A∞–coalgeb a s uc u es on he Zp-homology o
Eilenbe g-Mac Lane spaces
Ainhoa Be ciano and Ped o Real
May 2004
Abs ac
We s udy he e he A(∞)-coalgeb a s uc u e o he homology H∗(K(π, n); Zp) o an
Eilenbe g-Mac Lane space K(π, n), whe e πis a ini ely gene a ed abelian g oup and
nis a posi i e in ege . Using di e se echniques o homological pe u ba ion, we ge
ha he componen s ∆i(p−2)+2 o deg ee i(p−2) (wi h i≥0) a e he only (possibly)
non-null mo phisms o said s uc u e.
1 Ex ended Abs ac
An Eilenbe g-Mac Lane space K(π, n), depending on a g oup πand a non-nega i e in ege
nis a simplicial se ha ing only wo non-null in ege homo opy g oups π0(K(π, n)) = Zand
πn(K(π, n)) = π(excep he case n= 0, which only ha e one non-null homo opy g oup in
deg ee 0 π0(K(π, 0)) = π). We only conside he e K(π, n) spaces being πan ini ely gen-
e a ed abelian g oup. In his case, he chain complex C∗(K(π, n)) canonically associa ed
o an Eilenbe g-Mac Lane space ha e a Hop -algeb a s uc u e, ha is, C∗(K(π, n)) is en-
dowed wi h an algeb a and a coalgeb a s uc u es ha a e compa ible in some s ong sense.
Hen i Ca an de e mines in his comple e s udy [1] o he homology o Eilenbe g-Mac Lane
spaces ou ypes o commu a i e di e en ial g aded augmen ed algeb as, called elemen a y
complexes, which can be made by using ex e io and di ided powe algeb as. The i s is he
ex e io algeb a E(u, 2n−1) wi h one gene a o uo deg ee 2n−1; he second is he di ided
powe algeb a Γ( , 2n) wi h one “gene a o ” o deg ee 2n; he hi d is he wis ed enso
p oduc Γ(u, 2n)˜
⊗ρ±p E( , 2n+ 1) wi h di e en ial ope a o de ined by ρ±p ( ) = ±p u;
and he ou h is he wis ed enso p oduc E(u, 2n−1) ˜
⊗ρ±p Γ( , 2n) wi h di e en ial op-
e a o de ined by ρ±p ( ) = ±p u. Tenso p oduc s o hese elemen a y complexes desc ibe
he in ege homology algeb a H∗(K(π, n)) o any πand n≥1. Alain P ou e in his b il-
lian Ph. D. Thesis “Alg´eb es di e en ielles o emen homo opiquemen associa i es” [8]
s udies he Zp-homology algeb a ( he ield Zpbeing he g ound ing) o K(π, n) and s a s
he wo k o de e mining i s A(∞)-coalgeb a s uc u e. In he case H∗(K(π, n); Z2), his
comul iplica i e s uc u e is educed o a simple coalgeb a. Consequen ly, his homology
has a Hop -algeb a s uc u e. I p6= 2 (pp ime), he si ua ion changes d ama ically. To
he di icul p oblems o con ol as he A(∞)-s uc u es by enso p oduc e ol e, he ac
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ha he A(∞)-coalgeb a s uc u e explodes in he case p6= 2 joins up. P ou ´e only gi es
pa ial esul s o elemen a y algeb as embodied in his con ex , ha hey pe mi howe e
o conjec u e aspec s o na u e o his complica ed s uc u e. In his a icle, we in end o
ex end P ou ´e’s wo k and o de e mine he o al g oup o componen mo phisms possibly
non ze o o he A(∞)-coalgeb a s uc u e o H∗(K(π, n); Zp).
Only enso p oduc s o wo elemen a y complexes appea in H∗(K(π, n); Zp). They
a e he ex e io and di ided powe algeb as. The Real’s wo k [10] is ou s a ing poin ,
whe e, using homological pe u ba ion, speci y a con ac ion c(π,n);p(pa icula homo opy
equi alence) among C∗(K(π, n); Zp) and H∗(K(π, n); Zp). This s ong compa ibili y is
possible basically hanks o he ollowing elemen a y con ac ions:
•A con ac ion CW B “connec ing” he no malized chain complex C∗(W(G)) o he
geome ic classi ying space o an abelian g oup G, and he educed ba cons uc ion
B(C∗(G)) o he no malized chain complex o G[9].
•An isomo phism om he educed ba cons uc ion o an ex e io algeb a o a di ided
powe algeb a.
•A con ac ion om he educed ba cons uc ion B(Γ(u, 2n)) o an in ini e enso
p oduc o algeb as o he kind E( , 2npi+ 1) ⊗Γ(w, 2np(i+1) + 2).
A ending o he e minology deli e ed in [10], all hese h ee con ac ions a e semi- ull
algeb a con ac ions and, in pa icula , ha means ha he algeb a s uc u e is ans e ed
co ec ly and o simple manne o he homology H∗(K(π, n); Zp). Rega ding coalgeb a’s
s uc u es, wo i s con ac ions espec o his s uc u e, being he las he one and only
con ac ion ha does no show a p ope y o compa ibili y. The explosion o coalgeb a’s
s uc u e in he homology is p ecisely p o oked by his las con ac ion. As said con ac-
ion is gene a ed s a ing om elemen a y con ac ions cBQ among B(Qp(u, 2n)) (being
Qp(u, 2n) = P(u, 2n)/(up)) and he enso p oduc E( , 2n+ 1) ⊗Γ(w, 2np + 2) , a i s
s ep o examine he s ongly homo opy associa i i y o he homology H∗(K(π, n); Zp) is o
deal i s wi h his ques ion in elemen a y con ac ions.
P ou ´e’s s udy was limi ed, o he p e iously easons expounded, o he de e mina-
ion o he A(∞)-coalgeb a’s s uc u e o he 1-homology o Qp(u, 2n). Such s uc u e
(∆2,∆3,∆4, ...) only shows wo non-null mo phisms: ∆2and ∆p. Ou s a egy is o discuss
his wo k om he pe spec i e o Homological Pe u ba ion Theo y and o y o ex end
his echnique o mo e complex algeb as.
Fi s , gi en a con ac ion c={N, M, , g, φ}among a di e en ial g aded coalgeb a
(N, dN,∆) and a chain complex (M, dM), he componen mo phisms o he A(∞)-coalgeb a
s uc u e on M ollow (up o signs) he ollowing o mulae ( om now on, we use Real’s
no a ion [10]):
∆i= ⊗i(∆[(i−1)]φ[c,i−1] · · · (∆[2]φ[c,2])∆g, ∀i≥1.(1)
The de elopmen o his o mula in e ms o he cop oduc ∆ : N→N⊗Nand he
homo opy ope a o φ:N∗→N∗+1, gi es us a sum o composi ions
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⊗i(∆[i−1,ji−1]φ[c,i−1,ki−1])· · · (∆[2,j2]φ[c,2,k2])∆g, 1≤j , k ≤ . (2)
An elemen a y wo k is o p o e ( we would be able o do i using banally In e sion’s
Theo y [10, 2]), he ollowing ule:
“The mo phism (2) will be ze o i j 6=k , o some .”
Now, le us ocus in he con ac ion cBQ ={B(Qp(u, 2n)), E( , 2n+ 1) ⊗Γ(w, 2np +
2), BQ, gBQ, φBQ}.
We deno e an elemen o B(Q(p)(u, 2n)) o he o m [u 1|. . . |u m] by [ 1|. . . | m], whe e
0≤ i< p. F om now on, E( , 2n+ 1) ⊗Γ(w, 2np + 2) will be deno ed by H o sho .
The explici mo phisms o cBQ a e he ollowing:
BQ[ 1| 1|. . . | m| m] = {Qn
k=1 δp, k+ k}γm(w),
BQ[ 1| 1|. . . | m| m|l] = δ1,l{Qn
k=1 δp, k+ k} ·γm(w),
whe e he symbols δi,j a e de ined by:
δi,j =0i6=j
1i=j
The mo phism gBQ :H→¯
B(Q(p)(u, 2n)) is de ined o e he gene a o s as ollows:
gBQ( ) = [1],
gBQ(γk(w)) = [1|p−1|k imes
. . . |1|p−1].
The homo opy ope a o φBQ is de ined by:
φBQ1 = 0; φBQ[1] = 0;
φBQ[x] = −[1|x−1] 1 < x < p;
φBQ[x|y] = −[1|x−1|y];
φBQ[x|y|z] = −[1|x−1|y|z]−δp,x+y[1|p−1|φ(z)]
whe e z∈¯
B(Q(p)(u, 2n)).
The coassocia i e cop oduc on B(Q(p)(u, 2n)) is
∆([a1| · · · |a ]) =
X
i=0
[a1| · · · |ai]⊗[ai+1| · · · |a ].
The ollowing p ope ies can be easily e i ied:
I ∆(x) = P∆1(x)⊗∆2(x), being xan elemen o B(Q(p)(u, 2n)),
φBQ(∆1)ig= 0, i ≥1
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φBQ∆1φBQ = 0, i ≥1.
Le us conside he submodule So B(Q(p)(u, 2n)) gene a ed by he elemen s [a1|a2| · · · |a ]
such ha o else a2i+1 = 1 o all io else a2i= 1 o all i.
Again, i is easy o p o e he ollowing p ope ies:
φBQ(S)⊂S,
∆(S)⊂S⊗S,
Im gBQ ⊂S.
Now, we will use In e sion Theo y o ge an economical o mula ion ( in e ms o numbe
o summands embodied) o he mo phisms ∆i:H→H⊗H.We will say ha an elemen
[a1|a2| · · · |a ]∈B(Q(p)(u, 2n)) has in e sions i he e exis s di e en couples (a2i−1, a2i)
o he o m (1, a), whe e a∈ {1,2, . . . , p −3, p −2}.
Wi h his de ini ion in hand, we can e i y he ollowing p ope ies:
φBQ(elemen wi h i-in e sions) = Xelemen s wi h a leas (i+1)-in e sions,
∆(elemen wi h i-in e sions) = Xelemen s wi h a leas (i−1)-in e sions,
BQ(elemen wi h i-in e sions) = 0,∀i≥1.
The e o e, in he economical o mula o ∆i(see (1)) de i ed om his echnique, he
cop oduc ∆ o B(Q(p)(u, 2n)) will be only applied o elemen s wi h 1-in e sions and he
homo opy ope a o φBQ will be applied o elemen s wi h 0-in e sions only. Using his
a gumen , i is possible o deduce ha only he mo phisms ∆2and ∆pa e non-ze o in he
A(∞)-coalgeb a s uc u e o e H.
This wo k can be ex ended wi hou oo many p oblems o educed ba cons uc ions o
enso p oduc s o Q(p)(u, 2m). In his way, we show he main esul o he pape :
Theo em. Le πbe a ini ely gene a ed abelian g oup, n≥1be a posi i e in ege and
p6= 2 be a p ime. The (possibly) non-null mo phisms o he A(∞)-coalgeb a s uc u e o
H∗(K(π, n); Zp)a e ∆i(p−2)+2,∀i≥0.
Re e ences
[1] Ca an H.: Alg`eb es d’ Eilenbe g-MacLane. S´eminai e H. Ca an, 1954/55, (expos´e 2
al 11), 1956.
[2] CHATA g oup. Compu ing “small” 1–homological models o CDGAs. P ep in Dep o.
Ma . Apl. I, Uni . de Se illa.
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[3] V. K. A. M. Gugenheim. On he chain complex o a ib a ion. Illinois J. Ma h. 3, pp.
398-414, 1972.
[4] V. K. A. M. Gugenheim and L. Lambe. Pe u ba ion heo y in Di e en ial Homological
Algeb a, I. Illinois J. Ma h 33, pp. 56-82, 1989.
[5] V. K. A. M. Gugenheim, L. Lambe and J. S ashe . Pe u ba ion heo y in Di e en ial
Homological Algeb a, II. Illinois J. Ma h. 35 n. 3, pp. 357-373, 1991.
[6] V. K. A. M. Gugenheim and J. S ashe . On Pe u ba ions and A∞–s uc u es. Bull.
Soc. Ma h. Belg., ol. 38, pp. 237-246, 1986.
[7] J. Huebschmann and T. Kadeish ili. Small models o chain algeb as. Ma h. Zei . .
207, pp. 245-280, 1991.
[8] A. P ou ´e. Alg`eb es di ´e en ielles o emen homo opiquemen associa i es. Ph. D. he-
sis, Uni e si ´e Pa is VII, 1984.
[9] Real P.: Algo i mos de c´alculo de homolog´ıa e ec i a de los espacios clasi ican es, Ph.
D. hesis, Se ille Uni e si y, 1993.
[10] P. Real. Homological Pe u ba ion Theo y and associa i i y, Homology, Homo opy and
Applica ions, . 2, pp. 51-88, 2000.
Ainhoa Be ciano Alca az
Depa amen o de Ma ema ica Aplicada, Es adis ica e In es igacion Ope a i a.
Facul ad de Ciencia y Tecnologia.
Uni e sidad del Pais Vasco-Euskal He iko Unibe si a ea.
Ba io Sa iena s/n. 48940, Leioa (Spain)
Tel´e ono: 34.946.013.368; e-mail: mepb[email p o ec ed]u.es
Ped o Real Ju ado
Dp o. Ma em´a ica Aplicada I
Escuela T´ecnica Supe io de Ingenie ´ıa In o m´a ica.
Uni e sidad de Se illa
A da. Reina Me cedes, s/n, 41012, Se illa (Spain)
Tel´e ono: 34.954.556.921; e-mail: [email p o ec ed]
P´agina de ed: h p://www.us.es/g ocoma
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