Cohomology operations and H-spaces

Pub . Ma . UAB Vol . 26 Nó 3 Des . 1982 1 . INTRODUCTION COHOMOLOGYOPERATIONS AND H-SPACES A . Zab odsky The heo y o cohomology ope a ions and he heo y o H-spaces we e in e locked h cuqhou hei a ious s ages o de elopmen : The . i s sys ema icapp oach o he heo y o (high o de ) cohomology . ope a ions is due o J . F . Adams ([Adams]) . In ha celab a ed pape a solu ion was gi en o a ques ionwhose one o mula ion is he ollowing : Wha sphe es suppo con inuous mul iplica ions wi h uni s (i .e . H-s uc u es)? The cohomologyope a ions o he ope a ions . These we e ied oge he Bocks ein spec al sequence which was dimensional H-spaces . [Zab odsky] 1,2,3' [ Kane ]1 , [Li ]1,2,3 and o he sused high o de ope a ions o . a he analyze he cohomology o ini eH-spaces . In pa icula , [Lin]1,2p o ed he classical "loop space congec u e" : The homologyo he loop space o a ini e dimensional H-space is o sion ee . [Hubbuck] 1 ~ 2,3 used k- heo yope a ions o s udy he cohomology and opologyo ini e H-spaces . He ound es ie ions on hei possible ypes and hei Pon iagin ings . Amongo he heo ems he p o ed ([Hubbuck] 1 ) ha a simples ype a e he Bocks ein by B owde ([B owde ] 1 2 ,3 ) o o m he used o s udy he cohomology o ini e 215 homo opycommu a i e ini e H-space has he homo opy ype o a o us . Finally [Kane12,3 ecen ly used BP ope a ions o s udy he cohomology o H-spaces . Going in he o he di ec ion, he heo y o H-spaces was used in he cbns uc ións and-e alua ions o high o de ope a ions . in he ollowing lec u es I shall y o demons a e by some examples hese ela ions be ween he wo heo ies . 2 . BASIC DEFINITIONS Wle usually assume spaces o be o . he homo opy ype o CW complexes wi h a (non-degene a e) base poin . Maps and homo opies a e base poin p ese - ing . Thus, an H-space could be assumed o be a space X wi h a mul iplica ion so ha he base poin *xo is an ac ual'uni : u(x,x 0 ) = x = u(xo,x) . The de ini iono a cohomology ope a ion has a ious deg ees o abs ac ions . . One o he mos gene al o m is he ollowing : A cohomology o e a ion ¿ consis so h ee .spaces and wo maps _ <K o , E, K 1 , , h> : : E - KoY h : E- " K 1 : de ines a "na u al ans o ma ion" om im([ , E] - "[ , Ko ]) o he amily o subse s o [ , K 1 ] . In a mo e di ec e ms : Fo any space Xm de ines a unc ion om a subse o he se [X, Ko ] o homo opy classes o maps X - "K o o he se o subse s o [X, K1] in he ollowingway : The domain o 0 is he se .im( ., : [X, E] -> [X, Ko ]) whe e * is he le composi ion wi h  : * ([ ] = [ . ]  ([u]  he homo opy class o  u) .  Hence, [ ] E [(,,K0] . is in he domain o o i and only i [ ] "li s" o [ ] . E [X, E], o i' . (see diag am D1) .  