] , y ] , txt − = x − , tyt − = y − E When k = 1, we have Γ = Γ. The group Γ k − is a central extension with center theorder (2 k − subgroup generated by [ x, y ] and with quotient group isomorphic to Γ.For later convenience, we denote this subgroup and quotient group by (cid:0) k − Z (cid:1) / Z and (cid:0) k − Z t (cid:1) × Z , respectively. That is, Γ k − is the extension:1 −→ (cid:0) k − Z (cid:1) / Z −→ Γ k − −→ (cid:0) k − Z t (cid:1) ⋊ Z −→ φ k − : Γ → Γ k +1 by t t , x x k +1 , y y k +1 . One easily seesthat this assignment gives us a well-defined homomorphism: by the identities[ ab, c ] = a [ b, c ] a − [ a, c ] , [ c, ab ] = [ c, a ] a [ c, b ] a − we have [ x, y ] [ x k +1 , y k +1 ] = [ x, y ] (2 k +1) , and using this, the relations are verified.Similarly, for 2 k − | ℓ −
1, we define a homomorphism Γ k − → Γ ℓ − given by t t, x x (2 ℓ − / (2 k − , y y (2 ℓ − / (2 k − . Then the groups Γ k − together with these homomorphisms form a direct system.Recall that the (classical) localization of Z at the prime 2 is denoted by Z (2) = { a/b ∈ Q | b is odd } . Theorem 3.1.
Suppose R is a subring of Z (2) . Then for any k , the R -homology local-ization of Γ k − is the colimit lim −→ k Γ k − endowed with Γ k − → lim −→ k Γ k − . In particular Theorem 3.1 gives that the homology localization of Γ = Γ is lim −→ k Γ k − . IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 9
By Theorem 3.1, we have the following commutative diagram, where vertical mapsare the limit maps, and Z (2) / Z is a central subgroup of b Γ:1 / / (cid:0) k − Z (cid:1) / Z / / (cid:127) _ (cid:15) (cid:15) Γ k − / / (cid:127) _ (cid:15) (cid:15) (cid:0) k − Z t (cid:1) ⋊ Z / / / / (cid:127) _ (cid:15) (cid:15) / / Z (2) / Z / / b Γ / / ( Z t (2) ) ⋊ Z / / Proof of Theorem 3.1.
By abelianizing the above presentation, we see that H (Γ k − ) ∼ =( Z / × Z generated by x , y , t . Since we need it later, we give a proof of the followingstronger fact: the commutator subgroup [Γ k − , Γ k − ] is generated by x , y . In fact, therelations txt − = x − , tyt − = y − implies [ t, x ] = x − and [ t, y ] = y − . The commutator[ x, y ] is also contained in the subgroup generated by x , y since [ x, y ][ x , y ] k − k =[ x, y ][ x, y ] k − k = [ x, y ] (2 k − = 1. In addition the subgroup generated by x and y isa normal subgroup since tx t − = x − and yx y − = [ y, x ] x = [ y, x ] x = [ x, y ] − x .This verifies the claim. It follows that φ k − : Γ → Γ k − induces an isomorphism on H ( − ).Also, it is verified that H (Γ k − ) ∼ = Z and φ k − induces an isomorphism on H ( − ),by computation using the Lyndon-Hochschild-Serre spectral sequence twice, first for thecentral extension 1 −→ (cid:0) k − Z (cid:1) / Z −→ A k − −→ (cid:0) k − Z t (cid:1) −→ A k − is the subgroup of Γ k − generated by x and y , and then for1 −→ A k − −→ Γ k − −→ Z −→ . More details are as follows. First observe that the relation [[ x, y ] , t ] of Γ k − is redundant,since txt − = x − and tyt − = y − imply t [ x, y ] t − = [ x − , y − ] and since we obtain[ x a , y b ] = [ x, y ] ab for any integers a , b by combining the relations [[ x, y ] , x ] = 1 = [[ x, y ] , y ]with the identities for [ ab, c ] and [ c, ab ] as above. Therefore Γ k − is an HNN extensionof the subgroup A k − = (cid:10) x, y (cid:12)(cid:12) [ x, y ] (2 k − , [[ x, y ] , x ] , [[ x, y ] , y ] (cid:11) by the infinite cyclic group generated by t with the action txt − = x − , tyt − = y − . Thisis the second extension given above. Also, the first extension above follows immediatelyfrom our presentation of A k − .The first extension is central and has spectral sequence with E p,q ∼ = H p (( k − Z t ) ) ⊗ H q ( k − Z / Z ). We have E ∞ , = 0, E ∞ , = E , = ( k − Z t / Z ) , and E ∞ , is the kernelof d , : H (cid:0) ( k − Z t ) (cid:1) −→ H (cid:0) k − Z / Z (cid:1) . Viewing d , as the transgression map, one sees that d , is identical with the projection k − Z → k − Z / Z . It follows that H ( A k − ) is given by0 −→ ( k − Z t / Z ) −→ H ( A k − ) −→ Z −→ . It is easily seen that H ( A k − ) = ( k − Z t ) from the presentation. The expression of Γ k − as an extension of A k − by Z gives a spectral sequence with E p,q = H p ( Z ; H q ( A k − )). From this we obtain the Gysin sequence H ( A k − ) t ∗ − −−−→ H ( A k − ) −→ H (Γ k − ) −→ H ( A k − ) t ∗ − −−−→ H ( A k − ) . Combining this with our computation of H , H of A k − it follows that H (Γ k − ) = Z as claimed.Now, from the above it follows that φ k − : Γ → Γ k − , and consequently Γ k − → Γ ℓ − for 2 k − | ℓ −
1, are in Ω R for any R . Therefore, the homomorphism Γ → Γ k − induces an isomorphism on localization, and it suffices to show that lim −→ k Γ k − is the R -homology localization of Γ.For any R -local group K and a homomorphism f : Γ → K , there is a unique homo-morphism g k − : Γ k − → K such that f = gφ k − since φ k − ∈ Ω R . It follows thatthe limit g = lim −→ k g k − is the unique homomorphism making the following diagramcommute: Γ lim −→ k Γ k − K / / (cid:15) (cid:15) f z z ttttttttt g Now, by the following proposition, lim −→ k Γ k − is the R -homology localization of Γ. (cid:3) Proposition 3.2. If R is a subring of Z (2) , then lim −→ k Γ k − is R -local. This follows from the fact that Z t (2) is a local module over Z [ t ± ] in the sense of Cohnand Vogel. For completeness, we present a proof of Proposition 3.2 in Appendix A.The following shows that b Γ, and for k >
1, Γ k − have nontrivial ω -term in the lowercentral series. Lemma 3.3.
