Seifert orbifolds and their unitary tangent space

Pub . Ma . UAB Vol . 28 2-3 Se . 1984 In oduc ion SEIFERT ORBIFOLDS AND THEIR UNITARY TANGENT SPACE i .e . in a sec o Ma ía del Ca menGazólazA e a In hispape we de ineSei e o bi olds as geome ic 2~ R i s uc u eson su aces, allowingsingula i ies o angles n . i e le  lao . small  2~  ni p . 1 n . we iden i y wo pa s  e ¡o  and  e  1  and we call his co- cien  p /G i  (Gi  is he g oup gene a ed by  i , gi en as 21 R i (x)  = x .  e  al  ) . We o o e ha heses uc u es can be de ined, in a na u al way, ompolygons in H 2 , IR 2 o S2 wi h he same angles and ha he uni a y angen bundle o a polygonp ojec s o a Sei e o bi old . This pape dealswi h a geome ic in e p e a- ion o he cons uc ion o olia ions on Sei e bundles as i appea sin "Folia ions on Sei e Bundles", Ph .D . Disse a ion, Uni e si y o Chicago 1977 o he au ho ; and a p ep in o he same i le in 1980, wi hsome co ec ions on he compu a ions and mo e gene al esul s . The same p oblem o exis ence o ans e se olia ions is app oached wi h di e en me hods in he pa- pe "T ans e se Folia ions o Sei e Bundles and Sel Homeomo - phisms o he Ci cle", D . Eisenbud, U . Hi sch, W . Newmann, Co men . Ma h . Hel 56, 1981, 638, 660 . I .- Sei e o bi olds and hei uni a y angen spa ce 1 .1 . De ini ion .- A Sei e o bi old is a closedsu ace o ge- nus g,B 9 , wi h he ollowingaddi ionals uc u e : a) Bg is co e ed by open se s {U i }, whe e each Ui is asso- cia ed o a homeomo phism o i : Ui --, i/Gi . . 0  2II(3 . whe e P . = { e l , 0 6 9  1 , > 0 being a small numbe } n . 1 o some pai o in ege s (R i , n i )  wi h 0 < Qi n i ,  and G i 2IIp is he g oup gene a ed by he a ion ola angle o b) Whene e U i U j , he e is an injec i ehomomo phism ij : G i - G j and an isome ic embedding p ese ing he canonical o ien a ion so ha he ollowing diag am commu es : pi/Gi  ` p j / ij (G i ) Gi en a Sei e o bi old s uc u e on  Bg ,  we can choose an open co e  {U a }  o  B g  and a subcollec ion  {U i } ¡=l, . . .,N' such ha  na = 1  o  a q!  {l, . . . . N},  ni * 1  o  i  = 1, . . . . N . We deno e he Sei e o bi oldby  B (g,{(n l ,R i )},N)' wi h he con en ion ha  g  will be posi i e i  Bg  is o ien a bleand g will be nega i e i B g is no o ien able . Le us suppose ha P is a geodesicpolygon, ha can be ealized in  H 2 ,  IR2 o  S2 ,  wi h sides  {Si}i=l, . . . . 2N+4g posi i ely o ien ed and angles ai , be ween S i and Si+l' wi h alues 2TW 1  2I1 nl 0 4g+N . 2 W 2 2 P N 2II n 2  n N _  '  4g+N' . . . o pai s o in ege s  (ni ,(3 i )  wi h  0 < Qi < ni . Le us deno e he polygon by  P(g, (nl'Rl)' . . .' (nN'RN) andle  { i } i , =1, . . . . N  be an isome y sending  S 2i-1 o  S 2i while we shall deno e by g i . he isome ysending -1 S2 N +4j +i+2 .  j=l, .. . . g,  i  = 1,2 . Simila ly we can conside a polygon S 2N+4j +i o P(g . (n1 , R 1 ) p . . .. (N .R N )) 9 1 wi h angles  {a i } _ { 2 ~ 1 ,  2g+N ,  2~ 2 , 2g n N , . . .,  2~ N ,  2 g+Ñ . . .} i=1, . . .N+2g .  The  { i }  a e de ined as abo e and  g i sends S2N+2j+l o S 2N+2j+2 j-0, . . .,g-1 . Le G be he g oup gene a ed by { i ,g i 1 o a gi en polygon P(g ;(ni .0l)~ . . .,(nN#RN))The isome ies i a e con~ juga ewi h o a ions and he isome ies g i a e hype bolic unc ions ; hey can be conside ed also as di eomo phisms o he ci cle es ic ing o he bounda y o H 2 (in he caseo S2 o IR2  hey a e o a ions) .  The e o e hey ac on he bounda y o P and on he ib e o hesepoin s i.e . G ac s on he uni a y angen space o he polygon T 1 (P (g ; (n1,pl) , . . . . (nNYON) ). 1 .2 . Theo em .- llowing conmu a i e diag am : 9 2 Gi en a polygon  P(g,(n1,Q1), . . . . (n N '0 N )  we ha e he o T  T 1 (P) 1 íP) --~  G 1) p is a Sei e bundlewi h Sei e in a ian s, [3], (0,  o,  g,  N-2+2g,  n i ,  Ql, . . . . n N ,  O N ) o m lo i + i ni = 1,  0 < Si < n i i  g > 0 . 1') p is a Sei e bundle wi h Sei e in a ian s : (0, n ., -1 g1 , N-2+1 g1 , n 1 1 11 . . . l o N )  o  m¡oi + 7ini = 1, 0 < , i < ni i  g < 0 . 2) II  gi es o  P/ G  an s uc u e o Sei e o bi old B (g, { (ni, Ri) } ,  N) T (P) G ~ D 3)3)  G  ~>  admi s a connec ion on he Sei e bundle . 1 .3 . De ini on .- A connec ion in a Sei e bundle :M --> B, g > 0,  is a one o 0 E A'(M) -sa is ying : 2) Rg- e = 6 whe e R g is gi en by he ac ion o g on M, and a X : IR --> TXM is he induced map on he angen spaces o he map S1 ---, M, gi en by he ibe in a poin . I B is a non-o ien ableclosed su ace, TI : B' --> B is he o ie i ed doubleco e wi h g oup G o co e ing ans- o ma ions, hen a connec ion o a Sei e bundle  (M  P "" B) is a one o m 9 wi h wis ed coe icien s A'(M ;IR{G]) such ha 11 * (j) is a connec ion o he double Sei e bundle o o e B' . We call he image o aX e ical ec o s o he connec ion and he ke nel o e X ho izon al ec o so he connec ion . p oo o 1 .2 .- a)  Suppose ha  E R 1 <N - 2  + 2g (E~ i <N - 2 + I g1 )  he e n .  n .  - 1  1 9 3 o e P can be ealized in hype bolic o euclideangeome y . G is a subg oupo Moebius ans o ma ions o H 2 and he e o e hei s es ic ion o he bounda y is a subg oup o TOPS 1 (homeomo phisms o  S1 ) . i)  We claim ha  [a2g ,9 2g-1 1  ...  [ g2' g1" "'012" l is a ansla ion by  N-2 +2g i O<- i (x) -x < l,  0'!!! 1 (X) -x <l . ü)  1G  is_a Sei e bundlewi h Sei e in a ian s iii)  b  = [ g2g'g2g-l1  o . . . o 20 1 To p o e ii) obse e ha , i B E is a smallball in he e ex  2 ~ R 1 ,  hen  BE n P  is isome ic o a smallball in he 2  ie 21W i cen e o  H  wi h poin s  e  , 0 <,O-< n ..  Le  - he equi- alence ela ion (0,  o,  b,  n 11 (3 1, . . . ,Q N ) . 2IW ,  2IT(i ( e e ,ele)  ^, ( ei (e +  ni l) ,  e l (0 +  nli) ) . . We cons uc a co e ing map om a solid o us o 1 (B E ñ P)  as ollows : (B FnP) x S1 T -~ ( ele,e( e ie,Ri'  ei(~ - wi h e=el + 2M ose , <2 1 n i n i The g oupo co e ing ans o ma ions is gene a ed by : 2 n "i 2II i P(ni,m i )( e le ,  e le)  _  ( e  (e  +  n i ) "  e + n l ) ) whe e0imi + Tini = 1 o in ege s 1 ~< m i < ni . b)  I  ~ > N - 2 + 2g ( ñ i > N - 2 + I gl ) i  i hen he polygon can be ealized in S2 as in case a) . The p oo o i) and iii) is jus a i ial geome ic a - gumen . Thenwe ha ep o ed 1) o 1') . 2) is i ial and 3) is gi enby he p oyec ion o he s anda d connec ion o H2, I2 1 .4 . Dé ini ion .- In he condi ions o heo em 1, 2, we call T 1 (P)/G  he uni angen bundle o he Sei e o bi old B(g,{(ni,pi)},N)- and he p oo wo ks I g=0, N=1, o g =0, N=2, n 1 * n2 we de ine he uni a- y angen bundle o  B (0,{(n i , pi )},N)  as he Sei e bundle o e S2 wi hSei e in a ian s (0, o, o, - 1, (n1 ,p 1 )) and (0,o,o,o,(n1,p1),(n2,p2)) 1 .5 . De ini ion .- A Sei e o bi old B (g,{(n i , p i )},N) associa ed numbe has an ha we call Eule ch ac e is ic o he Sei e o bi old . Obse e ha 1 .5 is wellde ined by heo em 5 in Sei e 's pa- pe 131 . Lis o e e ences .- [1] Magnus, W . : "Non euclidean essala ions and hei g oups" . AcadenucO ess (1974) . [2] Kobayashi and Nomizu : "Funda ionso di e en ial geome y" . In e science . [3] Sei e , H . : "Topologie d eidimensionale ge ase e Ráume" . Ac . Ma h ., 60 (1933) . [4] Siegel, C .L . : "Topic in complex unc ion heo y, 2" . Wiley (1969) . [5] Thu s on, W .P . : "Non cobo dan olia ions" . Bull . Ame . Ma h . Soc . 78 (1982), 511-514 . Rebu el 26 d'abn U de¡ 1984 SPAIN R . -(N-2+1g1 -  ñ l ) . g < 0 . i Uni e sidad Au ónoma de Mad id Di isión de Ma emá icas CiudadUni e si a ia de Can o Blanco Mad id