On handlebody-links and Milnor's link-homotopy invariants (Intelligence of Low-dimensional Topology)

On

_{handlebody‐links}

and \mathrm{M}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{s}

_{link‐homotopy}

invariants Yuka Kotorii

Graduate School of Mathematical

_Science,

The

_University

of

_Tokyo

1 Introduction

This is a_{survey of the}jointwork

[13]

withAtsuhiko Mizusawa.

A _{handlebody‐link}

_{[11, 27]}

is a disjoint union of embeddings of handlebodies in the

3‐sphere S^{3}

_(Figure

_1). Two_{handlebody‐links} are equivalent if there is an ambient iso‐

FIGURE1. A_{handlebody‐link.}

topy which transforms one tothe other. An HL_‐homotopyis an equivalencerelation on

handlebody‐links, which is _analogous to _{link‐homotopy} oflinks. _{Here, link‐homotopy}is

generated by ambientisotopies and_{self‐crossing changes.} In

_[22]

,Mizusawa and Nikkuni

showed that the _{HL‐homotopy} classes of _{2‐component handlebody‐links} were classified

completely by the _linking numbers for _{handlebody‐links,} which was defined by Mizu‐

sawa in

[21].

In

[13],

we construct _{HL‐homotopy}invariants for _{handlebody‐links by} us‐ ing Milnor’s _{\overline{ $\mu$}}‐invariants for links. We then give a _{necessary and sufficient condition}of

that a handlebody‐link is HL‐homotopic to a separable one _by the extended Milnor’s

\overline{ $\mu$}‐invariants. _Here, a handlebody‐link is _separable if there exists a disjoint union of 3‐ balls such that each _component ofthe _{handlebody‐link} is contained ina distinct 3‐ball.

Moreover, we _give a bijection between the set of _{HL‐homotopy} classes of n‐component

handlebody‐linkswithsomeassumptionandaquotientofatensorproduct of\mathbb{Z}‐modules

2 Preliminaries

J. Milnor defined a family of invariants for an ordered oriented link in S^{3} as a _gen‐

eralization of the _{linking numbers,} in

_{[19, 20].}

These invariants are called Milnor’s _\overline{

$\mu$}-invariants. For anordered orientedn_‐componentlink L, Milnor’s\overline{ $\mu$}‐invariant isspecified

by a _{sequence I}ofindices in

\{

_{1, 2,}..._,

n\}

and denotedby\overline{ $\mu$}_{L}(I). If the sequence is with

distinct _indices, then this invariant is also _{link‐homotopy} invariant and called Milnor’s

link‐homotopy invariant.

We introduce thedefinition ofMilnor’s_{link‐homotopyinvariants,} andtogiveinvariants

for_{handlebody‐links,} weshowthat these areadditive underabundsum for_components.

Let _{L=L_{1}\cup\cdots\cup L_{n}}beanordered orientedn_‐componentlink inS^{3}. Consider the link

group

$\pi$=$\pi$_{1}(S^{3}\backslash L_{1}\cup\cdots\cup L_{n-1})

ofL_{1}\cup\cdots\cup L_{n-1} anddenote thei‐thmeridianbym_{i}

for

_{i(1\leq i\leq n-1)}

.

Given afinitely generated_group G, the reduced group\overline{G}is definedtothe quotientof G

byits normal_{subgroup generated by}

_{[g, hgh^{-1}]}

for any g,h\in G, where

[a, b]

means the

commutatorofa and b. Then\overline{ $\pi$} is generated bythe meridiansm_{1}, m_{2}_,..._,m_{n-1}.

Let

_{\mathbb{Z}[[X_{1}, . . . , X_{n-1}]]}

be the non‐commutative formal power series ring generated by X_{1},..._,X_{n-1}. Denoteby\hat{Z}itsquotient ringbythetwo‐side ideal generated byallmono‐

mials inwhichatleastoneofthe_{generatorsappear at}least twice. TheMagnusexpansion $\varphi$ is a homomorphism from the free _group

F(m_{1}, \ldots, m_{n-1})

generated by m_{1}, ..._,m_{n-1}

into

_{\mathbb{Z}[[X_{1}, . . . , X_{n-1}]]}

, definedby sendingm_{i} to 1+X_{i}and

m_{i}^{-1}

to

1-X_{i}+X_{i}^{2}-\cdots

. It

induces ahomomorphism from

\overline{F(m_{1},\ldots,m_{n-1})}

into \hat{Z}. Let

w_{n}\in F(m_{1}, . . . , m_{n-1})

be a

word_{representing L_{n}}in\overline{ $\pi$}. We thendefine

$\mu$_{L}(i_{1}i_{2}\ldots i_{r}n)

fordistinct indicesi_{1}, i_{2}_,. .._,i_{r},n

asthecoefficient ofthe _{Magnus expansion}of_{w_{n}} in\hat{Z}:

$\varphi$(w_{n})=1+\displaystyle \sum$\mu$_{L}(i_{1}i_{2}\ldots i_{r}n)X_{i_{1}}X_{i_{2}}\ldots X_{i_{r}},

where the summation isover all sequences _{i_{1}i_{2}\ldots i_{r}} with distinct indices between 1 and

n-1. Similarly, we define

$\mu$_{L}(i_{1}i_{2}\ldots i_{s})

for any distinct indices between 1 and n. We

define

_{\overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{r}n)}

as the residue class of

$\mu$_{L}(i_{1}i_{2}\ldots i_{r}n)

modulo the indeterminacy

\triangle_{L}(i_{1}i_{2}\ldots i_{r}n)

whichis the _greatest commondivisor of

$\mu$_{L}(j_{1}j_{2}\ldots j_{s})' \mathrm{s}

, wherej_{1}j_{2}\ldots j_{s}

ranges over all _sequences obtainedby deleting at least one of theindices i_{1}, i_{2},..._,i_{r},n

and _permuting the _remaining ones cyclicly. Moreover we define

\triangle_{L}(i_{1}n)=

O. Similar

to_this, for _anyn_‐component link L, we candefine

\overline{ $\mu$}_{L}(I)

for anysequence I of distinct

indices in

_\{

1,2,..._,

n\}

Theorem 2.1

_([19,

20 IfL and L' are link‐homotopic, then

\overline{ $\mu$}_{L}(I)=\overline{ $\mu$}_{L},(I)

for _any sequence I with distinct indices.

Lemma 2.2

_([20]).

Let L be an ordered oriented link. Then thefollowing relations hold.

(1) \overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{m})=\overline{ $\mu$}_{L}(i2. . . i_{m}i_{1})

(2)

Ifthe orientation _ofthek‐th _componentofL is_reversed, then

_{\overline{ $\mu$}_{L}(i_{1}i_{2}\ldots i_{m})}

ismulti‐

plied by+1 or-1 _according asthe sequence_{i_{1}i_{2}\ldots i_{m}} containsk aneven orodd number

oftimes.

Thefollowing lemma is used for _Proposition 3.4. This lemma is showed _{by using}the

definition of Milnor’s _{link‐homotopy}invariants.

Lemma 2.3. LetL=L_{1}\cup L_{2}\cup\cdots\cup L_{n-1} be an

(n-1)

_‐componentlink inS^{3}. Let K and

K' be _disjoint knots in

_{S^{3}\backslash L}

. Let I be a sequence with distinct indices in

\{

1,2,..._,

n\}.

IfI contains the index_n,

$\mu$ L\cup(K_{b}\# K')(I)\equiv$\mu$_{L\cup K}(I)+$\mu$_{L\cup K'}(I)

mod

_{\mathrm{g}\mathrm{c}\mathrm{d}(\triangle_{L\cup K}(I), \triangle_{L\cup K'}(I))}

,

where _{K\#_{b}K'} is a band sum ofK and K' with _{respect to any} band, and

L\cup(K\#_{b}K')

,

L\cup K andL\cup K' are n_‐component links whose n‐th _components are K\#_{b}K', K andK', respectively.

Remark 2.4. _By a_propertyof the_{\overline{ $\mu$}‐invariant,}we canobtain thesameresult for aband

sum ofthei‐th_componentinstead of the n‐th_component.

Remark 2.5. In

_[14],

V. S. Krushkal showed Milnor’s _{\overline{ $\mu$}}‐invariants are additive under

connected sumfor links whichareseparated bya 2‐sphere.

3

_{\mathrm{M}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{s}\overline{ $\mu$}}

‐invarinats for

_{handlebody‐links}

Inthis_section,wedefinetheHL‐homotopy,whichisanequivalencerelationonhandlebody‐

links and construct _{HL‐homotopy} invariants for _{handlebody‐links by using} Milnor’s _\overline{

$\mu$}-invariants.

Definition 3.1

_{(HL‐homotopy).}

Let H_{0} be n handlebodies and

H_{i}(i=1,2)

two n‐

component handlebody‐links obtained_{by embedding H_{0}} to S^{3} by f_{i}. Twohandlebody‐

linksH_{1} andH_{2} arecalled HL‐homotopicif there ishomotopyh_{t} fromf_{1} to_{f_{2}}where the

components of

_{h_{t}(H_{0})}

aremutually disjoint at any0\leq t\leq 1.

