VIRTUAL DOMINATION OF 3-MANIFOLDS III HONGBIN SUN Abstract. We prove that for any oriented cusped hyperbolic 3-manifold M

2025-05-06 0 0 826.94KB 63 页 10玖币

侵权投诉

VIRTUAL DOMINATION OF 3-MANIFOLDS III

HONGBIN SUN

Abstract. We prove that for any oriented cusped hyperbolic 3-manifold M

and any compact oriented 3-manifold Nwith tori boundary, there exists a

ﬁnite cover M0of Mthat admits a degree-8 map f:M0→N, i.e. Mvirtually

8-dominates N.

1. Introduction

In this paper, we assume all manifolds are compact, connected and oriented,

unless otherwise indicated. By a cusped hyperbolic 3-manifold, we mean a compact

3-manifold with nonempty tori boundary, such that its interior admits a complete

hyperbolic structure with ﬁnite volume, unless otherwise indicated.

For two closed oriented n-manifolds M, N and a map f:M→N, a natu-

ral quantity associated to fis its mapping degree. The mapping degree of fis

d∈Zif f∗([M]) = d[N] for oriented fundamental classes [M]∈Hn(M;Z) and

[N]∈Hn(N;Z). The notion of mapping degree can be generalized to proper maps

between manifolds with boundary. For two compact oriented n-manifolds Mand

Nwith boundary, a map f:M→Nis proper if f−1(∂N) = ∂M holds. The

mapping degree of a proper map f:M→Nis d∈Zif f∗([M, ∂M]) = d[N, ∂N ]

for oriented relative fundamental classes [M, ∂M ]∈Hn(M, ∂M ;Z) and [N, ∂N ]∈

Hn(N, ∂N;Z). In either of the above cases, fis a non-zero degree map if the degree

of fis not zero. If the mapping degree d6= 0, we say that M d-dominates N, and

we say that Mdominates Nif M d-dominates Nfor some non-zero integer d. In

this paper, we will work on 3-manifolds with nonempty boundary, and all maps

f:M→Nbetween such 3-manifolds are proper, unless otherwise indicated.

Roughly speaking, if Mdominates (or 1-dominates) N, then Mis topologically

more complicated than N. For certain invariants of manifolds, e.g. ranks of funda-

mental groups, betti numbers, simplicial volumes, representation volumes, etc, this

impression on behavior of topological invariants under non-zero degree maps (or

degree-1 maps) are classical results. However, for some other invariants, e.g. Hee-

gaard genera and Heegaard Floer homology of 3-manifolds, it is unknown whether

the above impression is correct.

In [CT], the authors asked the following question: Whether there is an easily

described class Cof closed oriented n-manifolds, such that any closed oriented n-

manifold is dominated by some M∈C. In [Gai], Gaifullin proved that, for any

positive integer n, there exists a closed oriented n-manifold M0, such that any closed

oriented n-manifold is dominated by a ﬁnite cover of M0(virtually dominated by

Date: October 4, 2022.

2010 Mathematics Subject Classiﬁcation. 57M10, 57M50, 30F40.

Key words and phrases. hyperbolic 3-manifolds, non-zero degree maps, good pants construc-

tion, quasi-isometric embedding.

The author is partially supported by Simons Collaboration Grants 615229.

arXiv:2210.00402v1 [math.GT] 2 Oct 2022

2 HONGBIN SUN

M0), i.e. we can take Cto be the set of all ﬁnite covers of M0. In [Sun2,LS,Sun5],

the author and Liu proved the following result.

Theorem 1.1. [Sun2,LS,Sun5]For any closed oriented 3-manifold Mwith positive

simplicial volume and any closed oriented 3-manifold N, there exists a ﬁnite cover

M0of Mthat admits a degree-1map f:M0→N.

So for any closed oriented 3-manifold Mwith positive simplicial volume, we can

take Cto be the set of all ﬁnite covers of M.

Note the condition that Mhas positive simplicial volume is necessary for Theo-

rem 1.1, since a manifold with zero simplicial volume does not dominate any man-

ifold with positive simplicial volume, and the simplicial volume has the covering

property.

In this paper, we generalize the above virtual domination result from closed 3-

manifolds to 3-manifolds with tori boundary. The following theorem is the main

result of this paper.

Theorem 1.2. For any oriented cusped hyperbolic 3-manifold Mand any compact

oriented 3-manifold Nwith tori boundary, there exists a ﬁnite cover M0of Mthat

admits a proper map f:M0→Nwith deg(f) = 8.

