a Xi :0707.1892 2 [ma h.KT] 19 Oc 2007
MALTSINIOTIS’S FIRST CONJECTURE FOR K1
FERNANDO MURO
Abs ac . We show ha K1(E) o an exac ca ego y Eag ees wi h K1(DE) o
he associa ed iangula ed de i a o DE. Mo e gene ally we show ha K1(W)
o a Waldhausen ca ego y Wwi h cylinde s and a sa u a ed class o weak
equi alences ag ees wi h K1(DW) o he associa ed igh poin ed de i a o
DW.
In oduc ion
Fo a long ime he e was an in e es in de ining a nice K- heo y o iangula ed
ca ego ies such ha Quillen’s K- heo y o an exac ca ego y Eag ees wi h he K-
heo y o i s bounded de i ed ca ego y Db(E). Schlich ing [Sch02] showed ha
such a K- heo y o iangula ed ca ego ies canno exis . I was hen na u al o
ask abou he de ini ion o a nice K- heo y o algeb aic s uc u es in e pola ing
be ween Eand Db(E).
The bes known in e media e s uc u e is Cb(E), he Waldhausen ca ego y o
bounded complexes in E, wi h quasi-isomo phisms as weak equi alences and co i-
b a ions gi en by chain mo phisms which a e le elwise admissible monomo phisms.
The de i ed ca ego y Db(E) is he localiza ion o Cb(E) wi h espec o weak equi -
alences. The Gille -Waldhausen heo em1, ela ing Quillen’s K- heo y o Wald-
hausen’s K- heo y, s a es ha he homomo phisms
τn:Kn(E)−→ Kn(Cb(E)), n ≥0,
induced by he inclusion E⊂Cb(E) o complexes concen a ed in deg ee 0, a e
isomo phisms.
The ca ego y Cb(E) is conside ed o be oo close o Eso one would s ill like o
ind an algeb aic s uc u e wi h a nice K- heo y in e pola ing be ween Cb(E) and
Db(E). The no ion o a iangula ed de i a o [G o90, Mal07] seems o be a s ong
candida e.
Mal sinio is [Mal07] de ined a K- heo y o iangula ed de i a o s oge he wi h
na u al homomo phisms
ρn:Kn(E)−→ Kn(DE), n ≥0,
1991 Ma hema ics Subjec Classi ica ion. 18E10, 18E30, 18F25, 19B99, 55S45.
Key wo ds and ph ases. K- heo y, exac ca ego y, iangula ed de i a o , Pos niko in a ian ,
s able quad a ic module.
The au ho was pa ially suppo ed by he Spanish Minis y o Educa ion and Science unde
MEC-FEDER g an s MTM2004-03629 and MTM2007-63277, and a Juan de la Cie a esea ch
con ac .
1The p oo due o Thomason-T obaugh [TT90, Theo em 1.11.7] co ec s Gille ’s [Gil81, 6.2]
and uses an ex a hypo hesis on E. This hypo hesis is no s ic ly necessa y, since he gene al case
ollows hen om co inali y a gumen s, see [Cis02].
1
2 FERNANDO MURO
whe e DEis he iangula ed de i a o associa ed o an exac ca ego y E, con-
s uc ed by Kelle in he appendix o [Mal07]. Cisinski and Neeman p o ed he
addi i i y o iangula ed de i a o K- heo y [CN05]. Mal sinio is also conjec u ed
ha ρnis an isomo phism o all n. He succeeded in p o ing he conjec u e o
n= 0.
The ollowing heo em is he main esul o his pape .
Theo em A. Le Ebe an exac ca ego y. The na u al homomo phism
ρ1:K1(E)∼
=
−→ K1(DE)
is an isomo phism.
In o de o ob ain Theo em A we use echniques in oduced in [MT07]. The e we
gi e a p esen a ion o an abelian 2-g oup D∗Wwhich encodes K0(W) and K1(W)
o a Waldhausen ca ego y W, and mo eo e he 1- ype o he K- heo y spec um
K(W) whose homo opy g oups a e he K- heo y g oups o W. This p esen a ion
is a highe dimensional analogue o he classical p esen a ion o K0(W). He e we
simila ly de ine an abelian 2-g oup Dde
∗Wwhich models he 1- ype o he K- heo y
spec um K(DW) o he igh 2poin ed de i a o DWassocia ed o a Waldhausen
ca ego y Wwi h cylinde s and a sa u a ed class o weak equi alences, such as W=
Cb(E). The K- heo y o his kind o de i a o s, mo e gene al han iangula ed
de i a o s, was de ined by Ga kusha [Ga 06] ex ending he wo k o Mal sinio is
[Mal07]. The e a e de ined compa ison homomo phisms
µn:Kn(W)−→ Kn(DW), n ≥0.
These homomo phisms canno be isomo phisms in gene al, as shown in [TV04].
Ne e heless we he e p o e he ollowing esul .
Theo em B. Le Wbe a Waldhausen ca ego y wi h cylinde s and a sa u a ed class
o weak equi alences. The na u al homomo phism
µ0:K0(W)∼
=
−→ K0(DW),
µ1:K1(W)∼
=
−→ K1(DW),
a e isomo phisms.
In Rema k 5.3 we commen on he case whe e he hypo hesis on he sa u a ion o
weak equi alences is eplaced by he 2 ou o 3 axiom, which is a weake assump ion.
Theo em A is ac ually a co olla y o he Gille -Waldhausen heo em and Theo-
em B, since DCb(E) = DEand he na u al homomo phisms ρn ac o as
ρn:Kn(E)τn
−→ Kn(Cb(E)) µn
−→ Kn(DE), n ≥0.
We assume he eade ce ain amilia i y wi h exac , Waldhausen and de i ed
ca ego ies, wi h simplicial cons uc ions and wi h homo opy heo y. We e e o
[Wei, GM03, GJ99] o he basics.
Acknowledgemen s. I am e y g a e ul o G igo y Ga kusha o sugges ing he
possibili y o using [MT07] in o de o ackle Mal sinio is’s i s conjec u e in di-
mension 1. I also eel indeb ed o Denis-Cha les Cisinski, who kindly indica ed how
o ex end he esul s o a p elimina y e sion o his pape o a b oade gene ali y.
2The e e ences [G o90, Cis03] and [Ga 06, RB07] ollow a di e en con en ion wi h espec o
sides. He e we ollow he con en ion in [G o90, Cis03], so wha we call a ‘ igh poin ed de i a o ’
is he same as a ‘le poin ed de i a o ’ in [Ga 06].
MALTSINIOTIS’S FIRST CONJECTURE FOR K13
1. The bounded de i ed ca ego y o an exac ca ego y
In his sec ion we ou line he wo-s ep cons uc ion o he de i ed ca ego y Db(E)
o an exac ca ego y E. This cons uc ion is a special case o he homo opy ca ego y
Ho Wo a Waldhausen ca ego y Wwi h cylinde s sa is ying he 2 ou o 3 axiom,
Db(E) = Ho Cb(E).
De ini ion 1.1. AWaldhausen ca ego y is a ca ego y Wwi h a dis inguished ze o
objec 0 and wo dis inguished subca ego ies wWand cW, whose mo phisms a e
called co ib a ions and weak equi alences, espec i ely. A mo phism which is bo h
a weak equi alence and a co ib a ion is said o be a i ial co ib a ion. The a ow
֌s ands o a co ib a ion and ∼
→ o a weak equi alence.
•All mo phisms 0 →Aa e co ib a ions. All isomo phisms a e co ib a ions
and weak equi alences.
•The push-ou o a mo phism along a co ib a ion is always de ined
A////

