UNIT SPHERE FIBRATIONS IN EUCLIDEAN SPACE DANIEL ASIMOV FLORIAN FRICK MICHAEL HARRISON AND WESLEY PEGDEN Abstract. We show that if an open set in Rdcan be ﬁbered by unit n-spheres then

2025-05-06 0 0 946.82KB 11 页 10玖币

侵权投诉

UNIT SPHERE FIBRATIONS IN EUCLIDEAN SPACE

DANIEL ASIMOV, FLORIAN FRICK, MICHAEL HARRISON, AND WESLEY PEGDEN

Abstract. We show that if an open set in Rdcan be ﬁbered by unit n-spheres, then

d≥2n+ 1, and if d= 2n+ 1, then the spheres must be pairwise linked, and n∈ {0,1,3,7}.

For these values of n, we construct unit n-sphere ﬁbrations in R2n+1.

1. Motivation and statement of results

A classical theorem states that R3can be decomposed as a union of disjoint unit circles.

This statement may be folklore, but it seems to have been popularized by Conway and Croft,

who gave a nonconstructive proof using transﬁnite induction [10]. Since then, a number of

fascinating questions, regarding the possibility of decomposing a ﬁxed space as the union of

disjoint copies of another space, have been studied from set theoretic, topological, and geo-

metric perspectives. A collection of decomposition questions appear in an expository article

by Martin Gardner [11], whereas a collection of geometric constructions using transﬁnite

induction can be found in [19]. Similar questions appear every so often on MathOverﬂow;

see [20,22,23,24]. It seems to be unknown whether there exists a Borel decomposition of

R3into unit circles; see [21].

With diﬀerent topological or geometric constraints, versions of this question have appeared

in a wide range of articles [2,3,5,4,6,8,7,9,18,26,31], and several explicit constructions

have appeared. For example, Szulkin constructed an explicit covering of R3using geometric

(round) circles of varying radii, which fails to be a foliation only on the union of a countable

discrete set of concentric circles [29], whereas foliations of R3by topological circles were

thoroughly studied by Vogt [30]. Bankston and Fox [1], generalizing a construction of

Kakutani, gave a decomposition of Rn+2 by tamely embedded, unknotted, pairwise unlinked

(topological) n-spheres, but it seems unknown whether there exists any decomposition of

Rn+2 by unit n-spheres for n > 1.

Here we address the existence question for continuous unit sphere decompositions of

open subsets Ein Euclidean space Rd. A decomposition of Eby unit n-spheres is called

continuous if the map taking p∈Eto its containing sphere is continuous; that is, if the

sphere centers and their normal spaces vary continuously with p. In this case we say that

Eis ﬁbered by unit n-spheres. Global ﬁbrations of Rdby unit spheres cannot exist for

topological reasons, as can be seen from the long exact sequence associated to the ﬁbration.

FF was supported by NSF grant DMS 1855591, NSF CAREER grant DMS 2042428, and a Sloan Research

Fellowship. MH was supported by NSF grant DMS 1926686. WP was supported by NSF grant DMS 2054503.

arXiv:2210.13981v1 [math.GT] 25 Oct 2022

2 DANIEL ASIMOV, FLORIAN FRICK, MICHAEL HARRISON, AND WESLEY PEGDEN

However, the existence question for local ﬁbrations, has, to the authors’ surprise, turned out

to be nontrivial.

A simple example of a unit circle ﬁbration can be described as follows. Consider an

open torus T∈R3with major and minor radii both equal to 1(the boundary of Thas a

singularity in the center). The interior of this torus may be foliated by a collection of tori Tr,

each with major radius 1, but with minor radii rranging from 0to 1. Now each individual

torus admits two foliations (a left-handed one and a right-handed one) by Villarceau circles.

The radius of a Villarceau circle is equal to the major radius of the torus, and hence all such

circles have radius 1. Choosing the right-handed foliation on each torus yields a collection

of pairwise linked unit circles which ﬁll the interior of T. See Figure 1.

Figure 1. Fibration of a toroidal region in R3by linked unit circles

Remark 1.1. We note that the stereographic projection of the Hopf ﬁbration of S3by unit

circles produces an image which looks very similar to that of Figure 1. While the circles

there are also Villarceau circles on tori, they are not unit circles.

In Section 3we construct explicit unit n-sphere ﬁbrations in R2n+1, for n∈ {1,3,7}.

When n= 1 this construction produces the ﬁbered torus described above. The dimensions

of our construction are sharp, as shown by our main result:

Theorem 1.2. Suppose that there exists a unit n-sphere ﬁbration of an open subset E⊂Rd.

Then d≥2n+ 1, and if d= 2n+ 1, then n∈ {0,1,3,7}.