The proof of Theorem 1.2 can also be applied to prove a similar result on certain

mixed 3-manifolds. Here a mixed 3-manifold is a compact oriented irreducible 3-

manifold with empty or tori boundary, such that it has nontrivial JSJ decomposition

and at least one hyperbolic JSJ piece.

Theorem 1.3. For any compact oriented mixed 3-manifold Mwith tori boundary

such that a hyperbolic piece of Mintersects with ∂M, and any compact oriented

3-manifold Nwith tori boundary, there exists a ﬁnite cover M0of Mthat admits

a proper map f:M0→Nwith deg(f) = 8.

At ﬁrst, we can not prove virtual 1-domination for Theorems 1.2 and 1.3. Al-

though we can prove virtual 1-, 2, or 4-domination in certain special cases, we do

need to state our result as virtual 8-domination.

For technical reason, we can not prove Theorem 1.3 for other mixed 3-manifolds

with tori boundary, although we do expect the virtual domination result still holds

in that case. To fully resolve this problem, it remains to study mixed 3-manifolds

such that all of their boundary components are contained in Seifert pieces.

Question 1.4. Let Mbe a compact oriented 3-manifold with nonempty tori bound-

ary and positive simplicial volume. Does Mvirtually (1-)dominate all compact

oriented 3-manifolds with tori boundary?

For a statement as Theorem 1.2, we do not have to restrict to compact oriented

3-manifolds with tori boundary, and we ask what happens for all compact oriented

3-manifolds with nonempty (possibly higher genus) boundary.

Question 1.5. Which compact oriented 3-manifold Mwith boundary virtually

dominates all compact oriented 3-manifolds with boundary?

Two necessary conditions for Questions 1.5 are: Mhas a boundary component

of genus at least 2, and the double of Mhas positive simplicial volume. If the

boundary of Monly consists of 2-spheres and tori, so does any ﬁnite cover M0

of M. Then M0does not dominate any 3-manifold with higher genus boundary,

VIRTUAL DOMINATION OF 3-MANIFOLDS III 3

by considering the restriction map on the boundary. Moreover, if Mvirtually

dominates Nand both manifolds have boundary, then D(M) virtually dominates

D(N). Since we can choose Nso that D(N) has positive simplicial volume, then

so does D(M).

Before we sketch the proof of Theorem 1.2, let’s ﬁrst recall the proof of virtual

domination results (Theorem 1.1) of closed 3-manifolds in [Sun2], [LS] and [Sun5].

All these three proofs roughly follow the same circle of ideas, and we sketch the

proof of the most general result in [Sun5] here. At ﬁrst, by [BW], we can assume

the target manifold Nis a closed hyperbolic 3-manifold, and we take a geometric

triangulation of N. Since Mhas positive simplicial volume, let M0be a hyperbolic

JSJ piece of a prime summand of M. Then we construct a map j1:N(1) →M0from

the 1-skeleton N(1) of Nto M0, such that j1maps the boundary of each triangle

∆ in Nto a null-homologous closed curve in M0. For each triangle ∆ in N, we

construct a compact orientable surface S∆with connected boundary and a map

S∆#M0that maps ∂S∆to j1(∂∆), so that S∆is mapped to a nearly geodesic

subsurface in M0. Then the maps j1:N(1) →M0and {S∆#M0}together

give a map j:Z#M0from a 2-complex Zto M0. If we construct the maps

{S∆#M0}carefully enough, j:Z#M0induces an injective homomorphism

on π1. Since j∗(π1(Z)) < π1(M0)< π1(M) is a separable subgroup in π1(M) (by

[Sun3], which generalizes Agol’s celebrated result on LERFness of hyperbolic 3-

manifold groups in [Agol2]), the map j:Z#Mlifts to an embedding j0:Z →M0

into a ﬁnite cover M0of M. A neighborhood of Zin M0is a compact oriented

3-manifold Zwith boundary, and is homeomorphic to the manifold obtained from

a neighborhood N(N(2)) of N(2) in N, by replacing each ∆ ×Iby S∆×I. Then

there is a proper degree-1 map g:Z → N(N(2)) that maps each S∆×I⊂ Z

to ∆ ×I⊂ N(N(2)). This proper degree-1 map g:Z → N(N(2)) extends to a

degree-1 map f:M0→N, by mapping each component of M0\ Z to the union of

some components of N\ N(N(2)) (each component is a 3-ball) and a ﬁnite graph

in N.

In the context of manifolds with boundary, the above proof fails in the last step,

but we need to ﬁx it from the very ﬁrst step. For example, if we apply the above

approach to manifolds with boundary, it is possible that some component Cof

M0\Z does not intersect with ∂M 0, but a component of N\N(N(2)) intersecting

g(∂C) may contain some component of ∂N. In this case g:Z → N(N(2)) does

not extend to a proper map f:M0→N. Moreover, even if each component Cof

M0\ Z intersects with ∂M0, it is also diﬃcult to construct the desired extension

f:M0→N. So we need to take a more careful construction for proving Theorem

1.2, which is sketched in the following.

In Section 4, we reduce the proof of Theorem 1.2 to 3-manifolds Mand N

satisfying the following extra assumptions.

•Mhas two components T1, T2, such that the kernel of H1(T1∪T2;Z)→

H1(M;Z) contains an element with nontrivial components in both

H1(T1;Z) and H1(T2;Z).

•Nis a ﬁnite volume hyperbolic 3-manifold with a single cusp.

In Section 5.1, we take a geometric cellulation of a compact core N0of Nwhich

has extra edges than a geometric triangulation, such that each triangle contained

in ∂N0is almost an equilateral triangle. In Section 5.2, we construct two maps

j(1)

s:N(1) →Mfor s= 1,2 such that the following hold:

4 HONGBIN SUN

(1) For each triangle ∆ of N0contained in ∂N0,j(1)

s(∂∆) bounds a geodesic

triangle in M.

(2) For each s= 1,2, the union of geodesic triangles in Mbounded by j(1)

s(∂∆)

in item (1) gives a mapped-in torus T→Mhomotopic into Ts.

(3) For each triangle or bigon ∆ of N0not contained in ∂N0,j(1)

1(∂∆)∪j(1)

2(∂∆)

is null-homologous in M.

For each triangle ∆ of N0as in item (3), we construct a compact orientable surface

S∆and a nearly geodesic immersion S∆#Mbounded by two copies of j(1)

1(∂∆) ∪

j(1)

2(∂∆). Then two copies of j(1)

s:N(1) →Mwith s= 1,2, two copies of the tori in

item (2) and the maps {S∆#M}together give a 2-complex Zand a map j:Z#

M. In Section 6, we prove that if the construction is done carefully, j:Z#M

is π1-injective. After this step, the construction of the virtual domination (proper)

map is similar to the closed manifold case. We ﬁrst use Agol’s result ([Agol2]) that

j∗(π1(Z)) < π1(M) is a separable subgroup to lift Zto an embedded 2-complex in

a ﬁnite cover M0of M, and take a neighborhood of Zin M0denoted by Z. Then

we have a proper degree-4 map g:Z → N(N(2)), such that the following hold.

•For the component T0=∂N0of ∂N(N(2)), each component of g−1(T0) is

a torus in M0parallel to a component of ∂M0.

This key property implies that gcan be extended to a proper degree-4 map f:

M0→N, as desired (see Section 5.3).

Note that the π1-injectivity of j:Z#Mcan not be proved by exactly the

same way as in [Sun2,LS,Sun5]. In [Sun2,LS,Sun5], we equipped Zwith a

natural metric and proved that the map ˜

j:˜

Z→˜

M=H3on universal covers is

a quasi-isometric embedding. However, in the current case, ˜

j:˜

Z→˜

Mis not a

quasi-isometric embedding anymore, since j(Z) contains some tori homotopic into

∂M . To prove the π1-injectivity of j, we modify Zas following. For each torus

Tin Zas in item (2) above (that is homotopic into a horotorus in M), we add

the cone of Tto Zwith the cone point deleted, and get an ideal 3-complex Z3(a

3-complex with certain vertices deleted). The map j:Z#Mextends to a map

j1:Z3#Mthat maps ideal vertices of Z3to corresponding ends of M. In Section

6, we prove the π1-injectivity of j:Z#Mby proving that ˜

j1:˜

Z3→˜

M=H3is

a quasi-isometric embedding.

Although the above description of j:Z#Mis mostly topological, we actually

need geometric methods to construct it. Our main geometric tool for constructing

various geometric objects is the good pants construction. Roughly speaking, the

good pants construction is a tool box that uses so called good curves, good pants

and other good objects to construct geometrically nice objects in hyperbolic 3-

manifolds. The good pants construction was initiated by Kahn and Markovic in

[KM], for constructing nearly geodesic π1-injective immersed closed subsurfaces in

closed hyperbolic 3-manifolds, with good pants as building blocks. Then in [KW],

Kahn and Wright generalized Kahn-Markovic’s work to construct nearly geodesic

π1-injective immersed closed subsurfaces in cusped hyperbolic 3-manifolds. These

geometrically nice subsurfaces of Kahn-Wright are basic pieces for constructing our

2-complex j:Z#Min cusped hyperbolic 3-manifolds. More details on the good

pants construction can be found in Section 2.

VIRTUAL DOMINATION OF 3-MANIFOLDS III 5

Now we summarize the organization of this paper. In Section 2, we review the

good pants construction in closed and cusped hyperbolic 3-manifolds, including

works in [KM,LM,KW,Sun4]. In Section 3, we review and prove some elementary

geometric estimates in hyperbolic geometry. In Section 4, we prove preparational

results that reduce the domain and target manifolds in Theorems 1.2 and 1.3. The

technical heart of this paper is in Sections 5and 6. In Section 5, we construct the

mapped-in 2-complex j:Z#Mand the virtual domination map from Mto N,

modulo the π1-injectivity of j:Z#M(Theorem 5.17). The π1-injectivity of j

will be proved in Section 6.

2. Preliminary on the good pants construction

In this section, we review the good pants construction on ﬁnite volume hyper-

bolic 3-manifolds, including constructions of nearly geodesic subsurfaces ([KM] and

[KW]), works on panted cobordism groups ([LM] and [Sun4]) and the connection

principle of cusped hyperbolic 3-manifolds ([Sun4]).

2.1. Constructing nearly geodesic subsurfaces in ﬁnite volume hyperbolic

3-manifolds. In [KM], Kahn and Markovic proved the following surface subgroup

theorem. This work initiates the development of the good pants construction, and

it was the ﬁrst step of Agol’s proof of Thurston’s virtual Haken and virtual ﬁbering

conjectures ([Agol2]).

Theorem 2.1 (Surface subgroup theorem [KM]).For any closed hyperbolic 3-

manifold M, there exists an immersed closed hyperbolic subsurface f:S#M,

such that f∗:π1(S)→π1(M)is injective.

The immersed subsurface of Kahn and Markovic is geometrically nice, and it is

built by pasting a large collection of (R, )-good pants along (R, )-good curves in a

nearly geodesic way. These terminologies are summarized in the following.

We ﬁx a closed oriented hyperbolic 3-manifold M, a small number  > 0 and a

large number R > 0.

Deﬁnition 2.2. An (R, )-good curve is an oriented closed geodesic in Mwith

complex length satisfying |l(γ)−2R|<2. The (ﬁnite) set consisting of all such

(R, )-good curves is denoted by ΓR,.

Here the complex length of γis deﬁned by l(γ) = l+iθ ∈C/2πiZ, where l∈R>0

is the length of γ, and θ∈R/2πZis the rotation angle of the loxodromic isometry

of H3corresponding to γ. In this paper, we adopt the convention in [KW] that

good curves have length close to 2R, instead of the convention in [KM] that good

curves have length close to R.

Deﬁnition 2.3. We use Σ0,3to denote the oriented topological pair of pants. A

pair of (R, )-good pants is a homotopy class of immersion Σ0,3#M, denoted by Π,

such that all three cuﬀs of Σ0,3are mapped to (R, )-good curves γ1, γ2, γ3∈ΓR,,

and the complex half length hlΠ(γi) of each γiwith respect to Π satisﬁes

|hlΠ(γi)−R|< .

We use ΠR, to denote the ﬁnite set of all (R, )-good pants.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

VIRTUALDOMINATIONOF3-MANIFOLDSIIIHONGBINSUNAbstract.Weprovethatforanyorientedcuspedhyperbolic3-manifoldMandanycompactoriented3-manifoldNwithtoriboundary,thereexistsanitecoverM0ofMthatadmitsadegree-8mapf:M0!N,i.e.Mvirtually8-dominatesN.1.IntroductionInthispaper,weassumeallmanifoldsarecompact,connect...

展开>> 收起<<

VIRTUAL DOMINATION OF 3-MANIFOLDS III HONGBIN SUN Abstract. We prove that for any oriented cusped hyperbolic 3-manifold M.pdf

共63页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

VIRTUAL DOMINATION OF 3-MANIFOLDS III HONGBIN SUN Abstract. We prove that for any oriented cusped hyperbolic 3-manifold M

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: