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修士学位論文

題名

A genera1ization for【Bott tower  from the viewpoint of Seifert

fiber space and app1ication to

tOruS aCt1OnS.

Bott towerのSeifert fiber空間からの

一般化とトーラス作用への応用(英文)

指導教授   神島芳宣 教授

平成  23年  1月 10目  提出

首都大学東京大学院

理工学研究科   数理情報科学専攻

学修番号   10878314

氏 名   中山雅友美

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A genera1ization for Bott tower from the viewpoint of Seifert iber space

       and apP1ication to torus a.ctions.

      Bott towerのSeifert丘ber空間からの一般化と

      トーラス作用への応用(英文)

      Author:Mayumi Nakayama

   In this note we sha11exp1ain a notion of81一仰rθdη乞肥。批亡。ωeヅwhich is a genera1ization of rea1Bott tower.A rea1Bott manifo1d is an orbit space M(λ)of a free action of(Z2)η

on P associated to a Bott matrix A We can check that M(λ)is di丘eomorphic to a Riemannian且at manifo1d R肌/F.(Here F≦R肌×O(η)is a discrete uniform subgroup).

And M(λ)occurs as an iterated81−bund1e which is ca11ed a rea1Bott tower:

       〃=M肌_今M1η_ユ_今.、、_今Mユ_今{ptト       (0.1)

That is,for each乞,M1乞→M乞_1is an31−bund−1e which induces a group extension of fundamenta王groupsη:7rユ(ル4);

       1→Z→η→η一。→1.       (0.2)

More precise1y,1et X乞:R乞be the universa1cover of M乞,then we have a principa1bund1e

R一→X{→Xづ_1and so we have two free actions ofπ乞and R on X壱.By the condition of

ni1Bott tower,クr壱norma1izes R.Then it is easy to check that the31−bund1e M;of each

stage is viewed as a3e仰れ仰εヅ8ραce:

       1一一一→Z一一一→7r4一一→η_1一一>1

↓ ⊥  !

R一一→X圭一→X{_1

(0.3)

! !  /

31一→弘一→Mト。.

   In Section1,we d−e丘ne an31一μrεd〃3o批to胱r as an itera.ted81−bund1e;

       M:M、→仏一1→...→Mユ→{Pt}.     (0.4)

where each M{is a c1osed aspherica1manifo1d.And−this tower can be regarded as an

iterated8e伽竹〃θr3ραoθ.Its top space is ca11ed an31一五bred ni1Bott manifo1d.

   一In Section3and4,we observe the fo11owing(cf.[11)、Let7r be a virtua11y ni1potent

group i,e.πcon七ains a torsionfree discrete ni1potent subgroup of丘nite index△.SupPose 人〔s a simp1y comected ni1potent Lie group containing△as discrete cocompact subgroup

and form the semidirect prod−uct E(人ブ)=〃×Iκ(κis a compact subgroup of Aut(人r))、

Then

1

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In particu1ar, /ρ(7r)is an infrani1manifo1d.

   In Section5,we prove our main theorem(See[31):

Theorem(Rigidity).Any81−ibred ni1Bott manifo1d is d倣。mo叩肋。 to an infrani1mani−

fo1d.

   In a proof of this theorem,we show丘rst of a11that the fundamenta1group of an31一

五bred早i1Bott manifo1d−is加地α吻η伽。加肌ムUsing Theorem0.1and Seifert rigidity,we

can obtain the theorem above.Masuda and Lee[2]have a1so proved the above resu1ts by a di丘erent method.Next,to study31一五bred ni1Bott manifo1ds further,we d−e丘ne two c1asses of31−ibred ni1Bott manifo1d.s as伽伽勿ρe or抗力η伽勿ρe a㏄ordingto the group extension

(0.2)ofeach stage.We prove that M is of丘nite type ifand on1y ifM is di丘eomorphic to

a Riemannian iat manifo1d一.Kamishima and Nazra have proved that a rea1Bott manifo1d

M「iλ)admits a homo1ogica11y injective maxima1抄一action.And we show that8ユーibred

ni1Bott manifo1ds of丘nite type do so.

   In Section6,we c1assify the di丘eomorphism c1asses of3−d−imensiona131一五bred ni1Bott manifo1d−s M「.On the other hand the di冊eomorphism c1asses of rea1Bott manifo1ds have

been c1assi丘ed up to dimension5,by Kamishima,Nazra,Masuda ahd Choi.We canshow

that there are3−dimensiona181一五bred.ni1Bott manifo1ds of丘nite type which d−o not be1ong

to the c1ass of rea1Bott manifolds.And any3−dimensiona131一五bred ni1Bott manifo1d of

in丘nite type is either a Heisenberg ni1manifo1d N/△(た)or a Heisenberg infrani1manifo1d

N/F(ん).In section7,we show a one−to−one correspond−ence between3−dimensiona181一

旦bred ni1Bott manifo1ds an(i group extensions ofZ by Q.Here Q is the fmd−amenta1group of a.K1ein Bott1e K or a torus T2.As a consequence given a group extension of Z by Q,

we can formu1ate that any possib1e31−ibred ni1Bott tower of d.epth3.

Re胎rences

[1]Y.Kamishima,K.B.Lee and F.Raymond,珊θ8吻θ竹。oη8か刎。物ηα棚伽αρ〃。α一    伽η8右0伽伽棚mαηが0zd3,Quart.J.Math.,Oxford(2),34(1983),433−452.

[2]J.B.Lee,and−M.Masuda,刀。ρo1o卿。ゾ物m右ed81−6刎棚1θ8,Preprint,arXiv:1108.0293

   math.AT(2011).

エ3]M.Nakayama,8θ伽竹。o肌5君mc考乞。η!oグηψo右e械gヅ。ψ8α〃α〃Z乞。α亡乞。 oαη31一〃rθ∂

   η畑・批乃ω・・(t・・pP・a・).

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A GENERALIZATION FOR BOTT TOWER FROM THE

   VIEWPOINT OF SEIFERT FIBER SPACE AND

      APPlLICATION TO TORUS ACTIONS.

MAYUMI NAKAYAMA

      1.INTRODUCTION

 Let M be a c1osed aspherica1manifo1d which is a top space of an

iterated31−bund1e over a point:

(1,1)  〃=仏→仏一。→_→M。→{Pt}.

SupPose X is the universaユ。overing ofハ∫and each X乞is the universa.1

・・v・・ing・f払・ndputπユ(M4)=η(づ=1,_,上1)・ndπ・(〃);π、

De丘nition1.1.An31イうredη畑。批亡。ωeザis a sequence(1.1)which

satis丘es I,II and−III be1ow.The top space〃is sa.id−to be an81一ガ伽ed

η畑・批m㎝伽町・μ・〃ηノ.

   I.Each M1台is a丘ber space over M1包_1with丘ber31.

  II.For the group extension

(1.2)    1→Z→πド→π1一・→1

    associated−to the丘ber space I,there is an equivariant principa1

    bund1e:

(1.3)    R→石ムX壱.。.

  III.Ea.chπ壱norma1izes思.

 The purpose of this paper is to prove the fo11owing resu1ts.

Theorem1.2. 舳〃08e伽t M 乞8αη314う棚η畑。批m㎝ψo〃.

  (Iけθ岬・・ψθ・ゾ巧(πH;Z)ω肋伽桝・θη1・αgグ・ψθ漁一

    8乞。η(1.2)乞50f丘nite order,枕θηルτ乞8流がθomoτρん4c亡。α月4θ一

    m㎝η乞㎝刀α左m㎝榊d.

 Dα士ε:Jan亡ary10.2012.