The transfinite lower central subgroup (Γ k − ) ω is equal to the subgroup ( k − Z ) / Z of Γ k − , and (Γ k − ) ω +1 is trivial. Consequently b Γ ω = Z (2) / Z and b Γ ω +1 is trivial.Proof. As previously shown, (Γ k − ) = [Γ k − , Γ k − ] is the subgroup generated by x , y . By an induction using the same argument, (Γ k − ) q +1 is the subgroup generatedby x q , y q . Therefore (Γ k − ) ω lies in the kernel of Γ k − → ( k − Z t ) ⋊ Z , which isthe subgroup ( k − Z ) / Z .We claim that the generator [ x, y ] of ( k − Z / Z ) lies in (Γ k − ) ω . Since x q ∈ (Γ k − ) q +1 , it follows that [ x q , y ] = [ x, y ] q lies in (Γ k − ) q +2 . Since [ x, y ] has oddorder, it follows that [ x, y ] ∈ (Γ k − ) q +2 . Since this holds for any q , [ x, y ] ∈ (Γ k − ) ω asclaimed. It follows that (Γ k − ) ω = ( k − Z ) / Z .Since the generator [ x, y ] of (Γ k − ) ω is central, (Γ k − ) ω +1 is trivial.The conclusion on b Γ is obtained by taking the direct limit. (cid:3)
In the remaining part of this paper, we will always use the integral homology local-ization of Γ, which we denote by b Γ. IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 11
The homology localization b Γ is not in Strebel’s class D ( R ) for R = Z , Q or Z /p , while weneed (amenable and) D ( R )-groups to apply the main result of [CO12]. In this subsection,we construct a representation of b Γ to an amenable D ( Z /p )-group, by separating p -torsionelements from those with order coprime to p .For the above purpose, we consider the mixed-type coefficient commutator series {P n G } of a group G which was introduced in [Cha]. Suppose P = ( R , R , . . . ) be asequence of commutative rings with unity. Then for a group G , the series {P n G } isdefined inductively by P G = G, P n +1 G = Ker n P n G −→ P n G [ P n G, P n G ] −→ P n G [ P n G, P n G ] ⊗ Z R n o . The subgroup P n G is a characteristic normal subgroup of G .We use the case P = ( Z , Z , Z ( p ) ) applied to b Γ. (In this case we define and use P n b Γonly for n ≤ Lemma 3.4.
Suppose p is a prime. For P = ( Z , Z , Z ( p ) ) , the following hold: (1) P b Γ is the subgroup of b Γ generated by Z (2) / Z and (2 Z t (2) ) , and b Γ / P b Γ ∼ =( Z / × Z . (2) P b Γ is the subgroup Z (2) / Z , and b Γ / P b Γ ∼ = ( Z t (2) ) ⋊ Z . (3) P b Γ is the subgroup ( Z (2) ∩ Z ( p ) ) / Z , of P b Γ = Z (2) / Z . Also, P b Γ / P b Γ ∼ = ( if p = 2 , Z [ p ] / Z otherwise . Consequently, if p = 2 , then b Γ / P b Γ ∼ = ( Z t (2) ) ⋊ Z . If p is odd, then we have thefollowing central extension: −→ Z [ p ] / Z −→ b Γ / P b Γ −→ ( Z t (2) ) ⋊ Z −→ Proof.
From our choice of P , it follows that P k ( − ) is the ordinary derived series for k ≤ P Γ k − = [Γ k − , Γ k − ] is thesubgroup generated by x , y . Therefore, by the direct limit construction of b Γ, it followsthat P b Γ = [ b Γ , b Γ] is the subgroup generated by Z (2) / Z and (2 Z t (2) ) , and b Γ / P b Γ ∼ =( Z t (2) / Z t (2) ) ⋊ Z ∼ = ( Z / × Z . This proves (1).Now, again from the presentation of Γ k − , P Γ k − = [ P Γ k − , P Γ k − ] is gener-ated by [ x , y ] = [ x, y ] , which is equal to the subgroup generated by [ x, y ] since [ x, y ]has odd order in Γ k − . Therefore P b Γ = [ P b Γ , P b Γ] is seen to be the subgroup Z (2) / Z ,and b Γ / P b Γ ∼ = ( Z t (2) ) ⋊ Z . This proves (2).For (3), recall that P b Γ is the kernel of the map Z (2) / Z → ( Z (2) / Z ) ⊗ Z ( p ) . It followsimmediately that P b Γ = { elements in Z (2) / Z with order coprime to p } = { b/a + Z ∈ Z (2) / Z | ( a, b ) = 1 = ( a, p ) } = ( Z (2) ∩ Z ( p ) ) / Z . Also ( Z (2) / Z ) / P b Γ = Z (2) / ( Z (2) ∩ Z ( p ) ). One easily verifies that for any odd prime p theinclusion of rings Z [ p ] → Z (2) gives rise to an isomorphism Z [ p ] / Z ∼ = Z (2) / ( Z (2) ∩ Z ( p ) ) . Since b Γ / P b Γ is a central extension of P b Γ / P b Γ by b Γ / P b Γ, we obtain the exact sequencestated in (3). (cid:3)
Remark 3.5.