If there exists a unit n-sphere ﬁbration of an open region E⊂Rd, then there exists a

unit n-sphere ﬁbration of E×R⊂Rd+1, by “stacking” unit n-sphere ﬁbrations of E. Thus

it would be interesting to determine:

Question 1.3. Given n∈N, do there exist dand an open subset E⊂Rdsuch that Eis

ﬁbered by unit n-spheres? If so, what is the minimum such dimension d?

The answer to this question is known for two other types of geometric ﬁbrations, and it

would not be surprising if the answer to Question 1.3 matches one of these described below:

UNIT SPHERE FIBRATIONS 3

Agreat sphere ﬁbration is a sphere bundle with total space Sdand ﬁbers which are great

n-spheres. The standard example is the Hopf ﬁbration of S3by unit circles, which arises by

choosing an orthogonal complex structure on R4and intersecting S3with all of the complex

lines. Similar constructions with quaternions or octonions yield Hopf ﬁbrations with ﬁbers

S3or S7. Algebraic topology imposes strong restrictions on the possible dimensions of great

sphere ﬁbrations; in particular the ﬁbers must have dimension 0,1,3, or 7. With this in

mind, it would not be surprising if only these ncould serve as the ﬁber dimension of a unit

sphere ﬁbration. See [12,13,14,25] for more information about great sphere ﬁbrations.

Askew ﬁbration is a vector bundle with total space Rdand ﬁbers which are pairwise skew

aﬃne n-planes. In [27], Ovsienko and Tabachnikov showed that a skew ﬁbration of Rdby

n-planes exists if and only if n≤ρ(d−n)−1, where ρis the Hurwitz–Radon function,

deﬁned as follows: Decompose q∈Nas the product of an odd number and 24a+b, where

0≤b≤3, then ρ(q) = 2b+ 8a. It follows from unboundedness of ρthat for every nthere

exists dsuch that Rdmay be ﬁbered by skew aﬃne n-planes. With this in mind, it would

not be surprising if every ncould serve as the ﬁber dimension of a unit sphere ﬁbration. See

[15,16,17,28] for more information about skew ﬁbrations.

Remark 1.4. In the proof of Theorem 1.2, we will demonstrate a correspondence between

linked unit sphere ﬁbrations and skew ﬁbrations. In particular, we will use the fact that

skew ﬁbrations of R2n+1 by n-planes exist if and only if n+ 1 = ρ(n+ 1), which occurs if

and only if n∈ {0,1,3,7}.

Although we have found some relationships among unit sphere ﬁbrations, great sphere

ﬁbrations, and skew ﬁbrations, we have found that the techniques used to study unit sphere

ﬁbrations are somewhat diﬀerent from those used to study the other types of ﬁbrations.

For example, the collection of n-spheres in a great sphere ﬁbration of Sdcorresponds to a

submanifold of Grn+1(d+ 1), and an important component of studying great sphere ﬁbra-

tions is understanding the topology of the Grassmann manifold. Similarly, the study of skew

ﬁbrations requires understanding the topology of the aﬃne Grassmann and spaces of non-

singular bilinear maps. By contrast, geometry plays a large role in the study of unit sphere

ﬁbrations, and it seems unlikely that their study can be completely reduced to questions of

topology or linear algebra.

2. Proof of the main result

The proof of the main result uses the notion of linkedness for unit n-spheres. Recall

that two disjoint topological n-spheres S1and S2in Rdare linked if S1is nontrivial as an

element of πn(Rd−S2). Linking of topological n-spheres can potentially occur in dimensions

n+ 2 ≤d≤2n+ 1, but linking for unit n-spheres is much more restrictive.

Lemma 2.1. Let S1and S2be two disjoint unit n-spheres in Rd,n+ 2 ≤d≤2n+ 1, and

let Pibe the aﬃne (n+ 1)-plane containing Si. The spheres S1and S2are linked if and only

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：

UNITSPHEREFIBRATIONSINEUCLIDEANSPACEDANIELASIMOV,FLORIANFRICK,MICHAELHARRISON,ANDWESLEYPEGDENAbstract.WeshowthatifanopensetinRdcanbeberedbyunitn-spheres,thend2n+1,andifd=2n+1,thenthespheresmustbepairwiselinked,andn2f0;1;3;7g.Forthesevaluesofn,weconstructunitn-spherebrationsinR2n+1.1.Motivationand...

展开>> 收起<<

UNIT SPHERE FIBRATIONS IN EUCLIDEAN SPACE DANIEL ASIMOV FLORIAN FRICK MICHAEL HARRISON AND WESLEY PEGDEN Abstract. We show that if an open set in Rdcan be ﬁbered by unit n-spheres then.pdf

共11页,预览3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！

UNIT SPHERE FIBRATIONS IN EUCLIDEAN SPACE DANIEL ASIMOV FLORIAN FRICK MICHAEL HARRISON AND WESLEY PEGDEN Abstract. We show that if an open set in Rdcan be ﬁbered by unit n-spheres then

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: