On the computability of the Steenrod squares

Ann. Uni . Fe a a - Sez. VII - Sc. Ma . Vol. XLII, 57-63 (1996) On he Compu abili y o he S een od Squa es. PEDRO REAL (*) SUNTO - In ques o la o o o iamo in modo esplici o le o mule di una se ie de mo ismi che con oUano la mancanza di commu a i i ~ del p odo o cup a li ello di coca eni, supponendo di la o a e con insiemi sempliciali; ques e o mule si s abliliscono in e mini di mo ismi componen i di una con azione di Eilenbe g-Zilbe da a. Di con- seguenza, nel caso in cui l'insieme sempliciale sia ini o in ogni dimensione, o e e- mo un algo i mo di calcolo di quad a i di S een od. ABSTRACT - We gi e explici ely he o mulas o a sequence o mo phisms which mea- su e he ailu e o commu a i i y o he cup p oduc on he cochain le el, p o ided ha we wo k wi h simplicial se s; hese o mulas a e es ablished in e ms o he componen mo phisms o a gi en Eilenbe g-Zilbe con ac ion. As a consequence, in he case in which he simplicial se is ini e in each dimension, we ob ain an algo- i hm o calcula ing S een od squa es. 1. - In oduc ion. Recen ly, many au ho s ha e ied o e o mula e se e al concep s om Algeb aic Topology in an e ec i e way, achie ing me hods o p o ide algo- i hms compu ing hose concep s ([13], [14], [12], [9]). We a e in e es ed he e in exhibi ing an e ec i e solu ion o he cons uc ion and compu a ion o he S een od squa es ope a ions [16]. I is well-known ha a classical p ocedu e o de ine he S een od squa es is based on he cons uc ion o a amily o mo phisms ha measu e he ail- u e o commu a i i y o he cup p oduc on he cochain le el [2, sec . 6.2A]. (*) Indi izzo dell'au o e: Uni . de Se illa, Fac. de In o m~ ica y Es adis ica, Dp o. de Ma em~ ica Aplicada I, A da. Reina Me cedes s/n, 41012 Se illa (Spain); Fax: 34-5- 4557878; e-maih [email p o ec ed] Ma h. Subjec Classi ica ion (1991): 55S05. 58 PEDRO REAL F om now on, ollowing [1], his sequence o mo phisms will be called highe diagonal app oxima ion. In he li e a u e, he exis ence o his highe diago- nal app oxima ion is always gua an eed by he acyclic models me hod [4]. In a semi-simplicial con ex [8], he echnique o acyclic models can be consid- e ed as an ac ual algo i hmic me hod (see [11]). In consequence, o each sim- plicial se X, we can de e mine a highe diagonal app oxima ion in his way. In his pape , we de elop ano he me hod o explici ely gi e he gene al o mulas o a highe diagonal app oxima ion, p o ided ha we wo k wi h simplicial se s. Conc e ely, hese o mulas a e es ablished in e ms o he componen mo phisms o a gi en Eilenbe g-Zilbe con ac ion. The e o e, in he case in which he simplicial se X is ini e in each dimension, we ob ain an algo i hm o calcula ing cochains o X by he S een od squa es. Finally we conclude ha he p oblem o he compu abili y o cochains by hese cohomol- ogy ope a ions exhibi in gene al an exponen ial complexi y. In iew o he plen i ul li e a u e exis ing abou me hods o de ining S een od ope a ions (see [6] o ob aining a non-exhaus i e lis ), a ema k seems us necessa y. Ou elemen a y p esen a ion can be use ul o a majo unde s anding o he unde lying s uc u es which allow us o de e mine he homo opy ype o spaces, as he same ime as i can be a s a ing poin o an algo i hmic app oach o cohomogical ope a ions. 2. - P eliminai es. Fi s , we will assume ha we a e wo king o e a g ound ing R which is commu a i e wi h uni . We will use he le e s A, B .... o designa e di e - en ial g adua ed modules o e R o DG-modules, and he le e s X, Y, ... o deno e simplicial se s; he ace and degene ancy ope a o s o a simplicial se will be deno ed by 3i and si, espec i ely. Finally, C. (X) and C* (X) will be he no malized chains and cochains o he simplicial se X espec i ely. Eilenbe g and MacLane de ined in [5] a con ac ion o a DG-module A on- o a DG-module B as a iple ( , g, q~) in which : A -, B ( he p ojec ion o he con ac ion) and g: BoA ( he injec ion o he con ac ion) a e mo - phisms o DG-modules while ~b: A --*A is a mo phism o g adua ed modules o deg ee 1 ( aising dimensions by 1). I is equi ed ha (cl) g = 1B; (C2) ~b = 0; (C3) ~bg = 0; (C4) ~d + d~ = g - 1A ; (c5) ~ = o. ON THE COMPUTABILITY OF THE STEENROD SQUARES 59 An Eilenbe g-Zilbe con ac ion is a con ac ion o C,(X x Y) on o C,(X) | C,(Y), whe e X and Y a e simplicial se s. The e exis s a leas one: THEOREM 1 [3]. Le X and Y be simplicial se s. The Alexande -Whi ney ope a o AW: C, (X x Y) --> C, (X) | C, (Y), he Eilenbe g-MacLane ope a- o EM: C, (X) | C, (Y) -~ C, (X x Y) and he Shih ope a o SHI: C,(X x Y) --~C.+i(Xx Y) o X and Y a e de ined by he ollowing o mulas: AW(xn x y~) = ~ ~ + 1"" ~n Xn @ ~0." ~i - lYn , i=0 EM(xp| = ~ (-1)sg(a,~)(S~q...s~lXp x s,p...s, lyq) , (a, ~) ~ {(P, q) - shu les} SHI(x~ x y~) = n-p-q+sg(a, l) = - ~(-1) (S lq +n-p-q'" S ll+n-p-qSn-p-q-I ~n-q+l"" ~nXn X X Sap +1 ~-T -P-q ~176 Sal ~-~-P-q ~-P-q'~ ~n-q-1 Y~), whe e he las sum is aken o e all he indices 0 <<. q <<. n - 1, 0 ~< p <~ n - - q - 1, (a, l) e {(p + 1, q) - shu les} and sg(a, l) = ~ (ai - (i - 1)). The iple (AW, EM, SHI) is a con ac ion o C,(Xx Y) on o C, (X) | C, (Y). The explici o mula o he Shih ope a o is gi en in [10]. 3. - The de e mina ion o a highe diagonal app oxima ion. I is well-known ha i is no possible o cons uc an Eilenbe g-Zilbe con ac ion wi h commu a i e p ojec ion (see, o example, [7, sec . 8.5]). We p esen in his sec ion a new ela ion be ween he lack o commu a i i y o he Alexande -Whi ney ope a o and he cons uc ion o a ~,highe cop od- uc ,, which allows us o de ine he S een od squa es. We will ob ain his e- sul , by p o ing he ollowing heo em: THEOREM 3.2. Le A and B be wo DG-modules. Le ( i g, ~) be a con- ac ion o A on o B. Le h: A -~ A be an idempo en mo phism o DG-mod- ules. Le us suppose ha he ollowing ela ion holds (1) ~phg = O. 60 PEDRO REAL Then he e exis s a sequence o mo phisms { i }i ~ o , i : A ~ B o deg ee i~ such ha (2) o= , dBj~-(-1)i idA=h' i_l +(-1)i i_lh i i~ l whe e h ' = hg. Mo eo e , an explici o mula o he mo phisms i can be gi en (3) ~ = (h~) ~ Vi >10. Now, we apply his heo em o he case in which he da a a e he Eilen- be g-Zilbe con ac ion (AW, EM, SHI) and he au omo phism o ansposi- ion ,: C,(X xX)~C,(X x X) whe e (x x y) = (y x x). F om now on, we will deno e he di e en ials o C, (X), C, (X x )X and C, (X)| C,(X) by d, d• and d| espec i ely. We easily can es ablish ha SHI , EM = 0; we ob ain, hus, a sequence o mo phisms {~}i~0 ii : C, (X x X) ~ C, (X) | C, (X) o deg ee i. wi h ~=(AW) ( ,SHIP. e i ying (4) d| - ( - 1)~j~ d • = Tj~ _ 1 + ( - 1)~j~ _ 1 ,, i i~>l, whe e T: C, (X) | C, (X) ~ C, (X) @ C, (X) is de ined by T(a| = (-1)Pqb| wi h a e Cp (X), b e Cq (X). Now, we can cons uc he highe diagonal app ox- ima ion {Ai}i~0. whe e he mo phism Ai: C,(X)----)C,(X)| is gi en by (5) A~ =A oA . and he mo phism A: C,(X)-->C,(X x X) is de ined by A(a)= a x a, Va ~ X. The mo phism A 1 is a mo phism o DG-modules ( he Alexande - Whi ney diagonal app oxima ion) and he maps A i (i >>-2) sa is y ela- ions: (6) d~Ai - (-1)iAid = TAi-1 + (-1)iAi- 1 9 I is well-known ha a amily o mo phisms e i ying (6) a e used o cons uc he U-p oduc s and he S een od squa ing ope a ions (see, o ON THE COMPUTABII~ITY OF THE STEENROD SQUARES 61 example, [15, sec . 5.9]). Now, he de ini ion o he cohomology ope a ion Sqi: Hq(X) "-> Hq§ is: (7) {~ |215 sii<q, Sqi(c)(x) = si i > q, whe e c e Hom( Cq (X), Ze ) e x e Xq + i . Then, i is clea ha , a leas in he case in which X is ini e in each dimen- sion, his explici o mula ion cons i u es an ac ual algo i hm. Mo eo e , his de ini ion shows ha he compu a ion o cochains by Sq~ ca ies always he use o he Shih ope a o , which equi es, in gene al, exponen ial ime o gi e an answe . In ac , he sum ha de ines his ope a o is aken o e he shu - les (special ype o pe mu a ion). Roughly speaking, his mo phism e lec s in his case he passing om Geome y o Algeb a. To sum up, compu ing S een od squa es is a genuinely compu a ionally di icul p oblem. We conjec u e ha he measu e o he complexi y o he compu a ion o hese cohomology ope a ions is exponen ial. 4. - P oo o he Theo em 2. To p o e he esul , i su ices o e i y he ollowing condi ion: (8) ~dA hCphd~ = - ~h~hdA ~b . In ac , i he condi ion (8) is ue, we can easily deduce he equali- y: (9) ~dA(h~ = (-l~-l(~bh~dA~, Vj >I 1, and his ela ion will allows us o show (2). He e, we i s ly p o e he equali ies (2), supposing ue he o mulas (9); inally, we will es ablish he ela ion (8). Fo i = 1, we do no need he ela ion (9) o ob ain (2)1. Using he p ope - y (c4) o a con ac ion, we ha e dB h~ + h~dA = ds]h~ + h(g - 1A - dA = Since and h a e mo phisms o DG-modules, = ds h~ + hg - h - hdn~b = hdA + ( hg) - h - hdA = ( hg) - h 62 PEDRO REAL Now, we ha e o p o e he ela ion (2)i, i i I> 2. Fi s , we use he de ini- ion (3), dB i -- (--1)i idA = dB (hd~) i + (-1) i+ l (hdp)idA = = dB (h~))i + ( _ 1)i + i h(~h)i - 1 ~)dA = We make use o he p ope y (c4) o a con ac ion, = dB (hCp)i + ( _ 1)i + i h( - 1 (el__ 1A -- dA d~) = By (1) and (9), we ha e = dB (h~)i + (_l)i + l h(~h)i - ig + (_l )i h( )i - 1 + (_l)i h(~h)i - 1 dA = = hdA d l( hdp )i - 1 + (_ l )i ( h~ )i - 1 h + hd~dA ( h ) i - 1 = = h(dAd p + + (_l)i (hgp)i-1 h = We conside now he p ope ies (c3) and (c4) o a con ac ion, he idem- po ency o he DG-mo phism h and he o mulas (3); = h(g - 1A )(h~) i - ~ + ( - 1)i2~ _ i h = = ( ag) (hd~) i - 1 _ ~ h~(h~)~ - 2 + ( _ 1)i i _ i h = = h' i-1 + (-1)ij~-i h. This comple es he p oo o (2)i, i I> 1. Now we shall show ha he ela ion (8) holds. 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SPANIER, Algeb aic Topology, McG aw-Hill Book Comp., New Yo k (1966). [16] N. E. STEENROD, Reduced powe s o cohomology classes, Ann. Ma h., 56 (1952), pp. 47-67. Pe enu o in Redazione il 23 gennaio 1996. In o ma de ini i a il 21 o ob e 1996