Remark 3.2. In

_[22],

the notation of _{neighborhood homotopy} of _{spatial graphs}was in‐ troduced. A _{spatial graph} is an embedding of graph in S^{3}. We can represent the HL‐

homotopyof_{handlebody‐links by}the_{neighborhood homotopy}of_{spatialgraphs.}

Let H=L_{1}\cup\cdots\cup L_{n} be an n_‐component handlebody‐link with genus g_{i} for each i. Let

_{\{e_{1}^{i}, . . . , e_{g}^{i_{i}}\}}

be a basis of

H_{1}(L_{i};\mathbb{Z})

and

\mathcal{B}=\{e_{1}^{1}, . . . , e_{g_{1}}^{1}, . . . , e_{1}^{n}, . . . , e_{g_{n}}^{n}\}

. We can

regardanelementofBas anembeddedclosed oriented circle inS^{3}. Sothe disjointunion

e_{k_{1}}^{1}\cup e_{k_{2}}^{2}\cup\cdots\cup e_{k_{n}}^{n}

canberegarded as anordered orientedlink for each

k_{i}(1\leq k_{i}\leq g_{i})

.

Let Ibea _{sequence of}length

m(m\leq n)

with distinctindices in

\{

1, 2,. .._,

n\}

. For each

I,wedefineanelement

M_{H,\mathcal{B}}(I)

oftensorproductspace

(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1}}\otimes\cdots\otimes(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n}}

as

\mathbb{Z}/\triangle_{I}\mathbb{Z}

‐module defined _by

M_{H,B}(I):=\displaystyle \sum_{k_{1},.,k_{n}=1}^{g_{1}.'.\cdot,g_{n}}\overline{ $\mu$}_{e_{k_{1}}^{1}\cup\cdots\cup e_{k_{n}}^{n}}(I)e_{k_{1}}^{1}\otimes\cdots\otimes e_{k_{n}}^{n},

where

_{\overline{ $\mu$}_{e_{k_{1}}^{1}\cup\cdots\cup e_{k_{n}}^{n}}(I)}

is in

_{\mathbb{Z}/\triangle_{I}\mathbb{Z},}

\triangle_{I} isthe _greatestcommon divisorofall

\triangle_{e_{k_{1}}^{1}\cup\cdots\cup e_{k_{n}}^{n}}(I)

for allk_{1},..._,k_{n}_, where

\triangle_{e_{k_{1}}^{1}\cup\cdots\cup e_{k_{n}}^{n}}(I)

is _{indeterminacy}of the _original Milnor’s invariant for the link

_{e_{k_{1}}^{1}\cup e_{k_{2}}^{2}\cup\cdots\cup e_{k_{n}}^{n}}

and

_{e_{k_{i}}^{i}}

is the canonical basis

(0, \ldots, 0,\check{1}, 0, \ldots, 0)k_{i}

of

(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{9i}

as

\mathbb{Z}/\triangle_{I}\mathbb{Z}

‐module.

Remark 3.3. If the first _homologygroup ofeach component ofH is \mathbb{Z}, the

M_{H,B}(I)

is

identifiedwith the_original Milnor’s_{link‐homotopy}invariant foralink, essentially.

We consider a natural action of

GL(g_{1}, \mathbb{Z})\times\cdots\times GL(g_{n)}\mathbb{Z})

on

(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1}}\otimes\cdots\otimes

(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n}}

and denote_by

_{M_{H}(I)}

theresidueclass of

_{M_{H,I3}(I)}

_bythe action for

_{(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{1}}\otimes}

...

\otimes(\mathbb{Z}/\triangle_{I}\mathbb{Z})^{g_{n}}.

Proposition 3.4. Let H bean n_‐componenthandlebody‐link. Then

M_{H}(I)

isindependent

ofa basis\mathcal{B} of

H_{1}(H, \mathbb{Z})

and anHL‐homotopyinvariant.

Proof. The_proofis_byinductiononthe lengthmof_{sequence I}. Wecanshow itby using

properties of_{\overline{ $\mu$}}‐invariantsforlinks

_(Lemma

2.2 and

_2.3).

See

_[13]

fordetails. \square

Example 3.5. Let H be ahandlebody‐linkwhicharetheregularneighborhoodofgraph

illustrated in _Figure 2. Let I=123. Then,

\triangle_{e_{1}^{1}\cup e_{1}^{2}\cup e_{1}^{3}}(I)=\triangle_{e_{1}^{1}\cup e_{1}^{2}\cup e_{2}^{3}}(I)=2

and

\triangle_{e_{k_{1}}^{1}\cup e_{k_{2}}^{2}\cup e_{k_{3}}^{3}}(I)=0

inothercases. So \triangle_{I}=2and

M_{H}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}\in(\mathbb{Z}_{2})^{2}\otimes(\mathbb{Z}_{2})^{2}\otimes(\mathbb{Z}_{2})^{2}

We canshow the following corollary by using clasper theoryintroducedby Habiro

[8].