The alue ~D([ ])  is hen he se {[h o ]1 o % } c [X, K 1 ] (Dl) In case E = Ko , = 1 he ope a ion is called ima and is simply he igh composi ion wi h h . The domain o o is hen all o [X,K 0 ] and i s alues a e single ons, i .e . : elemen s o [X, K 1 ] . This is a gene al o mula ion which is no e y use ul i one does no es ic onesel . o some speciaL cases .  No mallywe conside cohomology ope a ions eTa ed o .(gene alized) cohomology heo ies (hence he name) . Al] cohomologyope a ions he e will be gi en in e ms o si-spec a : An .  9 .  spec um is a sequence . .E * = {E n , `pn}n=o  whe e  E n  a e spaces and a e homo opy equi alences, (p n : En a  n+l' The cohomology heo y E* associa ed wi h he sa-spec um E is a . L, .,  sequence  oi  u  . . nco  s a . . . . . .  l . L n , I  _  L  ,  E n l J  a 1 . . . a he  -  ~  wia  i s :  Fo á  space X - E n(X) = [X, E ' . Fo a .map : X - " Y E n ( ) : E n (Y) - E n (X) is he igh composi ion wi li : E n ( )[u] = .[u o ] (u : Y - E n ) . As E n a e double loop spaces (and much mo e) . E n (X) a e .abelian g oups and E n ( ) a e homomo phisms . 3 . PRIMARY OPERATIONS . STABLE OPERATIONS An elemen a y p ima y ope a ion o ype m,n in he cohomology heo y E* is an elemen  [a] E  E n - E n .  I de ines a p ima y ope a ion m = <Ko = E m = E, K 1 = En , = l, h = a> which is ob iously he le compesi - ion wi h a . The se o all p ima y ope a ionso ype m,n is he se [E m , E n ] .  As an ope a ion a is a unc ion E m (X) - . En(X) . A s able elemen a y p ima y ope a ion o deg ee k is a sequence a = {a n E [ E n'  En+k]ln=o  ( o k < o we conside E = poin o < o) .  These a e ela ed by he ollowing (homo opy) commu a i e diag am : i  a n En En+k (D2)  (D nl  1 O n+k Sia .n+l í2 E n+l  --, í? E n+l+k In his case a n : E n (X) - . E n+k (X) a e homomo phisms . The se o s able cohomology ope a ions in he heo y E  o ms a g aded . ing :  One can add any wo ope a ions o he same deg ee as  [E n , E n + is an abelian  g oup .  The p oduc  is gi en by :  (n" " a') n = a~ +k o a~ i a' is o deg ee k . The deg ee o a" " a' is he sum o he deg ees o a'  and a" .  These de ini ions a e consis en wi h he de ining ela ions o a s able ope a ion (D2) . Exam le :  The S een od Algeb a .  Le  E n = K(Z/pZ,n) - he Eilenbe g MacLanespaces, p-a p ime,(E,, is hen called he Eilenbe g MacLane spec um K(Z/pZ)) . The ing o elemen a y s able cohomology ope a ions is called he S een od algeb a a(p) . Fo p = 2 a(2) is gene a ed by ope a ions Sq~ o deg ee i  (Sq o .= 1) subjec o ela ions known as he Adem ela ions . Re e ence : [S een od-Eps ein] . 219 A non elemen a y p ima y cohomology ope a ion in E  is a map a : E(0) - E(1) whe e E(i) = n E(') i = 0,1 . One can easily see how o de ine a non elemen a y s able p ima y ope a ion . Such an ope a ion is gi en by a ma ix whose en ies (a ij ) a e elemen a y s able ope a ions wi h he p ope y : deg ee a .  - deg ee a .  . is independen o j . * Fix he cohomology heo y E . By E(i) we always deno e a p oduc o e ms  ni _Ek~ ) . (E(i,j) will deno e o he ypes o spaces as will be seen j = 1 j in he sequel) . Gi en (non elemen a y and no necessa ily s able) ope a ions 01 0 : E(0) ~ E(1), a l : E(1) - " É(2) . " A ela ion among p ima y ope a ions is a ela ion o he ype a 1 , no i * . (*- he cons an ma P) " (I a l a e s able, and he e o e gi en by ma ices, his ela ion desc ibes o dina y ela ions in he ing o s able ope a ions) . The ela ion a l o no % * induces a commu a i e diag am : 4 . SECONDARY OPERATIONS ASSOCIATED WITH A RELATION n E(2) whe e E(i,i+1) is' he homo opy ibe o a i , i =0,1 . j1 is he inclusion o he ibe o 1 . 01 0,1 exis s since a l ono % * . ao,1,2 is induced by el oj .  no,1  and  a o,1,2  a e uniquelyde e mined by he choice o he homo opy E(0) x I + E(2), * i a l o 01 0 . The ope a ion o =< E(0), E(0,1), n E (2), 0, n o,1,2 >  is called a seconda y ope a ion associa edwi h he ela ion The abo e ope a ion  o depends on he choice o a o,l , o as ema ked, on he choice o he homo opy * i a l o no . The di e ence be ween choices o such homo opies is gi en by a map  w : E(0) --  2 E(2) .  The di e ence be ween he wo maps ao,l .2 and ao,l,2 induced by he wo choices o homo- opies is hen gi en by ao,l,2 -a o,l,2 = w , o . Gi en a space X and a cohomologyclass  x E [X, E(0)] . (x is n . ac ually a " ec o " o cohomology classes x n . (0)E E J(X)) . x is in he do- J _ main o o ( o any o, induced by any null homo opy * La l . a o ) i and only i  a ox = 0 .  The  alue  o(x)  is  hen  [a o,1,2  ° x]  whe e  x  :  X  , E(0,1) is a "li ing" o x : X - E(0), o o x i x .  l a', <D" co espond o wo di e en homo opies * i a l , ao whose di e ence, as abo e, is w : E(0) , 9 E(2)  hen  [ao,l,2 o x] - [ao~,] , 20 X] = [w o o o X] = Lw o x] . Hence, D"(x) is ob ained by ansla ing <D'(x) by w . x whe e w E [E o , 2 E(2)] is a p ima y ope a ion . This could be o mula ed as ollows : A ela ion  a lo a o i * among p ima y ope a ions induces seconda y ope a ions {m} . Any wo .such ope a ions di e by a p imá y ope a ion . 5 . MASSEY PRODUCTS - TODA BRACKETS . HIGH ORDER OPERATIONS Le a o : E(0) -- E(1), a l : E(1)  E(2), a2 : E(2) y E(3) be p ima y ope a ions and suppose a l o ao i *, a 2 o a l i * . Again, E(i) a e p oduc s ni Eñ e) . Ex end diag am (D3) o ob ain : J=1 E(i,i+1) - he homo opy ibe o a i , 3 i - he inclusion o he ib e . ao,l, a l,2' a l,2,3' ao,1,2 exis as a l o ao a 2 o a l * . They a e uniquely de e mined by choices o ` homo opies * i a l o a o , * i a 2 a a l . The class [(1 1,2,3 - ao,l ] E [E(0), 2 E(3)] is a p ima y ope a ion . Two di e en choices o he homo opy * i a l oao will yield' wo maps ao ,l , a~ ,l . These maps a e ela ed by [a l,2,3 - a o,l] -[x 1,2,3  [í2a 2 ° wo]' whe e w o E [E(0), o E(2)] measu es he di e ence be ween he wo choices o homo opies  * i a l  o ao .  (No e ha he di e ence  (1 1,2,3 0 a o,1  - a ]  2,3 ° ao,l is independen o he choice o he homo opy  * i a2 a a l and i s induced map (11,2,3') (R)  implies  [a l  o n o ] = Sq 4n+1 Sq 1  + Sq2Sq4n = Sq4n+2 = 0  ( he la e anishes by he non s abili ycondi ion  Sg4n+2[1 E(o) ] = 0  as 1 E(o) :  E(0) -> E(0)  is an elemen o  H 4n+l (E(0),  Z/2Z)) . We shall in es iga e he alue o a seconda yope a ion i associa ed wi h * i a l ~ no on a p imi i e  class  x E H 4n+l (X, Z/2Z) in he domain o (We shall conclude ha he e is no such class and he e o e Sg 4n x ~ 0 o .any p imi i e elemen o dimension 4n+l .) H de ia ions : Le X,p, Y, _ p ' be H-space . Gi en a map : X } Y he e exis s a map D : X n X - . Y called he H-de ia ion o wi h he ollowing p ope íes . (Compa e wi h [Zab odsky] 4 , Chap e 1 whe e D is deno ed by H D( , , p')) : i)  The wo maps X  X:Y gi en by x, y - (x .y) and x, y - D (x,y) " [ (x) " (y)] a e homo opic (he e ( ) " ( ) deno es bo h p oduc s u and u') o in a unc ional no a ion : o u i po{D n x [u .( - )]1 -, áxxx whe e á X-X (x,y) ` (x,y, x,y) is he .diagonal map and n : .X x .X y X n X - he p ojec ion . ii) is an H-map i and only i D i * . iii) Le X o " X i be H-spaces and X2 - a loop space . Gi en maps 0 : X o - X 1 , : X 1 ~ X 2 . Suppose D lo (D 0 Al) : X0 n . X 0 n X l ~X2 is null  homo opic .  Then  [D 1 o 0 ]  = [D 1o ( 0 n o )] + [ l ,D ] . 0 In pa icula : I l is an H-map D a % l o D and i 1 0  0 o is an H-map D , i D , ( on o) . 1 0 1 Now conside again he ope a ion 1 associa ed wi h (p) and he ollowingcommu a i e diag ams : K(Z/2Z, 8n_+3) Bj (D7)  B E(1,2) B  1 B 1  g K(Z/2Z, 4n+2) - o-, K(Z/2Z, 4n+3) x K(Z/2Z, Sn+2) -- 1 - K(Z/2Z, 8n ;l4) Bao is gi en by p l ° Bao = Sq l : K(Z/2Z, 4n+2) - K(Z/2Z, 4n+3), p2 ° Bao = Sg4n : K(Z/2Z, 4n+2) - . K(Z/2Z, 8n+2) . Ba 1 = Sg4nó1p,  + Sq 2 °  p2 ,  hence  2 Ba o % CC o ,  s2 Ba l  i a l .  E -  he homo opy ibe o S g 4 n+ 2 , B E(1,2) - he homo opy ibe o Ba l , sz B E(1,2) = E(1,2) as in he (D4) diag am de ining m . Loop he abo e diag am and obse e ha 9 E _ K(Z/2Z, 4n+1) x K(Z/2Z, 8n+2) and he e o e =s2 B : 9 É  K(Z/2Z, 4n+1) admi s a le in e se x : K(Z/2Z, 4n+1) ° x i 1 . Oné can see ha he choices o such in e ses (also called c oss sec ions) a e in 1-1 co espondence wi h li ings ao,l : E(0) = K(Z/2Z, 4n+1) - E(1,2) o diag am 04 o D . Thus, looping (D7) one ob ains : (D8) RBj=j Sa É SZ & 1 112 E(O)= K(Z/2Z, 4n+1) 1 ; 1 E(1,2) 1 . 1 E(1) I 1 (any choice!) is an H-map one can use some obs uc ion heo y o show ha hen x is indeed a loop map . Hence, B admi s a c oss sec ion B2,  B , Bl  1 BE(o)'  This will  imply ha  Sg4n+2 :  K(Z/2Z, 4n+2) -- K(Z/2Z, 8n+4) is null homo opic which is alse . I ollows ha x is no an H-map and D : X E(0) A E(0) -" 2E is no null homo opic . Now, [E(0) n E(0) = K(Z/2Z, 4n+1) A K(Z/2Z, 4n+1), 2 E a K(Z/2Z, 4n+1)%K(Z/2Z, 8n+2)] a H 4n+1 (K(Z/2Z, 4n+1) A K(Z/2Z, 4n+1), Z/2Z) + H 8n+2 (K(Z/2Z, 4n+1) A K(Z/2Z, 4n+1), Z/2Z) . The i s summand is ze o (E(0) A E(0) is 8n+1 connec ed) he second equals Z/2Z . . Hence, he only non i ial map in  [E(0) A E(0), 2 E]  is gi en by w K(Z/2Z, 4n+1) A K(Z/2Z, 4n+1)  - K(Z/2Z, 8n+2) -~-, n E and DX  wd (w d.is also deno ed by '4n+1 (x) `4n+1)'  By he p ope ies o H-de ia ions [p ope y (iii)] [Da  ]_ [s2 a o D ic ] , _ En a o J o w o ] = [J l owo ] 0,1 And again, his is ue o any choice o  a o ~ l . Now conside he (D4) diag am o D and i s e alua ion on a p imi i e class  x E H4n+1 (X, Z/2Z), xE ke Sg4n ,  (x E ke - Sg l by ema k a)) . (D9) P E(1)  s ~-+ 2 E(2) CL o,1,2 ~1 Sg4n x=0, Sg l x=0 implies a 0 . x i * and x  exis s . Now, x, 0 . a e H-maps hence : 0  =  ED x]  =  ID .  0 X]  =  E 0  0  DX],  hence,  . 0 DX : X ,E0,1 li s o a map w : X n X - . 9 E(1), DX i jo 0 w . Ej l  0  Da ]  =  ID .  a  ]  =  IDa  0 ( 0  A  0)]  =  Ej 1  0  w 0  .( o  n  o ] . 0,1 ,2  ~l° 0,1 .2  0,1 Now, E(0) n E(0) is 8n+l connec ed, n i (sz E(1))= 0 o i hence EE(0) n E(0), si E(1)] = 0 and consequen ly j l* : EE(0) n E(0), sZ E(2)] - . EE(0) n E(0), E(1,2)] is injec i e and 0 ( n ) . e¿ o .1 .2  w 0  0  0 Now, Da  0 (Dx A 1) ,, w o0( 0 ^ 0 )0(Dx n 1) _ o,1,2 On he o he hand, conside  w ' E EX n X, 2 E(])] a 1 . j0 . w o0( o 0 DX n 0 ) -n . *as o 0 DX i D x Hence he condi ions in p ope y (iii) o H-de ia ion hold and 8n, ED "o,1,2 0x] - Eao .l .2 0Dx] + ED ao l 2° (x n x)] = [ao,1,2 0 j o w] + . . +Ew0 0 ( 0 ^ 0 ) 0 (x n x)] = Esza l 0 w] + N oo (x n x)] . Ew 0] - 1 4n+l 0 '4n+1' Ew0 0 (x n x)] =x i . x E H* (X n X, Z/2Z), * Now, he image o  x O x  in  H (X x X, Z/2Z)  is o algeb a il a ion 2 (and no o il a ion > 2) as  x © x = (x 0 1) "  (1 U x), and  x  is in- decomposable . H4n+1 (X ^X, Z/2Z) + H8n(X n X, Z/2Z) .  Fo dimension easons he image o  w in EX x X, sa E(1)] mus ha e algeb a il a ion a leas .4 : The image o H * (X n X,  Z/2Z) - . H* (X x X,  Z/2Z)  has il a ion  > 2 .  As all  gene a o s in H*(X x X, Z/2Z)  a e o cong uency  s 1  (mod 4)  elemen s o dimension mod 4 ha e il a ion > 4 . Elemen so dimension il a ion > 1 mus ha e il a ion %5 . - H * (X x X,  Z/2Z)  is  injec i e) sequen ly  [D  n,] $ x 0 x D  i ~ * and a o,l ,2 ' x he e a e no algeb a gene a o s in hese dimensions . * . As his holds, o all choices o a o,l , 0 1 o(x) . Mo eo e , one can use Hop algeb a p ope ies o The conclusion is he e o e ha he e a e no p imi i e elemen s o 1 mod 4 and o As he S een od algeb a p ese e il a ion (and H *(X A X, Z/2Z) [ga l ow] has il a ion > 4 and con- mod F4H * (X xX, Z/2Z)) . In pa icula , * H (X, Z/2Z) and he abo e e alua ion o D  i o conclude ha he elemen s in e, o,1,2  -  x m(x)  a e al]  gene a o s .  This is impossible o  P(x) c H 8n+2 (X,  Z/2Z)  and in  H *(X .  Z/2Z)  in he domain o  o .  As  Sq l x = 0  o e e y  x E H* (X,  Z/2Z) Sg 4n x # 0  o e e y p imi i e el .emen  x  in  H 4n+1 (X, Z/2Z) . Rema k : The e a e H-spaceswi h his ype o cohomology : I Sp is he simplec ic g oup hen Sp p 52 2X .  Bo h X and he uni e sal co e ing space o 9 2 Sp a e H-spaceswi h . .cohomology o he ype desc ibed in he heo em . We shall show he e how he heo yo cohomology ope a ionsuses H-space heo y . Conside he Adem ela ion (R 1 )  Sq2Sq 2 + S g lS q 2 Sq 1 = 0 R 1 induces a seconda y ope a ion o 1 desc ibed by he (D4) ype diag am as ollows : a PE(2) = K(Z/2Z,N+3) K(Z/2Z,N) =E(O) - O ~ E(1) = K(Z/2Z,N+2) - K(Z/2Z,N+3) -~ E(2) = K(Z/2Z,N+4) o¡ 0is gi en by p l e a o= Sq2 , P2 , ao a l is gi en by [a l ] = [Sg2 " p l ] + [Sql a l  o  lo i  *  by  (R l ) . 7 . H-SPACES AND COHOMOLOGYOPERATIONS Sg2Sg 1 P2] he .p e ious chap e [P l a a o o Sg 4n ] =[Pl o ao o .Sg4n o p] = [Sq 2 o Sg4n o p] _ _ [Sg 4n+ o p + Sg4n+l .e Sq l o p] . (Di 0) Conside he composi ion Sg 4n as in he las chap e ° 4n K(Z, 4n+1) p - . K(Z/2Z, 4n+1) - S g---~ K(Z/2Z, 8n+1) whe e p is induced by he educ ion Z - Z/2Z . Sg 4n is in he domain o i ( o N = 8n+1) . Indeed, by (R) o loop maps . Now, Sq 4n+2 [p] = 0 as p is o dimension 4n+1 (using he non s abili ycondi ion o he S een od algeb a) . [Sg l op] = 0 as H 4n+2 (K(Z,4n+1),M) = 0 o any coe icien s module M . Consequen ly, [p l oa o oSg 4n ] = 0, [p 2 oa° oSg4 n ] = [Sg2SgluSq4noa] . Using Adem ela ions one has Sg2Sg1Sq4n = Sg4n+2Sg1 and as Sq l [ p] = 0, [P2oao,Sg4n] =0 and [Sg 4n ] E Ke a o . We shall e alua e o[[Sg 4n ] : As in he p e ious chap e : D& _ [n&~ °) 0x ] _ [a&~ °) oj o ow o ] = [j ooi l ow o] 0  0 K(Z,4h+1) Sg ~  K(Z/2Z 8n+1)  a°  . E ( l j  a  -- . E(2) w o= p 0 p E [K(Z,4n+1) n K(Z,4n+1), K(Z/2Z,8n+2)] . ¡Pl SQ4n+2 1 ° ----------------- . K(j/2Z,8n+3) d o' o , xo , nE o a e analogous o j, , x, siE in (D8) and sha e simila p ope ies . All spáces and maps excep o xo and &° a e loop spaces and 4n _ whe e As  a o,1,2  is an H-map  Da o,1,2' a0 =  [a o,l,2 ] `  Da0 = _[a o11,2  j o ~i 1 1w 0 1 = [na l ~ i l ow 0 ] = Sq 2 o w0 (2al`i1 = Sq 2 * K(Z/2Z,8n+2) ., K(Z/2Z,8n+4)) . Using he Ca an o mula ([S een od Eps ein]) one ob ains o any li ing a0 o Sg04n : Da  a  = Sq 2 w0 = Sq2 (pop) = Sg 2 p op + p © Sg2 p  (Sq ' p=D) . 0,1,2` o s u : K(Z,4n+1) -> K(Z/2Z,8n+4) is being gi en algeb aically by [u] = [p]-S9~[p] (o "geome ically" by he composi ion K(Z,4n+1) A y K(Z,4n+l) - K(Z,4n+1) -px - P --> K(Z/2Z,4n+1) X K(Z/2Z,4n+1) Sq 2 xl  _ K(Z/2Z,4n+3) i K(Z/2Z,4n+1) -"-+ K(Z/2Z,4n+3) A K(Z/2Z,4n+1) K(Z/2Z,8n+3)]whe e ® ep esen s he nene a o o H 8n+3 (K(Z/2Z,4n+3) A K(Z/2Z,4n+1), Z/2Z = Z/2Z) .  Then  D u = Sg2 p 0p + p ® Sg2p o a cohomology class u o an H-space X is he educed cop oduc in (D u he Hop algeb a H (X,Z/2Z)) . l ollows easily ha i  = [ a 0,1,2' ao ] EP,(Sg4n )  is any elemen -u is p imi i e . Now, one can show ha á 0 can be chosen so ha =u and [p] " Sq2 [p] E o l (Sg4n ) . (Ou ]ine 1oopino o p oo :,,(Dll) wice one ob ains 22(Sg0on) i * and o some z : K(Z/2Z,4n-1) -> n3 E(1) = K(Z/2Z,8n) x K(Z/2Z,8n+1) jo H-map as  92 % % n2  o z  is  an  H-map,  0 =  [92 jo]  o  Dz , n K(Z/2Z,4n-1), 23 E(o) = K(Z/2Z,8n-2)] = 0 (2 2 jo ) * on 3 A K(Z/2Z,4n-1), 0 E(1)]  is injec i e, D z= 0 .  Any H-map be ween 2 2 6, 0% . 22 j o ,z z mus be an [K(Z/2Z,4n-1) [K(Z/2Z,4n-1) Eilenbe g MacLane spaces is an -loop map o any   and  z i n 2 z  o some and as z : K(Z/2Z,4n+1) -,- s2E(1) . Use z o change he homo opy i ao o Sq o and hen, o he new ao one has 2 [no,1,2oáo] = 0, * i 22 i áa 2 ( -u) (as P2u i *, since 51 2 A `, * in he "geome ic" de ini ion o u) .  Bu  one  can see ha  o 2 : H 8n+4 (K/Z,4n+1), Z/2Z)  + H8n+2(K(Z,4n-1),Z/2Z) is injec i e on p imi i es (Eilenbe g Moo e spec al sequence) and u . i ) . Consequen ly : 01(Sgo n ) _ [p] oSq2[p] + im Sq 2 + Co olla y I :  Le  x E H 4 n +1 (X,Z) be any class > X - any space . I 4n 4n  22 1  8n+2 Sqo x = Sq  px = 0  hen  px " Sq px = Sq yl  +Sq y2  o some  yl  E H  (X ,Z/2Z), y 2EH 8n+3 (X,Z/2Z) . P oo o I : Conside he ollowing i  J oI , a o,1,2 1-1 E(0,1) im Sq l . X - X " K(Z,4n+l)- o -- " K(Z/2Z,8n+1) 2) a o' °`o,1,2 as in D11, d o chosen so ha a o,1,2 ,ao = [p] Sq2[p] . As Sg 4n x = 0 ao  x = j o , y o some y : X , iE(1) d K(Z/2Z,8n+2) - K(Z/2Z,8n+3) . Pu yi = p i y and hen px " Sg2px = [a o,l,2o aoox] _ [pa lo y] = Sg2yl + Sg1y2 . Co olla y II : The e is no space X wi h H* (X,Z/2Z) being he ex e io algeb a on- x  and  Sq 2 x,  dim x = 4n+1 .  (I .e .  X sa is ies :  H i (X,Z/2Z)  ~ 0 only i i=0, 4n+1, 4n+3, 8n+4, and in hese dimensions H i (X,Z/2Z) s :d Z/2Z wi h non ze o elemen s 1, x, Sg 2 x and x " Sg 2x o i=0, 4n+l, 4n+3 and 8n+4 espec i ely1 P oo o Co olla y II : In such a space Sgó nx = 0 (as H 8n+1 (X,Z/2Z) = 0) bu 0 ~ x o Sg2x = Sg2 y 1 + Sg l y 2 is impossible o he e a e no elemen s in he dimensions o yl and y 2 .