push
B

X////X∪AB
and he lowe map is also a co ib a ion.
•Gi en a commu a i e diag am
X
∼

A
∼

oo////B
∼

X′A′
oo////B′
he induced map X∪AB∼
→X′∪A′B′is a weak equi alence.
No ice ha cop oduc s A∨B=A∪0Ba e de ined in W.
A unc o W→W′be ween Waldhausen ca ego ies is exac i i p ese es co i-
b a ions, weak equi alences, push-ou s along co ib a ions and he dis inguished ze o
objec .
Example 1.2.Recall ha an exac ca ego y Eis a ull subca ego y o an abelian
ca ego y Asuch ha Econ ains a ze o objec o Aand Eis closed unde ex ensions
in A. A sho exac sequence in Eis a sho exac sequence in Abe ween objec s
in E. A mo phism in Eis an admissible monomo phism i i is he ini ial mo phism
o some sho exac sequence. The ca ego y Eis a Waldhausen ca ego y wi h
admissible monomo phisms as co ib a ions and isomo phisms as weak equi alences.
In o de o comple e he s uc u e we ix a ze o objec 0 in E.
We deno e by Cb(E) he ca ego y o bounded complexes in E,
· · · → An−1d
−→ And
−→ An+1 → · · · , d2= 0, An= 0 o |n| ≫ 0.
A chain mo phism :A→Bin Cb(E) is a quasi-isomo phism i i induces an iso-
mo phism in homology compu ed in he ambien abelian ca ego y A. The ca ego y
Cb(E) is a Waldhausen ca ego y. Weak equi alences a e quasi-isomo phisms and
co ib a ions a e le elwise admissible monomo phisms.
The e is a ull exac inclusion o Waldhausen ca ego ies E⊂Cb(E) sending an
objec Xin E o he complex
· · · → 0→X→0→ · · · ,
4 FERNANDO MURO
wi h Xin deg ee 0.
De ini ion 1.3. The homo opy ca ego y Ho Wo a Waldhausen ca ego y is a ca -
ego y equipped wi h a unc o
ζ:W−→ Ho W
sending weak equi alences o isomo phisms. Mo eo e , ζis ini ial among all unc-
o s W→Csending weak equi alences o isomo phisms, so Ho Wis well de ned up
o canonical isomo phism o e W. This ca ego y can be cons uc ued as a ca ego y
o ac ions, in he sense o [GZ67], by o mally in e ing weak equi alences in W.
The class o weak equi alences is sa u a ed i any mo phism :A→Bin W
such ha ζ( ) is an isomo phism in Ho Wis indeed a weak equi alence :A∼
→B.
Example 1.4.Weak equi alences in Cb(E), i.e. quasi-isomo phisms, a e sa u a ed
since hey a e de ec ed by a unc o H∗:Cb(E)→AZ, he cohomology unc o
om bounded complexes in E o Z-g aded objec s in A, see [CF00, P oposi ion
1.1].
The homo opy ca ego y always exis s up o se heo e ical di icul ies which do
no a ise i Wis a small ca ego y, o ins ance. This is no a ha m ul assump-
ion i one is in e es ed in K- heo y since smallness may also be equi ed in o de
o ha e well de ined K- heo y g oups. The homo opy ca ego y can howe e be
cons uc ed in a mo e s aigh o wa d way i he Waldhausen ca ego y Wsa is ies
u he p ope ies.
De ini ion 1.5. A Waldhausen ca ego y Wsa isi ies he 2 ou o 3 axiom p o ided
gi en a commu a i e diag am in W
C
A
@@






