Power-cancellation of CW-complexes with few cells

Publicacions Ma emá iques, Vol 36 (1992), 715-724 . A bs ac POWER-CANCELLATION OF CW-COMPLEXES WITH FEW CELLS IRENE LLERENA Dedica ed o Pe e Menal, in memo iam In his pape , we use he ac ha he ings o in ege ma ices ha e he powe -subs i u ion p ope y in o de o ob ain a powe - cancella ion p ope y o homo opy ypes o CW-complexes wi h one cell in dimensions 0 and 4n and a ini e numbe o cells in dimension 2n . In oduc ion Cancella ion in homo opy is closely ela ed o he genus . Recall ha he genes o a homo opy ype X is he se o all homo opy ypes Y such ha he p-localiza ions o X and Y a e homo opy equi alen o all p imes p : X p - Y p [9] . I X and Y a e in he same genus, we shall w i e X -Y . All he known examples o non-cancella ion occu o homo opy ypes in he same genus . A ele an esul in his line is he ollowing . Theo em 1 (Wilke son [12]) . Le X, Y, W be nilpo en co-H- complexes wi h ini ely gene a ed homology . Then i) X V Y- X V W  implies Y-W,  and k  k ii) X - V Y  implies X - Y . In [10] Mislin cha ac e ized he genes o nilpo en co-H-spaces wi h ini ely gene a ed homology in he ollowing way X -Y  i and only i  X V V Sn ; -Y V V Smj , whe e he dimensions o he sphe es a e de e mined by he homo opy g oups o X and Y . A simila esul had been ob ained by Molna [111 71 6  1 . LLERENA o CW-complexes wi h wo cells, Sn U en' ., wi h a aching map o ini e o de in 7 ,,_ 1 (S') . See also [7] . In [1] Boko de elops ma icial me hods in o de o deal wi h spaces S2n U el, such ha he a aching map is o in ini e o de . This lead him o p o e he ollowing . Theo em 2 (Boko [1]) . Le xi = s2n U i and only i i) h=h' H H ii) V xi - V Y i Theo em 3 . Le Yi, = S2n U 9i e 4n ' H H V x i = V Y , H H VXi - VYi k Suppose ha i , gj ep esen elemen s o in ini e o de in 7 4 n _ 1 (VS 2 n) i and only i i < h and j < h' .  Then h+l h+l iii)  The e exis s a pe mu a ion T o {I, . . . , h} such ha X i -- Y,(¡) o alli<h . In [8] we ied o ex end Boko 's esúl o complexes wi h one cell in dimensions 0 and 4n and k cells in dimension 2n . Fo n >_ 2, he esul u ned ou no o be ue . Howe e , i we es ic ou sel es o he p-local case, he heo em holds . The p oo uses a echnical lemma ha is also used in he p oo o a cancella ion p ope y o he ing M k (7L ) o ma ices o e he p-localiza ion 7L o he ing 7L ([4, Th .2]) . Fo ou pu pose, we s a e he ollowing e sion o he main heo em in [8] . k  k x i = V S2n U i e 4n  1 i - V S2n U9i e4n  i = 1) . . ., H . Suppose ha i, g7 ep esen elemen s o in ini e o de in 7F4n_1(Vk S2n) i and only i i < h and j < h' .  Then i and only i i) h=h' h h ii) VXi _ VY, H H VX i -VY i . h+l h+l In his pape we use a p ope y o he ings M k (ZL), he (le ) powe - subs i u ion p ope y, o p o e he ollowing heo ems . Theo em I . Le POWER-CANCELLATION  717 k  k X=VS-U e-, Y=VSnU9e- . k Assume ha ,g a e suspension elemen s o ini e o de in 7,,_1(VSn) . Then i X - Y, he e is a posi i e in ege such ha  VX-VY . Obse e ha he con e se holds by Theo em 1 . Theo em II . Le k  k Xi - V_  S2n u , e4n  Yi = V S2n U9i e4n  i = 1 . .. , H . Assume ha i, g j ep esen elemen s o in ini e o de in 7 4n_ 1(V k S2n) i and only i i < h,  j < h' .  Then i o all i < h, and H H V Xi -V Yi we ha e h = h' and he e exis posi i e in ege s s, 1, . . . , h and a pe - mu a ion T o {1, . . . , h} such ha i  i V X i - V Y , (¡) s H  s H /( / Xi) - V(V h+l h+l 718  I . LLERENA Powe -subs i u ion A ing E sa is ies he "(le ) powe -subs i u ion p ope y" i gi en xa+ b = 1 in E he e exis a posi i e in ege and a ma ix T E M (E) such ha I a+Tb is an uni in M (E) . He e I is he iden i y x ma ix . All sub ings o he a ionals Q and in pa icula 7L, as well as all he ma ix ings M,,,(ZL), n >_ 1, sa is y powe -subs i u ion ([5, (2 .9) and (3 .4)]) . So, gi en XA+B=I n in M n (ZZ), he e exis a posi i e in ege and a ma ix T E M n (ZZ) such ha I ®A+ T (I ® B) is in e ible . He e ® deno es he K onecke p oduc A 0 . . . 0 0 A . . . 0 I ®A= (0 0 . . . Powe -subs i u ion implies powe -cancella ion in he ollowing sense . I A, B, C a e igh R-modules and End R (A) has he powe -subs i u ion p ope y, hen A® B =A®C implies B - C o some posi i e in ege . P oo o Theo em I Since and g a e suspension elemen s, hey ep esen elemen s in ®7 -  ,_1(Sn) C 7 , n _ 1 (V k Sn) . So, and g a e de e minad by column ma ices x( ) and x(g) wi h en ies in 7 ,-1(S') . I was shown in [7] ha he mapping ones o and g, X and Y, a e in he same genus i and only i he e exis s an in ege ma ix A such ha x(g) = Ax( ) and (de A, l ) = l, whe e l deno es he o de o . Le B be an in ege ma ix such ha AB = de A I k , and ake , s such ha de A + sl = 1 . Since 7L has he powe -subs i u ion p ope y, he e exis an in ege c and a ma ix C such ha I, de A+C 1  is in e ible . Then POWER-CANCELLATION  719 Ik®(I~de A)+(Ik®Cl)=(1,0B)(I,®A)+(Ik®Cl)=V is in e ible and, by he powe -subs i u ion p ope y o M,k(7Z), we ob- ain an in ege d and a ma ix D such ha Id 0(I~®A)+D(Id®V-1(Ik®Cl))=U has an in e se . So applying powe -subs i u ion once mo e, we ob ain an in e ible ma ix o he o m Q=Ie®(Icd®A)+E(Ie®U -1 D(Id®V -1 (Ik0CI)))=I ®A+TI whe e = cde and T= E(I, ®U -1 D(Id ® V -1 (I k (D C))) . Now líen e P oo o Theo em II k Fi s o all, ecall ha he homo opy ype o each X i = V S2n U i e4n is de e mined by he k xk in ege ma ix H( i) o he associa ed Hil on- Hop quad a ic o m and one one-column ma ix x(~  he en ies o which a e suspension elemen s in 7 4 n (S?n +1 )[1] . Mo eo e , V H Xi - Y i and only i he e exis s a homo opy commu a i e diag am H V Son-1 19 H V S4n-1  X-V Y . h . . . H H k 1 ( S2n) H k 91V .VgH` V(V S2n) =x(gV . . .Vg) . 72 0  1 . LLERENA wi h d and cp homo opy equi alences . Le B i j be he deg ee o H H Son-1 a  V S4n-1  19  Son-1 ~ S4n-1 . The ma ix (Bi ) = 0 cha ac e izes he homo opy class o 19 . Le now Oil be he ma ix o k  H k  H k  k S2n 2nj -> V(V S2n) --W-+ V(V S2n) -P i> V S2n . The homo opy commu a i i y o he p e ious diag am is equi alen o he ollowing condi ion [1], [8] . O7i H ( i)oji = e7iH(g9)  o all  i, j (*)  OjiH( i)Oli = 0  i  l :,A j O9ix(1 : i) = e7ix(Egj)  o all  i, j whe e Oji is he anspose o Oji . We know ha i is o ini e o de i and only i H( i) = 0 . Hence, B ; i = 0 o j < h' , i > h . Thus, de 0 = 1 implies h >_ h' and by symme y h = h' . Now w i e he ma ices o í9 and cp in he o m They a e in e ibles . Thus = C 61 0)  C 4 >1 'P2 / 63 64  _ ~D3 ~D4 C1-P1+CA3=I 034)2+C4'P4 = I whe e  C3  C4  =0 -1 . By he powe -subs i u ion p ope y o Mkh(Z) and Mk(H-h) (7L), we can ind posi i e in ege s , s and ma ices R, S such ha A= I, .®<D 1 +R(I,0CA 3 ) B= I s ®'D4 + S(I s ® C3~D2) a e in e ibles . Conside he diag ams and POWER-CANCELLATION  72 1 h  h  (V i)   h  k V(V S 4n-1 ) _ V(V(V S2n)) V (V S 4n-1 ) V  V(V(V S 2n )) SH V(V S4n-1) h+l s H h H V(V i) s H h+1 SH k V (V (V S2n)) h+l la V(V gi) V ( V S 4n-1 )  V(V (V S 2n )) h+l  h+l SH k whe e (, ~, a and 0 a e homo opy equi alen es wi h ma ices I 01 , I,, ® 194, A and B espec i ely . By checking he ma icial condi- ions men ioned abo e (*), we can see ha hese diag ams a e homo opy commu a i e . (See [81 o he de ails in a qui e simila case) . So, hese diag ams induce homo opy equi alen es h  h  SH  SH V(VXi) =V(VYj) and V(V X i ) -V(V Yi) h+l h+l We may now assume ha all he gi en spaces Xi and Y¡,  i = 1, . . . , H , ha e a aching maps o in ini e o de .  Le be he maxi- mum ank o he ma ices H( i ) and H(gi) and assume ha he ank o  H(g1) is .  Since de e =± l,  0,1 :7~ 0, o some l and, hence, 01,H( l)oil = 011H(g1) has ank . Thus kH( l ) = . Now, om he es o he ma icial condi ions (*), one ge s 9jl = 0 i j :~ 1, and 0,1 = l . Fo simplici y we suppose l = 1 . Le us w i e he ma ices o 99 and ~p in he o m _  011  C3 )  0 _ ( 011  < 1 > 2 0 84  < D3 'D4 722  I . LLERENA and le 0 -1 =  Cl  C2  . Thus ~!3 ~!4 0 1011 + CA3 = I  and  C3>2 +C 4 - 4 = I . As be o e, we can ind in e ible ma ices o he o m A=I,00,1+R(I ®C2'P3) and B=I s ®<D4+S(I s ®C3)2) . Conside homo opy commu a i e diag ams V S4n-1 V S4n-1 s H V(V S4n-1) 2 s H V(V S4n-1) 2  VXl=VY, and V 1   k V (V S2n) s H V( 2) V(V(V S 2n )) 2 s H V( g2) V(V(V S2n)) 2 10 whe e (, ~, a and,3 a e homo opy equi alen es wi h ma ices B11I , , I5 04 , A and B espec i ely . These diag ams induce homo opy equi a- lences s H  s H V(VX i)=V(VY) . 2  2 This, by induc ion, comple es he p oo o Theo em II . Rema k . The ollowing ques ion na u ally a ises in he s udy o can-  cella ion : is X - Y equi alen o V X - V Y? (See [61 o an algeb aic esul o his kind) . Theo em I and Theo em 1 gi e a posi i e answe o some co-H-spaces wi h ew cells . Fo mapping ones X and Y o POWER-CANCELLATION  72 3 elemen s o in ini e o de in 7 4n_1(VkS2n), i ollows om [8] Theo-  em 1 ha V X _ V Y implies X -Y .  In his case, howe e , he con e se is no always ue . I , o ins ance, and g a e such ha H( )= (  0) , H(g) = (0  2pq) , E =0 and L .g=0, 0 2q whe e p :,~ q a e p ime in ege s, hen X -Y and X 9~ Y (see [2]) . Bu , since H( ) and H(g) a e o maximum ank, i ollows om [8] Theo em 2  ha V X 9~ V Y o all in ege s . Re e ences 1 .  BOKOR, L, On genus and cancella ion in homo opy, Is ael J . o Ma h . 73 (1991), 361-379 . 2 .  BOKOR, I ., On he connec ion be ween he opological genus o ce - ain polyhed a and he algeb aic genus o hei Hil on-Hop quad a ic o ms, Publicacions Ma emá iques 34 (1990), 323-333 . 3 .  BOKOR, L, On mapping cones o suspension elemen s o ini e o de in he homo opy g oup o a wedge o sphe es, P oc . Am . Ma h . Soc . ( o appea ) . 4 .  EVANS, E . G ., K ull-Schmid and cancella ion o e local ings, Pac . J . o Ma h . 46, no . 1 (1973), 115-121 . 5 .  GOODEARL, K . R ., Powe -cancella ion o g oup and modules, Pac . J . o Ma h . 64, no . 2 (1976), 387-411 . 6 .  GURALNICK, R . M ., The genus o a module, J . o Numbe Theo y 18 (1984), 169-177 . 7 .  LLERENA, I ., Wedge cancella ion and genus, P ep in . 8 .  LLERENA, I ., Wedge cancella ion o ce ain mapping cones, Comp . Ma h . 81 (1992), 1-17 . 9 .  MISLIN, G ., "The genus o an H-space," Lec . No es in Ma h . 249, Sp inge , 1971, pp . 75-83 . 10 . MISLIN, G ., Cancella ion p ope ies o H-spaces, Comm . Ma h . Hel . 49 (1974), 195-200 .