 2000M一α伽mα地88吻ec士αα5蛎。α肋㎝.53C55,57S25,51M10(this丘1e draft

ni1tOWer).

 Kεgωorゐαη∂ρ伽α5e8.AsphericaI manifo1d−s,Bott tower,Infrani1manifo1ds,

Seifert丘ber spaces.

       1

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      eπ舌eη8づ。η(1.2)づ80f in丘nite order,流eηルτ乞8流がeoη70τρ肋。亡。

      ㎝伽亦㎝伽㎝伽〃.∫ηαd肋0η,〃Cαηη0れe励炊0m0ψ乞C左0

      ㎝V肋emαηηづ㎝刀α古mα吻0〃.

  As a consequence,we have the fo11owing c1assi丘。ation.(See Propo−

sition6.1and Proposition6.4.)

Proposition1.3.乃e3一流mθ几84oηαZ81一〃ヅθ∂η4棚。批mαη伽〃80ゾ 伽加物θαヅe伽5e0〃工,92,β1,β2,β3,β4.

Proposition1.4.λημ3一励meη8乞。ηα231一ガ6ヅe6η乞肥。材mαη伽 oゾ加一 ガη批e勿ρe乞5θ批んθrα∬θづ8eηろeγgη4Zmαη幼。〃N/△(ん)oザαη∬θ乞8θη6εγg

乞化かαη4Zηzαηψo〃N/I「(ん).

  Rea1Bott manifo1ds consist of91,92,β1,β3amgng these91,92,

β1,β2,β3,β4.

  (Refer to tbe c1assi丘。ation of3−d−imensiona1Riemannian且at mani−

fo1ds byWo1f[18].Wequotethenotations9づ,β壱there.)

  Masud−a and Lee[11]have a1so proved the above resu1ts.

  By(1.2)ofDe丘nition1.1,a3−dimensiona131一五bred nilBott manifo1d

M gives a group extension:

       1一一一→Z一一→π1(〃)→Q一→1

where Q is the fundamenta1group of a K1ein Bott1e K or a torus T2.

Then this group extension gives a2−cocyc1e in the group cohomo1ogy

弔(Q,Z肌)with a homomorphismφ:Q→Z・Converse1y every co−

cyc1e of H多(Q,Zη)can be rea1ized as a di丘eomorphism c1ass of an

81一五bred ni1Bott manifo1d.Then we correspond−the group extension to3−dimensiona131一五bred−ni1Bott manifo1d.

C0NTENTs

1. Introduction

Part1. Ni1Bott Tower

  2.Pre1iminaries   2.1  Topo1ogica1group

  2.2  GLspace

  2.3  Infra.homogeneous space

  2.4 Group cohomo1ogy

  2.5. Group extension   2.6  Pu11−back and Pushout   3. Seifert丘bering

1

 3  3  4  5  7  8  9

12 13

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3.1

4.

5.

5.1

6.

6.1 6.2

7.

7.1

712

 Existence of the Seifert construction

Infrani1manifo1d

31一五bred ni1Bott tower

 Torus actions on3ユ.丘bred ni1Bott manifo1d−s

3−dimensiona131一五bred−nilBott towers

 3−dimensiona131一五bred ni1Bott manifo1ds of丘nite type.

 3−d−imensiona13ユー五bred ni1Bott manifo1ds of in丘nite type.

Rea1ization

 Rea1iza.tion of31一五bration over a K1ein Bott1e K  Rea1ization of3ユー丘bration over r2

Part2. Appendix

  8.Torus actions on a c1osed aspherica1manifo1d   8.1 Covering spaces

  8.2 Actions of connected groups

  8,331−a・ti・nonanasphe・i・・1mnifo1d   9. C1assica1Beiberbach,s theorems

  9.1  Bieberba.ch丘rst theorem

  9.2 Bieberbach second−theorem

References

13 19 21 25

28 28

29 33 33

38

39

39 40

41

44 47 47

49 53

Part1.Ni1Bott Tower

       2.PRELIMINARIES

  Notations

Let G be a simp1y comected−Lie group、

    ○H≦G:H is a subgroup of G.

    ●Aut(G):The group of a11autmorphisms of G.

    ●Inn(G):The group of a11imer autmorphisms of G.

    ○Out(G):Aut(G)/Inn(G).

    ●A(η):The a舐ne transformation group Rη×GL(η,R肌).

    ●Suppose H is norma1of G.Thenμ(α)is a conjugation map by       α∈G.i.e.μ(α)∈Aut(∬)de丘ned byμ(α)( )=α α一ユ.

Le声G be a group(not necessary a Lie group)、

    ○Z(G):The center of G.

    ●ρ,G]:The commutator of G.

    ・0。(G)=G,0ん(G):[G,0た一1(θ)1.

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2.1.Topo1ogica1group.

De丘nition2.1.G be a topo1ogica1group,if

  (又)(;三is a group,

  (2)Gis・t・p・1・gi・・1・p…,

  (3)A mapμ:G×G→G,μ( ,μ)=ガ1g is continuous.

Ac1osedsubgroupisasubgroupwhichisc1osed−subset.Simi1ar1y,a

discrete(resp.compact)subgroup is de£ned.

lLemma2.2.ωGうeα吻。log乞。α1gヅ㎝ρ.Foヅ㎝V舳65ε o∫G,

      λ二∩、∈γγλ.

此re∩、∈γ乞5Hη乞㎝0!αllη吻肋0rん00∂0∫e,

P…ヅIf ∈λ,th・・γ ユ・i・…ighb・・h・・d・fパ…11n・ighb・・h・・d

γofεso thatγ一 ∩λ≠②.Hence ∈γλi.e.λ⊂∩、∈γγλ.

Converse1y,ifπ≠λ,then there is a neighborhoodひ。f such that

σ∩A=②.Take a neighborhoodγof e satisfyingγ一1エ⊂σ.Then

γ一1 ∩λ=②.And−so ≠γλ.Therefore∩、∈γγλ⊂λ.

       □ Lemma2.3.〃G6θα老。ρolog乞。αl groψ,Fαc108ed舳う5θ亡㎝d0

αCOmραC左舳う3εt.4F∩0=②,伽η伽ヅe乞5αη吻肋0ヅん00dγ0μ

舳。ん仇α老γF∩γ0:②.

PmoナLetひbe a neighborhood of e and put F(ひ)=ひ一1ひF.Then F(σ)⊂σひ■1σF by Lemma2.2.Noting that there is a neighborhood H7of e such thatσσ一1ひ⊂η7,

        ∩、∈σ(F(σ)∩0)⊂∩、∈σσひ.1σF∩0

      ⊂∩、∈WWF∩0       =F∩0

      =F∩0:②.

Since O is compact,∩匙1(F(叫)∩0)二②.Putγ:∩匙1騎,γ■1γF∩

0=②,that is Fγ∩0γ=②.

       口

 Let G be a topo1ogica1group and∬a subgroup.Then the coset

・p…G/H={9H19∈G}h・・th・q・・ti・ntt・p・1・gyf・・th・p・・j・・ti・n ρ:G→G/∬,ρ(g)二g∬.This means thatρis continuous and open.

Then

Proposi抗。n2.4.〃H≦G乞8αcZ08θ68ωgroψ,G/H48∬α仙8doげ。

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Pザ。qブLet[ ],[μ]∈G/H,[ ]≠[ひ].Then ≠μ亙,that is ∩μH:②.

Using Lemma2.3,γ ∩γμH二・②for some neighborhoodγofe.From

thisρ(γ ∬)∩ρ(γV∬)=②,a.nd so G/亙is Hausdor任.

