Ca an’s con uc ions and he wis ed Eilenbe g-Zilbe heo em 1
V. ´
Al a ez, J.A. A ma io, M.D. F au and P. Real
Dp o. Ma ema ica aplicada I, Uni e sidad de Se illa, Spain
Dedica ed o P o esso To nike Kadeish ili o his 60 anni e sa y
Abs ac
Le G×τG0be he p incipal wis ed Ca esian p oduc wi h ib e G, base Gand
wis ing unc ion τ:G0
∗→G∗−1whe e Gand G0a e simplicial g oups as well as G×τG0;
and CN(G)⊗ CN(G0) be he wis ed enso p oduc associa ed o CN(G×τG0) by he
wis ed Eilenbe g-Zilbe heo em. He e we p o e ha he pai (CN(G)⊗ CN(G0), µ)
is a mul iplica i e Ca an’s cons uc ion whe e µis he s anda d p oduc on CN(G)⊗
CN(G0). Fu he mo e, assuming ha a con ac ion om CN(G0) o HG0exis s and
using echniques om homological pe u ba ion heo y, we ex end he o me esul o
o he “ wis ed” enso p oduc s o he o m CN(G)⊗HG0.
Key Wo ds: Simplicial g oups; Twis ed ca esian p oduc ; Eilenbe g-Zilbe heo em; Ca an’s
cons uc ion; Con ac ion; Homological pe u ba ion lemma.
MSC: 55R20; 18D99.
1 In oduc ion
The wis ed Eilenbe g-Zilbe heo em [B o67, Shi62] es ablishes a con ac ion (a special
chain homo opy equi alence) om he no malized canonical chain complex CN(F×τB) o
he wis ed ca esian p oduc F×τB o he wis ed enso p oduc (in he sense o [B59])
CN(F)⊗ CN(B). As a module, CN(F)⊗ CN(B) is he o dina y enso p oduc o CN(F)
wi h CN(B); bo h o which a e DGA-algeb as, when Fand Ba e simplicial g oups. In
a ecen pape [AAFR07], he au ho s p o ed ha i F, B and F×τBa e g oups, hen
CN(F)⊗ CN(B) is a DGA-algeb a wi h espec o he module map
µ:CN(F)⊗ CN(B)⊗CN(F)⊗ CN(B)→CN(F)⊗ CN(B)
by µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1) whe e µCN(F)and µCN(B)a e he p oduc s in CN(F)
and CN(B) espec i ely, and T(x⊗y)=(−1)|x| |y|y⊗x.
Ca an in oduces in [Ca 56] he no ion o cons uc ion as an impo an ool o ho-
mology compu a ions and o he s udy o cohomology ope a ions ( o u he de ails, see
[Moo76]). I is well-known ha i Fis a g oup hen CN(F)⊗ CN(B) is a Ca an’s con-
s uc ion. Fu he mo e, P ou e [P o84] p o ed ha i Gis an abelian simplicial g oup hen
CN(G)⊗ CN(W(G)), associa ed o no malized chain complex o he uni e sal G-bundle
1Add ess co espondence o P o . Jos´e And ´es A ma io, Depa amen o de Ma em´a ica Aplicada I, ETSII,
Uni e sidad de Se illa, A da. Reina Me cedes, S.N. 41012 Se illa (Spain); Fax: +34-954557878; E-mail:
[email p o ec ed]
1
G×τW(G) (see [May67, p.88]) by he wis ed Eilenbe g-Zilbe heo em, is a mul iplica-
i e Ca an’s cons uc ion. He e we ex end his esul o a wide class o p incipal wis ed
ca esian p oduc o simplicial g oups (TCP). Be o e s a ing his esul we ecall he no ion
o TCP. Conside wo simplicial se s F,Band a simplicial g oup Gwhich ope a es on F
om he le . A Twis ed Ca esian P oduc Ewi h ib e F, base Band s uc u al g oup G
consis s o a simplicial se En=Fn×Bnand
∂0( , b)=(τb ∗∂0 , ∂0b)
∂i( , b) = (∂i , ∂ib), o i > 0
si( , b) = (si , sib), o i≥0;
as ace and degene acy ope a o s. He e ∗:G×F→Fis he ac ion o Gon Fand τis a
wis ing unc ion, i.e., τn:Bn→Gn−1, n ≥1 sa is ies
∂0τ(b)=[τ(∂0b)]−1·τ(∂1b)
∂iτ(b) = τ(∂i+1b), o i > 0
siτ(b) = τ(si+1b), o i≥0
τ(s0b) = en,
whe e endeno es he iden i y elemen o he co esponding g oup Gn. We w i e E=F×τB.