From the above computation, it follows that an element g ∈ b Γ has finiteorder if and only if g ∈ P b Γ = Z (2) / Z . If p is odd, the quotient map P b Γ → P b Γ / P b Γ isexactly the map eliminating all the q -primary factors of Z (2) / Z for which ( p, q ) = 1. Inparticular, if g ∈ b Γ has order p k r with ( p, r ) = 1, then its image in b Γ / P b Γ has order p k . Lemma 3.6.
The group b Γ / P b Γ is amenable and in D ( Z /p ) .Proof. By the exact sequence in Lemma 3.4 (3), it follows that b Γ / P b Γ admits a subnormalseries with successive quotients either torsion-free abelian or p -torsion abelian, namely, Z , ( Z t (2) ) , and Z [ p ] / Z . By applying Strebel’s work [Str74] and known properties ofamenable groups (see [CO12, Lemma 6.8]), we obtain the desired conclusion. (cid:3) In this paper we mainly use the case of odd prime p . We remark that if p = 2, then b Γ / P b Γ ∼ = b Γ / P b Γ is PTFA.
Recall that two closed 3-manifolds M and N are (topologically) R -homology cobordant by a topological 4-manifold R -homology cobordism W if ∂W = M ∪ − N and H ∗ ( W, M ; R ) = 0 = H ∗ ( W, N ; R ) . We give a special case of a key result in [CO12]:
Theorem 3.7 (Cha-Orr [CO12, Special case of Theorem 1.1]) . Suppose R is a subring of Q or Z /p . Suppose W is an R -homology cobordism between closed 3-manifolds M and N .Suppose G is an amenable group lying in D ( R ) and φ : π ( M ) → G and ψ : π ( N ) → G are homomorphisms that extend to a common homomorphism π ( W ) → G . Then, the L -Betti numbers b (2) i ( M, φ ) and b (2) i ( N, ψ ) are equal for any i , and L -signature defects ρ (2) ( M, φ ) and ρ (2) ( N, φ ) are equal. For the definitions of b (2) i ( M, φ ) and ρ (2) ( M, φ ), see, e.g., [CO12, Section 7].Combining the Z /p version of Theorem 3.7 with our computation, we obtain thefollowing result: Theorem 3.8.
For a closed 3-manifold M satisfying \ π ( M ) ∼ = b Γ , let φ M be the compo-sition φ M : π ( M ) −→ \ π ( M ) ∼ = b Γ −→ b Γ / P b Γ where P = ( Z , Z , Z ( p ) ) . Then the i -th Betti number b (2) i ( M, φ M ) and L -signature defect ρ (2) ( M, φ M ) are homology cobordism invariants. That is, if another 3-manifold N ishomology cobordant to M , then \ π ( N ) ∼ = b Γ , and we have b (2) i ( M, φ M ) = b (2) i ( N, φ N ) , and ρ (2) ( M, φ M ) = ρ (2) ( N, φ N ) . IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 13
Proof.
Suppose M and N are as in the statement of the theorem, and W is a homologycobordism between M and N . Then, since π ( M ) → π ( W ) and π ( N ) → π ( W ) areintegral homology 2-connected, we have b Γ ∼ = \ π ( M ) ∼ = \ π ( W ) ∼ = \ π ( N ). Also it followsthat φ M and φ N are restrictions of the composition π ( W ) −→ \ π ( W ) ∼ = b Γ −→ b Γ / P b Γ . Since W is also a ( Z /p )-homology cobordism and b Γ / P b Γ is in D ( Z /p ) by Lemma 3.6,the ( Z /p )-version of Theorem 3.7 implies b (2) i ( M, φ M ) = b (2) i ( N, φ N ) and ρ (2) i ( M, φ M ) = ρ (2) i ( N, φ N ) . (cid:3)
4. Group localization and homology cobordism of 3-manifolds
In this section, R is always a subring of Q . Theorem 4.1.
Suppose M is a closed 3-manifold. If α : π ( M ) → G is a homomor-phism into a finitely presented group G which is R -homology 2-connected, then there is an R -homology cobordism W between M and a closed hyperbolic 3-manifold M satisfyingthe following: (1) α extends to a surjection π ( W ) → G . (2) The inclusion induces a surjection π ( M ) → π ( W ) .Consequently, the induced map π ( M ) → G is a surjection which is R -homology 2-connected. It is known that for any finitely presented group G , the R -homology localization b G isthe direct limit of a sequence of homomorphisms on finitely presented groups G = G −→ G −→ · · · −→ G n −→ · · · which are R -homology 2-connected [Cha08, Theorem 2.6]. From this we obtain thefollowing consequence of Theorem 4.1: Corollary 4.2.
Suppose M is a closed 3-manifold with G = π ( M ) . For any finitelygenerated subgroup H in the R -homology localization b G , there is a closed hyperbolic 3-manifold M satisfying the following: (1) M is R -homology cobordant to M . (2) There is a 2-connected homomorphism f : π ( M ) → G such that H ⊂ Im { π ( M ) −→ \ π ( M ) f −−→ ∼ = b G } . To prove Theorem 4.1, we will construct a homology cobordism of M by attaching 1- and2-handles according to a homological description encoded through certain equations overthe group G . For this purpose, we first recall a definition from [Cha08, Definition 4.1].We denote by d ( R ) the set of denominators of reduced fractional expressions of elementsin R . Definition 4.3. A system of R -nullhomologous equations over a group π in n variables x , . . . , x n is a collection S of n expressions x ei = w i ( x , . . . , x n ), i = 1 , . . . , n , where e ∈ d ( R ) and each w i = w i ( x , . . . , x n ) is an element in the free product of G and thefree group F generated by the x i which lies in the kernel of the projection G ∗ F → F → H ( F ) = F/ [ F, F ].Levine first defined his group closure in [Lev89a, Lev89b], to extend link invariantsin [Orr89]. Levine’s group closure gave an alternative description of the group localizationof Vogel, and was inspired, in part, from a similar notion of group closure defined byGuti´errez in [Gut79]. Shortly thereafter, Farjoun and Shelah rediscovered an analogousconstruction to describe Bousfield homology localization of groups [DFOS89]. Variations,for instance, with untwisted and twisted coefficient systems, appear in various sources,for instance, [Cha08] and [Hec09].For a system of nullhomologous equations S = { x ei = w i ( x , . . . , x n ) } over π , weassociate a group π S defined as follows: π S = h π, z , . . . , z n | z ei = w i ( z , . . . , z n ) , i = 1 , . . . , n i This group π S is endowed with a natural map π → π S and can be viewed as obtainedfrom π by adjoining a solution { z i } of the system S . We need the following fact: Lemma 4.4. (1)
For any π and any system of R -nullhomologous equations S over π , the map π → π S is R -homology 2-connected. (2) If α : π → G is a homomorphism which is R -homology 2-connected and G isfinitely generated, then there is a system of R -nullhomologous equations S over π and a surjection α S : π S → G making the following diagram commute: π / / α (cid:15) (cid:15) π Sα S ~ ~ ⑤⑤⑤⑤⑤⑤⑤ G For a proof of Lemma 4.4, see [Cha08, p. 245, proof of Theorem 5.2]. (See also [Bou77],Corollary 2.17, for a related result to part (2) above.)
We construct a homology cobordism from R -nullhomologous equations. Proof of Theorem 4.1.
Suppose M is a closed 3-manifold with π = π ( M ), and α : π → G is R -homology 2-connected. By Lemma 4.4, we obtain a system of R -nullhomologousequations S = { x ei = w i } i =1 ,...,n over π such that the given α extends to a surjection π S → G . We start with M × [0 , V be the cobordism from M to another3-manifold, say M , which is obtained by attaching n M × [0 , x i . Identify π ( V ) with the free product of π and the free group generatedby the x i . Since M has dimension 3, we can choose a collection of disjoint simple closedcurves γ i on M which represent the elements x − ei w i ∈ π ( V ). By attaching n γ i , we obtain a cobordism, say V , between M and a new 3-manifold, say N .We will verify the following properties of ( V, M , N ) constructed above:(0) V is an R -homology cobordism between M and N . IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 15 (1) The given α : π ( M ) → G extends to a surjection π ( V ) → G .(2) The inclusion induces a surjection π ( N ) → π ( V ). π ( M ) / / α % % ❏❏❏❏❏❏❏❏❏❏ π ( V ) (cid:15) (cid:15) (cid:15) (cid:15) π ( N ) o o o o G First note that H ∗ ( V, M ; R ) can be computed from the handlebody structure of V :the cellular chain complex vanishes in dimensions other than 1 and 2, and the image ofthe i th 2-handle under the boundary map ∂ is determined by the word x − ei w i . Sincethe equations are nullhomologous, it follows that ∂ is represented by an n × n diagonalmatrix with all diagonals − e . Since e is a unit in R , it follows that H ∗ ( V, M ; R ) = 0.By duality, H ∗ ( V, N ; R ) = 0. That is, V is a homology cobordism between M and N .From the construction, π ( V ) can be identified with π S . By our choice of the equations(i.e., by Lemma 4.4), α extends to a surjection π ( V ) = π S → G .Reversing the handle decomposition, V is obtained from N by attaching 2- and 3-handles, and therefore π ( N ) → π ( V ) is surjective. This completes the verification ofthe above properties (0), (1), (2).Note that our N is not necessarilly hyperbolic. To obtain a hyperbolic 3-manifoldsatisfying the same properties, we need the following result of Ruberman: Theorem 4.5 (Ruberman [Rub90, Theorem 2.6]) . For any closed 3-manifold N , there isa homology cobordism U between N and a closed hyperbolic 3-manifold M and a retraction r : U → N such that the composition M ֒ → U r −→ N is a degree one map. Applying Theorem 4.5 to our N and by attaching the resulting homology cobordism tothe above cobordism V , we obtain a new R -homology cobordism, say W , between M andthe resulting hyperbolic 3-manifold M . Since a degree one map induces a surjection onthe fundamental group [Hem76, Lemma 15.12], ( W, M , M ) satisfies the above properties(0), (1), (2).The last sentence of Theorem 4.1 follows from the above properties since the inducedmaps of π ( M ), π ( M ) into π ( W ) are all R -homology 2-connected, and therefore induceisomorphisms on the R -homology localization. (cid:3)
5. 3-manifolds with local hidden torsion
Throughout this section, we fix an odd prime p . Several objects constructed in thissection depend on p , but for simplicity, we omit p in our notation. Construction of a “seed” 3-manifold.
Let Y be the twisted torus bundle over S withfundamental group Γ defined in Section 3. (For a description of Y , see the beginning ofSection 3.) Recall that the Z -homology localization b Γ is given as the central extension0 −→ Z (2) / Z −→ b Γ −→ ( Z t (2) ) ⋊ Z −→ . Let M be a hyperbolic 3-manifold obtained by applying Corollary 4.2 to Y and thesubgroup H generated by p + Z ∈ Z (2) / Z ⊂ b Γ. Then our M is homology cobordant to Y , and for π = π ( M ), there is a 2-connected surjection π → Γ inducing b π ∼ = b Γ. Under thisisomorphism we identify b π with b Γ; then by our choice of M , p + Z ∈ b Γ = b π is the imageof some nontrivial element [ α ] ∈ π = π ( M ), where α is a simple closed curve in M .Since π is torsion-free, [ α ] has infinite order in π but its image in b π = b Γ has order p .That is, [ α ] ∈ π is a local hidden torsion.Since Z (2) / Z ⊂ b Γ = b π lies in b π ω by Lemma 3.3, the pre-image of Z (2) / Z in π lies in π ω by Stallings’ Theorem [Sta65] π/π ω ֒ → b Γ / b Γ ω . Furthermore, [ α ] is not contained in π ω +1 since the image of [ α ] in b Γ is nontrivial and b Γ ω +1 is trivial by Lemma 3.3. Summarizing,we have proven the following: Lemma 5.1.
