NONPOSITIVE TOWERS IN BINGS NEIGHBOURHOOD MAX CHEMTOV AND DANIEL T. WISE Abstract. Every 2-dimensional spine of an aspherical 3-manifold has the nonpositive

2025-05-01 0 0 793.76KB 7 页 10玖币

侵权投诉

NONPOSITIVE TOWERS IN BING’S NEIGHBOURHOOD

MAX CHEMTOV AND DANIEL T. WISE

Abstract. Every 2-dimensional spine of an aspherical 3-manifold has the nonpositive

towers property, but every collapsed 2-dimensional spine of a 3-ball containing a 2-cell

has an immersed sphere.

1. Introduction

Deﬁnition 1.1. A 2-complex Xhas nonpositive immersions if for every combinatorial

immersion Y→Xwith Ycompact and connected, either χ(Y)≤0 or Yis contractible.

A 2-complex Xhas nonpositive towers if for every tower map Y→Xwith Ycompact

and connected, either χ(Y)≤0 or Yis contractible.

There are many variations: For instance, one can generalize to combinatorial near-

immersions, or relax to π1Y= 1 or χ(Y)≤1, and there also variations requiring χ(Y)≤

−c|Y|for some “size” |Y|of Y. Note that nonpositive immersions implies nonpositive

towers. The main consequences of nonpositive immersions hold for nonpositive towers.

E.g. if Xhas nonpositive towers then π1Xis locally indicable. These ideas have promise

as a contextualizing framework towards Whitehead’s asphericity conjecture, as well as

towards understanding coherence.

In [8] it was shown that every aspherical 3-manifold with nonempty boundary has a spine

with nonpositive immersions. This utilized that there exists a spine with no near-immersion

of a 2-sphere [1].

In this note, we observe the following failure, which is a special case of Proposition 3.5:

Theorem 1.2. Every collapsed spine of a simply-connected 3-manifold containing a disc

has an immersed sphere.

Bing’s “house with two rooms” provides such a spine. It is obtained from a 3-ball divided

into two rooms by a pair of collapses, corresponding to entering the left room from the

right side of the house and entering the right room from the left side. See Figure 1.

W. Fisher also found examples of the failure of nonpositive immersions in other con-

tractible 2-complexes: the Miller-Schupp balanced presentations of the trivial group [2].

There are thus two sources of counter-examples to the conjecture that contractible 2-

complexes have nonpositive immersions [8, Conj 1.7].

Date: February 9, 2023.

2020 Mathematics Subject Classiﬁcation. 57K20, 57N35.

Key words and phrases. 3-manifolds, asphericity.

Research supported by NSERC.

arXiv:2210.01395v2 [math.GT] 7 Feb 2023

NONPOSITIVE TOWERS IN BING’S NEIGHBOURHOOD 2

Figure 1. An immersed sphere (left) in Bing’s house (right).

In [3] it is shown that for n≥3, every PL n-manifold Mwith ∂M 6=∅has a spine

Xsuch that ∂M →Xis an immersion, and moreover, such spines are generic among all

spines. So Theorem 1.2 is a variant of the simplest instance of their result. However it is

a counterpoint to the following statement (see Theorem 2.11) which is our main motive:

Theorem 1.3. Every 2-dimensional spine of an aspherical 3-manifold has nonpositive

towers.

Thus nonpositive immersions does not always hold for a natural family of contractible

complexes which nevertheless have nonpositive towers, so we are motivated to refocus on:

Conjecture 1.4. Every contractible 2-complex has nonpositive towers.

For instance:

Proposition 1.5. A contractible 2-complex with two 2-cells has nonpositive towers.

Proof. As a contractible 2-complex Xadmits no nontrivial connected covering map, any

tower map Y→Xmust begin with a subcomplex X0⊂X, so X0has at most one 2-cell.

But X0has nonpositive immersions (hence nonpositive towers) by [7, 5]. 

Acknowledgement: We are extremely grateful to Grigori Avramidi for helpful com-

ments and the valuable reference [3]. We tremendously appreciate the referee’s useful

feedback in improving the exposition.

2. Thickenings and nonpositive towers

Deﬁnition 2.1. Let Xbe a connected 2-complex. A tower map Y→Xis a ﬁnite

composition of covering maps and subcomplex embeddings, such that Yand the domain

of each embedding are compact connected 2-complexes.

Tower maps arose in Papakyriakopolous’ classical 3-manifold proofs, and also arose

naturally in one-relator group theory [6].

Deﬁnition 2.2. A 2-complex Xhas nonpositive towers if, for any tower map Y→X,

either χ(Y)≤0 or Yis contractible.

The goal of this section is to prove the nonpositive tower property for an aspherical

2-complex Xembedded in a 3-manifold M. The idea of the proof is to consider a manifold

“thickening” Tof Xin M, which deformation retracts to X. The asphericity of Xensures

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：

NONPOSITIVETOWERSINBING'SNEIGHBOURHOODMAXCHEMTOVANDDANIELT.WISEAbstract.Every2-dimensionalspineofanaspherical3-manifoldhasthenonpositivetowersproperty,buteverycollapsed2-dimensionalspineofa3-ballcontaininga2-cellhasanimmersedsphere.1.IntroductionDenition1.1.A2-complexXhasnonpositiveimmersionsiffore...

展开>> 收起<<

NONPOSITIVE TOWERS IN BINGS NEIGHBOURHOOD MAX CHEMTOV AND DANIEL T. WISE Abstract. Every 2-dimensional spine of an aspherical 3-manifold has the nonpositive.pdf

共7页,预览2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！

NONPOSITIVE TOWERS IN BINGS NEIGHBOURHOOD MAX CHEMTOV AND DANIEL T. WISE Abstract. Every 2-dimensional spine of an aspherical 3-manifold has the nonpositive

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: