Equivariant hyperbolization of 3 -manifolds via homology cobordisms
aa r X i v : . [ m a t h . G T ] A p r EQUIVARIANT HYPERBOLIZATION OF -MANIFOLDSVIA HOMOLOGY COBORDISMS DAVE AUCKLY , HEE JUNG KIM , , PAUL MELVIN , AND DANIEL RUBERMAN , Abstract.
The main result of this paper is that any 3-dimensional manifold with a finite groupaction is equivariantly, invertibly homology cobordant to a hyperbolic manifold; this result holdswith suitable twisted coefficients as well. The following two consequences motivated this work. First,there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide classof finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolichomology 3-sphere. The theorem has other applications, including establishing the existence of aninfinite number of hyperbolic homology spheres with a free Z p action that does not extend to anycontractible manifold. A non-equivariant version yields an infinite number of hyperbolic integerhomology spheres that bound integer homology balls but do not bound contractible manifolds. Inpassing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric. Introduction
The 3-dimensional smooth homology cobordism group Θ H is rather complicated, and not fullyunderstood despite many advances coming from 4-dimensional gauge theory; see e.g. [17, 20]. Itappears in the theory of higher-dimensional manifolds, and also features prominently in the studyof smooth 4-manifolds. The Rohlin invariant gives an epimorphism from this cobordism groupto Z , and for a while this was all that was known about it. With the advent of gauge theorytechniques [13] it was shown that Θ H is infinite [17] (e.g. it is an easy consequence of Donaldson’sdiagonalization theorem [14] that the Poincare homology sphere represents an element of infiniteorder), indeed infinitely generated [20, 18]. There have been many results since on the structure ofthis group, and on its applications, including Manolescu’s spectacular resolution of the triangulationconjecture [33].It is interesting to explore how homology cobordism interacts with geometric structures on3-manifolds. For example, there exist homology 3-spheres that are not homology cobordant toSeifert fibered homology spheres; see [48], although the question of whether Seifert fibered spaces generate the homology cobordism group is still unsolved. In contrast, Myers [39] proved that every3-manifold is homology cobordant to a hyperbolic manifold, and this result was later refined byRuberman [46] to show that such cobordisms can be taken to be invertible; the latter result hasbeen applied to construct exotic smooth structures on contractible 4-manifolds [1].In this paper it is shown that any 3-manifold with a finite group action is equivariantly invertiblyhomology cobordant, with twisted coefficients, to a hyperbolic manifold. Even if the group is trivial,this refines the earlier work of Myers and Ruberman since it applies simultaneously to all coveringspaces. As will be seen, this result has applications to 4-dimensional smooth topology, to the3-dimensional space form problem, and may also be of interest in spectral geometry (cf. [5]).Throughout we work implicitly in the category of smooth, compact, oriented manifolds; all groupactions will be assumed to be effective and to preserve the given orientations. To state our mainresult, recall that a homology cobordism is a cobordism whose inclusions from the ends induceisomorphisms on integral homology. For non-simply connected manifolds there is a stronger notionof homology cobordism with twisted coefficients in any module over the group ring of the fundamental All of the authors were supported by an AIM SQuaRE grant. Supported by NRF grant 2015R1D1A1A01059318and BK21 PLUS SNU Mathematical Sciences Division. Partially supported by NSF Grant 1506328. uckly, Kim, Melvin and Ruberman group. Also recall (see e.g. [49, 50, 46]) that a cobordism P from M to N is invertible if there is acobordism Q from N to M with P Y N Q – M ˆ I ; see Section 2 for details, and the Appendix fora proof that invertible homology cobordism is a partial order on 3-manifolds. Theorem A.
Any closed -manifold M with an action of a finite group G is equivariantly invertibly Z r π p M qs -homology cobordant to a hyperbolic -manifold N with a G -action by isometries. We were led to this theorem by a question in 4-dimensional smooth topology. Consider thefamily of all finite groups that can act effectively on the boundary of some compact contractible4-dimensional submanifold of R ; these include all finite subgroups of SO p q . In a recent paper [4]we constructed for any such group G a compact contractible 4-manifold C with a G -action onits boundary and an embedding of C in a closed 4-manifold X such that removing C from X and regluing by distinct elements of G produces distinct smooth 4-manifolds; related results wereobtained by Tange [51] for G finite cyclic and Gompf [21] for G infinite cyclic. We call such a gadgeta G -cork . In our construction B C is reducible, and it was natural to ask if there are G -corks withirreducible or even hyperbolic boundaries. We refer to the latter as hyperbolic G -corks . Tange [52]has recently shown that his cyclic corks have irreducible boundaries, and (by computer calculationswith HIKMOT [27]) that some are hyperbolic. As a consequence of Theorem A we will deduce: Corollary B.
There exist hyperbolic G -corks for any finite group G that acts on the boundary ofsome compact contractible -dimensional submanifold of R . The proof will be given in Section 6, along with the following applications to low dimensionaltopology. We start with a hyperbolic version of a non-extension result for group actions due toAnvari and Hambleton [2].
Corollary C.
For any Brieskorn homology sphere Σ p a, b, c q and prime p not dividing abc , there isa hyperbolic homology sphere N p a, b, c q with a free action of Z p such that N p a, b, c q and Σ p a, b, c q are Z p -equivariantly homology cobordant, and the action of Z p does not extend over any contractible -manifold that N p a, b, c q might bound. We apply Theorem A in a non-equivariant setting to show that the difference between boundingan acyclic and contractible 4-manifold occurs for hyperbolic homology spheres.
Corollary D.
There are an infinite number of hyperbolic integer homology spheres that boundinteger homology balls but do not bound any contractible manifold.
The class of groups G that can act on some homology 3-sphere – hyperbolic or not – include thefinite subgroups of SO p q , but also a infinite proper subclass of the generalized quaternion groupsof period 4 (as shown by Milgram [35] and Madsen [32]; see also [12, p.xi], [29]). It has been an openquestion since the early 1980s to determine exactly which groups lie in G , and to say somethingabout the geometric nature of the homology spheres on which they act. Theorem A sheds light onthis last question, especially for free actions. The constructive part of this corollary is related torecent work of Bartel and Page [5]. Corollary E.
Any finite group that acts on a homology -sphere also acts on a hyperbolic homology -sphere with equivalent fixed-point behavior. In particular, there exist infinitely many finite groupsthat are not subgroups of SO p q , and so by geometrization do not act freely on the -sphere, butthat do act freely on some hyperbolic homology -sphere. In outline, the proof of Theorem A is similar to the proofs of the analogous theorems in [39] and[46]. Start with a Heegaard splitting of M (of genus ě
2) with gluing map h . Then replace eachhandlebody, viewed as the exterior of a trivial tangle in the 3-ball, with the exterior of an invertiblynull-concordant hyperbolic tangle. To build the cobordism, glue the two concordances together bythe map h ˆ id. The top of the cobordism will be hyperbolic by Thurston’s gluing theorem. quivariant hyperbolization To make this construction G -equivariant requires some modifications of this argument, even inthe case of a free action. In any case it is natural to start with a Heegaard splitting of the orbitspace M { G , and then to replace the handlebodies with copies of the tangle exterior as in the outlineabove. One thus obtains an invertible cobordism from M { G to a hyperbolic 3-manifold. Now ifthe action is free, then the induced G cover is an invertible cobordism P from M to a hyperbolic3-manifold N that is equivariant with respect to the G -action. However, there is no reason that P should be a homology cobordism, or indeed that N should have the same homology as M . Theissue is that while the tangle exteriors are Z -homology equivalent to handlebodies, they are notnecessarily homology equivalent with arbitrary (in this case Z r G s ) coefficients. This is of coursefamiliar from knot theory; a covering space of a homology circle such as a knot complement neednot be a homology circle. The resolution of this issue is to further decompose each handlebody into0 and 1-handles. These handles will be replaced with ‘fake’ 0 and 1-handles that will be hyperbolictangle exteriors. These are no longer homology handles, but rather homology handlebodies, butnow one has control over their lifts.We will begin with some standard tangle exteriors, referred to as atoms , then glue these togetherby a bonding process to make the fake handles, and finally glue these fake handles together to makefake handlebodies and relative cobordisms. This localization will ensure that the replacement ishomology cobordant to a handlebody H (with coefficients in Z r π p H qs ) and again Thurston’sgluing theorem will be used to create a closed hyperbolic manifold. With some additional work,this argument extends to the case when G has some fixed points. In this setting the quotient M { G will be an orbifold, and we will essentially be working with an orbifold Heegaard splitting.In our proof we need tangles that are doubly slice and simple (a.k.a. hyperbolic ). Furthermorethe tangles should retain these properties as they are suitably glued together. These notions willbe made precise in the next section. An elementary four-component tangle in the 3-ball with theseproperties, denoted R , is displayed in Figure 1a. Its n -component generalization R n is the liftof the generating arc α in the 3-ball shown in Figure 1b to the n -fold branched cover along the diameter δ perpendicular to the page. We will refer to the R n as atomic tangles ; they were the keyplayers in the last author’s construction of invertible homology cobordisms in [46]. The followingtechnical proposition is extracted from Theorem 2.6 of [46].a) The atomic tangle R b) Generating arc α αδδ Figure 1.
Atomic tangles
Proposition 1.1.
The atomic tangles R n are doubly slice for all n , and simple for n ě . The proof is reviewed in the next section in the process of analyzing the more complicated tanglesthat arise in our constructions. uckly, Kim, Melvin and Ruberman Technical background
Tangles.
In this paper, a tangle refers to a union T of finitely many disjoint arcs properly embedded ina 3-manifold M ; closed loops are not allowed. Two tangles (necessarily with the same endpoints)are equivalent if they are isotopic rel boundary. A marking of T is a collection of disjoint arcs in B M joining the endpoints of each of its strands. Note that markings are not generally unique; twomarkings for R are shown in Figure 2, where the part of the tangle inside the ball is drawn intastefully muted tones. In subsequent pictures of tangles T Ă M , the part inside M will be drawnschematically or omitted entirely, but a marking may be drawn to indicate which strands are pairedup inside. Figure 2.
Two markings for the atomic tangle R A tangle T Ă M is trivial if it is boundary parallel, meaning the strands of T together withthe arcs in a suitable marking A of T bound disjoint disks in M meeting B M in the markings.More generally, a tangle T with n components is a boundary tangle if the strands of T togetherwith a marking bound n disjoint surfaces in M meeting B M in the markings. The union of thesesurfaces will be called a Seifert surface for T , with outer boundary A and inner boundary T ; itwill be a trivial Seifert surface if all components are disks. For example, the atomic tangles R n introduced above (and all other tangles we construct in this paper) are boundary tangles, with theobvious Seifert surfaces consisting of n genus one surfaces. In fact, these Seifert surfaces satisfy anadditional condition, captured in the following definition of an ‘elementary tangle’. Definition 2.1.
Let T Ă M be a boundary tangle that has a Seifert surface F with a geometricsymplectic basis (embedded curves α , β , . . . , α n , β n representing a basis for H p F q with | α i X β i | “ δ ij ) satisfying the two conditions a) the α curves bound disjoint disks in M that intersect F only in arcs transverse to the β curves, b) the β curves bound disjoint disks in M that intersect F only in arcs transverse to the α curves.Equivalently, the α curves should have zero linking number with their pushoffs in F , and shouldbound disjoint disks in M that lie in the complement of a collection of arcs γ i in F from α i X β i to the outer boundary of F , and similarly for the β curves. Then we say that the tangle T Ă M is elementary . For example, as noted above, the atomic tangles R n Ă B are elementary. Tangle Sums.
Tangles can be added together in a variety of ways. For the present purposes, the followingnotion of a ‘tangle sum’ will suffice.
Definition 2.2.
Given a pair of tangles T i Ă M i for i “ ,
1, choose gluing disks D i Ă B M i containing an equal number of tangle endpoints, all at interior points of the D i . Then glue M to M by a diffeomorphism h : D Ñ D that identifies these endpoints without creating any loops.The result is the tangle sum T Y h T , a tangle in the boundary connected sum M Y h M . Thecommon image of the D i under the gluing is a properly embedded disk D Ă M Y h M called the quivariant hyperbolization splitting surface for the sum. More generally we allow the D i , and thus D , to be unions of morethan one disk.To propagate hyperbolic structures on tangles to their tangle sums (see Proposition 2.12 below),we will use a restricted class of ‘simple’ tangle sums. Definition 2.3.
A tangle sum T “ T Y h T as defined above is simple if each gluing disk in D i contains at least two tangle endpoints, and each component of B M i ´ D i contains at least twotangle endpoints if it is a sphere and one if it is a disk. In other words, D i ´ T i contains no diskcomponents, and pB M i ´ D i q ´ T i contains no sphere or disk components.An example is shown in Figure 3, where the thickened arcs indicate both the pairing of tangleendpoints inside the 3-manifold and the new marking of the tangle sum. Note that in general, themarking is only well-defined up to Dehn twists about the boundary of the splitting disk. In thetangle sums performed in Section 4, this issue does not arise, as we have a preferred gluing. M M D D h M Y h M Figure 3.
A simple tangle sum T Y h T Ă M “ M Y h M Invertible cobordisms and doubly slice tangles.
Invertible cobordisms of manifolds and knots have been studied since the 1960s; see for example[19, 23, 49, 50]. Recall first that if if M and N are manifolds of the same dimension (with boundariesidentified by a diffeomorphism h if they are nonempty), then a cobordism from M to N is a manifold P with B P “ ´ M \ N (or in the bounded case, B P “ ´ M Y h N , in which case P can be viewedas a relative cobordism from M to N so that the vertical part of B P is diffeomorphic to B M ˆ I ,extending h at the top). This cobordism P is said to be invertible if there is a cobordism Q from N to M such that P Y N Q – M ˆ I . We then say that M ˆ I is split along N , and call Q an inverse of P ; this inverse need not be unique, nor invertible (see Remark 2.4 below). Familiar examplesof 3-dimensional invertible cobordisms arise from homology 3-spheres M that bound contractible4-manifolds W whose doubles are the 4-sphere; the complement in W of an open 4-ball is then aninvertible cobordism from the 3-sphere to M .Similar language applies to concordances of knots, links, and tangles, where for tangles theconcordance is required to be a product along the boundary. In particular, a concordance from S to T is invertible if it can be followed by a concordance from T to S to produce a productconcordance from S to itself. If S is an unknot, unlink or trivial tangle, then T is said to be invertibly null-concordant or doubly slice . Remark 2.4.
The relations of invertible cobordism and concordance are clearly reflexive andtransitive, but generally not symmetric. For example, any sphere is invertibly cobordant to adisjoint union of two spheres, but not conversely, and analogously an unknot is invertibly concordantto a two component unlink, but not conversely. In fact, for closed manifolds of dimension 3 orless, invertible cobordism is an antisymmetric relation, and thus a partial order. For hyperbolic3-manifolds this follows from degree and volume considerations (cf. [6, Theorem C.5.5]) and ageneral proof for 3-manifolds is given in the appendix (where it is also noted that antisymmetryfails in higher dimensions). uckly, Kim, Melvin and Ruberman The focus here is on tangles. The first part of the following proposition is straightforward fromthe definitions, while the second part is a relative version of a well known result of Terasaka andHosokawa [54]; compare [46, Proof of 2.6].
Proposition 2.5. a) Tangle sums of doubly slice tangles are doubly slice. b) Elementary tangles p and in particular all atomic tangles R n q are doubly slice.Proof of b) . Let T Ă M be an n -stranded elementary tangle, and F be a Seifert surface for T with outer boundary A and geometric symplectic basis t α i , β i u as in Definition 2.1. View F as n disjoint disks with bands attached along the geometric basis. Removing the β bands from F yieldsa surface F Ă M that can be capped off with (parallel copies of) the disks bounded by the α curves, provided by 2.1b, to form a trivial Seifert surface E Ă M for a trivial tangle U . Similarlyform F Ă M by removing the α bands from F , and then cap off with disks bounded by the β curves to produce another trivial Seifert surface E Ă M for a trivial tangle U . By construction, E and E have the same outer boundary A as F .Now build a 3-dimensional cobordism P Ă M ˆ r , { s from E Ă M ˆ F Ă M ˆ {
2, withouter lateral boundary A ˆ r , { s , as follows. Start with P “ F ˆ r , { s with 2-handles attachedambiently in F ˆ r´ { , s along the α i Ă F ˆ
0. To arrange for P to lie in M ˆ r , { s , push it upfrom its bottom level E . After this adjustment, a top down movie of the inner lateral boundary P of P (the closure of the complement in B P of the union of A ˆ r , { s , E and T ) is described asfollows: Start with T . Then perform saddle moves along the cocores of the α bands, tracing outa genus zero cobordism from T to B F . Finish the movie by capping off the α i with disjoint disks.Note that P is a concordance from U to T .Similarly build a cobordism Q Ă M ˆ r { , s from F Ă M ˆ { E Ă M ˆ A ˆ r { , s and inner lateral boundary Q , a concordance in M ˆ r { , s from T to U . Then P Y Q is a product cobordism. Indeed, since | α i X β j | “ δ ij , the 1-handles (upsidedown 2-handles) in P are cancelled by the 2-handles in Q , and so P Y Q is in fact a union of 3-balls.It follows that P is the desired null-concordance of T , with inverse Q . (cid:3) Remark 2.6.
Note that each tangle component has a preferred meridian, defined as the boundaryof a normal disk. Each component J of a marked boundary tangle also has a longitude, determinedby a Seifert surface and consisting of two arcs u Y v with u lying along the marking and v runningalong J . Now suppose that two marked tangles T and T are concordant in M ˆ I , via a tangleconcordance that is homologous (rel boundary) to the outer lateral boundary A ˆ I . Fix a component J of T and its corresponding component J in T . Note that the longitude u Y v for J is freelyhomotopic to the longitude u Y v for J along a cylinder part of which is parallel to the concordanceand the other part of which travels along the boundary. In particular, a longitude on one end ofthe concordance determines a preferred longitude on the other end.A key consequence of this remark, to be used in the proof of Theorem A in Section 5 is thatfor concordant tangles T and T in M , as above, there is a canonical identification between theboundary of M ´ int N p T q and that of M ´ int N p T q . Homology cobordisms.
Recall that a homology cobordism is a cobordism for which the inclusions from the ends inducehomology isomorphisms. It is a standard and very useful observation that a concordance betweenknots or links induces a homology cobordism between their complements; see for instance [22]. Wenote a somewhat stronger property for the concordances constructed in the preceding subsection.Let X be the exterior of a tangle T in a 3-manifold M . The inclusion of X ã Ñ M inducesa homomorphism π p X q Ñ π p M q . Thus any module V over Z r π p M qs is also a module over Z r π p X qs , so we can consider the twisted homology H ˚ p X ; V q . quivariant hyperbolization v u B M ˆ Iv u J J Figure 4.
Longitude for J coming from a longitude for J . Lemma 2.7.
Let T and T be tangles in a compact -manifold M , with exteriors X and X , and C be an invertible concordance in M ˆ I from T to T , with exterior X . Then X is an invertiblehomology cobordism from X to X , with twisted coefficients in any module V over Z r π p M qs .In particular, any cobordism from a handlebody H to the exterior of a doubly slice tangle in the -ball induced by an invertible null-concordance of the tangle is an invertible homology cobordismwith twisted coefficients in any Z r π p H qs -module.Proof. The idea is implicit in [9], but here is a quick proof for the reader’s convenience. Byhypothesis, X “ M ˆ I ´ int p N q and X i “ X X p M ˆ i q for some tubular neighborhood N of C . Set N i “ N X p M ˆ i q , and note that the restriction of the coefficient system V to B N (andsimilarly for the B N i ) is trivial, because it extends over N . Then for i “ ¨ ¨ ¨ ÝÑ H ˚ p X X N, X i X N i q ÝÑ H ˚ p X, X i q ‘ H ˚ p N, N i q ÝÑ H ˚ p M ˆ I, M i q ÝÑ ¨ ¨ ¨ , with V -coefficients understood throughout, in which all the groups except H ˚ p X, X i q are obviouslyzero. Thus H ˚ p X, X i q “ (cid:3) Thurston’s hyperbolization and simple tangles.
To show that the 3-manifolds we produce are hyperbolic, we will use Thurston’s hyperbolizationtheorem for Haken 3-manifolds [55, 28, 40, 41] and standard techniques for checking that 3-manifoldsobtained by gluing satisfy the hypotheses of his theorem.To state Thurston’s theorem and the relevant gluing results in a unified way, we will use the phrase essential surface in a -manifold M to mean a compact, connected, incompressible, nonboundary-parallel, properly embedded surface in M (recall that all manifolds are assumed oriented). With thislanguage, a 3-manifold that is compact, irreducible and boundary-irreducible (see Waldhausen [58])is Haken if it contains an essential surface, and simple if it contains no essential tori, annuli or disks;we also refer to a tangle as simple or Haken if its exterior has that property. Thurston’s theorem for closed
Theorem 2.8. (Thurston)
Any closed simple Haken manifold admits a complete hyperbolic metric.
The hyperbolization result of [46] relied on proving that the atomic tangles R n described in § † We need a way to see that certain manifolds built from these atomsare also simple. The necessary gluing results can be found in Myer’s work [38, 39]. † Here is a sketch of the argument from [46] (it would also be nice to have a direct proof using Lemma 2.10 below):Recall from the end of § R n is a branched cover of a solid torus V “ B ´ δ branched along a knotted arc α .Let X “ V ´ int p N q , where N is a tubular neighborhood of α , and set P “ B X X N . Gluing results from [36] can be uckly, Kim, Melvin and Ruberman Definition 2.9.
Let M be a compact, irreducible 3-manifold and F be a compact surface in B M .The pair p M, F q is simple if it satisfies the following three properties: a) F and B M ´ F are incompressible in M . b) F contains no torus, annulus or disk component. c) M contains no essential tori, annuli disjoint from B F , or disks intersecting F in a single arc.The pair is very simple if it satisfies a), b) and d), where d) extends c) by also disallowing essentialdisks in M that intersect F in two disjoint arcs. Note that M is simple if and only if p M, Hq issimple, or equivalently p M, Hq is very simple.These notions of simple and very simple are exactly Myers’ Properties B and C , and feature inthe following result [39, Lemma 2.5], proved in [38]: Lemma 2.10. If p M , F q is very simple, p M , F q is simple, and h : F Ñ F is a homeomorphism,then M Y h M is simple and Haken. To identify new gluing regions in the boundary of a simple manifold, we will use the following:
Lemma 2.11. If M is simple and F is a compact surface in B M , then p M, F q is very simple ifand only if F has no torus, annulus or disk components, and B M ´ F has no disk components.Proof. For the forward implication, note that a disk component in B M ´ F gives a compressionof F in B M , and thus in M , contradicting property a) in Definition 2.9 of ‘very simple’. For theconverse we first verify 2.9a, so let D be a properly embedded disk in M with boundary in F orin B M ´ F . Since M is simple, D is inessential in M , so B D bounds a disk E in B M . But then E must lie entirely in F or B M ´ F , since neither has disk components. Thus F and B M ´ F areincompressible in M . It remains to verify 2.9b, which is immediate from the hypotheses, and 2.9d,which follows from the fact that M is simple, precluding the existence of any essential tori, annulior disks whatsoever. (cid:3) We illustrate the use of these gluing techniques in two situations, the first when the gluingsurfaces have boundary, and the second when they are closed. For the bounded case, consider a simple tangle sum T “ T Y h T of a pair of simple tangles T i Ă M i , as defined above, with gluingdisks D i Ă B M i . Write X i for the exterior of T i in M i , and set Y i “ D i X X i . By Definition 2.3,this means that Y i has no annulus or disk components, and B X i ´ Y i has no disk components. Thusboth pairs p X i , Y i q are very good by Lemma 2.11, and so the exterior X Y h X of in M Y h M issimple (and Haken) by Lemma 2.10. This proves the first part of the following result; the secondpart is an immediate consequence of Lemmas 2.10 and 2.11 and Thurston’s theorem. Proposition 2.12. a)
Any simple sum of simple tangles is a simple tangle. b) Any closed 3-manifold obtained by gluing together a pair of simple 3-manifolds is hyperbolic. (cid:3)
Our proof of Theorem A will rely on this proposition in the following way: Starting with aHeegaard splitting H Y h H of a 3-manifold M , we will apply Propositions 2.12a and 2.5a repeatedlyto construct simple, doubly slice ‘molecular’ tangles in H and H , with an equal number of strands.Then gluing their exteriors H and H together by a natural map h hyp induced by h will yield ahyperbolic manifold N , by Proposition 2.12b. If the Heegaard splitting of M is equivariant withrespect to the action of a finite group G on M (in a strong sense explained in the next section),and the simple tangles in the H i are suitably chosen, then M and N will be invertibly homologycobordant. The details of this construction will be explained in the next three sections. used to show that X is hyperbolic in a certain sense, in particular the pair p X, P q is a ‘pared manifold’. The resultthen follows by standard arguments about incompressible surfaces in branched covers, using equivariant versions ofthe loop, sphere, annulus and torus theorems (see Theorem 2.10 in [46]). quivariant hyperbolization To complete the proof of Theorem A we will need to show that the orbifold M { G is hyperbolic.There are two routes to this: one could either expand the discussion above to include a definitionof simple orbifold pairs, and argue that the gluing results hold in this more general setting, or onecould make use of Thurston’s orbifold theorem [8, 11]. We follow the latter route. In fact, we needonly the following special case (see for example [6, Theorem C.5.6]). Theorem 2.13.
Any action of a finite group G on a closed hyperbolic -manifold M is conjugateto an action by isometries, and so M { G is a hyperbolic orbifold. Equivariant Heegaard splittings
Given a closed 3-manifold M with an action of a finite group G , we seek to replace M with ahyperbolic manifold with a G -action. We assume without loss of generality that M is connected.The strategy is to find a G -equivariant Heegaard splitting H Y H of M (the goal of this section),and then to replace each handlebody H i with a fake handlebody H i with a G -action, chosen sothat the glued up manifold H Y H is hyperbolic (treated in the next section). This replacementprocess will require a further decomposition of the H i into 0 and 1-handles that will be regardedas part of the structure of the Heegaard splitting. Our exposition will be facilitated by passingback and forth between M and its quotient M { G , and so for clarity and notational economy wehenceforth denote the image of any subset K of M under the quotient map M Ñ M { G by K . Inparticular M “ M { G .If G acts freely, then we could simply lift a Heegaard splitting of the quotient manifold M withan arbitrary handle structure on the two sides. When G has fixed points, the quotient M is anorbifold, albeit a good one and so still a 3-manifold. In this case we will need the Heegaard splittingof M and the associated handle structures of the sides to be adapted to the orbifold structure, cf.[34, 59] for a related discussion of orbifold handlebodies . This splitting is constructed as follows.Observe that the G -action on M is locally linear (since it is smooth) and orientation preserving(by hypothesis). Thus the stabilizer G x of any point x P M is isomorphic to a finite subgroup ofSO p q , so is either cyclic, dihedral, or one of the three symmetry groups of the Platonic solids,acting linearly on a 3-ball about x . It follows that the singular set ∆ of all points in M withnontrivial stabilizers forms a graph in M , which may include edges with endpoints identified andcircle components with no vertices. The vertices of ∆ are the points with noncyclic stabilizers, andeach (open) edge is made up of points with the same nontrivial cyclic stabilizer. To record this factmore precisely, we assign labels to these vertices and edges. Since the noncyclic finite subgroups ofSO p q are all triangle groups (the dihedral group D n is ∆ p , , n q , while the tetrahedral, octahedraland icosahedral groups are respectively ∆ p , , q – A , ∆ p , , q – S and ∆ p , , q – A ), assignthe integer triple p p, q, r q to each vertex x of ∆ with G x – ∆ p p, q, r q , and assign the integer n toany edge whose stabilizer is isomorphic to C n .Now consider the image ∆ of ∆ in the quotient orbifold M . This is also a graph, with labelsinherited from ∆. Since it is locally the singular set of a finite linear quotient of the 3-ball, ∆ is infact a trivalent graph, with each vertex labeled by the triple of labels on the edges incident to it.We call ∆ the branch locus as M is the branched cover of M along ∆, with branching indices givenby the labels. The quotient map ∆ Ñ ∆ is illustrated in Figure 5 near a tetrahedral vertex in ∆.To build the Heegaard splitting of M , first extend the branch locus ∆ to a larger trivalent graph∆ Ă M whose complement is an open handlebody. To accomplish this, add new 1-labeled edgesto ∆ corresponding to all the 1-handles of a relative handlebody structure of the complement of aregular neighborhood of ∆, with endpoints chosen to lie at interior points of the edges in ∆. Ofcourse some edges of ∆ may be subdivided in this process. If e is such an edge, with label n , thenlabel each new edge of ∆ lying in e with n , and each new vertex lying on e with p , n, n q . Notethat M is still a branched cover of M along ∆ , so we call ∆ the extended branch locus . uckly, Kim, Melvin and Ruberman ∆ Ă M ∆ Ă M
32 3 p , , q Figure 5.
The picture of ∆ Ñ ∆ near a point with tetrahedral stabilizerNow let H be a regular neighborhood of the extended branch locus ∆ , built as a handlebodywith 0 and 1-handles corresponding in the usual way to the vertices and edges of ∆ . The closure H of the complement of H in M is another handlebody of the same genus, which we can decomposeinto 0 and 1-handles using an arbitrarily chosen trivalent ‘trivially’ labeled spine (label all the edgeswith 1 and all the vertices with p , , q ). This gives an orbifold Heegaard splitting M “ H Y H with ∆ Ă H , in which each of the handlebodies is equipped with a specific handle structurereflecting the orbifold structure on M . The lifts of the H i will then be equivariant handlebodies H i , equipped with their lifted handle structures, giving the desired equivariant Heegaard splitting M “ H Y H . In the next section, the orbifold Heegaard splitting will be used as a template to build a stabilized,equivariant ‘fake’ Heegaard splitting H Y H of the desired hyperbolic 3-manifold.4. Replacement handlebodies
In this section we describe how to insert equivariant doubly slice hyperbolic tangles T and T into the handlebodies in the equivariant Heegaard splitting M “ H Y H constructed in §
3. These‘molecular’ tangles will be built up using tangle sums from ‘atomic’ tangles placed in the 0-handlesand 1-handles in the decompositions of H and H described in the last section. We view theexteriors of the tangles in each of H and H as a replacement for those handlebodies (as in [46]).Each exterior is a simple and therefore hyperbolic homology handlebody that comes equipped withan equivariant invertible cobordism from a genuine handlebody. In § M and then lift allof the constructions back up to M . To this end we first describe how to place atomic tangles inthe orbifold 1-handles, then explain the somewhat more complicated tangle sums that are insertedin the orbifold 0-handles, next show how to assemble the orbifold tangles into a single tangle T i ineach orbifold handlebody H i , and finally lift these tangles up to the handlebodies in M . Orbifold -handles. From the discussion above, an orbifold 1-handle of degree n is a pair p D ˆ I, D ˆ B I q wherethe orbifold singularity is the n -labeled arc t u ˆ I . For any such 1-handle, insert a copy of thetangle R (defined in the introduction) so that the singular set corresponds to the diameter δ inFigure 1a, and so that the endpoints of two of the tangle components lie on the disk D ˆ t u ofthe attaching region while the endpoints of the other two components lie on D ˆ t u . Thus anylift of this 1-handle in H or H has a copy of R n inserted with half its endpoints on each disk ofthe attaching region of the handle. quivariant hyperbolization Orbifold -handles. An orbifold 0-handle is a 3-ball neighborhood of a singularity corresponding to a triangle group,as described above. The intersection of the extended branch locus (see §
3) with the sphere onthe boundary is 3 points, which form the vertices of a triangle K on the sphere. The edges of K are simply 3 arcs joining these points, as drawn below in Figure 6. We remark that a similarconstruction can be done with an arbitrary graph K on the boundary of a 3-manifold, and willmake use of this generalization when we glue handles below. K Figure 6.
The triangle K lying on the surface to an orbifold 0-handle.Now insert a 12-stranded tangle R K in a boundary collar of the 0-handle, as follows. Place acopy of R near each vertex of K . The ball in which R lies is drawn as a prism over a bigon, asshown on the left side Figure 7, with its top face on the surface of the 0-handle, and its bottomface on the inner boundary of the collar. The tangle is placed in this prism so that the diameter δ coincides with the singular set, and so that its strands run from the top bigon to the vertical sidesinside the 0-handle, two to each side. Similarly, each edge is surrounded by a long thin rectangularprism, with a copy of R positioned so that its strands run from the top rectangle to the shortsides, two ending on each side to match up with the ends of the vertex tangles. This is drawn onthe right side of Figure 7, where the markings on the surface designate as usual how the strandspair up inside the prism. Figure 7. R tangles in a neighborhood of a vertex and edge.The resulting tangle R K , shown schematically in Figure 8, has twelve strands, two joining eachvertex prism to each of its adjacent edge prisms. Figure 8.
Assembling tangles along the triangle K . Lemma 4.1.
The tangle R K created in this fashion is doubly slice and simple. uckly, Kim, Melvin and Ruberman Proof.
This follows from Proposition 2.5 a) and Proposition 2.12a. In particular, to show that R K is simple, first order the six simplices in K (three vertices and three open edges) so that the union K i of the first i of them is connected for each i “ , . . . ,
6. Then for i ą
1, the tangle R K i , definedin the obvious way, is obtained from R K i ´ by a simple tangle sum. The result follows by applyingProposition 2.12a repeatedly. (cid:3) Assembling the orbifold tangles T i Ă H i . The tangles T i Ă H i are now formed by gluing together the tangles in their 0 and 1-handles.Every orbifold 0-handle has 3 attaching bigon regions. Each 1-handle will then be a product of abigon with an interval, containing an R tangle as above. When we attach it to the 0-handles, weare performing a simple tangle sum. The final result is thus a pair of doubly slice simple tangles T i in H i for i “ ,
1, by Propositions 2.5 and 2.12. The exterior of T i in H i , denoted H i , is thehomology handlebody that replaces H i . Remark 4.2.
Each 0-handle in H i contributes 6 components to T i , since the 12 components of R K are glued up in pairs in the 1-handles. If the H i have genus g , then there are 2 g ´ H i . Thus T i has12 g ´
12 components.
Lifting the orbifold tangles to T i Ă H i . When the orbifold tangles T i Ă H i are lifted to equivariant tangles T i Ă H i , the picture isexquisitely embellished, as in the creation of folded paper sculptures. Fortunately, the proof thatthese lifted tangles are doubly slice and simple is essentially the same as in the orbifold case. Themodel for the 1-handles above a degree n orbifold 1-handle is now a prism over a 2 n -gon, with a copyof the atomic tangle R n inserted so that half the strands end on the top and half on the bottom ofthe prism. For the 0-handles, the triangle K lifts to a 1-complex K (namely the 1-skeleton of thefirst barycentric subdivision of the corresponding dihedron, tetrahedron, octahedron or icosahedron)and the proof that the tangle R K is simple proceeds exactly as in the proof of Lemma 4.1. Thesetangles now assemble into a pair of doubly slice simple tangles T i Ă H i for i “ ,
1, whose exteriors H i are the homology handlbodies that replace H i . Remark 4.3.
We could have used many other tangles in place of the atomic tangle R as the basisof our construction. The only properties that were needed for a tangle T in the 3-ball to give riseto doubly slice, simple, equivariant tangles in the handlebodies of M , are that it should be doublyslice and simple, and that all its cyclic branched covers (along a suitable diameter of the 3-ball)should also be doubly slice and simple.5. Gluing replacement handlebodies and the proof of Theorem A
Proof of Theorem A.
Starting from an action of G on M , we constructed in the last section anequivariant handlebody decomposition M “ H Y H by lifting an orbifold handle decompositionof the quotient M “ H Y H . Then we removed neighborhoods of doubly slice simple tangles T i Ă H i and their lifts T i Ă H i to obtain the replacement homology handlebodies H i Ă H i coveredby H i Ă H i . The remaining step is to describe how to glue these replacement handlebodies togetherto create the equivariant homology cobordism that proves the theorem.We begin by working in the quotient orbifold. Recalling that T i is doubly slice, choose a boundaryparallel tangle U i in H i that is invertibly concordant to T i . Removing a neighborhood of U i from H i has the effect of stabilizing H i , i.e. adding 1-handles to H i . For our purposes, and in particular toproperly specify how to glue H and H together to form a hyperbolic manifold homology cobordantto M “ M { G , we need to make this more precise. quivariant hyperbolization From this data and an ordering of the n “ g ´
12 (see Remark 4.2) components of U i and T i consistent with their identification by the concordance C i , Remark 2.6 gives rise to preferreddecompositions B H i “ B H i n p S ˆ S q where the k th torus summand is chosen so that the first S factor is identified with the preferredlongitude of the k th component of T i , while the second S factor is the meridian of that component.From this, the boundary of the vertical part of the exterior of the tangle concordance in H i ˆ I acquires a preferred diffeomorphism with(1) pB H i ˆ I q I p S ˆ S ˆ I q where I denotes the connected sum along a vertical arc.Now we glue H to H via a diffeomorphism of their boundaries that identifies correspondingtori in such a way that their meridians and longitudes are interchanged, but that is otherwise theidentity. This yields a hyperbolic orbifold N . Because of our choice of the meridian/longitude pair,a similar construction with T i replaced by U i simply stabilizes the orbifold Heegaard splitting of M , and does not change the resulting orbifold. Gluing the exterior of the concordance C in H ˆ I to the exterior of C in H ˆ I then gives an orbifold homology cobordism P from M to N . Byrepeating the construction with the inverses of the concordances C i to obtain the inverse orbifoldhomology cobordism Q and applying Lemma 2.7 we see that P is in fact an invertible cobordism.Finally we pass to the orbifold covers. The cobordism that has been constructed is automaticallyinvertible and equivariant, and so it remains to show that the orbifold cover N is hyperbolic, with G acting by isometries, and to check the homological properties of the invertible cobordisms P and Q . That N is hyperbolic follows from the fact that the tangles T i Ă H i are simple, as noted at theend of §
4, together with Proposition 2.12b. By Theorem 2.13, we can in fact assume that G acts byisometries. The decomposition (1) lifts to a similar decomposition of the boundary of the verticalpart of the exterior of the preimage of the tangle concordance in H i ˆ I . Because of the interchangeof meridian and longitude, this implies that both P and Q are Z r π p M qs homology cobordisms.The final statement about the fundamental group is a general property of maps induced on thefundamental group of invertible cobordisms. Its proof is postponed to Appendix A, where we makeuse of similar arguments about invertible cobordisms of 3-manifolds. (cid:3) Applications of hyperbolization
In this section, we supply proofs for the corollaries of Theorem A listed in the introduction.
Hyperbolic G -corks. As mentioned in the introduction, our original motivation was to show the existence of (effective)hyperbolic G -corks, and we start there. Proof of Corollary B.
The main result of our earlier paper [4] asserts that if G is a finite groupthat acts smoothly on the boundary of some compact contractible 4-dimensional submanifold of R , then there exists a 4-manifold X and a compact contractible submanifold C Ă X , with a G -action on its boundary, such that the 4-manifolds X C,g “ p X ´ int p C qq Y g C for g P G are allsmoothly distinct. We say that C (with its boundary G -action) is an effective G -cork in X . Nowby Theorem A, there is a G -equivariant invertible homology cobordism P from B C to a hyperbolichomology sphere N , with inverse the equivariant homology cobordism Q . Claim. C “ C Y B C P is an effective G -cork in X , with boundary B C “ N .To see this, note first that C “ p C Y B C P q Ă p C Y B C P Y N Q q – C, uckly, Kim, Melvin and Ruberman and this induces an embedding of C in X . Now the G -equivariance of P and Q implies that X C ,g – X C,g for every g P G . Since the manifolds X C,g are smoothly distinct as g runs over G , thesame is true for X C , g . (cid:3) Non-extendible group actions.
The hyperbolization results in [39, 46] have been used to show that results proved about homologycobordisms and knot concordance can apply to hyperbolic examples; the next application is anequivariant version of this principle. Building on work of Kwasik-Lawson [30], Anvari-Hambleton [2]have shown that for any Brieskorn sphere Σ p a, b, c q and prime p ∤ abc , the natural Z p action onΣ p a, b, c q does not extend over any contractible manifold that it bounds. (Note that while not allBrieskorn spheres bound contractible manifolds–see for instance [17, 20], there are infinite families[10, 47, 16] that do.) We now show that Theorem A gives non-extension results examples withhyperbolic boundaries. Corollary C.
For any Brieskorn homology sphere Σ p a, b, c q and prime p not dividing abc , there isa hyperbolic homology sphere N p a, b, c q with a free action of Z p such that N p a, b, c q and Σ p a, b, c q are Z p -equivariantly homology cobordant, and the action of Z p does not extend over any contractible -manifold that N p a, b, c q might bound. We remark that for many choices of p a, b, c q , the conclusion can be strengthened to say thatthe action of Z p does not extend over any acyclic N p a, b, c q might bound. Thisis shown via the method of Kwasik and Lawson taking into account that Donaldson’s definitemanifolds theorem applies to non-simply connected manifolds; see [14]. Kwasik and Lawson [30,Proposition 12] give a list of examples to which this method applies. Proof of Corollary C.
The condition that p does not divide abc implies that the action of Z p onΣ p a, b, c q is free. By Theorem A there is a Z p -equivariant invertible homology cobordism P fromΣ p a, b, c q to a hyperbolic manifold N p a, b, c q . By construction, the action of Z p on both cobordismsis free. If the action on N extends over a contractible manifold W , then the manifold W Y N P isa homology ball over which the Z p action on Σ p a, b, c q extends. By Proposition A.2, this homologyball is simply-connected, and hence contractible, contradicting [2]. (cid:3) Acyclic versus contractible.
An important consequence of Taubes’ periodic ends theorem [53], observed by Akbulut is thatthere are reducible homology spheres that bound homology balls, but do not bound contractiblemanifolds; the original example was Σ p , , q ´ Σ p , , q . We show that one can in fact choosethe homology sphere to be hyperbolic. Corollary D.
There are an infinite number of hyperbolic integer homology spheres that boundinteger homology balls but do not bound any contractible manifold.Proof.
Let Σ be any integer homology sphere that bounds a smooth 4-manifold X with non-standardnegative definite intersection form. (For example the Poincar´e homology sphere.) We first describehow to generate one example, and then describe the modifications required to detect infinitelymany distinct examples. By our main theorem, there is an invertible homology cobordism P fromΣ ´ Σ to a hyperbolic 3-manifold N . Furthermore, by Proposition A.2 the fundamental group of P is normally generated by the fundamental group of N . One sees that W “ I ˆ p Σ ´ int p B qq Y Σ ´ Σ P is an integer homology ball with boundary N .Now assume that N bounds a contractible manifold Z . Adding a 3-handle to Z Y N P alongthe sphere separating Σ and ´ Σ results in a simply-connected (since π p P q is normally generated quivariant hyperbolization by π p N q ) acyclic 4-manifold V with boundary Σ Y ´
Σ. This contradicts [53, Proposition 1.7], aconsequence of Taubes’ periodic ends theorem.To show that there are infinitely many distinct examples, we iterate this process. Add a copy of P to X to get a smooth 4-manifold Y with non-standard negative definite intersection form andboundary N . Let Y k denote the boundary connected sum of k copies of Y , with boundary equal to N k “ k N . Note that the intersection form of Y k is also non-standard; this is readily verified usingElkies’ criterion for diagonalizablity of a unimodular form [15]. Repeating the argument above with Y k in place of X , we obtain a series of hyperbolic manifolds M k that bound acyclic 4-manifoldsbut not contractible ones. By construction, there are homology cobordisms from N k to M k , andhence degree one maps from M k to N k . In particular the Gromov norm of M k is at least that of N k , which is in turn k times the (non-zero) norm of M . It follows that the Gromov norms of the M k are unbounded, so an infinite sequence of them are distinct. (cid:3) Finite groups acting on homology spheres.