The element [ α ] chosen above is local hidden torsion of π ( M ) and liesin π ( M ) ω − π ( M ) ω +1 . Remark 5.2.
It follows that π ( M ) has lower central series length > ω . Since M ishomology cobordant to Y and Γ = π ( Y ) has lower central series length ω , this provesTheorem 1.3 stated in the introduction. Twisting around local hidden torsion.
For a knot K in S , we define M ( α, K ) to be the3-manifold obtained by removing a tubular neighborhood of α from M and then fillingit in with the exterior of K in such a way that a meridian of K is identified with aparallel copy of α and a longitude of K is identified with a meridian curve of the tubularneighborhood. It is well known that the resulting M ( α, K ) is always homology equivalentto M . Since we have to use it again later in this paper, we state it as a lemma. Lemma 5.3 (Well-known) . For a knot K , there is a homology equivalence h K : M ( α, K ) → M which induces a surjection h K ∗ : π ( M ( α, K )) → π ( M ) . In fact, for the exterior E K of a knot K , there is a homology equivalence ( E K , ∂E K ) → ( S × D , S × S ) which induces a surjection on π ( − ), and our h K is obtained byglueing this with the identity map of the exterior of α ⊂ M (e.g., see [Cha10, Proof ofProposition 4.8]).Now, applying Theorem 4.5 to our M ( α, K ), we obtain a hyperbolic 3-manifold M K homology cobordant to M ( α, K ) and a homology equivalence g K : M K → M ( α, K ) in-ducing a surjection on π ( − ). Note that if K is unknotted, we may assume that M K isequal to M . Proposition 5.4.
For any knot K , the following hold: (1) There is a homology equivalence f K : M K → M which induces a surjection onthe fundamental group. (2) The homology localization of π ( M K ) is isomorphic to b Γ . (3) There are local hidden torsion elements of π ( M K ) lying in π ( M K ) ω .Proof. (1) Composing g K with h K given in Lemma 5.3, we obtain a desired homologyequivalence f K : M K → M .(2) Since f K ∗ : π ( M K ) → π ( M ) is 2-connected, \ π ( M K ) ∼ = \ π ( M ) = b Γ.(3) Since f K ∗ is surjective, there is an element, say [ β ], in π ( M K ) which is sent to[ α ] ∈ π ( M ). Since π ( M K ) is torsion-free and [ α ] is a local hidden torsion elementby Lemma 5.1, it follows that [ β ] is a local hidden torsion element in π ( M K ). Since f K induces an isomorphism π ( M K ) /π ( M K ) ω ∼ = π ( M ) /π ( M ) ω by Stallings’ Theo-rem [Sta65] and [ α ] lies in π ( M ) ω by Lemma 5.1, we have [ β ] ∈ π ( M K ) ω . (cid:3) IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 17
Now, Theorem 1.1 in the introduction is an immediate consequence of Proposition 5.4. L -signatures We continue to use the notations of the previous subsection. In particular p is a fixedodd prime and M and M K are the hyperbolic 3-manifolds given in Section 5.1.We consider the L -signature invariants discussed in Section 3.3: recall that for any3-manifold N with \ π ( N ) ∼ = b Γ, we define φ N to be the composition φ N : π ( N ) −→ \ π ( N ) −→ \ π ( N ) / P \ π ( N ) ∼ = b Γ / P b Γwhere P n G denotes the mixed-coefficient commutator series associated to P = ( Z , Z , Z ( p ) ).Using the key fact that the image of [ α ] in the homology localization has finite order p , we show the following formula for the L -signature of M K . We denote the Levine-Tristram signature function of a knot K by σ K ( w ) = sign(1 − w ) A + (1 − w ) A T for w ∈ S ⊂ C , where A is a Seifert matrix for K . Lemma 5.5.
For our M and M K , the following holds: ρ (2) ( M K , φ M K ) = ρ (2) ( M, φ M ) + 1 p p − X k =0 σ K ( e πk √− /p ) Proof.
By Theorem 3.8, ρ (2) ( N, φ N ) is a homology cobordism invariant. Therefore, inour case, we may replace ( M K , φ M K ) by ( M ( α, K ) , φ M ( α,K ) ) since M K is homologycobordant to M ( α, K ).By [COT04, Proposition 3.2] (see also [CHL09, Lemma 2.3]), for any φ : π ( M ) → G ,we have ρ (2) ( M K , φ ◦ h K ∗ ) = ρ (2) ( M, φ ) + ρ (2) ( N K , φ ′ )where N K is the zero-surgery manifold of K , φ ′ : π ( N K ) → G is the map that factorsthrough H ( N K ) = Z and sends any meridian of K to φ ([ α ]), and h K : M ( α, K ) → M is the homology equivalence given in Lemma 5.3.For our purpose, consider the case of φ = φ M . Then φ M ◦ h K ∗ is equal to φ M K (up toan automorphism of b Γ / P b Γ) since h K gives rise to an isomorphism on \ π ( − ). Therefore ρ (2) ( M K , φ M ◦ h K ∗ ) = ρ (2) ( M K , φ M K ).Also, since the image of [ α ] ∈ π ( M ) in \ π ( M ) = b Γ has order p , φ M ([ α ]) has order p in b Γ / P b Γ by Lemma 3.4. (See also Remark 3.5.) Therefore, by the L -induction propertyand [CO12, Lemma 8.7], we obtain ρ (2) ( N K , φ ′ ) = 1 p p − X k =0 σ K ( e πk √− /p ) . From these the conclusion follows. (cid:3)
Theorem 5.6.
There is an infinite sequence of knots K = unknot, K , K , . . . in S ,for which (1) R S σ K i ( w ) dw = 0 for any i , and (2) M K i and M K j are not homology cobordant whenever i = j . We remark that the conclusion on the signature function integral will be used in thenext subsection.