I F=G hen we say ha his PCT is p incipal. He e a e ou main esul s.
Theo em 1.1. Le Fand Bbe simplicial g oups and τ:B→Fbe a wis ing unc ion,
such ha , he p incipal wis ed Ca esian p oduc (PTCP), F×τB, wi h ib e Fand base
Bis a simplicial g oup. Then, he pai
(CN(F)⊗ CN(B), µ)
associa ed o CN(F×τB)by he wis ed Eilenbe g-Zilbe heo em, is a mul iplica i e Ca an’s
cons uc ion.
I we also suppose ha Fis educed and he e exis a con ac ion c om CN(B) o a
DGA-module HB,
c:φ:CN(B)
*
)
gH(B).
Then, using, he echniques o homological pe u ba ion heo y, i is able o cons uc (see
[LS87, AAFR09]) a con ac ion
φ:CN(F×τB)
*
)
g(CN(F)⊗HB, D)
whe e Ddeno es he di e en ial o he complex on he le and has he o m d⊗1+1⊗d+d ∩
(“ e ms o highe o de ”).
In he case ha HB is small enough so ha he compu a ion o i s homology can
ac ually be ca ied ou , we say ha he pai (c, HB) is a homological model o B. Le us
obse e ha , in his si ua ion, (CN(F)⊗HB, D) is simila o he dual o Hi sch complex
[Hi 53].
2
Theo em 1.2. Unde he hypo heses o he heo em 1.1 and assuming ha Fis educed,
HB is a DGA-algeb a and cis a semi- ull algeb a con ac ion (a no ion o con ac ion
be ween algeb as weake han algeb a con ac ion). I is able o s a e ha
CN(F)⊗H(B), D
is a mul iplica i e Ca an’s cons uc ion as well.
2 The p oo o he heo em 1.1
Fi s ly, we will quickly e iew basic no ions o Homological Algeb a, and in oduce he
no a ion and e minology ha we use h oughou he emainde o his a icle. Mo e de ails
can be ound in [McL95]. Le Λ be a commu a i e ing wi h non ze o uni , aken hence o h
as g ound ing and ixed h oughou , and Abe an augmen ed di e en ial g aded algeb a
o e Λ, b ie ly a DGA-algeb a. The di e en ial, p oduc , augmen a ion and coaugmen a ion
o Awill be deno ed espec i ely by dA,µA,Aand ηA. Ne e heless, we will some imes
w i e hem simply as d,µ,and ηwhen no con usion can a ise. In wha ollows, he
Koszul sign con en ions will be used. A mo phism ρ:A∗→A∗−1is called de i a ion i i
is compa ible wi h he algeb a s uc u es on A. The deg ee o an elemen a∈Ais deno ed
by |a|. In addi ion, we ecall ha i Bis also a DGA-algeb a, hen A⊗Bhas canonically
associa ed an algeb a s uc u e by means o he mo phism µA⊗B= (µA⊗µB)(1A⊗T⊗1B),
whe e T(b⊗a) = (−1)|b| |a|a⊗b. I he DG-algeb a Ais connec ed, ha is A0= Λ and
d1:A1→A0is ze o, hen he e is a canonical augmen a ion A= 1Λ:A0→Λ. I A
is a DG-algeb a, hen A]will deno e he g aded algeb a ob ained om Aby se ing he
di e en ial o Aequal o ze o (i.e. o ge ing he di e en ial), and i Mis an A-module,
hen M]will deno e A]-module ob ained by se ing he di e en ial equal o ze o.
We will use he e he wis ed enso p oduc s uc u e. Le Abe a DG-algeb a and C
be a DG-coalgeb a (we deno e by ∆Ci s cop oduc ). A wis ing cochain is a mo phism o
g aded modules :C∗→A∗−1such ha
dA + dC+ ∪ = 0, A = 0, ηC= 0;
whe e ∪ =−µA( ⊗ )∆C. I is well-known ha d =dA⊗1 + 1 ⊗dC+ ∩is a di e en ial
on A⊗C, whe e he mo phism ∩is de ined by:
∩= (µA⊗1)(1 ⊗ ⊗1)(1 ⊗∆C).(1)
The DG-module (A⊗C, d ) is called he wis ed enso p oduc (o TTP) o Aand C
along . We will also use he no a ion A⊗ C o such a DG-module.
A cons uc ion is a iple (A, N, M) whe e
1. Ais a DGA-algeb a.
2. Mis an augmen ed A-module.
3
3. Nis a DGA-module such ha N=M= Λ ⊗AM=M/I(A)Mwhe e I(A) is he
augmen a ion ideal o A.
sa is ying ha M]=A]⊗N].
Example 2.1. The wis ed enso p oduc A⊗ Cgi es ise o a cons uc ion (A, C, A⊗ C).
Amul iplica i e cons uc ion is a cons uc ion (A, N, M) oge he wi h he s uc u e o
an algeb a on Mand also on Nsuch ha A]⊗N]→M]is an isomo phism o algeb as.
Hence, he p oo o Theo em 1.1 ollows a once om he ac ha he mo phism
µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1)
endows o CN(F)⊗ CN(B) o a DGA-algeb a s uc u e (see [AAFR07, Theo em 3.9.]).
3 The p oo o he heo em 1.2
We assume h oughou his sec ion ha Mand Ndeno e wo DGA-modules such ha a
con ac ion om N o Mexis s.
We ecall ha a con ac ion (see [EM53], [HK91]) is a da a se c:{N, M, , g, φ}whe e
:N→Mand g:M→Na e mo phisms o DGA-modules ( espec i ely, called he p o-
jec ion and he inclusion) and φ:N→Nis a mo phism o g aded modules o deg ee +1
(called he homo opy ope a o ). These da a a e equi ed o sa is y he ules: (c1) g = 1M,
(c2) φdN+dNφ+g = 1N(c3) φφ = 0, (c4) φg = 0 and (c5) φ = 0. The las h ee a e
called he side condi ions [LS87]. In ac , hese may always be assumed o hold, since he
homo opy φcan be al e ed o sa is y hese condi ions [GL89]. These o mulas imply ha
bo h chain complexes Nand Mha e he same homology. We will also deno e a con ac ion
cby φ:N
*
)
gM. The Eilenbe g-Zilbe heo em [EZ53] p o ides he mos classic example o
a con ac ion o chain complexes.
Now we add an addi ional s uc u e: Nis a DGA-A-module wi h p oduc µN:A⊗N→
N. No such assump ion is made on M(Mis a DGA-module) bu he ques ion will a ise
i (M, µM) whe e µM=φµN(1 ⊗g): A⊗M→Mbecomes a DGA-A-module. Unde he
hypo hesis ha φµN(1 ⊗g) = 0 we gi e an a i ma i e answe . Mo eo e , he injec ion gis
A-lineal. The p oo o his esul is a simple inspec ion.
Now, we ecall he concep o a pe u ba ion da um. le :N→Nbe a mo phism
o g aded modules. The mo phism is poin wise nilpo en i o all x∈N(x6= 0), a
posi i e in ege nexis s (in gene al, he numbe ndepends on he elemen x) such ha
n(x) = 0. A pe u ba ion o a DGA-module Nis a mo phism o g aded modules δ:N→N
o deg ee −1, such ha (dN+δ)2= 0 and Aδ1= 0. A pe u ba ion da um o he con ac ion
c:{N, M, , g, φ}is a pe u ba ion δo he DGA-module N e i ying ha he composi ion
φδ is poin wise nilpo en .
4
AT ans e ence P oblem consis s o a con ac ion c:{M, N, , g, φ} oge he wi h a pe -
u ba ion δo he DGA-module N. The p oblem is o de e mine new mo phisms dδ, δ, gδ
and φδsuch ha cδ:{(N, dN+δ),(M, dM+dδ), δ, gδ, φδ}is a con ac ion.
The Basic Pe u ba ion Lemma ([B o67, GL89, GLS91, Rea00]) gi es an explici solu ion
o he T ans e ence P oblem, assuming ha δis a pe u ba ion da um o c.
Theo em 3.1. (BPL)
Le c:{N, M, , g, φ}be a con ac ion and δ:N→Na pe u ba ion da um o c. Then,
a new con ac ion
cδ:{(N, dN+δ),(M, dM+dδ), δ, gδ, φδ}
is de ined by he o mulas: dδ= δΣδ
cg; δ= (1 −δΣδ
cφ);gδ= Σδ
cg;φδ= Σδ
cφ; whe e
Σδ
c=X
i≥0
(−1)i(φδ)i= 1 −φδ +φδφδ − · · · + (−1)i(φδ)i+· · · .
Le us no e ha Σδ
c(x) is a ini e sum o each x∈N, because o he poin wise nilpo ency
o he composi ion φδ. Mo eo e , i is ob ious ha he mo phism dδis a pe u ba ion o
he DG-module (M, dM).
The wis ed Eilenbe g-Zilbe heo em can be seen as an impo an example o he use-
ulness o his lemma (see [Shi62]). I sol es he T ans e ence P oblem o wis ed ca esian
p oduc s.
In he heo em below we assume ha Nis a DGA-A-module. This heo em gi es
condi ions unde which he BPL wo ks p ese ing he DGA-A-module ca ego y.
Theo em 3.2. Le δ:N→Nbe a pe u ba ion da um o φ:N
*
)
gMsuch ha δis compa ible
wi h he A-module s uc u e on N(i.e., µN(1 ⊗δ) = δµN). I
φµN(1 ⊗g) = 0 and φµN(1 ⊗φ)=0,
hen he DGA-module Mδ= (M, d+dδ), ob ained by applying BPL, is a DGA-A-module wi h
ega ds o he module map µMδ= µN(1 ⊗g)and he injec ion o he pe u bed con ac ion,
gδ, is A-lineal.
P oo . This is again seen by inspec ion.
Le us ecall ha he DGA-module A⊗Nhas a i ial s uc u e o A-module wi h
ega ds o he module map
µA⊗N:A⊗(A⊗N)→A⊗N
a1⊗(a2⊗n)→µA(a1⊗a2)⊗n.
F om φ:N
*
)
gM, i is well-known ha we can es ablish his new con ac ion
1⊗φ:A⊗N(1⊗ )
*
)
(1⊗g)A⊗M. (2)
5

I may be eadily e i ied ha he ollowing iden i ies hold:
(1 ⊗φ)µA⊗N(1 ⊗(1 ⊗g)) = 0,(1 ⊗φ)µA⊗N(1 ⊗(1 ⊗φ)) = 0.
µA⊗M= (1 ⊗ )µA⊗N(1 ⊗(1 ⊗g)).
Wi h hese iden i ies a hand, we can w i e he ollowing consequence o Theo em 3.2.
Co olla y 3.3. I δ:A⊗N→A⊗Nis a pe u ba ion da um o (2) such ha i is
compa ible wi h he A-module s uc u e on A⊗N, hen he DGA-module (A⊗M)δ=
(A⊗M, d +dδ), ob ained by applying BPL, is a DGA-A-module wi h ega ds o he module
map µA⊗Mand he injec ion o he pe u bed con ac ion, (1 ⊗g)δ, is A-lineal.
In he sequel, we ocus on he case A=CN(F) and N=CN(B) whe e Fand Ba e
simplicial g oups. I we assume ha a con ac ion, c,
φ:CN(B)
*
)
gH(B) (3)
exis s, hen we can es ablish he ollowing con ac ion
CN(F)⊗CN(B)*
)CN(F)⊗HB. (4)
Le τ:B→Fbe a wis ing unc ion, such ha , he p incipal wis ed Ca esian p oduc
(PTCP), F×τB, wi h ib e Fand base Bis a simplicial g oup. The complex CN(F)⊗
CN(B) deno es he TTP associa ed o CN(F×τB) by he wis ed Eilenbe g-Zilbe heo em.
Lemma 3.4. The mo phism ∩= (µCN(F)⊗1)(1 ⊗ ⊗1)(1 ⊗∆CN(B))(see (1)) sa is ies he
ollowing p ope ies:
1. I Fis educed, hen ∩is a pe u ba ion da um o he con ac ion (4).
2. ∩is compa ible wi h he CN(F)-module s uc u e on CN(F)⊗ CN(B)(i.e., µF⊗B(1⊗
∩) = ∩µF⊗B).
P oo .
1. See [LS87, Lemma 3.4] o [AAFR09, p oposi ion 5.3].
2. This is again seen by inspec ion.
Hence, i Fis educed, we can pe u b he con ac ion (4) using ∩as a pe u ba ion
da um. Wi h hese inpu s, he BPL gi es as ou pu he ollowing con ac ion:
CN(F)⊗ CN(B)*
)(CN(F)⊗HB, D).(5)
whe e Ddeno es i s di e en ial.
Now, we can s a e:
6
P oposi ion 3.5. I Fis educed, hen (CN(F)⊗HB, D)becomes a DGA-CN(F)-module
wi h ega ds o he i ial module s uc u e on CN(F)⊗HB.
P oo .
The p oo ollows om Co olla y 3.3 and lemma 3.4.
Focusing only on he unde lying g aded module s uc u e. We ha e he ollowing iden-
i y:
I(CN(F))(CN(F)⊗HB) = I(CN(F)) ⊗HB.
I may be eadily e i ied ha he ollowing p ope ies hold:
1. ∩(I(CN(F)) ⊗CN(B)) ⊆I(CN(F)) ⊗CN(B).
2. The co esponden es ic ions o he mo phisms composing he con ac ion (5) o m
he ollowing con ac ion:
I(CN(F)) ⊗CN(B)*
)I(CN(F)) ⊗HB.
The o mula o he di e en ial o he complex (CN(F)⊗HB, D), gi en by he BPL, is
D=d⊗1+1⊗d+d ∩whe e
d ∩= (1 ⊗ ) ∩(1 ⊗g)−(1 ⊗ ) ∩(1 ⊗φ) ∩(1 ⊗g) + · · · (6)
As immedia e consequence o his o mula and he p ope ies abo e, we ha e
D(I(CN(F)) ⊗HB)⊆I(CN(F)) ⊗HB.
Hence, (I(CN(F)) ⊗HB, D) is a DGA-CN(F)-submodule o (I(CN(F)) ⊗HB, D).
P oposi ion 3.6. We ha e he ollowing iden i y o DGA-modules
HB = (CN(F)⊗HB, D)/(I(CN(F))(CN(F)⊗HB, D))
P oo .
On he one hand, i is easy o see ha his wo complexes a e isomo phic as g aded
module.
On he o he hand, aking in o accoun ha anishes on one-simplices since Fis
educed (see [May67]) and he o mula (6), we ha e
d ∩( ⊗b) = 0|b| ≤ 1
0 mod I(CN(F)) ⊗HB |b|>1
Hence, we ha e ha he di e en ial o bo h complexes a e he same.
P oposi ion 3.7. Unde he hypo heses o he P eposi ion 3.5 and 3.6. We can s a e ha
(CN(F)⊗HB, D)is a cons uc ion.
7
Finally, I we assume ha he con ac ion (3) is semi- ull, we will ge he desi ed esul .
Now, we ecall ha cis a semi- ull algeb a con ac ion ([Rea00]) i he injec ion, g, is a
mo phism o DGA-algeb as and he p ojec ion, , and he homo opy ope a o , φ, sa is y
he ollowing p ope ies:
µC(B)(φ⊗φ)=0, µC(B)(φ⊗g)=0, µC(B)(g⊗φ)=0,
φµC(B)(φ⊗φ)=0, φµC(B)(φ⊗g)=0, φµC(B)(φ⊗φ) = 0.
Unde hese condi ions we can say ha hese con ac ions (4) and (5) a e semi- ull (see
[Rea00, Theo em 4.18]). Mo eo e , (CN(F)⊗HB, D) is a DGA-algeb a wi h ega ds o
he s anda d p oduc CN(F)⊗HB. F om his ac , we conclude ha (CN(F)⊗HB, D) is
a mul iplica i e cons uc ion.
4 Conclusions
In his pape we ha e aced he p oblem o ans e ing he mul iplica i e cons uc ion
s uc u e up o con ac ion. The p oblem o ans e ing o he s uc u es (e.g. algeb a,
coalgeb a, TTP) up o con ac ion has been widely ea ed in he li e a u e [GLS91, Rea00,
AAFR05]. In gene al, we ha e ha (co)algeb a and TTP become A∞-(co)algeb a and A∞-
TTP ia con ac ion, espec i ely.
We p o ed ha he p ope y o being a mul iplica i e cons uc ion on CN(F)⊗ CN(B)
has been ans e ed o (CN(F)⊗HB, D) ia con ac ion. Howe e , i we ocus on he
s uc u e o TTP, we only ha e ha (CN(F)⊗HB, D) is an A∞- wis ed enso p oduc o
CN(F) (algeb a) and HB (A∞-coalgeb a) along ¯
= g (A∞- wis ed cochain), see [AAFR05,
Theo em 2.1].
Acknowledgmen
We would like o hank K is een Cheng o he eading o his pape . This wo k was
pa ially suppo ed by he PAICYT esea ch p ojec FQM–296 om Jun a de Andaluc´ıa
(Spain).
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[AAFR05] ´
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[AAFR07] ´
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[AAFR09] ´
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compa ison o he educed ba cons uc ion and he educed W-cons uc ion. To appea
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