      口

  Remark that if G is connected−and−1oca11y pathwise−comected,then 正r玉s a c1osed discrete subgroup if and on1y ifρ:G一→G/H is covering

(・f.S・・ti・n8.1).

2.2.G−space.Let X be a path−comected,1oca11y simp1y comected and1oca11y path−connected space.

De丘nition2.5.Suppose G is a topo1ogica1group.An ac七ion of G on X is a continuous map

       ψ:G×X→X

such that

   (i)ψ(9ん, );ψ(9,ψ, ))ξ…11g,ん∈G・nd ∈X,

   (ii〜(ε,・)ニバ…11・∈X,

The poin}(g, )is writtensimp1y as g or g. .We may denote this

・・ti・nby(G,X)・ndXi…11・d・G一・p….

  There is an ana1ogous notion of an action from the right(X,G);

       ψ:X×G→X

in which we denote byρ( ,g): g or .g.Thenエ(gん):( g)ん: gん.

De丘ni七ion2.6.An action(G,X)is a smooth action if G is a Lie group,X is a smooth manifo1d and.ψis a smooth血ap,

De丘nition2.7.Let X be a G space.An isotropy subgroup Gπof G at is de丘ned by

      G、={9∈G19 =外

If U皿∈xG ={e}(resp.∩ ∈xG、;{e}),then the G−action on X is sa・id to be a free(resp.e丘ective)action.

De丘nition2.8.(G,X)is亭proper action if

      ξ・(K)={9∈G19K∩K≠②}

is compact for a11compact subset K⊂X.In addition,if G is dis−

c1rete,thenξo(K)is丘nite set and the G−action is ca11ed a proper1y

discontinuous action.

Remark2.9.〃Qろeα卿。Zoglcαl gr㎝ρα〃Wαm㎜伽〃.4

(Q,W)1・αρ岬εψ伽㎝1切ω・・α・左1・η,挽㎝,∫・・α〃 ∈X,伽・・1・

αη吻肋0rん00dσπ舳Cん仇α亡

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   (i)叫Q、:σ、,

   (三i)4σ、∩α乙㌃≠⑦,妨εηα∈Q 、

PγooヅTake a1oca11y compact nさighborhoodσofπ∈W,thenξQ(σ)

is inite set becauseξQ(ひ)⊂ξQ(σ).So then we put

      ξQ(σ)={9・,...,9包,94。。,...,9m}

in which{g1,_,g{}=Q、.Take a neighborhoodひ二⊂ひ。f which

・・ti・丘・・㎎∩gμ=②(ゴ=乞十1,…m),th・・Q、=ξQ(叫)⊂ξQ(ひ)・

Furthermore if we take

      σ、:∩。∈Q皿σ二,

thenσ、Q、:ひ so thatξQ(σ、):Q .

      口

Theorem2.10.〃〃6eαm㎝伽Zd.互α8moo肋㏄劫㎝(G, )乞5

伽・αη〜・・ρ・ψ伽・・η肋㎜・刎・フ仇㎝伽・棚・ρα・θ〃/G6θ・・m・・α

m㎝榊∂.

〃。oヅFor any compact subset K⊂M「,since the(G,M)is proper1y

d−iscontinuous action we may putξG(K)={91,...,9几}.If z∈K,as the(G,ルC)is free and M−is Hausdor丘,then there is a neighborhood σ of such that for a.ny g壱∈ξo(K),σ、∩g壱σ、=②.It imp1ies that

for any g∈G,σω∩gひ。=②.Hence,noting that叫is di丘eomorphic

t・gひ、f・…yg∈G,th・・iti・…yt・・h・・kth・tM ム〃/Gi・・

covering space with丘ber G(cf.Section8.1)i.e.M「/G is a manifo1d.

      口

  Let X,γbe two G−spaces.

De丘nition2.11.A map∫:X→γis G−equivariant if it satis丘es

      ∫(9・・)=9・∫(

for a11g∈G and ∈X.Then the map∫is said to be a G−map.

In ad−dition,if the map∫is a homeomorphism,then∫is ca11ed a G−equiva1ence or G−isomorphism.A map∫:X一→γis weak1y G−

equivariant if there is a continuous autmorphismα∫of G such that

      ∫(9・ )=α∫(9)・∫(・)

for a11g∈0and ∈X.Furthermore if this map∫is a.homeomor−

phism,then∫is weak1y G−isomorphism.

  Let G/X,G/γbe orbit spaces. If I∫ :X → γis weak1y G一

・qui…i・nt,th・nthi・i・d・…プ:X/G→γ/G,∫([・1)=[ル)1.

Ifプis a G−isomorphism,then this∫is homeomorphism.

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2・3・Infrahomogeneous space・Let G be a(noncompact)simp1y

comected−Lie group,and Aut(G)denote the group of automorphisms ofG onto itse1f,Put A(G)=G×Aut(G).A(G)becomes a group;

       (9,α)・(ん,β)=(9・α(ん),α・β)

(9,ん∈G,α,β∈Aut(G)).λ(G)is ca11ed the a舐ne group of G.Here,

1etting X:G,an a田ne action(A(G),X)is obtained as fo11ows:

       ((9,α), )=9・α( ).

Let∬⊂Aut(G)be a compact subgroup(for examp1e,maxima1com−

pact subgroup,丘nite groups).Form asubgroup E(G)二G×∬⊂A(G).

Consid−er the action(E(G),X).If we note that∬is compact,then

I−emma2.12(Proper action).(E(G),X)乞8α〃。ρerαcれ㎝。

〃。ψLet K⊂X be a compact subset and

      ζE(G)(K)={(9,α)∈E(G)1(9,α)K∩K≠⑦}・

Given an in丘nite sequence{(9台,α{)}ofζE(G)(K),since(9{,α乞)K∩

K≠②,there exists a sequence{腕},{吻}∈一κsatisfying(94,α4)軌二

gρ仏)=μ乞.We may assume that!im叫= ∈K,1im跳=μ∈K.

Asα乞∈∬which is compact,it fo1王。ws that1imα{=α∈H.Noting X:G,91;μパα乞(外1)→μ・α(ガ1)this imp1ies thatζE(o)(K)is compact.

      口

  By Lemma2.12,ifπ≦E(G)is a discrete subgroup,we obtain a

proper1y discontinuous action (π,X)and the quotient space X/7r is

said to be an infrahomogeneous orbifo1d.Whenπhas no e1ements of

丘nite order,πis said to be torsionfree.

Propos三也。n2.13.47r乞8αむ。r5台。η介ee gro岬,抗θηπαc左5作θθ佃。η

X.肋θηX/π乞・・αM㎝伽伽ん・m・g㎝舳・m㎝州ゴ.

Pγり。アSupPose there is a g,for some工∈X,9 = ,then g㌦二 Since{吋⊂X is compact and(G,X)is a proper action,{9肌}η∈z is

丘nite.This is a contradiction.

      口

Examp1e(Riemamian且aもmanif〇三d).TakingthevectorspaceRηand

O(n)as G and H respective1y,we have the euc1idean group

       E(η)=Rη×O(η).

(Note that O(η)is the a maxima1compact subgroup of GL(η,R).)A

d−iscrete uniform subgroupπof E(η)is a crysta11ographic group.If

クr⊂E(Rη)is atorsionfree crysta11ographic group,thenπis said to be

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an Euc1idean space form,i.e.a Riemannian且at manifo1d.

Examp1e(Infrani1mnlfo1d).Let G be a simp1y comected ni1potent Lie group〃.A torsionfree discrete uniform subgroupπ⊂E(〃)is

ca11ed−the a1most crysta11ographic groups and〃/πan infrani1manifo1d.

  Wehavethefunda皿enta1c1assica1resu1tforcrysta11ographicgroups.

Theorem2.14(Bieberbach丘rst t}1eoreIn).〃π⊂E(η)6εαcrV5老α1−

Zogmρ肋。 gηouρ,挽θηRη∩7r皇Zれαη∂Zη45αηormαZ mα切mαZαろεZ乞αη

舳6g・㎝ρ伽π。M・・ε・砂・・π/Rη∩π1・α伽伽g・㎝ρ.μ…批㎝9。リ De丘nition2.15.The group7r/Rη∩クr is ca11ed the ho1onomy group.

Remark2.16.Zεけ:Rη×O(η)→O(η)6ε伽η痂mZ〃0畑亡40η。

r加η7r/Rη∩7rmαρ5伽加。亡伽e佃伽左。O(η)α棚工(7r)=π/Rη∩7r⊂O(η)。

  Theorem2.14is extend−ed to the a1most crysta11ographic groups.See

[41f・・i・・t・n・・.

Theorem2.17(Aus1ander−B三eberbach theorem).Zeオ7rうeα広。畑。η一

介eε肋8cヅθ乏e刎η幼。rm舳6grouρoプE(人ブ),仇θη〃∩πづ8αmα切mαZηormαZ

ψ・む・η之・吻・・ψ・方㎝∂π/〃∩π1・α伽伽g・㎝ρ、

2.4.Group cohomo1ogy.Letλ,Q be groups.Supposeλis a Q−

mod−u1e for which the Q−action is given by

       α =φ(α)(

(∀α∈Q,∀ ∈λ).Then a map∫:Qη→λis said to be an咋。ochain.

The set of a11作。ochain is an abe1ian group0η(Q,λ)with the group

1aW:

       (∫十9)(α1,.。.,αη)=プ(α1,...,α肌)十9(α1ゾ..,αη)。

Denote a homomorphismδη:0η(Q,λ)→0η斗1(Q,λ)by

  δη∫(α1,...,αη斗1)

       η

     一φ(α・)(∫(α・,…,α几・1))十Σ(一1)∫(α・,…,α糾・,…,αη十・)

      づ=1

     +(一1)肌十1プ(α1,...,αれ).

For examp1e,

(2.1)   δ1λ(α,β)=φ(α)(λ(β))一λ(αβ)十λ(α),

(2.2) δ2∫(α,β,7)=φ(α)(∫(β,7))一∫(α,β7)十∫(αβ,7)一∫(αラβ)

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We can seeδη十ユ。δη=O(η=1,2,3...).Hence{0η(Q,λ),δη}

becomes a chain comp1ex,and−we get a group of n−cocyc1e㌘(Q,λ),

a group ofη一。oboundary Bη(Q,λ)andη一。ohomo1ogy group:

∬茅(ρ,λ)一■∬岬,λ),柳ρ,λ)一zn(ρ,λ)/8肌(ρ,λ)・

Remark2ユ8.舳〃03θQづ8α伽伽gヅ。ψFフ士加n

(2・3)     ∬2(F;R肌)一0・

Pザ。oアLet∫:F x F→Rηbe a2−cocyc1e re1ative toφ:F→Aut(Rη).

Putん:F→R肌

(2・4)    ん(α)一Σ!(α,ア)

      τ∈F

Then

δ1ん(α,β)=φ(α)(ん(β))一ん(αβ)十ん(α)

       一Σ(φ(α)(∫(β,γ))一Σプ(αβ,τ)十Σ∫(α,τ)

         τ∈F         τ∈F      τ∈F

       一Σ(∫(αβ,τ)一∫(α,βτ)・∫(α,β))一Σ∫(αβ,ア)十Σ∫(αヨτ)

         τ∈F       τ∈F      ア∈F

       =1F1∫(α,β)

i…δ1

p一∫・1・imp1i・・th・・∬孝(F;Rη)一0・

      口

  Letλbe an abe1ian group.If we have a group extension;

       1→A→五ムQ→1

with a section q ofリwith g(1):1,then A becomes a Q−modu1e by a homomorphismφ:ρ→Aut(λ),αHμ(q(α))(cf.Section2.5).

Therefore we get a cohomo1ogy group H*(Q,λ).

2,5.Group extension.Let G,Q be groups andφ:Q→Aut(G)a

function.Suppose there is a.function∫:ρ×ρ→G which satis丘es that

   (互)φ(α)(φ(β)(η)):∫(α,β)φ(αβ)(η)∫(α,β)一1,

  (II)プ(α,1)=∫(1,β)=1,

 (In)φ(α)(∫(β,7))∫(α,β7):∫(α,β)∫(αβ,7),

whereη∈G andα,β,7∈ρ.The above∫defines a group亙which玉s the prod−uct G×Q with the group1aw:

(2.5)      (η,α)(m,β):(η・φ(α)(m)・∫(α,β),αβ).

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Then there is aφ一group extension1→G→亙」㌧ρ→1where

μ(η,α)=αand the group亙is denoted−by G×(∫,φ)Q.Remark that if

∫andφare trivia1maps,then G×(∫,φ)Q is ajust direct product G×Q・

  Converse1y,9iven a group extension1→G→亙」㌧ρ→1,we

can associate亙with aφ一9roup extension G x(∫,φ)Q・Choose a section

q:ρ→亙(μo q:。id)in which q(1)=1.Taking∀η∈G,∀α∈Q,

then we de丘ne(肌,α)∈亙byη.q(α).Moreover we can put

(2.6)      q(α)・q(β)二∫(α,β) q(αβ),

(2.7)      q(α)・η=φ(α,η) q(α).

Here∫:Q×Q→G andφ:Q x G→G.But

      (φ(α,ηm),α)=φ(α,ηm)・9(α)

      =q(α)・ηm

(2.8)       :(9(α)・η)・m       :φ(α,η)・9(α)・m       =(φ(α,η)φ(α,m),α)

(∀η,m∈G,∀α∈Q).Soφ:QxG→Gcan be regarded as the

φ:Q→Aut(G).Note thatφ(α)(肌)=g(α)ηq(α)一1.Then∫andφ

・・ti・fyth・・b・v・(I)(II)(III).Inf・・t

(2.9)   φ(αβ)(η)=q(αβ)ηq(αβ)一1:μ(∫(α,β)一1)φ(α)φ(β)(η).

         q(α)・(q(β)・q(7))=q(α)・∫(β,7) q(β7)

       :φ(α)(∫(β,7)) q(α)・q(β7)

(2.10)       :φ(α)(∫(β,7)) ∫(α,β7)・q(αβ7)

         (q(α)・9(β))・q(7)=プ(α,β)・q(αβ)・q(7)

       :プ(α,β)・∫(αβ,7)・9(αβ7)

From.(2.9)and(2.10),(I)and一(III)is obtained.By d−e丘nition,(II)is obvious・Therefore五is G x(プ,φ)Q,in which q(α)q(β):∫(α,β)q(αβ)

andφ(α)=μ(9(α)).

De丘nition2.19.A group extension亙=G×(プ,φ)Q is a semi−direct

prod−uct,if∫is trivia1i.e.there is a homomorphism section q:Q一→亙.

De丘nition2.20.A group extension1→G→亙→Q→1is a

centra1group extension,if G⊂Z(亙)。

Re1=nark2.21.Zε左1→λ一→亙→Q→16eαgrouρε批εη8包。η!or

ωん4cんλづ5αηαうεκαηgro刎ρ.!アφづ8炉伽乞α2,仇θη1一→λ一→五一合(フー→1

台8αCe伽αlgザ0岬e批㎝8乞㎝.

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Pザ。qブLet∀(m,β)∈∬ ∀(η,1)∈λ,then

      (m,β)(η,!)=(m+φ(β)(η)十∫(β,1),β)

      =(m+n,β)

      =(η,1)(m,β).

Henceλ≦Z(五).

De五nition2,22.Two group extensions1→G一→亙→ρ→1,

1→G→亙 →Q→1are congruent,if there is a isomorphism

λ :亙 _→ 亙 such that the restriction map 刈G :G → (;and the

ind−uced mapλ:Q→Q are identity maps.That is there is the fo11owing commutative diagram

      1→G一→亙→ρ→1

(・・1・)  !伽一/ ト・

      1一→G一→亙㌧一→Q一一→1.

  Let Opext(Q,G,φ)be the set of a11congruence c1a.sses ofφ一group

extensions.Then an e1emenリ∫1∈Opext(ρ,G,φ)is represented by an

extension1→G→五→Q→1with五=G×(プ,φ)Q.It is easy to

check tha.t[∫1]:[∫2]∈Opext(ρ,G,φ)if and on1y if there is a function 入:(;フー→Z(G)such that

(2.12)    !1(α,β)=δ1λ(α,β)・∫2(α,β) (∀α,β∈Q).

Hereδ1is de丘ned−byδ1λ(α,β):φ(α)(λ(β))入(α)λ(αβ) 1.For simp1ic−

ity,We Write it aS∫1=δユ入・∫2.

  In particu1ar,when G is an abe1ian groupλ,φ:Q→Aut(λ)is a homomorphism and henceλis a Q−modu1e.So there is the group

・・h・m・1・gyH彦(Q,λ)・・d∫i・・2一…y・1・by(11I),i…[∫1∈亙彦(Q,λ)・

Therefore any extension1→λ→亙→Q→1correspond−s to a

…y・1・[∫1∈H彦(Q,λ)・Th・・

Proposit三〇n2.23.8ψρ08ε肋αtλづ5αηα6ε24αηgヅ。ψ.丁加れ肋eヅε乞8 αo胱一老。−oηe come8ρo〃e肌eろε地eeη∬茅(Q,λ)αηd Opext(Q,λ,φ)・

Pザ。oヅLet[∫1],[∫2]∈Opext(Q,λ,φ).If[∫11=げ2],there is an iso−

morphismλ:λ×(∫王,φ)Q→λ×(∫、,φ)Q such that入Iλ:λ→λand一

λ:Q→Q are identity maps.We assume thatλ(1,α)=(λ1(α),α).

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Then

       λ((1,α)(1,β)):λ((∫1(α,β),αβ))

       :(∫1(α,β),1)(λ1(αβ),αβ)

       =(∫1(α,β)十λ1(αβ),αβ),

       λ(1,α)λ(1,β)二(λ1(α),α)(λ1(β),β)

       =(λ1(α)十φ(α)(λ1(β))十∫2(α,β)」,αβ),

so that(∫1一∫2)(α,β)=φ(α)(λ1(β))一λ1(αβ)十λ1(α).Therefore

[∫11:[∫21∈∬享(ρ,λ)・Simi1ar1y,converse is obtained・

      口

2.6.Pu11−back and Pushout.Let

(2.13)  1一→G一→五一→Q一→1

be a group extension represented−by[∫1∈Opext(Q,G,φ)andαa subgroup of Q.Given a mapφ:ρ→Aut(G),there is a restriction mapφ :ρ →Aut(G)satisfying(I),(II),(III)befGre.Thenφ makes

the group extension〃:G×(φ ,∫)Q which is said to be a pu11−back of

Q .Remark that〃≦亙.

      1一一→G一→五一一→ρ一→1

(・・1・)  ll l T

      1一一→G一→亙 一一→Q∫一一→1

  Letπbe a torsionfree group which contains△as a丘nite1y generated ni1potent norma1subgroup.If we put Q:π/△,then there exists

(2.15)  1一→△一→π一→Q一→1

which is represented by[∫1∈Opext(Q,△,φ).By Ma1 cev sθ挑一 加ηcθtheorem,there is a(simp1y comected一)ni1potent Lie group〃

。ontaining△as a discrete uniform subgroup and given a function φ:Q→Aut(△),Ma1 cev s unique extension theorem imp1ies that each automorphismφ(α):△→△extends unique1y to an.automor−

phismφ(α):〃→人ブ.In particu1ar,this gives a correspond−ence φ:Q→Aut(〃).Note that it is not necessari1y a homomorphism.In

genera1it satis丘es

(2.16)   φ(α)(φ(β)( ))=∫(α,β)φ(αβ)(∬)∫(α,β)一1(工∈人プ).

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Then the pushout π人ジ:{( ,α)レ∈人プ,α∈Q}can be constructed.

Its group1aw is de丘ned by( ,α)・(9,β)=( φ(α)(μ)∫(α,β),αβ);

       1__→人ブ_二_→π人ブ__→Q__→ 1

(・・1・)  T l l1

       1 → △ →  7r →Q→ 1.

This group(extension)7r人「is a1so represented by[∫1∈Opext(Q,人ブ,φ).

       3.SEIFERT FIBERING   Let

(3.1)  1一→△→π一一→F一一→1

be a group extension whereπis a torsionfree group,△is a。丘nite1y generated ni1potent norma1subgroup,and F is a丘nite group.i.e.πis

a virtua11y ni1potent group,Then there is a simp1y comected ni1potent

Lie group〃。ontaini㎎△as above,We1ike to prove the fo11owi㎎

rea.1ization theorem(cf.[5]).

Theorem3.1(Rea1izat1on).乃θrθθ挑オ3α伽。肋θ∫α肋〃rθ〃θ8昨

広αわ。ηρ:π→E(人ブ)醐。ん抗ατρ1△:id.伽ραヅ肋。ωαザフノ∫/ρ(π)48αη

4η伽η伽α吻0〃.

  This theorem is obtained as a specia1case of the existence of the

Seifert construction.In this section,we wi11review the Seifert con−

struction with typica1丘ber a ni1manifo1d.

3.1.Existence ofthe Seif;ert construction.Let H7be a contractib1e smooth manifo1d一.Suppdse that a group Q acts proper1y discont三nu−

ous1y on W such that the quotient space W/Q is compact.Given a group extension:

(3・2)  1→△一→π一㌧Q→1,

we sha11show that there is an a.ction of7T0n〃×W which is compatib1e with the1eft trans1ations of人プ.Let Di丘(人プ×W)be the group of a11

di丘eomorphisms of〃×W onto itse1f人〔s a subgroup ofDi冊(〃×W)

via an embedding:Z伽)(m,α)=(ηm,α).

  Let Di丘F(〃×W)denote the norma1izer of Z(〃)in Di丘(〃×Wっ

・ndp・・・・…th・丘b・…fth・p・i・・ip・1bund1・〃→〃×W→W.

SupPose Map(W,〃)is the set of smooth maps from W into人プ,then Map(W,〃)×(Aut(〃)×Di迂(W))becomes a group with

         (λ。,9。,ん。)(λ,9,ん)二((9工・λ・ん丁1)λ。,9.9,ん。ん)

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・ndf・・(π,ω)∈〃×W,

(3・3)   (λ,9,ん)( ,ω):(9(π)・λ(んω),んω)

de丘nes an action on人「×W.For a11η∈人グ,if we denoteη:W→人プ

by a constant mapPing toη,then(η,μ(η),4d)( ,ω)=Z(η)(工,仙)=

(η ,ω).We de丘ne Z:〃→Map(W,〃)oIm(〃)byηH(η,μ(η),乞d).

Proposition3.2.Di丘F(〃×W)co伽。肋舳肋伽grouρMap(W,〃)×

(Aut(〃)×Di冊(W))

片。oナLet(λ,g,ん)∈Map(W,〃)x(Aut(〃)x Di丘(W)).Then

         (λ,9,ん)(η,μ(η),乞d)(λ,9,ん) 1

         :(λ,9,ん)(η,μ(η),州9−1・λ・ん,9−1,パ)

         =(9・η,・人,9・μ(η),ん)(9−1・パユ・ん,9 1,ん一ユ)

         =(9・μ(η)・g−1・λ一1・9(η)・λ,μ(9・η),乞d)

         =(9(η),μ(9・η),乞d)

This shows that Z(〃)is norma1in Map(W,〃)x(Aut(〃)×Di程(W「)).

  Converse1y,1etん∈Di迂F(ノV ×W).Since Z(η)is norma1in Di丘F( ×

WF

j,there is a homomorphismラ:Di丘F(〃x W)→Aut(〃)d−e丘ned

by:

(3.4)    π・1(肌)・万一 =1(ず(万)(η))

and,from the丘ber preserving,form

(3・5)    ん(・,ω)=(ん1(・,ω),ん(ω)).

Since

(3・6)   ん・1( )(1,ω)=(す(ん)(・))・ん(1,ω),

if we putん(1,ω)=(λ(ん)(ω),ん(ω)),then

(3.7) ・  ん。(・,ω):(市)(・))・λ(ん)(ω).

Putず(ん)=g,λ(ん)=入,thenん=(λ,g,ん)∈Map(W,人プ)×(Aut(〃)x

Di丘(W)).

       口

De丘nition3.3.We ca11the set(△1π,Q,W)a smooth data for the

group extension(3.2).

De丘nition3.4.A continuous homomorphismΨ:π一→Di丘F(〃×W)

is the Seifert construction of the smooth data(△,π,Q,W),ifΨmakes

the fo11owing commutative diagram:

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△      一一→

⊥1

π   一→     ρ

⊥Ψ ⊥州

   Map(W,人ブ)o Inn(〃)一→Di任F(人ブ×W)F→Out(M)×Di舐(W).

  Conditions of the Seifert constructionΨ;Ψ(π)norma.1izes Z(△),

imp1y that we have an equivariant丘bration:

         (1(△),〃)一→(Ψ(π),〃・W)I一→(Q,W).

i.e,The projectionρ:〃×W→W ind−ucesρ:人プ×W/Ψ(π)一→

W/Ψ(π).HereΨ(π)=Ψ(π)/Z(△).

De丘nition3.5.人プx W/Ψ(π)4W/Ψ(π)1s sa1d to be the Se1fert

丘bering.And an inverse imageグ1(6)is ca11ed a丘ber.The丘berρ一1(6)

is a typica1丘ber ifρ一1(6);人プ/△.Otherwise,singu1a.r丘ber.

  The fo11owing theorem is obta.ined−in[5].

Theorem3.6 (Existence).工ε亡(△,π,Q,H7)6εα8moo仇dα亡α,仇εγム

抗ere e切5t8α8e伽れ。oη8かucれ。ηΨ:π→Di丘F(〃×W7).

  First of a11,we introduce the fo1Iowing1emma to prove above theo−

rern.

工emma3.7.〃(Wl,Q)わeα〃。ρeψ伽。㎝左づ舳。u8㏄尤乞。η㎝αc㎝一 か㏄肋Zθmα吻。〃∫o川肋。んW/Q乞3comραct,仇㎝〃(Q,Map(W,Rん));

0∫o川〃乞>0.

Pγり。∫First note,Map(W,Rゐ)is aρ一modu1e.Let叫be a neigh−

borhood of ∈W with the property0;Q =叫and such tha.t if

ひπ∩σ α≠②,thenα∈Q (see Remark2.9).一Let V;=σ、jV.{Take Map(K,R冶)⊂Map(以,R冶)to be the submodu1e of a11maps with sup−

port in篶.Simi1ar1y,Map(σ ,Rん)⊂Map(篶,Rゐ)⊂Map(W,Rκ)is the subspace of a11maps with support inひ、and Map(σ態,Rゐ)is a

Qπ一modu1e.Now

(3.8)   Homz(Q囮)(Zρ,Map(叫,Rん))豊Map(篶,Rん)、

The proof,based on the−we11−known fact that ZQ is a freeρガmodu王e with a basis given by choosing coset representa.tives.But using the

ShajPiro s1emma (see[21),we can say

(3.9) ∬・(Q ,M・p(σ、,剛) *(Q,M・p(篶,R冶)),

・・d・i…ρ i・・丘・it・g…p∬壱(ρ,M・p(篶,R冶))=0f・・1>O.

(cf.Remark2.18.)

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  To comp1ete the argument we1ook to a partition of unity.Since W/Q is compact there is a丘nite set of points 1,_, η∈W with neighborhoodsひ1,...,ひ肌as above whose images in W/Q cover this

space.Choose any partition of unity subord−inate to this open covering.

By composing each partition function with the quotient map W→

刊7/Q there resu1ts a partition of unityε1,...,εη on H7subordinate to Vl,.。.,Vみand−ea.ch partition function satis丘esεゴ(πα)…εゴ(π)for

a11α∈Q.Thus mu1tip1ication byεゴis a Q−modu1e homomorphism of Map(W,紗)into Map(巧.,R為).This imp1ies that a homomorphism

εゴ:Map(W,Rん)→Map(巧.,Rん)is de丘ned by∫→εプ∫,hereεプ∫(ω)=

εゴ(ω).∫(ω).And now we deineη:Map(W,R冶)一→Map(巧,R為),

ε二:Map(巧,R冶)→Map(W,Rん)byη(∫)二∫I巧・andε二(∫)=εゴ。∫一巧。

respective1y.Hereεゴ。∫■巧.(ω)=εゴ.∫(ω)or0,accord−ing to weather

ω∈巧。or not.We obtainin this fashion endomorphisms

         εラ:H*(Q,M・p(W,Rん))→H*(Q,M・p(W,R冶)),

         ぢ:∬*(Q,M・p(W,R烏))プ∬*(ρ,M・p(巧・,R冶))

         εザH*(Q,M・p(巧,Rた))→H*(Q,M・p(W,Rκ))

Then滅=ε三十 .十ε葛=Σεク。ぢ=O.HenceH*(Q,Map(W,Rた))=0.

      口

  We sha11review the proofofTheorem3.6which have shown in[5]一

Pヅ。oアUsing the pushout(2.16)in§2.6,if耐e show that there exists a

continuous homomorphismΦ:7r〃→Di冊F(〃×W)such that並一〃⊂

Z(人プ),then a Seifert constructionΨ:7r→Di丘F(人プ×W)is obtained as a restriction.Suppose there exists aΨ.For(η,α)∈7r人r,if we

putΨ(1,α)=(λ(α),g(α),ん(α))∈Map(W,〃)o(Aut(〃)×Di丘(W)),

thenΨ(η,α)=4(肌)Ψ(1,α)=(η,μ(η),棚)(λ(α),9(α),ん(α))= (η・

λ,μ(η)9(α),ん(α)).And we may put g=φ,ん:α.Then,

      Ψ(η,α)=(η・λ(α),μ(η)Oφ(α),α)

where入:Q→Map(W,〃)satis丘es

(3.10)  !(α,β)二(φ(α)o入(β)oα■1)・入(α)・入(αβ)一1 (α,β∈Q),

where∫be a function representing the group extension(3.2).We ca1−

cu1ateΨ(1,α)Ψ(1,β)二Ψ(∫(α,β),αβ)for obtain the condition(3.10).

Therefore to guarantee the existence of suchΨ,we have on1y to find a

map入satisfying the condition(3.10)。Remark that ifwe包ssume N is a vector spaceγ,then Map(W,γ)is a topo1ogica1group with Q−action

by

(3.11)   α・入(ω)二φ(α)(λ(ガ1ω)).

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S・w・h・…g…p・・h・m・1・gyH婁(QlM・p(W,γ))・Fi・・t・・t・th・t

亙募(ρlM・p(Wγ))一0f・…y…t…p…γ(…L・mm・3・7)・

  By induction,we supposethat the statement is true for any ni1potent Lie group whose dimension is1ess than dim人ブ.Let Z be the center of

人プand put大石:人r/Z,π人4=7r人ブ/Z.Consider the group extension

      1一→〃一→π〃」㌧ρ一→1

(・ユ・)  !1 !1

      1一→川一→π川」㌧Q一一→1,

with a section g1:ρo g ofμ1where q is a section toμ.The section q1

determines∫1一:Q×Q→M andφ1:Q→Aut(川)as in§2.5.We

suppose by induction on the d−imension of〃that there existsλ1:Q→

Ml・p(W,川)…hth・t

        ∫エ(α,β):(φ1(α)O人ユCβ)一〇α一ユ)・λユ(α)・人ユ(αβ) 1

Choose any1iftλ :Q→Map(W,ル)of人ユso that人ユ=ρo入.Put

         9(α,β)=(φ(α)○λ (β)Oα 1)・λ (α)・λ (αβ)】1,

then there exists an e1ement c(α,β)∈Map(刊7,Z)such that

       ∫(α,β)=C(α,β)・9(α,β).

Let check

       (φ(α)O g(β,7)Oα 1)・9(α,β7)=9(α,β)・9(αβ,7).

  (φ(α)O g(β,7)oα一1)・9(α,β7)

   =  (φ(α)Oφ(β)oλ (7)oβ■1oα一1・λ (β)oα市1・λ (αβ) 10α一1)

      (φ(α)・λ (β7)・α ユ)・λ (α)・λ (αβ7)・1

   =∫(α,β)・(φ(αβ)・λ (7)・(αβ)一1)・∫(α,β)一1

      (φ(α)O(入 (β)oα■1)・λ (α)・入 (αβ)T1・λ (αβ)・λ (αβ7)一1    =  ∫(α,β)・(φ(αβ)oλ (7)O(αβ)■1)一∫(α,β)^1

      9(α,β)・∫(α,β)■1・∫(α,β)・λ (αβ)・X(αβ7)一1

   = 9(α,β)・∫(α,β)一1・∫(α,β)・(φ(αβ)oλ (7)o(αβ)一1)・∫(α,β)一1

      プ(α,β)λ (αβ)・λ (αβ7).1

   =  9(α,β)・9(αβ,7).

Sin・・b・th∫・・dg・・ti・fy(iii)in§2.5,・i・・1… 2一…y・1・i.・.[・1∈

亙茅(ρlM・p(町1Z))whi・h…i・h・・b・・・…Zi・・…t…p・…S・

there is a functionη:Q_タMap(W,Z)such that

      C(α,β)=(φ(α)Oη(β)Oα一1)・η(α)・η(αβ)一1.

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Putλ:η.λ :Q→.Map(W,〃),thenλsatis丘es(3.10).

      □

Remark3.8.Ψ(π)αc左5㎝ ×Wわび

      Ψ(η,α)(π,ω):(η・λ(α),μ(η)Oφ(α),α)( ,ω)

(3.13)       .

       :(η・φ(α)( )・λ(α),αω)

・グ(3.3)

Remark3.9.工εポ1一→Z→πr→πH→1うθαgroψθ娩η84㎝α5

切(1.2).肋㎝π{α・左・・ηX1肋舳η加…α1…ぴ・∫必伽吻.λ舳mθ

挽α亡Ψづ:π壱→Di冊(X{)45流eグ印rθ8e耐ατ4oη一ん。momoヅp肋3mプ。r広肋5 α・れ㎝(π壱,Xづ)フ伽ηΨパη→Ψ4(π1)1・伽8・伽τ・・η・広川舳η・μ加

8moo肋d砿α(Z,7rゼ,7r壱_ユ,Xづ_1).

  Moreover the fo11owing has obtained in[51.

I−em1血a3.10(Uniqueness).λ8e伽r左。o耐用。向㎝Ψ∫orα8moo抗

伽α(△,π,Q,W)乞舳ηψe岬士。 coη加gα之乞。肌吻εleme舳。∫Map(W,〃)o

A・t(〃).

戸γり。ヅLetΨ,Ψ :π〃→Di丘F(〃 ×W)be homomorphisms for two

Seifert constructionsΨ,Ψ of a smooth data(△,7r,ρ,W)as before.

Assume thatΨ,Ψ correspond to someλ,λ :Q→Map(W,〃)with condition(3.10)respective1y.Thenthereis aξ∈Map(W,〃)suchthat

Ψ =μ(ξ)Ψif and on1y if there exists aξ∈Map(W,〃)with人 (α):

φ(α)oξ一1oα一1.λ(α).ξ.Simi1ar1y as in the proof of Theorem3.10,

we can obtain thatξ.We assume thatΨ,Ψ use the same embedding

ε:△一→人r in the pushout,thenΨ=μ(ξ)Ψ .SupPose thatΨ,Ψ use

the distinct embedd−ingsε,F :△→〃respective1y.Then there is a g∈Aut(〃)such thatε:μ(g)ε unique1γThen

         μ(9)・Ψ(η,1)=(〃,9,1d)(ε(η),〃,1d)(〃,グ1,1∂)

       二(μ(9)ε(η),棚,ld)

       :μ(ξ)・Ψ (η,1)、

HenceΨ :μ(ξ■1,9)oΨ.

      口

  Using this Lemma,we can obtain the Seifert rigid−ity which is used

to prove Theorem1.2.

Theorem3.11(Seifertrigidity)・〃△ルα伽。械舳吻。rm8吻ro岬

・∫α・lm〃・・㎜ε・尤・れ伽尤εη川εgη・刎ρ川(1=1,2)。〃Ψ。,Ψ・うε

8ε伽左・㎝・f用・左乞・η・伽・m・・挽∂α亡α(△。,π。,Q・,W。)フ(△。,π。,Q。,肌)

燗ρ・舳吻.3W・・6海・・ε洲・㎝1・・m・似1・mθ:π。→π。伽d㏄伽g

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乞・・m・ψ4・m・θ:△1→△・,θ:ρ・→Q・。凡肋θ㎜・・吋(Q。,W。)1・

θψ・αれ㎝吻鮒・・m・・〃・1・(ρ。,肌)ω肋欄岬用θフ流㎝S・if・・t

・igidity・ん・ω・仇αt(Ψユ(π。),川・W。)1岬舳wl㎜吻鮒・・m・ψい・

(Ψ。(π。),ル×肌).

〃。oヅLetん:W1→W:2be a di丘eomorphism such thatθ(α).ん(ω);

ん(αω) (∀α∈ρ1,∀ω∈W1).Nowθ:△1→△2induces the extension

d−i丘eomorphismん:川→人ら.PutΨ=μ(ん,ん)oΨ1oθ一1.Here

(ん,ん):川xW。→ル・W。,(・,ω)H(ん(π),ん(ω)).Sin・・Ψi・・

S・if・・t・・n・t・u・ti・・f・・(△。,π。,Q。,肌),th…i・・μ(ξ,9)∈M1・p(ル)・

Aut(ル)such thatμ(ξ,9)oΨこΨ2by Lemma3.10.And−so

      μ(ξ,9)■1・Ψ。=μ(ん,ん)・Ψ。・θ一1.

Gonsequent1y,for any(η,α)とπ1,

         Ψ20θ(η,α)(ξ,9・ん,ん);(ξ,9・ん,ん)OΨ1(η,α).

It is easyん=(ξ,9・ん,ん):人(×W1→人ら×W2is a d−i丘eomorphism。

       口

       4.INFRANILMANIFOLD

  In this section,we wi11show the proof of Theorem3.1.

Let(△,π,F,{が})be a smooth data with丘nite group F and一プa func−

tion representing the given group extension1 _→△→7r一→F_}1.

In the same way as the proof of Theorem3.6,we can obta.in a1−cha.in

X:F→〃・u・hth・t∫:δ1X;

(4.1)   ∫(α,β):φ(α)(X(β))X(α)X(αβ).1(α,β∈F).

Wesha11repeattheconstructionofXforouruse.Suppose∫:F×F→

人プ/Z is a function which represents1→人 →π人々→F→1,then

!;δ1λfor some functionλ:F→人ブ/Z.Choose a1ift入:F→

of入.Then it is ea.sy to see the function g=∫.(δ1入)一is a cocyc1e

1yi・gi・Zlth・ti・[91∈∬事(FlZ)・A・∬婁(FlZ)一0f・・m(2・3)lth…

is a mapμ:F→Z such thatδ1μ:g.巾henプ:δ1(μ.人)and the 1_chainX denoted byμ.入

  Now d−e丘ne an automorphism of〃ん(α):〃→人ブfor eachα ∈F

to be

       ん(α)(π)=X(α)一1・φ(α)(・)・X(α)(・∈〃).へ.

Using(2.16),we can prove thatん(αβ)=ん(α)ん(β)forα,β∈F.There−

foreん:F→Aut(〃)is a homomorphism−Since Aut(〃)is a noncom−

pact Lie group,it has a maxima1compact groupκ.Then the finite subgroupん(F)is conjugate to a subgroup ofκ.We can assume that

ん(F)⊂κ.

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  De丘neρ:π→万(人プ)to be

(4.2)    ρ((n,α))=(ηX(α),ん(α))(η∈△,α∈F)・

It is easy to check thatρis a homomorphism.We de丘ne an action of

πonノゾto be

(4.3)   ((η,α), )二ρ(η,α)( )=ηφ(α)( )X(α) ((η,α)∈π).

Theorem3.1is obtained by the fo11owing proposition.

Proposition4.1.丁加αcれ。η(π,〃)乞5α〃。ρぴ妙肋8co耐伽uo刎8作εe

αC肋0η.∫ηραヅれCu2αr,ρづ8α∫α批んア側Jreρre5εητα挽0η、

Pザ。oヅFirst note thatρ1△:〃,so△is contained inρ(π).Since△acts as1eft trans1ations of〃from(4.2),it acts proper1y discontinuous1y and free1y.Moreover since△is a丘nite index subgroup ofρ(7r)from(3ユ),

ρ(π)・・t・p・・p・・1ydi…ntinu・u・1y・・〃.

  Let(η,α)∈Kerρbe an e1ement ofπ。Then((η,α), ): (∀エ∈ ) by(4.3).Asπacts proper1y discontinuous1y,(η,α)is of丘nite order.

On the gther hand,7r is torsionfree,we obtain(η,α)=1and soρis

faithfu1.

      口

  Thefo11owingremarkshows thatρis aSeifert construction(cf.Theorem

3.6).

Remark4.2.〃A(〃)*うeαgroψω肋。んづ8腕〃。仇。左〃×Aut(〃)

ω舳伽gr0ψ1αωj.

       (η,α)・(m,β):(α(m)・η,α・β)

、戸。rη,m∈人ブ,αηdα,β∈Aut(人「).肋εαcτづ。η(A(〃)*,〃)づ80材α伽θd α5∫0〃0ω5j

      ((η,α),・)=α(・)・η

∫or r∈人ブ.Tんθη抗eザθづ8αη台50肌。r帥乞8mδ:A(人プ)*→A(人プ)dφ胱d 6μδ(η,α):(η,μ(η・1)(α)).∬ε肥μ:〃→Aut(人ブ)乞8流εcoη加gαれ。η ん・m・m・・〃・mjμ(η)(・):η・ガ1.〃・・α吻・加・ん・舳α1

       ((η,α), ):(δ(η,α), )

丁肋88ん。ω8流ω栃θαがηθαcれ。η(A(〃),〃)co伽。〃ε5ω挑ん肋εαろ。りθ

αωoη(A(〃)*,〃).

Remark4.3.肋erεづ5αcommu左洲ωε伽gmm.

         1一_→   人プ  _→E(人ブ)一_→κ_→ 1

(…)   T  T ・  {

         1一一1〃∩ρ(π)一→ρ(π)一→H一→1.

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助伽伽・ヅ・m・μ・・1㎝伽肋6ε柿㏄ん,〃∩ρ(π)1・αmα伽α1η・㎜α1

吻・むε伽吻・・ψ・∫ρ(π)。舳・物オ△⊂〃∩ρ(π),・・1ゾ△1・m伽1mα1,

伽η△=〃∩ρ(π)、

      5.31−FIBRED NILBOTT T0切ER

  SupPose that

(5・1)M−M1肌」1㌧M肌、1」㌧...」㌧〃1」㌧{。t/

is an81一五bred ni1Bott tower.By the de丘nition,there is a group exten−

sion of the丘ber spa.ce;

(5.2)    1→Z→π壱ムπ乞一。→1

for any乞.The conjugate by each e1ement ofπ4de丘nes a homomorphism

φ:π1−1→Aut(Z)={±1}.With this action,Z is aπづ一1−modu1e so that the group cohomo1ogy H喜(π4_ユ,Z)is de丘ned一.Then the above 9…p・・t…i・・(5・2)・・p・・…t・・2一…y・1・i・H茅(π1一・,Z)(・f・[131)・

Note thatπ4acts free1y on M「so that7r{is torsionfree by Smith theorem.

  The purpose of this paper is to prove Theorem1.2.Let reca11The−

orem1.2:

  Suppose thatルτis an31一五bred ni1Bott manifo1d.

   (互)If every cocyc1e of H2(π4_1;Z)which represents a group ex−

      tension(1.2)is o∫伽伽。r加r,then M is di任eomorphic to a       Riemannian且at manifo1d.

  (II)If there exists a cocyc1e of H孝(η_1;Z)which represents a group       extension(1.2)is o∫伽伽伽。r加r,then M is di丘eomorphic to       an infrani1manifo1d.In addition,〃。annot be di逓eomorphic to       any Riemannian且at manifo1d一.

PmoグGiven a group extension(5.2),we suppose by ind−uction tha.t there exists a torsionfree£nite1y generated−ni1potent norma1subgroup

△H・f丘・it・i・d・xi・η一。…hth・tth・i・d…d・xt…i・・△4i・・

Centra1eXtenSiOn:

      例

       1一一→Z一一一→η一→π乞_1→1

(…)  ll T T

       1一一一→Z一一一→△乞一→△づ_1一一→1.

It is easy to see that△づis a torsionfree丘nite1y generated−norma1sub−

group of丘nite index inη. Furthermore we sha11show that△ゼis

ni1potent. Let△乞_1be aん_step ni1potent group i.e.Cん(△{_1) = 1.

参照

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