Proof.
By [Cha, Proposition 4.12], there exists a knot K satisfying( ∗ ) p − X k =0 σ K ( e kπ √− /p ) = 0 and Z S σ K ( w ) dw = 0 . Furthermore, we can choose an infinite sequence of knots K , K , . . . for which ( ∗ ) holdsand the values of the sum P p − k =0 σ K i ( e kπ √− /p ) are all nonzero and mutually distinct.For example, we may choose as K n the connected sum of n copies of a fixed knot K satisfying ( ∗ ).By Lemma 5.5, we have ρ (2) ( M K i , φ M Ki ) = ( ρ (2) ( M, φ M ) if i = 0 ,ρ (2) ( M, φ M ) + p P p − k =0 σ K ( e kπ √− /p ) otherwise . By our choice of the K i , ρ (2) ( M K i , φ M Ki ) = ρ (2) ( M K j , φ M Kj ) whenever i = j . ByTheorem 3.8, it follows that M K i and M K j are not homology cobordant whenever i = j . (cid:3) From Proposition 5.4 and Theorem 5.6, it follows that the hyperbolic 3-manifolds M K i in Theorem 5.6 satisfy the conclusions (1) and (2) of Theorem 1.2 in the introduction.The remaining conclusions of Theorem 1.2, namely the fact that all the M K i look thesame to the eyes of previously known homology cobordism invariants, will be shown inthe next subsection. p -groups, iterated p -covers, and PTFA groups In this subsection we consider known homology cobordism invariants obtained from p -groups, iterated p -covers, and PTFA groups. We start by recalling known invariantsassociated to characters and representations. • For a character φ : π ( M ) → Z d of a closed 3-manifold M , the multisignature(= Casson-Gordon invariant = Atiyah-Singer α -invariant) is defined in [Wal70,CG78, AS68]. We denote it by σ ( M, φ ). • More generally, for a finite dimensional unitary representation θ : π ( M ) → U ( k ),Atiyah-Patodi-Singer’s (reduced) η -invariant ˜ η ( M, θ ) is defined in [APS75]. • For a representation θ : π ( M ) → GL( C , k ) and a homomorphism ψ : π ( M ) → H into a free abelian group H , a twisted torsion invariant τ θ ⊗ ψ ( M ) is definedin [CF].These are known to give homology cobordism invariants when the representationsfactors through p -groups and characters are of prime power order. (See [GL83, Rub84,Lev94, CF].) The following homology cobordism invariant obtained from iterated coversis also closely related to p -groups: • For a tower of iterated abelian p -covers M = M ← M ← · · · ← M n IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 19 (here M n → M is not necessarily abelian nor regular) and a character φ : π ( M n ) → Z d of prime power order, a Hirzebruch-type Witt-class-valued invariant λ ( M n , φ ) ∈ L ( Q ( ζ d )) is defined in [Cha10].Following [Cha09], we call a pair ( { M i } , φ ) as above a p -structure of height n for M .We continue to use the notations of Section 5.1. The following theorem says that theseinvariants related to p -groups do not distinguish the homology cobordism classes of our3-manifolds M K . The key fact used in the proof is that our hidden torsion element α isinvisible in the nilpotent completion. Proposition 5.7.
For any knot K , the following hold: (1) For any prime power order character φ : π ( M ) → Z d , σ ( M, φ ) = σ ( M K , φ ◦ f K ∗ ) . (2) For any unitary representation θ of π ( M ) factoring through a p -group, ˜ η ( M, θ ) =˜ η ( M K , θ ◦ f K ∗ ) . (3) For any representation θ : π ( M ) → GL( C , k ) factoring through a p -group andfor any homomorphism ψ : π ( M ) → H with H free abelian, we have τ θ ⊗ ψ ( M ) = τ ( θ ◦ f K ∗ ) ⊗ ( ψ ◦ f K ∗ ) ( M K ) . (4) The map f K gives rise to a 1-1 correspondences between p -structures of M and M K via pullback (i.e., f K is a p -tower map in the sense of [Cha10] ). If ( { M i } , φ ) and ( { M ′ i } , φ ′ ) are correponding p -structures of height n for M and M K , respec-tively, then λ ( M n , φ ) = λ ( M ′ n , φ ′ ) .Proof. Recall that M K is homology cobordant to M ( α, K ) via a homology cobordism U which admits a retraction r : U → M ( α, K ), and f K is the composition M K ֒ → W r −→ M ( α, K ) h K −−→ M . Since the invariants considered in Proposition 5.7 are all homologycobordism invariants as shown in [GL83, Rub84, Lev94, Cha10, CF], we may replace( M K , f K ) by ( M ( α, K ) , h K ). Now, since our [ α ] lies in π ( M ) ω , the image of [ α ] underany map of π ( M ) into a nilpotent group is trivial. Therefore [ α ] lies in the kernel of thegiven representation/character in (1), (2), (3).For the tower { M i } of iterated covers in (4), first observe that if we view π ( M i +1 ) asa subgroup of π ( M i ), then π ( M i ) ω is contained in π ( M i +1 ) ω since any p -group G isnilpotent. It follows that α always lifts to a loop in M n for any choice of a basepoint in M n , and any lift of α lies in the kernel of the given character φ of π ( M n ).Now, Proposition 5.7 is an immediate consequence of the following known fact, whichessentially says that tying a knot along a curve in the kernel of the given representa-tion/character does not change the concerned invariants. (cid:3) Lemma 5.8.
Suppose M is a closed 3-manifold, α is a simple closed curve in M , and M ( α, K ) is the 3-manifold obtained by tying a knot K along α . Let h K : M ( α, K ) → M be the homology equivalence described in Lemma 5.3. (1) If φ : π ( M ) → Z d satisfies [ α ] ∈ Ker φ , then σ ( M, φ ) = σ ( M ( α, K ) , φ ◦ h K ∗ ) . (2) If θ : π ( M ) → U ( k ) satisfies [ α ] ∈ Ker θ , then ˜ η ( M, θ ) = ˜ η ( M ( α, K ) , θ ◦ h K ∗ ) . (3) If θ : π ( M ) → GL( C , k ) and ψ : π ( M ) → H with H free abelian satisfy [ α ] ∈ Ker θ ∩ Ker ψ , then τ θ ⊗ ψ ( M ) = τ ( θ ◦ h K ∗ ) ⊗ ( ψ ◦ h K ∗ ) ( M ( α, K )) . (4) The map h K gives rise to a 1-1 correspondences between p -structures of M and M ( α, K ) via pullback. If ( { M i } , φ ) and ( { M ′ i } , φ ′ ) are correponding p -structuresof height n for M and M ( α, K ) , respectively, and any component of the pre-imageof α under M n → M is in Ker φ , then λ ( M n , φ ) = λ ( M ′ n , φ ′ ) . Proof.
It is known that the multisignature obtained from Z d -valued characters are equiv-alent to η -invariants associated to representations factoring through Z d . From this itfollows that (1) is a consequence of (2).The proof for (2) is similar to the arguments in [Cha10, Section 5.4]: for the knot tyingoperation, it is known (e.g., see [Cha10, Lemma 5.8]) that ˜ η ( M K , θ ◦ f K ∗ ) = ˜ η ( M, θ ) +˜ η ( N K , θ ′ ) where N K is the zero-surgery manifold of the knot K , and θ ′ : π ( N K ) → U ( k )is the representation that factors through H ( N K ) = Z and sends a meridian of K to θ ( α ). In our case, θ ′ is the trivial representation, since [ α ] ∈ π ( M ) ω and π ( M ) factorsthrough a nilpotent group. Therefore ˜ η ( N K , θ ′ ) = 0.(3) is shown by an argument similar to the above proof for (2), using the knot ty-ing formula given in [CF, Lemma 7.1] in place of [Cha10, Lemma 5.8]. For (4), by[Cha10, Lemma 3.7], h K induces a 1-1 correspondence between p -structures. Comput-ing λ ( M ′ n , φ ′ ) using the knot tying formula given in [Cha10, Lemma 4.6], we obtain thedesired conclusion. (cid:3) Now we show that L -signature invariants associated to torsion-free coefficient systemsdo not distinguish the homology cobordism class of M from that of M K (which is equalto that of M ( α, K )). First we consider the special case of coefficient systems factoringthrough the homology localization. Proposition 5.9. (1) If φ : π ( M ) → G is a homomorphism factoring through \ π ( M ) and G is torsion-free, then ρ (2) ( M, φ ) = ρ (2) ( M ( α, K ) , φ ◦ h K ∗ ) . (2) For Harvey’s homology cobordism invariant ρ n ( − ) defined in [Har08] , we have ρ n ( M ) = ρ n ( M ( α, K )) = ρ n ( M K ) for any n and K .Proof. As in the proof of Lemma 5.5, we have ρ (2) ( M ( α, K ) , φ ◦ h K ∗ ) = ρ (2) ( M, φ ) + ρ (2) ( N K , φ ′ )where N K is the zero-surgery manifold of K and φ ′ : π ( N K ) → G is the map that factorsthrough H ( N K ) = Z and sends any meridian of K to φ ([ α ]). In our case, φ ([ α ]) is trivialbecause of the following facts: the image of [ α ] in \ π ( M ) is torsion, φ factors through \ π ( M ), and G is torsion-free. That is, φ ′ is a trivial map. It follows that ρ (2) ( N K , φ ′ ) = 0.This shows (1).In [Har08], ρ n ( M ) is defined to be ρ (2) ( M, φ n ) where φ n : π ( M ) → π ( M ) /π ( M ) ( n ) H is the quotient map. Here π ( M ) ( n ) H denotes the torsion-free derived series definedin [CH05]. Due to [Cha08], there is a canonical injection j : π ( M ) /π ( M ) ( n ) H −→ \ π ( M ) / \ π ( M ) ( n ) H . By the L -induction property for ρ (2) , ρ n ( M ) is equal to ρ (2) ( M, j ◦ φ n ). Since j ◦ φ n factors through \ π ( M ) and h K induces an isomorphism on \ π ( − ), (2) follows from (1). (cid:3) Proposition 5.10.
Suppose K is a knot satisfying R S σ K ( w ) dw = 0 . If G is a torsion-free group, then for any φ : π ( M ) → G , we have ρ (2) ( M, φ ) = ρ (2) ( M ( α, K ) , φ ◦ h K ∗ ) .Proof. As in the proof of Proposition 5.9, we have ρ (2) ( M ( α, K ) , φ ◦ h K ∗ ) = ρ (2) ( M, φ ) + ρ (2) ( N K , φ ′ ) IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 21
Since G is torsion-free, φ ′ is either trivial or onto the infinite cyclic subgroup generatedby φ ([ α ]). In the former case, ρ (2) ( N K , φ ′ ) = 0. In the latter case, ρ (2) ( N K , φ ′ ) = R S σ K ( w ) dw by [COT04, Proposition 5.1]. From this the conclusion follows. (cid:3) Now, Theorem 1.2 in the introduction follows from our results in this section: for thehyperbolic 3-manifolds M K i given in Theorem 5.6, it follows that Theorem 1.2 (1), (2),(3), and (4) hold, respectively, from Proposition 5.4, Proposition 5.10, Proposition 5.7,and Theorem 5.6. Appendix A. Local groups and local modules
For completeness, we prove a result about local groups and modules used in Section 3 asProposition 3.2.We excerpt contents in this appendix from a manuscript in preparation by the au-thors [CO], which gives a much more thorough treatment of the theory of localization ofspaces, groups and modules and its role in the study of manifolds and knots.Throughout, R denotes a commutative ring with unity. Definition A.1.
A module A over the group ring RG is called a Cohn local module ifthe following holds: for any diagram F F A / / α (cid:15) (cid:15) f with F , F finitely generated free RG -modules of the same rank and α a homomorphisminducing an isomorphism 1 R ⊗ α : R ⊗ RG F → R ⊗ RG F , there is a unique homomorphism g : F → A making the diagram commute, i.e., f = gα .For a discussion related to the above definition, see [CO12, Section 2 and Appendix A]. Theorem A.2.
Suppose Γ is an R -local group and A is a Cohn local module over R Γ .Given any extension as below, −→ A −→ ˜Γ −→ Γ −→ then ˜Γ is an R -local group. One proves the following result, used in the proof of Theorem A.1, by a standardpartial chain contraction argument originally due to Vogel. (See also [Smi78], where aspecial case is proven using a different, but related, method.)
Lemma A.3 (Vogel [Vog82]) . Suppose A is a Cohn local RG -module and C ∗ is a chaincomplex over RG with C i finitely generated free for i ≤ n . If H i ( R ⊗ RG C ∗ ) = 0 for i ≤ n , then H i ( A ⊗ RG C ∗ ) = 0 = H i (Hom( C ∗ , A )) for i ≤ n . Since the argument has been presented and used in several papers (e.g, see [Vog82, Sec-tion 9.2], [Lev94, Proof of Propositon 3.2 on p. 95], [COT03, Proof of Proposition 2.10]),we omit the proof of Lemma A.3.
Proof of Theorem A.2.
Suppose0 −→ A −→ ˜Γ p −−→ Γ −→ R -local group and A is a Cohn local module over R Γ. Suppose α : π → G is a group homomorphism in Ω R , that is, π and G finitely presented and α is R -homology 2-connected. Suppose f : π → ˜Γ is given. We will show there is a unique g : G → ˜Γ satisfying f = gα .Let f ′ = pf . Since Γ is local, there is a unique g ′ : G → Γ such that f ′ = αg ′ . Takingpullback of f ′ and g ′ along ˜Γ → Γ, we obtain ˜ π → ˜Γ and ˜ G → ˜Γ which, together withother obvious maps, give the following commutative diagram with exact rows:0 → A ˜ G G → → A ˜ π π → → A ˜Γ Γ → / / / / (cid:15) (cid:15) g ′ / / ❄❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧⑧ / / ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ (cid:15) (cid:15) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ α (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ f ′ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ f / / / / p By the universal property of pullback, f gives rise to a splitting s : π → ˜ π . Similarly,homomorphisms g : G → ˜Γ satisfying f = gα are in 1-1 correspondence with splittings G → ˜ G which is compatible with s .Let c ∈ H (Γ; A ) be the element which correponds to the extension ˜Γ of Γ. Then, theabove extensions ˜ π and ˜ G corresponds to the images of c under the induced maps on H ( − ; A ): H ( π ; A ) H ( G ; A ) H (Γ; A ) o o α ∗ O O f ′∗ : : ttttttttttt g ′∗ Since ˜ π → π splits, f ′∗ ( c ) = 0. By Lemma A.3, H i ( G, π ; A ) = 0 for i ≤ α is R -homology 2-connected. Therefore α ∗ on H ( − ; A ) is injective. It follows that g ′∗ ( c ) = 0,that is, ˜ G → G splits and so ˜ G is a semidirect product of A and G .Now the splittings G → ˜ G are classified by H ( G ; A ). Again by Lemma A.3, α givesrise to an isomorphism H ( G ; A ) → H ( π ; A ). Therefore, splittings of ˜ π → π and thoseof ˜ G → G are in 1-1 correspondence. It follows that there is a unique splitting G → ˜ G which is compatible with s , and the induced homomorphism g : G → ˜Γ is exactly thedesired one. This shows that ˜Γ is R -local. (cid:3) For the application of Theorem A.2 that we needed in Section 3, the following twoexamples of Cohn local modules are useful.
Lemma A.4. If A is an RG -module with trivial G -action, then A is a Cohn local module.Proof. From the assumption it follows that A ∼ = R ⊗ RG A as RG -modules. If f and α are as in Definition A.1, then the composition g : F −→ R ⊗ RG F R ⊗ α ) − −−−−−−→ R ⊗ RG F R ⊗ f −−−→ R ⊗ RG A ∼ = A IDDEN TORSION, 3-MANIFOLDS, AND HOMOLOGY COBORDISM 23 is the required unique homomorphism. (cid:3)
Recall that Z t (2) is the abelian group Z (2) with the Z -action given by negation. Lemma A.5.
Suppose R is a subring of Z (2) . Then Z t (2) is a local R [ Z ] -module.Proof. Suppose f and α are as in Definition A.1. Let A ( t ) be a square matrix over R [ t ± ]representing α , where t is a generator of the group Z . Then A (1) is invertible in R since α induces an isomorphism R ⊗ R [ Z ] F → R ⊗ R [ Z ] F . Therefore det A (1) ∈ R is a unit,that is, the numerator of det A (1) is odd. Since det A (1) − det A ( −
1) is of the form 2 · r for some r ∈ R , it follows that det A ( −
1) has odd numerator too, that is, det A ( −
1) isa unit in Z (2) . Since A ( −
1) represents ¯ α := 1 Z t (2) ⊗ α : Z t (2) ⊗ R [ Z ] F → Z t (2) ⊗ R [ Z ] F , itfollows that ¯ α is an isomorphism.It is easily seen that the homomorphism Z t (2) → Z t (2) ⊗ R [ Z ] Z t (2) defined by a → ⊗ a is an isomorphism, with inverse a ⊗ b → ab . Now, the composition g : F −→ Z t (2) ⊗ R [ Z ] F α − −−→ Z t (2) ⊗ R [ Z ] F Z t (2) ⊗ f −−−−−→ Z t (2) ⊗ R [ Z ] Z t (2) ∼ = Z t (2) is a unique homomorphism such that f = gα . (cid:3) Proof of Proposition 3.2.
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