Blowing up fixed points

Pub . Ma . UAB N° 25 Juny 1981 BLOWING UP FIXED POINTS F ancisco Gómez Ruiz Facul adde Ciencias, Uni e sidad de San ande and Seccióde Ma emá iques,Uni e si a Au ónoma de Ba celona . Spain . Rebu el 15 de Feb e del 1981 This no e shows how o use he echnique o blowing up submani olds o gi e easie p oo s o some heo ems conce ning ixed poin se s o o al ac ions on smoo h mani olds . None o he esul s gi en he e is new . The main heo em ( heo em 3) as well as he applica ions (8), (9) and (10) a e well known . Ne e heless I belie e p oo s p esen ed in his no e a e no he usual ones . (1) Le us begin by desc ibing he equi a ian blowing up o in a ian subma- ni o1ds . Le G be a compac Lie g oup ac ing smoo hly on a smoo h mani old M and le B be a G-s able closed smoo h submani old o M . Conside he induced ac ion o G on he angen bundle TM o M . I is gi en by a . = (dT a ) x ( )  x E M, a EG, ET x (M) whe e TX(M) deno es he angen space o M a x and T a is he di eomo phism o M de ined by T a (x) = a .x . The abo e ac ion o G on T M es ic s o ac ions o GonT M IB and TB, since B is G-in a ian . Thus we ha e an induced ac ion o G on he no mal bundle, :E n+ B, o B in M . Le P( ) :P(E) -> B be he p ojec i e bundle associa ed o (i s i- b e o e x E B is he p ojec i e space associa ed o he quo ien TX(M)/TX(B)) The bundle P( )'inhe i s an ob ioús ac ion om he ac ion o G on . We alsoconside he canonical line bundle, É  P(E), on P(E) (i s ib e o e z E P(E) consis s o al] ec o s o z) . The abo eac ions o G on P(E) and E induce an ac ion o G on E such ha he map a :E - E, gi en by a(z, ) = , is G-equi a ian . Obse e ha a is su jec i e and es ic s o a di eomo phism É-P(E) ~ O ' E-B (we iden i y he base B o i s image unde he ze oc oss-sec- ion) . U o B in M oge he wi h a G-equi a ian di eomo phism ! E-P U such ha 82 I is no di icul o cons uc a G-in a ian open neighbou hood es ic s o he iden i y on B . Se a = sp oa :E } U C M . Thus a :E-P(E) , i U-B is a G-equi a ian di eomo phism . Le M be he space ob ained by a aching É o M-B ia hemap E-P(E) á . M-B (i .e . Ñ is ob ained om he disjoin union o É and M-B by iden i ying he poin s o É-P(E) o hei images on M-B unde o ) . Endow M wi h he ob ious smoo h s uc u e o which he inclusions É  M, M-B J+ M a e di eomo phisms on o hei images . I is clea ha he ac ions o G on É and M-B inducean ac iono G on M . De ine  a- :M -, M by  Q(i(x)) = o(x) i x E E and  á(j(x)) =x i x E M-B . Thé map Q is G-equi a ian and su jec i e . Fu he mo e á :M-P(E) M-B is a G-equi a ian di eomo phism . Obse e ha M is ob ained om M by bowingup B on o a G-in a ian hype su ace P(E) . In pa icula a poin o B has been blown up on o a eal p ojec i e space . In case B is a single poin , M is simply he connec ed sum o M and a eal p ojec i e space .  . Blowinp up he ixed póin se o a p oup ac ion . I is easily leen ha he connec ed componen eo he ixed poin se , F G (M), o he ac ion o G on M a e closed smoo h submani olds o M . Blowing up in u n each o hem we ob ain a smoo h mani old M, ac ed on by G, oge he wi h a G-equi a ina su jec i esmoo h map & :M - M . (2) Lemma . 0 Le G be he connec edcomponen o he uni in G and suppose ha 0 G/G has an odd numbe o elemen s . Then he ac ion o G on M has no ixed poin . P oo : Clea ly M -9 - 1 (FG(M)) does no ha e ixed poin s since i is G-equi- a ian ly di eomo phic o M-F G (M) . The e o e i is enough o show ha Q-1 (F G (M)) does no ha e ixed poin s . Le B be one o he connec ed componen e o F G (M) and le :E - u-> B deno e i s no mal bundle . Endow wi h a G-in a ian Riemannian me icand assume ha an elemen z E P(E) exi s such ha a .z = z o all a E G . Fix a ec o o no m 1 belonging o z . We mus ha e a . = E(a) . whe e E :G ~ {1,-1} is a g oup homomo phism, cons an on each connec ed componen e o G . 0 De ine E :G/G - {1,-1} by e(á) = E(a) (a deno ing he clase o a in G/G) . Se c-1 (1) = {a l , . . .,a }and E -1 (-1) = {Bl, . . .,bs} . We know ha -1 (-1) ~ 0 because i no one had a . = o all a E C and, i we iden i y E o a G-in a ian ubula neighbou hood o B in M, his would yield a con- adic ion, since B is a connec ed componen o FG(M) . The e o e E -1 (-1)  =  {a 1B1 , . .. ,a b 1 }  =  {B1' . . .,BS} .  Hence = s  and 0 G/G has an e en numbe o elemen scon adic ing he hypo hesis o he lemma . 83 (3) Theo em . Le G be a o us ac ing smoo hly on a compac smoo h mani old M . Then he Eule -Poinca é cha ac e is ic, x(FG(M)), o he ixed poin se o he ac iono G on M coincides wi h he Eule Poinca écha ac e is ic, x(M)', o M . P oo : Le B 1 , . . . ,B be he connec ed componen s o FG (M) wi h no mal bundles i :Ei  n i -  B i (i= 1, . . ., )  and  le w i :E i  + U i (i= 1, . . ., )  be G- equi a ian di eomo phismswhe e U i is a G-in a ian open neighbou hood o B . in M . Se U =  u U . . i=1 yield The co esponding Maye -Vie o is sequences o he open se s {M-F G (M), U} o M and he open se s {M-á-1(FG(M)) = M-FG(M)- c 1 (u) o M (4) x(M) + x(U-F G (M)) = x(U) + x(M-FG(M)) (5) x(M) + x(U-F G (M)) = x(° -1 (U)) + x(M-FG(M)) whe e x deno es he Eule -Poinca écha ac e is ic and Q :M ->- M is ob ained, as explained be o e, by bowing up in u n each o he B i .  Hence Gis G-equi- a ian and he ac iono G on M does no ha e ixed poin s . Choose nex an elemen h o he Lie algeb a o G such ha exp h is dense in G and le Z h be i s co esponding undamen al ec o ield in M . Explici ly Z h is gi en by Zh (x) = (dA x ) e (h) whe e A x :G 1M is gi en by A x (a) = a .x and e is he uni elemen o G . .The éc o ield Z h has no ze os since he ac ion o Gon M has no ixed poin . The e o e x(M) = 0 because o Hop heo em (see co olla y 3, page 399 o [11) . On he o he hand x(á -1 (U)) = E x(P(E j )) and since he ac iono i=1 G on P(E j ) has no ixed poin he same a gumen as be o e yield x(P(E j )) = 0 (i = 1, . , ) . 84 The e o e (5) can be w i en as ollows (6)  x(U-F G (M)) = X(M-FG(M)) . Finally we deduce om (4) and (6), using also he ob ious ac ha X(U) = X(FG(M)), (7)  X(M) = X(FG(M)) . We show now some applica ions o heo em (3) . (8) P oposi ion . Le Tbe a maximal o us o a compac connec ed Lie g oup G . Then X(G/T) is he numbe o elemen s o N(T)/T whe e N(T) is he no malize o T in G . P oo : The ixed poin se o he ob ious le ac ion o T in G/T, is gi en by F T (G/T) = {xT i x E N(T)} . The e o e FT (G/T) has he same numbe o ele- men s as N(T)/T . The p oo in now inished by applying (3) . (9) Co ollá y . Any wo maximal o i o a compac connec ed Lie g oup a e conjuga e . P oo : Le T,T' be maximal o i o a compac connec ed Lie g oup G and con- side he le ac ion o T' on G/T gi en by ' .xT = ( ' .x) .T( ' E T' xE G) . We know om p oposi ion (8) and heo em (3) ha X(FT,(G/T)) = numbe o elemen so N(T)/T . In pa icula FT' (G/T)  0 . The e o e he e exis s xEG such ha 'xT = xT o all ' E T' . Thus x-1 T'x = T . (10) Theo em . Le Kbe a closed connec ed subg oup o a compac connec ed Lie g oup G . Then X(G/K) =0 i ank K < ank G and )«G/K)= n G/n K i ank K= ank G (nG is he numbe .o elemen so N(T)/T o T being a maximal  o us o G and n K is he co esponding numbe o K) . 8 5 P oo : a) Suppose i s ha ank K c ank G and le T be a maximal  o us o G .  The le ac ion o T on G/K has no ixed poin s because i xK = xK o all E T hen x-1 Tx C K which is imposible since ank Kc ank G . We use hen heo em (3) o conclude ha x(G/K)= 0 . b) Assume now ha ank'K = ank G and le T be a maximal o us o K and hence o G . We ha e F T (G/K)  = {xK1 x-1Tx C K} .  Bu x-1Tx is a maximal  o us in K, i x-1Tx c K . Thus by co olla y (9)  he e exis s k E K such ha x-1 Tx = k-1 Tk . The e o e xk-1 E N G(T) (no maliza o T in G) . Hence xK = xk -1 .kK = xk-1 K wi h xk -1 E NG (T) . The e o e FT(G/K) = {yKly E NG (T)} = . NG(T)AN C (T) n K) = N G (T)/N K(T) = NG (T)/T NK T /T" B i b l i og- ~p~Y_ The e o e x(F T (G/K)) =nG /n Kand we inish now he p oo _by using (3) . 1 .- W . G eub, S . Halpe in, R . Vans one ; Connec ions, Cu a u eand Cohomolo- gy . Vol .I . Academic P ess .