Flat submersions and horizon immersions

Pub . Ma . UAB N ° 21 Oc . 1980 Ac es VII JMHL FLAT SUBMERSIONS AND HORIZON IMMERSIONS F .J . C a ei o de Ca alho Dp o . d e Ma emá ica Uni e sidade de Coimb a Abs ac The no ion o la subme sion is in oduced . We p o e ha no co e ing map, wi h o de g ea e han 1, is 1- la , ela e 2- la sukene sions M n+l ->S n o ho izon imme sions and show ha k- la  suJxm sions Mm _ Sn gi e ise o embeddings ~ n - Rpn+k . 1 . In oduc ion Th oughou his no e all he maní oldsa e C , Hausdo , compac , connec ed and bounda yless . All he maps a e C , unlesso he wise s a ed . I M and N a e mani olds, o dimensions m and n espec i ely, we shall deno e hem byMm  and  N n . Le  11 : N? --+ N n be a subme sion . De ini ion 1 : We say ha II is a k- la subme sion (k>m-n) i he e exis s a map F : M m -o Rk  such ha , o e e y y e Nn ,  F111 -1 (Y)  is an embedding . P oposi ion 1 : I 11 is a  k- la subme sion hen  ( í,F) :Mm --N n x Rk  is an embedding . P oo : I is clea ha (1I,F) is C~ and injec i e . The only hing o wo y abou is he ank o  (11,F) . Le x be any poin in M m . We p o e ha he ke nel o  (I1,F)*x : Tx J -- TII(x) Nn x T F(x) R k  is  O-dimensional . As ke (1T,F) *x = = ke R* X lke F *x , i  e ke (TI,F) *x we ha e  F *x ( ) = 0 . On he o he hand * AMS  subjec classi ica ion  57 D 40 ** Suppo ed in pa by Calous e GulbenkianFounda ion and INIC, Lisbon,Po ugal eke R*x,ke n *x = i *x (T x n-1(R(x))),whe e  i  is he inclusion  n 1 ( i(x)) --> M, and  FIT _1 (x)  is an embedding, he e o e   mus be ze o . 2 . Fla co e ingmaps In his sec ion we deal wi h he case  m=n . We shall conside a ,mo e gene al si ua ion . Le M be a compac , pa h-connec ed opological space and II :h -> N  be a co e ing map, whe e  D :  is also a opological space . P oposi ion 2 : Le  n :M ^ . A'  be a co e ing map such ha , o any  ycN, :W n -1 (y) > 1 . I  :M --> R  is a con inuous map hen he e exis  x  and  y  in he same ib e such ha  (x)  = (y) . P oo : We use some ac s abou co e ing spaces . E4] . Le x l be an absolu e maximum and  y l be an absolu e minimum o . Take a con inuous pa h y : E0,11 -+ M  such ha  y(O) = x 1 and  y(1) = y 1 .Choose  x 2 ~xl , wi h l(x 2 ) = i(x 1 ) and conside he li y' o POy such ha y'(0) =x2 . We de ine he map  ~ : E0,11 -> R  by aking  O( )  o be  oy( ) - oy'( ) . I 0(0) o ~(1) is'ze o hen he p oposi ion is p o ed . I no ,  mus ha e a ze o O , because ~(0) > 0  and . 0(l)< 0, and he p oposi ion ollows . An immedia e consequence is Co olla y 1 : I  :S n -- " R  is con inuous hen he e exis s  x ES n  such ha (x) = (-x) . O cou seco olla y 1 can also be , deduced om he Bo s u k-Ulam heo em L4- . 3 . Fla subme sions and ho izon imme sions We s a wi h he de ini ion o ho izon imme sion L2] . Le :M --> Rm+1 be an imme sion and L deno e an a ine line in Rm+1 . De ini ion 2 : We say ha  :b? n --> R m+1  is a ho izon imme sion wi h base-line L i , o e e y xEJ, he a ine angen space o  (M) a  (x) does no con ain L . In wha ollows we shall be conside ingsubme sions n :Mn+l -} Sn and we shall use he ollowing chain o di eomo phisms whe e X1 (x, l , 2 ) = (x e , 2 ), X 2 (x, l , 2 ) =( i x, 2 ) and R deno es he posi i e eals . The inclusion  i :R n+I {0} x R -+ Rn+2  will also be used and he composi ion io X 2 o X 1 will be deno ed by X . F om now on L will deno e he  a ine  line  { (0, . . . ,0, ) e R n+2 : e R} P oposi ion 3 : X  X S n x R 2 1-- .  S n x R+ x R ? > Rn+l {0} x R a) I 1I :j+l_  > Sn . is a 2- la subme sion hen X0(11J) is a ho izon embeddin g, wi h base-line  L,  such ha  (XO(II,F)) 1 (L) _ 0 . b) I :Mn+1 . -i Rn+2 is a ho izon embedding wi h base-line L such ha  -1 (L) = 0 hen  11 :M n+1 - S n gi en by subme sion . (11 11 deno es he s anda d no m in R  ) . P oo : a) I 111 is 2- la hen (11,F) is áns e se o { y} x R 2 , o e e y YES n .  I - is easy o check . ha X0(11J) is also ans e se o each  2-plane con aining L and he e o e no a ine angen (n+1)-plane o X0(II,F)(M) con ains L . Tha is o say, X0(11J) is a ho izon embedding . b) We eplace by ' :M n+1 = . Rn+1 {0} x R . Using X1 1 and X21 we ob ain a map M -> S n x R2 o which he i s componen is 11 and he second one has he equi edp ope y o 2- la ness . We ema k ha 11 has no c i ical poin s because no angen (n+1) - plane o (M) con ains L . 11(x) = ( 1(x), .. ., n+1(x)) /11 ( 1(X), . .1, n+1(x))11 is a 2- la n+2 Co olla y 2 : I 11 :0 +1 --> S n x S1 . P oo : Robe son and Chillingwo h ha e shown E2] 3] ha i  :M n+1 --* Rn+2, n > 1, is a ho izon imme sion wi h base-line  L, such ha  -1 (L) = 0 hen  M is di eomo phic o he Klein bo le o S n x S 1 .Howe e he Klein bo le canno admi a 2- la subme sion because i canno be embeddedin 3-space . S nis a 2- la subme sion hen M is di eomo phic o 4 . Embeddíngs in eal p ojec i en-space RP n . In El] i was shown ha i 11 :5 1 x S 1 --+ S 1 is  2- la i is possible o cons uc an embedding S 1 x S1 -+ RP 3 , ela ed o he Hop map h :S 3 -+ S2 . In his sec ion we show how o ob ain an embedding M m - . RP n+k  i we ha e a  k- la subme sion  1 :M m - . S n .  Le , us ake  F :~P ---u Rk , associa ed wi h 1I l necessa y, we eplace  F  by  F' :¿ n -+ Rk such ha  F'(M) G R+k . We ake T :S n x R k -+ RP n de ined by T(x,y) = Lx,y] ,  whe e Lx,y] is he 1-dimen_ sional subspace o Rn+k+l de e minad by (x,y) . The ank o Y' is n+k and i is s aigh o wa d  o check ha  Y Y 0(11,F)  is an embedding . Thanks a e due o S ewa A .Robe son o sugges ing he p oo o p o- posi ion 2, much simple han ou p e ious one . Re e ences Robe son,S .A . - Pe mu a ions associa ed wi h c i .ical poin s on su aces J .London Ma h .Soc .37 (1962), 329-337 . L2] - Robe son,S .A . - The-dual o a heigh unc ion, J .London Ma h .Soc . (2), 8 (1974), 187-192 . 13] - Robe son,S .A, and Chillingwo h, D .R .J . - Ho izon maps ( o appea ) . E4] - Spanie , E .H . - Algeb n .a,¿c Topozogy,  Ta a McG aw-Hill, New Delhi (1966)