Finally, we give an application of Theorem A to an aspect of the classical spherical space formproblem; see [12] and the discussion of problem 3.37 in [29].
Corollary E.
Any finite group that acts on a homology -sphere also acts on a hyperbolic homology -sphere with equivalent fixed-point behavior. In particular, there exist infinitely many finite groupsthat are not subgroups of SO p q , and so by geometrization do not act freely on the -sphere, butthat do act freely on some hyperbolic homology -sphere.Proof of Corollary E. The first part is a direct corollary of Theorem A, replacing an action of agroup G on a homology sphere by an action on a hyperbolic homology sphere. The second part,constructing free actions on homology spheres by groups that cannot act freely on the 3-sphere,requires results on the topological spherical space form problem dating to the 1970s and 1980s.The underlying principle is that there are homotopy-theoretic (finiteness) and surgery-theoreticobstructions, depending only on n modulo 8, for a finite group G to act freely on a sphere ofdimension n . If n is greater than 4, the vanishing of these obstructions is sufficient for the existenceof such an actions. In dimension 3, the vanishing of the obstructions implies only that G acts freelyon a homology 3 sphere; see for example [26, Remark 8.2].Work of Madsen [32], Milgram [35], and Bentzen [7] evaluated the finiteness and surgery obstruc-tions in number theoretic terms. Their results show that infinitely many generalized quaternionicgroups Q p p, q q (the smallest being Q p , q [7]) act freely on spheres in dimensions 8 k ` k ą S ; presumablyone could also verify that it cannot be Seifert-fibered. We deduce directly from Theorem A thatsuch a homology sphere can be taken to be hyperbolic. (cid:3) Corollary E is related to a recent paper of Bartel and Page [5] that constructs an action of afinite group G on a hyperbolic 3-manifold M such that the induced action on H p M ; Q q realizesany given finitely generated Q r G s module. This result and the main results of the current paper arerelated to some degree, as both construct actions of finite groups on 3-manifolds with prescribedhomological action. However, neither paper implies the results of the other; for instance [5] dealsonly with the action on rational homology, and does not provide a homology cobordism. On theother hand, our hyperbolization requires the existence of an action on some 3-manifold as a startingpoint. It would be of interest to establish a sharper result realizing a given Z r G s module (even onewith Z torsion) by an action on some 3-manifold; our hyperbolization procedure would then showthat this action is realized on a hyperbolic manifold. uckly, Kim, Melvin and Ruberman Appendix A. Antisymmetry of invertible homology cobordism of -manifolds We show that for closed oriented 3-manifolds, invertible homology cobordism is an antisymmetricrelation, and thus a partial order. This is false in higher odd dimensions, as seen from the existenceof h-cobordisms X with non-trivial Whitehead torsion, for which ´ X is the inverse cobordism;compare [45, Lemma 7.8]. Theorem A.1.
Let M and N be closed -manifolds. If there is an invertible homology cobordismfrom M to N , and one from N to M , then M and N are homeomorphic.Proof. Let P be the cobordism from M to N , and Q be the inverse cobordism from N to M , sothat P Y N Q “ M ˆ I . This gives a map f : N Ñ M , the composition of the inclusion N ã Ñ M ˆ I followed by the projection M ˆ I Ñ M . There is also another pair of cobordisms Q from N to M and P from M to N , so that Q Y M P “ N ˆ I , and this gives a map g : M Ñ N . Both of thesemaps have degree one, so their induced maps on π are surjective. Thus the composition g ˚ ˝ f ˚ : π p N q Ñ π p N q is surjective. But 3-manifold groups are Hopfian [3], which means that in fact this composition is anisomorphism. It follows that f ˚ is injective, so it is an isomorphism. We write π for π p M q – π p N q .Now computing the fundamental group of M ˆ I “ P Y N Q by van Kampen’s theorem yields apushout diagram:(2) π p Q q j Q $ $ ■■■■■■■■■ π p N q f ˚ / / i Q : : ✉✉✉✉✉✉✉✉✉ i P $ $ ■■■■■■■■■ π p M q – π p M ˆ I q π p P q j P : : ✉✉✉✉✉✉✉✉✉ Since f ˚ is an isomorphism, i P and i Q are injective, and j P and j Q are surjective. A standard resultabout pushouts says that in fact j P and j Q are injective, so all of these maps are isomorphisms.One can make a similar argument in homology, with arbitrary twisted coefficients, with theMayer-Vietoris sequence replacing the van Kampen pushout diagram. In particular, the inclusionsof M and N into P and Q induce isomorphisms on homology with coefficients in Z r π s , so byWhitehead’s theorem, those maps are homotopy equivalences. In particular, P is an h-cobordism.Now a theorem of Kwasik-Schultz [31] implies that the Whitehead torsions of p P, N q and p P, M q both vanish. (Their theorem was proved under the hypothesis that both M and N are geometric,which is now a consequence of the geometrization theorem.) In particular, M and N are simplehomotopy equivalent. By a theorem of Turaev (see [56, 57] as well as [31, Theorem 1.1]) M and N are homeomorphic. (cid:3) We remark that if M and N are hyperbolic manifolds, then there is an alternate (and perhapssimpler) route to this conclusion, based on the Gromov-Thurston proof of Mostow’s rigidity theo-rem [24, 25]. This proof implies directly that if there are degree one maps from M to N and from N to M , then M and N are homeomorphic.Finally, we establish the following general property of maps induced on the fundamental groupof invertible cobordisms that was used in Corollaries C and D. Proposition A.2.
Let P be an invertible cobordism from M to N , with inverse cobordism Q .Then the image of the map i P induced by the inclusion of N into P normally generates π p P q , andlikewise the image of i Q normally generates π p Q q . quivariant hyperbolization Proof.
We continue the notation from above. It suffices to show that the quotient groups G P “ π p P q{x im p i P qy and G Q “ π p Q q{x im p i Q qy are trivial, where x y denotes the normal closure. The natural maps k P : π p P q Ñ G P ˚ G Q and k Q : π p Q q Ñ G P ˚ G Q induce a unique map h : π p M q Ñ G P ˚ G Q such that h ˝ j P “ k P and h ˝ j Q “ k Q , which must be trivial since f ˚ is onto. This forces j P and j Q to be trivial, whichimplies that G P and G Q are trivial. (cid:3) References [1] S. Akbulut and D. Ruberman,
Absolutely exotic compact 4-manifolds , Comment. Math. Helv., (2016), 1–19.[2] N. Anvari and I. Hambleton, Cyclic group actions on contractible 4-manifolds , Geom. Topol., (2016), 1127–1155.[3] M. Aschenbrenner, S. Friedl, and H. Wilton, “3-manifold groups”, EMS Series of Lectures in Mathematics,European Mathematical Society (EMS), Z¨urich, 2015.[4] D. Auckly, H. J. Kim, P. Melvin, and D. Ruberman, Equivariant corks . Algebr. Geom. Topol. (2017), no. 3,1771–1783.[5] A. Bartel and A. Page, Group representations in the homology of 3-manifolds . https://arxiv.org/abs/1605.04866 , 2016.[6] R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry , Universitext, Springer-Verlag, Berlin Heidelberg,1992.[7] S. Bentzen,
Some numerical results on space forms , Proc. London Math. Soc. (3), (1987), 559–576.[8] M. Boileau, B. Leeb, and J. Porti, Geometrization of 3-dimensional orbifolds , Ann. of Math. (2), (2005),195–290.[9] S. Cappell and J. Shaneson,
The codimension two placement problem and homology equivalent manifolds , Annalsof Math., (1974), 277–348.[10] A. Casson and J. Harer, Some homology lens spaces which bound rational homology balls , Pacific J. Math., (1981), 23–36.[11] D. Cooper, C. D. Hodgson, and S. P. Kerckhoff, “Three-dimensional orbifolds and cone-manifolds”, vol. 5 ofMSJ Memoirs, Mathematical Society of Japan, Tokyo, 2000. With a postface by Sadayoshi Kojima.[12] J. F. Davis and R. J. Milgram, “A survey of the spherical space form problem”, vol. 2 of Mathematical Reports,Harwood Academic Publishers, Chur, 1985.[13] S. K. Donaldson, An application of gauge theory to four-dimensional topology , J. Differential Geom. (1983),no. 2, 279–315.[14] , The orientation of Yang–Mills moduli spaces and 4-manifold topology , J. Diff. Geo. (1987), 397–428.[15] N. D. Elkies, A characterization of the Z n lattice , Math. Res. Lett., (1995), 321–326.[16] H. C. Fickle, Knots, Z -homology -spheres and contractible -manifolds , Houston J. Math., (1984), 467–493.[17] R. Fintushel and R. J. Stern, Pseudofree orbifolds , Ann. of Math. (2), (1985),335–364.[18] ,
Instanton homology of Seifert fibred homology three spheres , Proc. London Math. Soc. (3), (1):109–137,1990.[19] R. H. Fox, A quick trip through knot theory , in “Topology of 3-manifolds and related topics (Proc. The Univ. ofGeorgia Institute, 1961)”, Prentice-Hall, Englewood Cliffs, N.J., 1962, 120–167.[20] M. Furuta,
Homology cobordism group of homology -spheres , Invent. Math., (1990), 339–355.[21] R. E. Gompf, Infinite order corks . Geom. Topol. (2017), no. 4, 2475–2484[22] C. M. Gordon, Knots, homology spheres, and contractible -manifolds , Topology, (1975), 151–172.[23] C. M. Gordon and D. W. Sumners, Knotted ball pairs whose product with an interval is unknotted , Math. Ann., (1975), 47–52.[24] M. Gromov,
Volume and bounded cohomology , Publ. I.H.E.S., (1982), 5–100.[25] U. Haagerup and H. J. Munkholm, Simplices of maximal volume in hyperbolic n -space , Acta Math., (1981),1–11.[26] I. Hambleton, Topological spherical space forms , in “Handbook of group actions. Vol. II”, vol. 32 of Adv. Lect.Math. (ALM), Int. Press, Somerville, MA, 2015, 151–172.[27] N. Hoffman, K. Ichihara, M. Kashiwagi, S., Masai, H., Oishi, and A. Takayasu,
Verified computations forhyperbolic 3-manifolds , Exp. Math. (2016), no. 1, 66–78.[28] M. Kapovich, “Hyperbolic manifolds and discrete groups”, Modern Birkh¨auser Classics, Birkh¨auser Boston, Inc.,Boston, MA, 2009.[29] R. Kirby, Problems in low–dimensional topology , in “Geometric Topology”, W. Kazez, ed., American Math.Soc./International Press, Providence, 1997. uckly, Kim, Melvin and Ruberman [30] S. Kwasik and T. Lawson, Nonsmoothable Z p actions on contractible -manifolds , J. Reine Angew. Math., (1993), 29–54.[31] S. Kwasik and R. Schultz, Vanishing of Whitehead torsion in dimension four , Topology, (1992), 735–756.[32] I. Madsen, Reidemeister torsion, surgery invariants and spherical space forms , Proc. Lond. Math. Soc., (1983), 193–240.[33] C. Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture , J. Amer.Math. Soc., (1):147–176, 2016.[34] D. McCullough, A. Miller, and B. Zimmermann, Group actions on handlebodies , Proc. London Math. Soc. (3), (1989), 373–416.[35] R. J. Milgram, Evaluating the Swan finiteness obstruction for periodic groups , in “Algebraic and geometrictopology (New Brunswick, N.J., 1983)”, vol. 1126 of Lecture Notes in Math., Springer, Berlin, 1985, 127–158.[36] J. W. Morgan,
On Thurston’s uniformization theorem for three-dimensional manifolds. , in “The Smith Conjec-ture”, H. Bass and H, Morgan, editors, Academic Press, 1984, 37–126.[37] J. Morgan and G. Tian, “The geometrization conjecture”, vol. 5 of Clay Mathematics Monographs, AmericanMathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2014.[38] R. Myers,
Simple knots in compact, orientable -manifolds , Trans. Amer. Math. Soc., (1):75–91, 1982.[39] , Homology cobordisms, link concordances, and hyperbolic -manifolds , Trans. Amer. Math. Soc., (1):271–288, 1983.[40] J.-P. Otal, Le th´eor`eme d’hyperbolisation pour les vari´et´es fibr´ees de dimension 3 , Ast´erisque, (1996), x+159.[41] ,
Thurston’s hyperbolization of Haken manifolds , in “Surveys in differential geometry, Vol. III (Cambridge,MA, 1996)”, Int. Press, Boston, MA, 1998, 77–194.[42] G. Perelman,
The entropy formula for the Ricci flow and its geometric applications . https://arxiv.org/abs/math/0211159v1 , 2002.[43] , Finite extinction time for the solutions to the Ricci flow on certain three-manifolds . https://arxiv.org/abs/math/0307245v1 , 2003.[44] , Ricci flow with surgery on three-manifolds . https://arxiv.org/abs/math/0303109v1 , 2003.[45] C. P. Rourke and B. J. Sanderson, “Introduction to piecewise-linear topology”, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69.[46] D. Ruberman, Seifert surfaces of knots in S , Pacific J. Math., (1990), 97–116.[47] R. J. Stern, Some more Brieskorn spheres which bound contractible manifolds , Notices Amer. Math. Soc., (1978), A448.[48] M. Stoffregen, Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert fibrations . https://arxiv.org/abs/1505.03234 , 2015.[49] D. Sumners, Invertible knot cobordisms , in “Topology of Manifolds”, J. Cantrell and C. Edwards, eds., Markham,1970, 200–204.[50] ,
Invertible knot cobordisms , Comm. Math. Helv., (1971), 240–256.[51] M. Tange, Finite order corks . Internat. J. Math. (2017), no. 6, 1750034, 26 pp.[52] , Boundary-sum irreducible finite order corks, https://arxiv.org/abs/1710.07034 [53] C. Taubes, Gauge theory on asymptotically periodic 4-manifolds , J. Diff. Geo., (1987), 363–430.[54] H. Terasaka and F. Hosokawa, On the unknotted sphere S in E , Osaka Math. J. (1961), 265–270.[55] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry , Bull. Amer. Math. Soc., (1982), 357–381.[56] V. G. Turaev, Homeomorphisms of geometric three-dimensional manifolds , Mat. Zametki, (1988), 533–542,575.[57] , Towards the topological classification of geometric -manifolds , in “Topology and geometry–Rohlin Sem-inar”, vol. 1346 of Lecture Notes in Math., Springer, Berlin, 1988, 291–323.[58] F. Waldhausen, On irreducible -manifolds which are sufficiently large , Ann. Math, (1), 1968, 56–88.[59] B. Zimmermann, Genus actions of finite groups on -manifolds , Michigan Math. J., (1996), 593–610. quivariant hyperbolization Department of MathematicsKansas State UniversityManhattan, Kansas 66506
E-mail address : [email protected] Department of Mathematical SciencesSeoul National UniversitySeoul 790-784, Korea
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