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. Fi s , using he p ope y
(c4) o he de ini ion o con ac ion and he idempo ency o he mo phism h,
we ha e
h(g -
dA ~) -- ~dA )
h
= 1A ;
and, hence, we ind
hg h - hdA ~h - h~dA h = 1A
Composing his equali y on he le and on he igh wi h he homo opy
ope a o q~, i u n ou
~hg h~ - ~hdA ~h~ - ~h~d.4 h~ = ~
ON THE COMPUTABILITY OF THE STEENROD SQUARES 63
Since h is a mo phism o DG-modules and keeping in mind he ela ions
(1) and (c5), we conclude
~)dA hCphdp = - ~bh~hdA .
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Pe enu o in Redazione il 23 gennaio 1996.
In o ma de ini i a il 21 o ob e 1996