//B
^^=
=
=
=
=
=
=
i wo a ows a e weak equi alences hen he hi d one is also a weak equi alence.
Gi en an objec Ain Wacylinde IA is an objec oge he wi h a ac o iza ion
o he olding map (1,1): A∨A→Aas a co ib a ion ollowed by a weak equi alence,
A∨A֌
iIA ∼
−→
pA.
We say ha Whas cylinde s i all objec s ha e a cylinde .
Example 1.6.The Waldhausen ca ego y Cb(E) has cylinde s. The cylinde o a
bounded complex Acan be unc o ially chosen as
(IA)n=An⊕An+1 ⊕An, d =

d−1 0
0−d0
0 1 d

: (IA)n−→ (IA)n+1.
Rema k 1.7.The 2 ou o 3 axiom is o en called he sa u a ion axiom. We do no
use his e minology in his pape in o de o a oid con usion wi h De ini ion 1.3.
Usually one conside s mo e s uc u ed cylinde s in Waldhausen ca ego ies, com-
pa e [Wei, De ini ion IV.6.8]. Fo he pu poses o his pape i is enough o conside
cylinde s as de ined abo e.
MALTSINIOTIS’S FIRST CONJECTURE FOR K15
Rema k 1.8.As one can easily check, a Waldhausen ca ego y wi h a sa u a ed class
o weak equi alences sa is ies he 2 ou o 3 axiom. This applies o Cb(E).
A Waldhausen ca ego y wi h cylinde s Wsa is ying he 2 ou o 3 axiom is an
example o a igh de i able ca ego y, in he sense o [Cis03], also called p eco i-
b a ion ca ego y in [RB07], see [Cis03, Example 2.23] o [RB07, P oposi ion 2.4.2].
In pa icula any mo phism in Wcan be ac o ed as a co ib a ion ollowed by a
weak equi alence which is le in e se o a i ial co ib a ion, see [RB07, P oposi-
ion 1.3.1]. Mo eo e , one can de ine a homo opy ela ion in Wand cons uc he
homo opy ca ego y Ho Wby a homo opy calculus o le ac ions as we indica e
below, see [Cis03, Sec ion 1] o [RB07, Sec ion 5.4].
Le Wbe a Waldhausen ca ego y wi h cylinde s sa is ying he 2 ou o 3 axiom.
As usual we say ha wo mo phisms , g:A→Bin Wa e s ic ly homo opic i
he e is a mo phism H:IA →Bwi h Hi = ( , g). The maps , g a e homo opic
≃gi he e exis s a weak equi alence h:B∼
→B′such ha h and hg a e s ic ly
homo opic. ‘Being homo opic’ is a na u al equi alence ela ion and he quo ien
ca ego y is deno ed by πW. The homo opy ca ego y Ho Wis ob ained by calculus
o le ac ions in πW. Objec s in Ho Wa e he same as in W. A mo phism A→B
in Ho Wis ep esen ed by a diag am in W,
A−→
α1
X∼
←−
α2
B.
Ano he diag am
A−→
α′
1
Y∼
←−
α′
2
B
ep esen s he same mo phism i he e is a diag am in W
X
>>
α1
}
}
}
}
}
}
}``
∼
α2
A
A
A
A
A
A
A
A Z
//oo∼

OOB
Y
α′
1
A
A
A
A
A
A
A~~
∼
α′
2
}
}
}
}
}
}
}
whose p ojec ion o πWis commu a i e. No ice ha , by he 2 ou o 3 axiom, he
e ical a ows in his diag am a e also weak equi alences. The composi e o wo
mo phisms A→
αB→
βCin Ho W ep esen ed by
A−→
α1
X∼
←−
α2
B−→
β1
Y∼
←−
β2
C
is de ined as ollows. I β1is a co ib a ion hen he push-ou
B
push
//β1
//
∼
α2

Y
∼¯α2

X//¯
β1
//X∪BY
is de ined, ¯α2is a weak equi alence, and βα:A→Cis ep esen ed by
A−→
¯
β1α1
X∪BY∼
←−
¯α2β2
C.

6 FERNANDO MURO
In gene al we can ac o β1as co ib a ion ollowed by a weak equi alence
β1:B֌
β′
1
Z∼
−→
Y
such ha he e is a mo phism s:Y∼
֌Zwi h s = 1Y. The diag am
Y
>>
β1
~
~
~
~
~
~
~``
∼
β2
@
@
@
@
@
@
@
B Z
//
β′
1
//oo∼
sβ2
OO
C
commu es in W, so β:B→Cis also ep esen ed by
B֌
β′
1
Z∼
←−
sβ2
C,
whe e he i s a ow is a co ib a ion, and we can use his ep esen a i e o de ine
he composi e βα:A→C.
The unc o
ζ:W−→ Ho W
is he iden i y on objec s and sends a mo phism :A→B o he mo phism
ζ( ): A→B ep esen ed by
A−→
B∼
←−
1B
B.
I :A∼
→Bis a weak equi alence hen ζ( ) is an isomo phism and ζ( )−1is
ep esen ed by
B−→
1B
B∼
←−
A,
hence a mo phism α:A→Bin Ho W ep esen ed by
A−→
α1
X∼
←−
α2
B
coincides wi h ζ(α2)−1ζ(α1) = α.
Rema k 1.9.I αabo e is an isomo phism in Ho W hen ζ(α1) = ζ(α2)αis also an
isomo phism. In pa icula i Whas a sa u a ed class o weak equi alences hen
α1:A∼
→Xis necessa ily a weak equi alence.
Rema k 1.10.Fo W=Cb(E) he ca ego y πW=Hb(E) is usually e med he
bounded homo opy ca ego y, while Ho W=Db(E) is called he bounded de i ed
ca ego y o E.
2. On Waldhausen and de i ed K- heo y
Recall ha a co ibe sequence in a Waldhausen ca ego y W
A֌B։B/A
is a push-ou diag am
A////

push
B

0////B/A
The e o e he quo ien B/A is only de ined up o canonical isomo phism o e B,
al hough he no a ion B/A is s anda d in he li e a u e.
MALTSINIOTIS’S FIRST CONJECTURE FOR K17
The K- heo ies we deal wi h in his pape a e cons uc ed by using he Wald-
hausen ca ego ies SnW ha we now ecall.
De ini ion 2.1. An objec A•• in he ca ego y SnW,n≥0, is a commu a i e
diag am in W
(2.2) Ann
.
.
.
OO
A22 //··· //A2n
OO
A11 //A12 //
OO
··· //A1n
OO
A00 //A01 //
OO
A02 //
OO
··· //A0n
OO
such ha Aii = 0 and Aij ֌Aik ։Ajk is a co ibe sequence o all 0 ≤i≤j≤
k≤n. No ice ha hese condi ions imply ha he whole diag am is de e mined,
up o canonical isomo phism, by he sequence o n−1 composable co ib a ions
(2.3) A01 ֌A02 ֌···֌A0n.
A mo phism A•• →B•• in SnWis a na u al ans o ma ion be ween diag ams
gi en by mo phisms Aij →Bij in W. The ca ego y SnWis a Waldhausen ca ego y.
A mo phism A••
∼
→B•• is a weak equi alence i all mo phisms Aij
∼
→Bij a e weak
equi alences in W. A co ib a ion A•• ֌B•• is a mo phism such ha Aij ֌Bij
and Bij ∪Aij Aik ֌Bik a e co ib a ions, 0 ≤i≤j≤k≤n. The dis inguished
ze o objec is he diag am wi h 0 in all en ies.
The ca ego ies SnWassemble o a simplicial ca ego y S.W. The ace unc o
di:SnW→Sn−1Wis de ined by emo ing he i h ow and he i h column, and
he degene acy unc o si:SnW→Sn+1Wis de ined by duplica ing he i h ow
and he i h column, 0 ≤i≤n. Faces and degene acies a e exac unc o s. Fo
he de ini ion o he simplicial s uc u e i is c ucial o conside he whole diag am
(2.2) ins ead o jus (2.3).
One can ob ain a poin ed space ou o he simplicial ca ego y S.Was ollows.
We es ic o he subca ego ies o weak equi alences wS.W, hen we ake le elwise
he ne e in o de o ge a bisimplicial se Ne wS.W, we conside he diagonal
simplicial se Diag Ne wS.W, and i s geome ic ealiza ion
|Diag Ne wS.W|.
This poin ed space, ac ually a educed CW-complex, is he 1-s age o he Wald-
hausen K- heo y spec um K(W) [Wal85], which is an Ω-spec um, hence he K-
heo y g oups o Wa e he homo opy g oups
Kn(W) = πn+1|Diag Ne wS.W|, n ≥0.
8 FERNANDO MURO
We now assume ha Whas cylinde s and sa isi ies he 2 ou o 3 axiom, so
ha he associa ed igh poin ed de i a o DWis de ined, see [Cis03, Co olla y
2.24 and he duals o Lemmas 4.2 and 4.3]. Then he Waldhausen ca ego ies SnW
also ha e cylinde s and sa is y he 2 ou o 3 axiom. We will nei he ecall he
no ion o de i a o no he de ini ion o he de i a o DWbu jus he K- heo y o
DW, we e e he in e es ed eade o [G o90, Mal07, Ga 06, RB07]. Fo his we
conside he homo opy ca ego ies Ho SnWand he subg oupoids o isomo phisms
iHo SnW. These g oupoids o m a simplicial g oupoid iHo S.Wand we can conside
he poin ed space
|Diag Ne iHo S.W|,
which is he 1-s age o Ga kusha’s de i ed K- heo y Ω-spec um DK(W).
Ga kusha [Ga 05] conside s de i ed K- heo y o W=Cb(E), and mo e gene ally
o Wa nice complicial biWaldhausen ca ego y, al hough he de ini ion immedia ely
ex ends o Waldhausen ca ego ies wi h cylinde s sa is ying he 2 ou o 3 axiom, as
indica ed he e. Mo eo e , Ga kusha shows ha he e is a na u al weak equi alence
DK(W)∼
→K(DW) be ween he de i ed K- heo y spec um o a nice complicial
biWaldhausen ca ego y Wand he K- heo y spec um o he associa ed de i a o
DW, compa e [Ga 05, Co olla y 4.3]. Ne e heless [Ga 05, Co olla y 4.3] only
uses he ac ha all mo phisms in W ac o as a co ib a ion ollowed by a weak
equi alence, compa e also [Ga 05, Lemmas 4.1 and 4.2], so we also ha e a na u al
weak equi alence DK(W)∼
→K(DW) o Wa Waldhausen ca ego y wi h cylinde s
sa is ying he 2 ou o 3 axiom, and he e o e
Kn(DW)∼
=πn+1|Diag Ne iHo S.W|, n ≥0.
The unc o s ζ:SnW→Ho SnW es ic o wSnW→iHo SnW. These unc o s
gi e ise o a map
|Diag Ne wS.W| −→ | Diag Ne iHo S.W|
which induces he compa ison homomo phisms in homo opy g oups,
µn:Kn(W)−→ Kn(DW), n ≥0.
This map is ac ually he 1-s age o a compa ison map o spec a
(2.4) K(W)−→ K(DW).
In he es o his pape we will be mainly conce ned wi h he s uc u e o he
bisimplicial se s X= Ne wS.Wand Y= Ne iHo S.Win low dimensions, ha we
now e iew mo e ho oughly.
Abisimplicial se Zconsis s o se s Zm,n,m, n ≥0, oge he wi h ho izon al
and e ical ace and degene acy maps
dh
i:Zm,n −→ Zm−1,n, sh
i:Zm,n −→ Zm+1,n,0≤i≤m,
d
j:Zm,n −→ Zm,n−1, s
j:Zm,n −→ Zm,n+1,0≤j≤n,
sa is ying some ela ions ha we will no ecall he e, compa e [GJ99]. An elemen
zm,n ∈Zm,n is a bisimplex o bideg ee (m, n) and o al deg ee m+n.
A gene ic bisimplex zm,n o bideg ee (m, n) can be depic ed as he p oduc o wo
geome ic simplices o dimensions mand nwi h e ices labelled by he p oduc
se
{0,...,m} × {0,...,n},
see Figs. 1 and 2. The ho izon al i h ace dh
i(zm,n) is he ace ob ained by emo ing
MALTSINIOTIS’S FIRST CONJECTURE FOR K19
(0,0) (1,0) z1,0
(0,1) (1,1)
(0,0) (1,0)
z1,1
(0,0) (1,0)
(2,0)



















z2,0
Figu e 1. Bisimplices o o al deg ee 1 and 2.
he in e io , he e ices (i, j), o all j, and he inciden aces o he bounda y.
Simila ly he e ical j h ace d
j(zm,n) is ob ained by emo ing he in e io , he
e ices (i, j), o all i, and he inciden aces o he bounda y.
(0,2) (1,2)
?
?
?
?
?
?
?
?
(0,1)
(0,0) (1,0)
(1,1)








z1,2z2,1
(0,1) (2,1)
?
?
?
?
?
?
?
?
?
?
(1,1)
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
(0,0)
?
?
?
?
?
?
?
?
?
?
(1,0)
(2,0)
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
z3,0
(0,0)
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
(1,0)
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
(2,0)
(3,0)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Figu e 2. Bisimplices o o al deg ee 3.
The bisimplicial se s Xand Ya e ho izon ally educed, i.e. X0,n =Y0,n a e
single ons o all n≥0, X1,0=Y1,0is he se o objec s in W,X1,1is he se o
weak equi alences in W,Y1,1is he se o isomo phisms in Ho W, and X2,0=Y2,0
is he se o co ibe sequences, see Fig. 3.
The se X1,2consis s o pai s o composable weak equi alences in W,Y1,2is
he se o composable isomo phisms in Ho W,X2,1is he se o weak equi alences
be ween co ibe sequences i.e. weak equi alences in S2Wwhich a e commu a i e
diag ams in W
(2.5) A′////B′////B′/A′
A////
∼
OO
B////
∼
OO
B/A
∼
OO
16 FERNANDO MURO
We hen ake he cop oduc o he i s and he second pai o degene a e bisim-
plices.
B
A∨B













A
OO
OO////
A
A∨B













B
OO
OO////
Finally we ake he di e ence be ween he co esponding gene a o s in D1W
−







B
A∨B













A
OO
OO////








+







A
A∨B













B
OO
OO////








.
The e is a non-abelian Eilenbe g-Zilbe heo em behind his o mula, compa e
[MT07, Theo em 4.10 and Example 4.13].
The main esul o [MT07] is he ollowing heo em.
Theo em 4.3. [MT07, Theo em 1.7] Le Wbe a Waldhausen ca ego y. The e is
a na u al isomo phism in Ho squad
D∗W∼
=
−→ λ0K(W).
This esul is meaning ul since λ0K(W) is huge compa ed wi h D∗W, while D∗W
is di ec ly de ined in e ms o he basic s uc u e o he Waldhausen ca ego y W.
As a consequence we ha e an exac sequence o g oups
K1(W)֒→D1W∂
−→ D0W։K0(W).
We now ex end Theo em 4.3 o de i ed K- heo y.
De ini ion 4.4. We de ine Dde
∗Was he s able quad a ic module gene a ed in
dimension ze o by he symbols
(DG1) [A] o any objec in W, i.e. he same as (G1),
and in dimension one by
(DG2) [A∼
=
→A′] o any isomo phism in Ho W,
(DG3) [A֌B։B/A] o any co ibe sequence in W, i.e. he same as (G3),
such ha he ollowing ela ions hold.
(DR1) ∂[A∼
=
→A′] = −[A′] + [A].
(DR2) = (R2).
(DR3) = (R3).
(DR4) [A1
→A] = 0.
(DR5) = (R5).
(DR6) Fo any pai o composable isomo phisms A∼
=
→B∼
=
→Cin Ho W,
[A∼
=
→C] = [B∼
=
→C] + [A∼
=
→B].

MALTSINIOTIS’S FIRST CONJECTURE FOR K117
(DR7) Fo any commu a i e diag am in Was (2.7) we ha e
[α:A∼
=
→A′] + [γ:B/A ∼
=
→B′/A′][A]=−[A′֌B′։B′/A′]
+ [β:B∼
=
→B′] + [A֌B։B/A].
He e α=ζ(α2)−1ζ(α1), β=ζ(β2)−1ζ(β1) and γ=ζ(γ2)−1ζ(γ1).
(DR8) = (R8).
(DR9) = (R9).
I Wis a Waldhausen ca ego y wi h cylinde s and a sa u a ed class o weak equi -
alences hen he p esen a ion o he s able quad a ic module Dde
∗Wis de e mined
by he bisimplicial s uc u e o Y= Ne iHo S.Wand he map ∨:Y×Y→Yin
o al deg ee ≤3, see Sec ion 2, exac ly in he same way as D∗Wis de e mined by
X= Ne wS.Wand ∨:X×X→X, see Rema k 4.2. The e o e eplacing Xby Y
in he p oo o [MT07, Theo em 1.7] we ob ain he ollowing esul .
Theo em 4.5. Le Wbe a Waldhausen ca ego y wi h cylinde s and a sa u a ed
class o weak equi alences. The e is a na u al isomo phism in Ho squad
Dde
∗W∼
=
−→ λ0K(DW).
As a consequence we ha e an exac sequence o g oups
K1(DW)֒→Dde
1W∂
−→ Dde
0W։K0(DW).
Rema k 4.6.As one can easily check, aking λ0in he compa ison map o spec a
(2.4) which induces µn:Kn(W)→Kn(DW) in homo opy g oups co esponds o
he na u al mo phism in squad,
¯µ:D∗W−→ Dde
∗W,
[A]7→ [A],
[ :A∼
→A′]7→ [ζ( ): A∼
=
→A′],
[A֌B։B/A]7→ [A֌B։B/A],
unde he na u al isomo phisms o Theo ems 4.3 and 4.5. In pa icula aking
π0and π1in his mo phism o s able quad a ic modules we ob ain µ0and µ1,
espec i ely. This ac will be used below in he p oo o Theo em B.
5. P oo o Theo em B
Theo em B is a consequence o he ollowing esul .
Theo em 5.1. Le Wbe a Waldhausen ca ego y wi h cylinde s and a sa u a ed
class o weak equi alences. The na u al mo phism in squad
¯µ:D∗W−→ Dde
∗W,
de ined in Rema k 4.6, is an isomo phism.
The key o he p oo o Theo em 5.1 is he ollowing lemma.
Lemma 5.2. Le Wbe a Waldhausen ca ego y wi h cylinde s sa is ying he 2 ou o
3 axiom. Two weak equi alences , g:A∼
→A′which a e homo opic ≃g ep esen
he same elemen in D1W,
[ :A∼
→A′] = [g:A∼
→A′].
18 FERNANDO MURO
P oo . Le IA be a cylinde o Aand
A∨A֌
iIA ∼
−→
pA
a ac o iza ion o he olding map, i.e. i i= (i0, i1) hen pi0=pi1= 1A. Since
bo h pand 1Aa e weak equi alences we deduce om he 2 ou o 3 axiom ha i0
and i1a e also weak equi alences. Mo eo e , o j= 0,1,
0(R4)
= [A1A
→A]
= [pij:A∼
→A]
(R6) = [p:IA ∼
→A] + [ij:A∼
→IA],
he e o e
[i0:A∼
→IA] = −[p:IA ∼
→A] = [i1:A∼
→IA].
Fu he mo e, ≃g, so he e is a weak equi alence h:A′∼
→A′′ and a mo phism
H:IA →A′′ such ha Hi0=h and Hi1=hg. Again by he 2 ou o 3 axiom H
is a weak equi alence, and
[h:A′∼
→A′′] + [ :A∼
→A′](R6)
= [h =Hi0:A∼
→A′′]
(R6) = [H:IA ∼
→A′′] + [i0:A∼
→IA]
= [H:IA ∼
→A′′] + [i1:A∼
→IA]
(R6) = [hg =Hi1:A∼
→A′′]
(R6) = [h:A′∼
→A′′] + [g:A∼
→A′],
hence we a e done. 
We a e now eady o p o e Theo em 5.1.
P oo o Theo em 5.1. We a e going o de ine he in e se o ¯µ,
¯ν:Dde
∗W−→ D∗W.
We i s show ha
¯ν0[A] = [A],
¯ν1[α:A∼
=
→A′] = −[α2:A′∼
→X] + [α1:A∼
→X],
¯ν1[A֌B։B/A] = [A֌B։B/A],
de ines a s able quad a ic module mo phism ¯ν. He e
A∼
−→
α1
X∼
←−
α2
A′
is a ep esen a i e o he isomo phism α. Fo his we a e going o p o e ha he
image o [α:A∼
=
→A′] does no depend on he choice o a ep esen a i e.
Suppose ha
A∼
−→
α′
1
Y∼
←−
α′
2
A′
MALTSINIOTIS’S FIRST CONJECTURE FOR K119
also ep esen s α. Then he e is a diag am in W
X
>>
α1
~
~
~
~
~
~
~
~``α2
A
A
A
A
A
A
A
A
A Z
//
1oo 2

g
OO
g′
A′
Y
α′
1
@
@
@
@
@
@
@
@~~α′
2
}
}
}
}
}
}
}
}
whe e all a ows a e weak equi alences and he ou iangles commu e up o ho-
mo opy, so
−[α2:A′∼
→X] + [α1:A∼
→X] = −[α2:A′∼
→X]−[g:X∼
→Z]
+[g:X∼
→Z] + [α1:A∼
→X]
(R6) =−[gα2:A′∼
→Z] + [gα1:A∼
→Z]
Lemma 5.2 =−[ 2:A′∼
→Z] + [ 1:A∼
→Z]
Lemma 5.2 =−[g′α′
2:A′∼
→Z] + [g′α′
1:A∼
→Z]
(R6) =−[α′
2:A′∼
→Y]−[g′:Y∼
→Z]
+[g′:Y∼
→Z] + [α′
1:A∼
→Y]
=−[α′
2:A′∼
→X] + [α′
1:A∼
→X].
Now we check ha he de ini ion o ¯νon gene a o s is compa ible wi h he
de ining ela ions. The only non- i ial pa conce ns ela ions (DR1), (DR6) and
(DR7). Compa ibili y wi h (DR1) ollows om
¯ν0∂[α:A∼
=
→A′] = ∂¯ν1[α:A∼
=
→A′]
=−∂[α2:A′∼
→X] + ∂[α1:A∼
→X]
(R1) =−(−[X] + [A′]) + (−[X] + [A])
=−[A′] + [A]
=−¯ν0[A′] + ¯ν0[A].
In o de o check compa ibili y wi h (DR6) we conside wo composable isomo -
phisms in Ho W
A∼
=
−→
αB∼
=
−→
βC
and we ake ep esen a i es o α,βand βα as in he ollowing commu a i e diag am
o weak equi alences in W
X∪BY
aa¯α2
C
C
C
C
C
C
C
C==
¯
β1
=={
{
{
{
{
{
{
{
Xaa
α2D
D
D
D
D
D
D
DBB
α1






push Y
β2
8
8
8
8
8
8
8
A B ==β1
==
{
{
{
{
{
{
{
{C
20 FERNANDO MURO
Then
¯ν1[βα:A∼
=
→C] = −[¯α2β2:C∼
→X∪BY] + [¯
β1α1:A∼
→X∪BY]
=−[¯α2β2:C∼
→X∪BY] + [¯α2β1:B∼
→X∪BY]
−[¯
β1α2:B∼
→X∪BY] + [¯
β1α1:A∼
→X∪BY]
(R6) =−([¯α2:Y∼
→X∪BY] + [β2:C∼
→Y])
+[¯α2:Y∼
→X∪BY] + [β1:B∼
→Y]
−([¯
β1:X∼
→X∪BY] + [α2:B∼
→X])
+[¯
β1:X∼
→X∪BY] + [α1:A∼
→X]
=−[β2:C∼
→Y] + [β1:B∼
→Y]
−[α2:B∼
→X] + [α1:A∼
→X]
= ¯ν1[β:B∼
=
→C] + ¯ν1[α:A∼
=
→B].
Le us now check compa ibili y wi h (DR7).
−¯ν1[A′֌B′։B′/A′]
+¯ν1[β:B∼
=
→B′]
+¯ν1[A֌B։B/A] = −[A′֌B′։B′/A′]−[β2:B′∼
→Y]
+[X֌Y։Y/X]−[X֌Y։Y/X]
+[β1:B∼
→Y] + [A֌B։B/A]
(R7) =−([α2:A′∼
→X] + [γ2:B′/A′∼
→Y/X][A′])
+[α1:A∼
→X] + [γ1:B/A ∼
→Y/X][A]
Rem. 3.2 =−[γ2:B′/A′∼
→Y/X]−[α2:A′∼
→X]
+[α1:A∼
→X] + [γ1:B/A ∼
→Y/X]
−h[A′], ∂[γ2]i+h[A], ∂[γ1]i
De n. 3.1 (2) and Rem. 3.2 =−[α2:A′∼
→X] + [α1:A∼
→X]
−[γ2:B′/A′∼
→Y/X] + [γ1:B/A ∼
→Y/X]
+h−∂[α2] + ∂[α1],−∂[γ2]i − h[A′], ∂[γ2]i+h[A], ∂[γ1]i
(R1) =−[α2:A′∼
→X] + [α1:A∼
→X]
−[γ2:B′/A′∼
→Y/X] + [γ1:B/A ∼
→Y/X]
+h−(−[X] + [A′]) + (−[X] + [A]),−∂[γ2]i
+h[A′],−∂[γ2]i+h[A], ∂[γ1]i
=−[α2:A′∼
→X] + [α1:A∼
→X]
−[γ2:B′/A′∼
→Y/X] + [γ1:B/A ∼
→Y/X]
+h[A],−∂[γ2]i+h[A], ∂[γ1]i
= +¯ν1[α:A∼
=
→A′] + ¯ν1[γ:B/A ∼
=
→B′/A′]
+h¯ν0[A], ∂¯ν1[γ:B/A ∼
=
→B′/A′]i
Rem. 3.2 = ¯ν1[α:A∼
=
→A′] + ¯ν1[γ:B/A ∼
=
→B′/A′]¯ν0[A].
This es ablishes ha ¯νis a well de ined mo phism o s able quad a ic modules.
MALTSINIOTIS’S FIRST CONJECTURE FOR K121
Le us now check ha ¯µ¯ν= 1Dde
∗Wand ¯ν¯µ= 1D∗W. Bo h equa ions a e ob ious
on gene a o s (G1) = (DG1) and (G3) = (DG3). Fo (G2)
¯ν1¯µ1[ :A∼
→A′] = ¯ν1[ζ( ): A∼
=
→A′]
=−[1A′:A′∼
→A′] + [ :A∼
→A′]
(R4) = [ :A∼
→A′].
I α:A∼
=
→A′is an isomo phism in Ho Wwe ha e he ollowing equa ion in Dde
1W,
0(DR4)
= [A1A
→A]
= [α−1α:A∼
=
→A]
(DR6) = [α−1:A′∼
=
→A] + [α:A∼
=
→A′],
so [α−1:A′∼
=
→A] = −[α:A∼
=
→A′]. Now o (DG2)
¯µ1¯ν1[α:A∼
=
→A′] = −¯µ1[α2:A′∼
→X] + ¯µ1[α1:A∼
→X]
=−[ζ(α2): A′∼
=
→X] + [ζ(α1): A∼
=
→X]
= [ζ(α2)−1:X∼
=
→A′] + [ζ(α1): A∼
=
→X]
(DR6) = [α=ζ(α2)−1ζ(α1): A∼
=
→A′].
The p oo o Theo em 5.1 is now inished. 
Rema k 5.3.Le Wbe a Waldhausen ca ego y wi h cylinde s sa is ying he 2 ou
o 3 axiom. We do no assume ha Whas a sa u a ed class o weak equi alences.
Howe e we can endow he unde lying ca ego y wi h a new Waldhausen ca ego y
s uc u e which does ha e a sa u a ed class o weak equi alences.
We conside he Waldhausen ca ego y Wwi h he same unde lying ca ego y as
W. Co ib a ions in Wa e also de same as in W. Weak equi alences in Wa e he
mo phisms in Wwhich a e mapped o isomo phisms in Ho Wby he canonical unc-
o ζ:W→Ho W. The e o e weak equi alences in Wa e also weak equi alences in
Wbu he con e se need no hold. This indeed de ines a Waldhausen ca ego y W
wi h cylinde s and a sa u a ed class o weak equi alences, and he ob ious exac
unc o W→Winduces an isomo phism on he associa ed de i a o s DW∼
=DW,
compa e [Cis03, dual o P oposi ion 3.16] and [RB07, Theo em 6.2.2]. Hence we
ha e a commu a i e diag am o n= 0,1,
Kn(W)µn//

Kn(DW)
∼
=

Kn(W)µn
∼
=//Kn(DW)
He e he lowe a ow is an isomo phism by Theo em B. Now we can use he ‘ ib a-
ion heo em’, [Wal85, 1.6.7] and [Sch06, Theo em 11], o embed he mo phisms
µn:Kn(W)→K(DW), n= 0,1, in an exac sequence. Mo e p ecisely, le W0
be he ull subca ego y o Wspanned by he objec s which a e isomo phic o 0 in
Ho W. The ca ego y W0is a Waldhausen ca ego y whe e a mo phism is a co i-
b a ion, esp. a weak equi alence, i and only i i is a co ib a ion, esp. a weak

22 FERNANDO MURO
equi alence, in W. The e is an exac sequence
K1(W0)−→ K1(W)µ1
−→ K1(DW)δ
−→ K0(W0)−→ K0(W)µ0
−→ K0(DW)→0.
The g oup K0(W0) has also been conside ed by Weiss in [Wei99]. Weiss de ines
he Whi ehead g oup o Was Wh(W) = K0(W0). Mo eo e , o any mo phism
:A→A′which becomes an isomo phism in Ho Whe de ines he Whi ehead
o sion τ( )∈Wh(W), which is he obs uc ion o o be a weak equi alence in
W. I :A→Ais an endomo phism which maps o an au omo phism in Ho W
hen one can check ha
δ[ζ( ): A∼
=
→A] = −τ( ),
he e o e an au omo phism in Ho Wcomes om a weak equi alence in Wi and
only i i s class in de i a o K1comes om Waldhausen K1.
Re e ences
[Bau91] H.-J. Baues, Combina o ial Homo opy and 4-Dimensional Complexes, Wal e de
G uy e , Be lin, 1991.
[BCC93] M. Bullejos, P. Ca asco, and A. M. Cega a, Cohomology wi h coe icien s in symme ic
ca -g oups. An ex ension o Eilenbe g-Mac Lane’s classi ica ion heo em, Ma h. P oc.
Camb idge Philos. Soc. 114 (1993), no. 1, 163–189.
[BF78] A. K. Bous ield and E. M. F iedlande , Homo opy heo y o Γ-spaces, spec a, and
bisimplicial se s, Geome ic applica ions o homo opy heo y (P oc. Con ., E ans on,
Ill., 1977), II, Lec u e No es in Ma h., ol. 658, Sp inge , Be lin, 1978, pp. 80–130.
[CF00] C. Casacube a and A. F ei, On sa u a ed classes o mo phisms, Theo y Appl. Ca eg.
7(2000), no. 4, 43–46.
[Cis02] D.-C. Cisinski, Th´eo `emes de co inali ´e en K- h´eo ie (d’ap `es Thomason), Semina
no es, h p://www.ma h.uni -pa is13. /~cisinski/co de .d i, 2002.
[Cis03] ,Ca ´ego ies d´e i ables, P ep in ,
h p://www.ma h.uni -pa is13. /~cisinski/k hw .d i, 2003.
[CMM04] P. Ca asco and J. Ma ´ınez-Mo eno, Simplicial cohomology wi h coe icien s in sym-
me ic ca ego ical g oups, Appl. Ca eg. S uc u es 12 (2004), no. 3, 257–285.
[CN05] D.-C. Cisinski and A. Neeman, Addi i i y o de i a o K- heo y, P ep in ,
h p://www.ma h.uni -pa is13. /~cisinski/k hde .dp , 2005.
[Con84] D. Conduch´e, Modules c ois´es g´en´e alis´es de longueu 2, J. Pu e Appl. Algeb a 34
(1984), no. 2-3, 155–178.
[Ga 05] G. Ga kusha, Sys ems o diag am ca ego ies and K- heo y II, Ma h. Z. 249 (2005),
no. 3, 641–682.
[Ga 06] ,Sys ems o diag am ca ego ies and K- heo y I, Algeb a i Analiz 18 (2006),
no. 6, 131–186, (Russian).
[Gil81] H. Gille , Riemann-Roch heo ems o highe algeb aic K- heo y, Ad . Ma h. 40
(1981), 203–289.
[GJ99] P. J. Goe ss and J. F. Ja dine, Simplicial Homo opy Theo y, P og ess in Ma hema ics,
no. 174, Bi kh¨ause Ve lag, Basel, 1999.
[GM03] S. I. Gel and and Y. I. Manin, Me hods o homological algeb a, second ed., Sp inge
Monog aphs in Ma hema ics, Sp inge -Ve lag, Be lin, 2003.
[G o90] A. G o hendieck, Les d´e i a eu s, Manusc ip ,
h p://www.ma h.jussieu. /~mal sin/g o h/De i a eu s.h ml, 1990.
[GZ67] P. Gab iel and M. Zisman, Calculus o F ac ions and Homo opy Theo y, Sp inge -
Ve lag, Be lin-Heidelbe g-New Yo k, 1967.
[Mal07] G. Mal sinio is, La K- h´eo ie d’un d´e i a eu iangul´e, Ca ego ies in algeb a, geom-
e y and ma hema ical physics (A. Da ydo e al., ed.), Con emp. Ma h., no. 431,
Ame ican Ma hema ical Socie y, 2007, wi h an appendix “Le d´e i a eu iangul´e as-
soci´e `a une ca ´ego ie exac e” by B. Kelle , pp. 341–368.
[MT07] F. Mu o and A. Tonks, The 1- ype o a Waldhausen K- heo y spec um, Ad ances in
Ma h. 216 (2007), no. 1, 178–211.
MALTSINIOTIS’S FIRST CONJECTURE FOR K123
[RB07] A. Radulescu-Banu, Co ib a ions in homo opy heo y, P ep in , a Xi :ma h/0610009 2,
2007.
[Sch02] M. Schlich ing, A no e on K- heo y and iangula ed ca ego ies, In en . Ma h. 150
(2002), no. 1, 111–116.
[Sch06] ,Nega i e K- heo y o de i ed ca ego ies, Ma h. Z. 253 (2006), 97–134.
[TT90] R. W. Thomason and T. T obaugh, Highe algeb aic K- heo y o schemes and de i ed
ca ego ies, The G o hendieck es sch i , a collec ion o a icles w i en in hono p he
60 h bi hday o Alexande G o hendieck, P og ess in Ma h. 88, ol. III, B ikh¨ause ,
Bos on, 1990, pp. 247–435.
[TV04] B. To¨en and G. Vezzosi, A ema k on K- heo y and S-ca ego ies, Topology 43 (2004),
no. 4, 765–791.
[Wal85] F. Waldhausen, Algeb aic K- heo y o spaces, Algeb aic and geome ic opology (New
B unswick, N.J., 1983), Lec u e No es in Ma h., ol. 1126, Sp inge , Be lin, 1985,
pp. 318–419.
[Wei] C. Weibel, An in oduc ion o algeb aic K- heo y, Book in p og ess a ailable a
h p://ma h. u ge s.edu/~weibel/Kbook.h ml.
[Wei99] M. Weiss, Hammock localiza ion in Waldhausen ca ego ies, J. Pu e Appl. Algeb a 138
(1999), no. 2, 185–195.
Uni e si a de Ba celona, Depa amen d’ `
Algeb a i Geome ia, G an ia de les
co s ca alanes 585, 08007 Ba celona, Spain
E-mail add ess: mu [email protected]