Corollary3.6. Ann‐componenthandlebody‐linkH is HL‐homotopictoaseparable handlebody‐

link_ifand _onlyif

_{M_{H}(I)=0}

_{for any I.}

Remark 3.7. T. _Fleming defined a numerical invariant

$\lambda$_{ $\Phi$}(H)

of a pair of a spatial

graph $\Phi$ and its _subgraph H under_{component homotopy}in

_[3].

_Now, we define $\Phi$ as a

handlebody‐linkinsteadofaspatial graphand Hasits_componentinsteadofasubgraph.

Wethen can naturally extendthis invariant to a pairofa handlebody‐link and itscom‐

ponent under_{HL‐homotopy. Then,}the valueof

_{$\lambda$_{ $\Phi$}(H)}

isthe_lengthof first_{non‐vanishing}

FIGURE2. _{Handlebody‐link}H.

4 Main Theorem

Let

_{\mathbb{H}[g_{1}, g_{2}, \cdots, g_{n}]}

betheset ofn_‐componenthandlebody‐linkswith genus g_{i} foreach 1\leq i\leq nsuch that its any

(n-1)

‐component subhandlebody‐linkis _{HL‐homotopic}to a

separable handlebody‐link. By Corollary 3.6, this condition is_equivalent tothat its any

M(I)

’s of_lengthless than nvanishes.

Let S be apermutation _group on

\{2, 3, . . . , n-1\}

. For any element a in S_, wedefine

I_{ $\sigma$} as a_sequence

1 $\sigma$(23\cdots n-1)n.

Theorem 4.1. For any element $\sigma$ in S, themap

$\varphi$:\mathbb{H}[g_{1}, \cdots , g_{n}]\rightarrow\oplus_{ $\sigma$\in S}(\mathbb{Z}^{g_{1}}\otimes\cdots\otimes \mathbb{Z}^{g_{n}})

H\mapsto(M_{H}(I_{ $\sigma$}))_{ $\sigma$\in S}

induces a bijection between the set _ofHL_‐homotopy classes _of

_{\mathbb{H}[g_{1}, g_{2}, \cdots, g_{n}]}

and the

residue class

_{of\oplus_{ $\sigma$\in S}(\mathbb{Z}^{g_{1}}\otimes\cdots\otimes \mathbb{Z}^{g_{n}})}

_{by diagonal}action _{of general} linear group.

We_give two_examples.

Example 4.2. Let I=123. Let H_{1} and H_{2} be two handlebody‐links which are the

regular neighborhoodof_{graphs depicted}in_Figure 3. _{Then, \triangle_{I}=0} and

M_{H_{1}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{3}^{2}\otimes e_{1}^{3}

+2e_{1}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+2e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+2e_{1}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}

\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

M_{H_{2}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}

1 e_{1}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}

\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

Wehave that

_{M_{H_{1}}(I)}

is transformedto

_{M_{H_{2}}(I)}

_bythe_diagonal action of_general linear group. ThereforeH_{1} and_{H_{2}} are_{HL‐homotopic.}

FIGURE3. _{Handlebody‐linksH_{1}} and_{H_{2}.}

Example 4.3. Let I=123. Let H3 and H_{4} be two handlebody‐links which are the

regular neighborhoodof_{graphs depicted}in _Figure 4. _{Then, \triangle_{I}=0} and

M_{H_{3}}(I)=1e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}

+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{3}^{2}\otimes e_{2}^{3}

\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

M_{H_{4}}(I)=2e_{1}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+2e_{2}^{1}\otimes e_{1}^{2}\otimes e_{1}^{3}+1e_{1}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}+1e_{2}^{1}\otimes e_{2}^{2}\otimes e_{2}^{3}

\in \mathbb{Z}^{2}\otimes \mathbb{Z}^{3}\otimes \mathbb{Z}^{2}.

Wecan show that H_{1} isnot _{HL‐homotopic}to _{H_{2} by using}someinvariantsforthe action

of_generallinear group

\mathrm{o}\mathrm{n}_{3}\mathrm{t}\mathrm{h}\mathrm{e}

tensor_{product space.} See

[13]\mathrm{f}\mathrm{o}\mathrm{r}3

details.

H_{3}:

Acknowledgements

Theauthor would like tothank Professor Tomotada Ohtsukifor_invitingmethe work‐

shop ‘

Intelligence of Low‐dimensional _Topology 2016 She would also liketo thank

Professor _{Sadayoshi Kojima}and Professor MitsuhikoTakasawa foryour advice.

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Graduate School of Mathematical Science

The _University of_Tokyo

Tokyo153‐8914

JAPAN

\mathrm{E}‐mail address: _{[email protected]‐tokyo.ac.jp}