Note on block invariants

Pub . Ma . UAB Vol . 26 Ns 3 Des . 1982 NOTE ON BLOCK INVARIANTS Shen Xin-Yao In [6], he T- o sions o an N-dimensional CW complex K a e in oduced . Using heseblock in a ian s, we can lis he gene a o s i and hei o de o he 2-p imany componen o cohomo opyg oups nN  1 (K) and nN - 2 (K) . Now, we gene a- lize hese blockin a ian s and discuss hei p ope y . Espe- cially, some necessa y condi ions a e ob ained o he exis en ce o a c oss-sec ion . 1 . Le K be an N-dimensional CW complex, n = N - 1, m = N - 2 . We ha e an exac couplewhich is based on he coho- mo opy exac sequence o he iple (K, Kg, K4  1 ) : (K,  Kg )  n (K, Kq  1 ) 7, (Kq Kq - 7  +  l (K,  K q)  . . . , whe e i and j a e he homomo phisms induced by he inclusions (K, Kq - 1 ) - (K, Kq) and (Kq Kq - 1) - (K, Kq  1 ) . espec i ely, and k is he cobounda y ope a o o he iple (K, Kg, Kg  l ) . De ine p ecisely, A ,q = and he homomo phisms A = Eq A ,q, c= E C ,q , C[ i ,ci : A ,q -, A ,q-1 . ,q : A ,a -, C ,q+1~ k ,q : C ,q -, A +1,q  + 1 S N  2  ; > NN±1 2 ' ,c,  +l  c  i  +1  1  N+1J_  l) c  = ~ ke (~  (K,  K -)  -'  n  (K,  Kq )),  =  2  , < C N 2 l, , a e he app op ia e  i,  j  and  k  o  >21,  when k , g is he inclusion, and he emaining homomo phisms and k a eall null homomo phisms . , j .T . is clea ha  <A, C ;  i,  j, k >  is an exac couple . 1) [N+1] s ands o he la ges in ege no exceeding N21 . 140 Conside he i s de i ed couple  < ,H ; i,j,b >  o he cohomo opy exac couple  <A, C ;  i, j, k >  . By de ini ion, .q = Im i ,q whe e  d .q = j +l,cT , k ,q : C ,q  C +l,q+l  and homomo phisms and ,a __ ke d 'q H  Im d -1'q-1 i .q : .q -~ .q-1 j .q : ,q -~ H ,q+1l b ,q : H ,a __, +l,q+1, a e induced by i .q - 1 , j .q(i ,q)-1 and k , q espec i ely . The g oup  (Kq, Kq-1 )  can be in e p e ed as he  q - h cochain g oupo K wi h coe icien g oup aq(S ) [1] . Thus H 'Q . Hq(K ;7 S (S )) . The homomo phism d,  =j° b :  Hq (K ;  s  (S -l )) -+  H q+2 (K ; i q+2 (S )) is a cohomology ope á ion [4) ; in pa icula , d o =jo b :  Hq (K7 . á(S q )) -  Hq +2 (K ; aa+2 (Sq+l )) d o =jo b : H q (K ;nq(S q-l )) --> Hq+ 2 (K . 7 q +2  (S q )) a e S een odsqua es omin eg alcoe icien s o coe icien s mod 2 and omcoe icien s mod 2 o coe icien s mod 2 . [3] ' ion as ollows . b ' q (a) EIm i +1,q+2 . Le 01E +1,q+2 imply iSc+1,q+2(P1) __ b , q(a) . Then J +l,q+2(R1) Eke d +1,q+3 . I is easily o e i y ha he class  h1 (a)  _ {7 +1,g+2(~')} E H +1,q+3  is indenenden o he choicemadeby he elemen (i 1 . Thus we ob ain he homomo phism h . . is he Adem'sseconda y ope a ion (D . On ke  dó'q,  we can de ine a seconda y cohomology ope a - h l : ke d , q --~ H +1,q+3 = ke d +1,q+3 Imd ,q+1 I  a Eke  d o 'q,  hen  J +l,q+ób ,q(a) =0,  hence In [5], he ollowing heo ems a e p o ed . Theo em 1 . The seconda y cohomology ope a ion hl : ke dq'q - Hq+1,q+3 Theo em  2 .  Fo  N >3,  we ha e a sho exac sequence n+1 0  H (K ;Z 2 )  7 n (K)  Hn (K ;Z) --> 0  (1) Sq 2 H n-1 (K ;Z) When  N >5,  o  i m (K)  we ha e 0 --' I' -i m (K) -> ke Sq 2 ( CH'(K ;Z)) - 0  (2) 0 --, Coke (D -> I' --> Coke Sq 2 , 0 Hm+2(K ;22)  2 Hm+1(K ;Z2) whe e Coke  <P =  - .Id coke  Sq  = Sq 2 H m (K ;Z 2 )+Im 4)  Sg 2 H m-1 (K ;Z) This heo em makes i clea ha , using cohomologyg oups and cohomologyope a ions, we may de e mine he s uc u e o he cohomology g oups n (K) and m (K) wi hinan ex ended limi o p ecision . 2 . Fo de e mining heseg oups exac ly, we i s no e Co olla y 3 . n (K) and Hn (K ;Z) ha e he same ank and odd p ima y componen s . Now, we conside he  2-p ima y componen o  i n (K) . Deno e he p-p ima ycomponen o g oup  G  by  G (p) and mG = { g : mg = 0} . Assume Hn(K ;Z)(2)_ Z211+ . . .+2211+2212+ . . .+Z212+ . . .+Z21 + . . .+Z21 . S 1  S 2  S 1 1 >1 2 > . . . >1 . 143 In (61, he cohomology ope a ions H n+l (K ; Z  ) 2 (1) (k) " 1 Hn(K ;Z) __3 . Coke Sq2 = T  . 2 k  Sq 2 Hn - l(K ;Z) a e de ined . Each ope a ion has he p ope ies : n (i)  Fo  { Z } E 21k H  (K ; Z) ,  we  can choose  e E , n (K)  such ha j e = {z} . Because 2 1 k{z} =0, hen j(2 l ke) = 21kj(e)= 21 k{z} =0 .  Le  FE Coke Sq 2 be an elemen such ha  iF=2 lke . Then T (1) (k) ({z}) = F . (No ice ha he elemen F is uniquely de e mined in Coke Sq 2 ) 144 (i i)  T (1) (k)  1  21 Hn(K ;Z)  =  0,  k > . Using heseope a ions,we can de e mine he cohomo opy g oup  n n(K) (2)  om  (1) .  In ac [2 ' 61 ,  we can econs uc e he g oup i n (K) (2)  by Hn(K ;Z)(2),  Coke Sq 2 and T (l) (k) as ollows .  Le  el+l , . . .,  eu +1  be a basis o Coke  Sq 2 ,  and el, . . ., es i ; es1+ .1, . . ., esl+s2 ; . . . ; e -1  , . . ., e  be he iEl si + l i E l si gene a o so Hn(K ;Z)(2~, he o de o ek_ 1 be 21 k . E si+j i=1 j  = 1, . . .,s k ,  k = 1, . . ., .  Then we can cons uc a g oup  E T(1) 0 (Coke Sq 2 , Hn(K ;Z)(2)) as ollows . Fi s , we ha e hese Coke Sg 2 xH n (K ;Z)(2) . No e he o de o he gene a o e k-l  , j - 1, . . ., sk, is 2 1 k, so he elemen o Hn (K ;Z)(2) i=1 has he o m k  n  ~~k <21k . k l j i l a j e k_l  ' 0  J ipl si+j Le  F E : Coke Sq 2 ,  hen he elemen o Coke Sq 2 xHn (K ;Z)(2) has he o m whe e sk sk (F, k l~ l j i- l aj ek_ 1 ) ipl si +j (F .kE .ak ek_1  ) + (F' ; ]¿ E 'ak ek_ l ) _ J j  ~j J iplsi+j  iE lsi+j e k 7 i i p cak < 21k . Using he ope a ion T (1) (k), de ine an addi ion o pai s as ollows . _  (F+F'+kEj  ekT (1) (k)  ek_ l , kE j (ak+'a _,k21k) ek-1  ). i E l s i j  i=lsi+ j ak + la k <21k JJ «k +'a~>21k . Thisaddi ion is associa i e, makes Coke Sg 2 x Hn (K ;Z) (2) a g oup . Tha is he g oup E T (1)(Coke Sq 2 , Hn(K ;Z)(2)) . Now, le u T (l) (k)  ek 1  =  X71 «j, el+l,  j  =  1, . . .,s k ,  k  =  1p . . .  , hen  én +l i 1, . . .,u,  ek l  =  (0,  ek 1  ), j!lsi+ j ip l si+j j = 1, . . .,s k ,k = 1, . . ., , is a se o gene a o so ET (1) (Coke Sq 2 ,  Hn (K ; Z) (2) ) ., .and he ela ions a e 21 i  1 =  0,  i  =  1, . . . . u, So we ha e Jis i +j _  u _ 2lkek-1  = E l 01 j ,e n+1 , j = 1, . . .,s k , k = 1, . . ., . 2 ; si+ j i=1 I is no ha d o show [61 n a  (K) (2) -ET (1)(Coke Sq 2 , Hn(K ;Z)(2)) . Theo em 4 . Le K be an N-dimensional CW complex . Hn(K ;Z)(2), Coke Sq 2 and hei gene a o s a e as abo e, hen he g oup n (K) (2) has a p esen a ion éln+1  é n+1  n  n  I  n+1 = 0 <  ,.. ., ,é l ,. . .,é  2'1 .n+1 i = 1, . . ., u, u ipl si _  u 21kek-1  p l a J  +1'  -1, . . .,s k , k-1,  > iYl si+7 whe e ,n+l, ék 1  a e co esponding o he gene a o s ipl si+ J en+l, ek-1  espec i ely, and (a ~ ) is he ma ix ep e- ¡ P lsi+ J sen a ion o T( 1 )(k) wi h espec o he o de ed bases { el , . . .,e  }  and { el+l~ . . .,eú+1) . i , -l s i Fo he g oup i m (K) (2) , we ha e simila esul s . Bu he si ua ion is mo e complica ed . In ác , ins eado T (1) (k), we ha e h ee g oups o homomo phisms, namely and T2 . 2(2) (k)- 21k ke Sq 2 -~ Coke Sq 2 , 2 ) (k) :  ke T (2) (K) -> Coke ,0  2 T(2) : Coke Sq 2 -~ Coke < , and call hem he S T (1) (k) o sions o K . Le K be an N-dimensional CW complex and le K n be i s,n-skel on . Le  n > 0  andle  Kn be he complex o med by shinking  K n-1  o a poin  e  which is no in  K-K n-1 and is he single  0-cello  K n .  Now we discuss he ela ion be ween s T (1) (k) o sions o K and o Kn , K n . and We i s conside Kn . Theo em 8 . ST(1)(k)  o siono  Kn = 0,  i  s >n-I . P oo . Ob iously, whe e im is induced by he iden ical map i : Kn'-~K . Whence 1s+1  =  Hs+l (K ;Z2)  Hs+1 (K ;Z2 ),  i  s+l <n, i : H (K ;Z) 25 H (K n ;Z),  i = s, s - 1, S  .  C n,, Cn+1 . S T (1) (k) o sion o Kn = s T (1) (k) o sion o K, i  s <,n-l, S T (1) (k) o sion o Kn = sT(1)(k) o sion o K, i s <n-1 .  On conside ing he no mal o m o he incidencema ix o whe e C m is he g oupo in eg al m-cochains in K, we see ha is monomo phism, and whe e Hn is a ee summand o Hn (K n ,Z) which a ises om 0 he basic cochain  c E C n such ha  Sc =  (21+1)c'  (1 >O)  and is he na u alhomomo phism . Since hen we ha e iñ : H n (K, Z 2 )  H n (Kn Z2 ) H n (K n ,Z 2 )  = in* Hn (K,Z 2 )+ PHó  , !i : Hn(Kn,Z) - Hn(Kn,Z2) i : H (K,Z) = H (K n ,Z), i = n-1, n-2, n-1 T (1) (k)  o sion o Kn = n-1 T(1) (k)  o síon o K .  Bu Hn(Kn .Z2)  n-1  n  Hn(K,Z 2 ) _ n-1 dim  íSq 2 H n-2 (Kn,Z)I -  °k  íSq o  K  > dim  2 H n-2(K,Z)  k  °k 1 o K . whence When  s + 1 >n,  hen Hs+l (Kn,Z2)  = 0, Coke  (  Sq 2  :  H S-1 (K n ,Z)  - H s+1 (K n ,Z 2 ))  = 0 . We ha e whence Now we conside K . n Theo em 9 . is an epimo phism, and ST(1) (k) o sion o K n = 0, i s+l>n . ST(1)(k)  o siono  K n = 0,  i s < n+l, ST (1) (k) o sion o Kn = S T (1) (k) o siono K, P oo . When  s <n, hen H S(K n ,Z) = 01 S T (1) (k) o sion o  K n = 0, i s <n . I s = n, hen H n (K n ,Z) is a ee Abelian g oupsince Kn has no (n-1)-dimensional o sion . So in his case n T (1) (k) o sion o K n = 0 . i  s > n+l . Le L : K -> Kn be heiden i ica ion map . Since LIK-K n-1 is a homeomo phism on o K n - e ° which mapseach cell o K-K n-1 on a cello K n - e ° , i ollows ha ñ : Hn (K n ,Z) -' Hn (K,Z)  (4) c s  :  HS (K n ,  G)  - H S (K, G) is a isomo phism i  s >n .  In he commu a i e diag am no e (4) is an epimo phism and (5) is an isomo phism, he homomo phism * induces isomo phism Then om he commu a i e diag am we ha e 2 S Hn(Kn,Z) ---, Hn +2 (K n ~z 2 ) s2  _ Hn (K, Z)  q - H n+2 (K,Z 2 ), * : H n+2 (KnPZ2) - Hn+2(K,Z2) Sg 2 Hn (K n ,Z) ~ Sg2Hn(K,Z) . Hn+l (Kn~Z)  n+1T(1)(k) .  H n+2 2  (K n 'Z 2 ) 2lk  Sq H (K n ,Z) n+l  n+1T(1)(k) . Hn +2(K'Z 2 ) 1 (TC,Z) 2 k  Sg2Hn(K,Z), n+1 T (1) (k) o sion o Kn = n+1T(1)(k) o sion o K . I  s >n+2, hen om  (5)  is an isomo phism, we ha e S T (1) (k) o sion o Kn = S T (1) (k) o sion o K . Le 7 : E'- B  be a map o  E  on o  B .  Map  F : B - E is called a sec ion o  7  i  i F : B - " B  is he iden i y o B . Whence Theo em 10 . I 7 : E' -- >B has a sec ion, hen k  k E  s° i  o  E >  E  S u¡  o  B,  k  =  1,  2,  . i=l  i=1 P oo . Le F : B' - - ' , E be a sec iono 7 , hen om 1 F = 1B , we ha e i *  :  H s (B,G) ' - H s (E,G) is a monomo phism . In ac , we ha e H s (E,G) = ke F * + Im s = ke F *+ 7 *HS(B,G) . H - (E, .Z 2 )  ke F .S . +2 . ± n .s + 2H - -(B,Z 2 ) Sg 2 H S (E,Z)  Sa 2 (ke F *+ asHs(B,Z) ke F s+2  17s+2Hs+2(B,Z2) _  + Sg 2 ke F * Sq 2 i * HS (B,Z) Hence om he commu a i e diag am Sq 2 HS(B,Z)  _ Hs+2(B,Z2) Sq 2 HS(E,Z) -> we ob ain we ha e * . Hs+2(B,Z 2 1  Hs+2(E,Z2) . Sg 2 H S (B,Z)  Sg 2 HS (E,Z) is a monomo phism . Then om he commu a i e diag am Hs+l(B,Z) S+1T(1)(k), HS+2(B,Z2) 21k l7 *  Sg2HsJB,Z) 4 i HS +2 +1 S+1T(1)(k),(E,Z .2) 1 H  (E,Z)  2 2  k  Sq  H s (E, Z) k  k E su . o E > E so, o B, K = 1,2, . . . i=1 1  i=1 b . Rema k . In he p e ious pa ag aphes we ha e discussed he gene aliza ion o T( 1 )(k) o sions and ob ained hei p o pe ies . We can use he same s a ege o gene alize he T (2 kk)- o sions, and ob aine he simila p ope ies . The de ails willbe published elsewhe e . REFERENCES 1 . S .-T . hu, homo opy heo y . Academic P ess, New Yo x and London . 1959 . 2 . S . MacLane, Homology . Sp inge , Be lin-Go ingen-Heidel- be g, 1963 . 3 . M . Nakaoka, Exac sequences Ep(K,L) and hei applica- ions, J . Ins . Poly ech . Osaka Ci y Uni . Se . A . Ma h . 3 (1952), 83-100 . 4 . F .P . Pe e son, Some esul s on cohomology g oups, Ame . J . Ma h ., 78(1956), 243-258 . 5 . X .-Y . Shen, High o de cohomology ope a ions and i n (K n+3 ), Ac a Ma hema ica Sinica,  14(1964),  559-570 . ChineséMa h ., 5 (1964), 602-614 . 6 . ---, On new homo opy in a ian s (I), (II), ibid, 21 (1978), 253-262, 327-346 . Ins i u e o Ma hema ics Chinese Academy o Sciences, Peking, China P esen add ess PDMMS Uni . o Camb idge Camb idge ENGLAND, CB2 lSB 160