Publ. Ma . 46 (2002), 97–118
STABLE RATIONAL COHOMOLOGY OF
AUTOMORPHISM GROUPS OF FREE GROUPS AND
THE INTEGRAL COHOMOLOGY OF MODULI SPACES
OF GRAPHS
C aig A. Jensen
Abs ac
I is no known whe he o no he s able a ional cohomology
g oups ˜
H∗(Au (F∞); Q) always anish (see Ha che in [5] and
Ha che and Vog mann in [7] whe e hey pose he ques ion and
show ha i does anish in he fi s 6 dimensions). We show ha
ei he he a ional cohomology does no anish in ce ain dimen-
sions, o he in eg al cohomology o a moduli space o poin ed
g aphs does no s abilize in ce ain o he dimensions. Simila e-
sul s a e s a ed o g oups o ou e au omo phisms. This yields
ha H5(ˆ
Qm;Z), H6(ˆ
Qm;Z), and H5(Qm;Z) ne e s abilize as
m→∞, whe e he moduli spaces ˆ
Qmand Qma e he quo ien s
o he spines ˆ
Xmand Xmo “ou e space” and “au e space”,
espec i ely, in oduced in [3] by Culle and Vog mann and [6]by
Ha che and Vog mann.
1. In oduc ion
Le Fndeno e he ee g oup on nle e s and le Au (Fn) and Ou (Fn)
deno e he au omo phism g oup and ou e au omo phism g oup, e-
spec i ely, o Fn.In[5] Ha che shows ha he in eg al cohomology
o he infini e symme ic g oup Σ∞is a di ec summand o he in e-
g al cohomology o Au (F∞). He men ions ha i is unknown whe he
o no he complemen a y summand is ze o and in pa icula whe he
o no ˜
H∗(Au (F∞); Q) is always ze o. In [6], Ha che and Vog mann
again pose he ques ion o whe he o no he s able a ional cohomol-
ogy g oups o Au (Fn) and Ou (Fn) all anish, and show ha i does
2000 Ma hema ics Subjec Classifica ion. P ima y: 05C25, 20F32, 20J05; Seconda y:
20F28, 55N91.
Key wo ds. G aphs, ee g oups, moduli spaces, ou e space, au e space.
98 C. A. Jensen
anish in dimensions 1 h ough 6. A ecen heo em o Madsen and
Tillman gi es (a e in e ing he p ime 2) a p oduc decomposi ion o
he plus cons uc ion BΓ+o he classi ying space o s able mapping
class g oups; howe e , i is cu en ly unknown o wha ex en his enables
one o answe he ques ion posed by Ha che and Vog mann.
Le ˆ
Xmbe he spine o ou e space (see Culle and Vog mann in [3])
and le ˆ
Qm=ˆ
Xm/Ou (Fm) be he co esponding moduli space o
g aphs. Simila ly, le Xmbe he spine o au e space (see Ha che and
Vog mann in [6]) and le Qm=Xm/Au (Fm) be he co esponding
moduli space o poin ed g aphs. In his pape , we show ha
Theo em 1. Le i∈{0,1}. Fo all posi i e in ege s k, ei he
H4k+i(Ou (F∞); Q)=0
o
H4k+i+1(ˆ
Qm;Z)ne e s abilizes as m→∞.
Theo em 2. Fo all posi i e in ege s k, ei he
H4k(Au (F∞); Q)=0
o
H4k+1(Qm;Z)ne e s abilizes as m→∞.
F om calcula ions in [7] ha
H4(Au (F∞); Q)=H4(Ou (F∞); Q)=H5(Au (F∞); Q)=0,
he abo e wo heo ems immedia ely show ha
Co olla y 3. The cohomology g oups H5(ˆ
Qm;Z)and H6(ˆ
Qm;Z)ne e
s abilize as m→∞.
Co olla y 4. The cohomology g oup H5(Qm;Z)ne e s abilizes as
m→∞.
The wo co olla ies a e ue because as minc eases, o sion om
inc easingly highe p imes is in oduced in H5(ˆ
Qm;Z), H6(ˆ
Qm;Z), and
H5(Qm;Z). The e a e na u al inclusions QmQm+1, and i is
known [6] ha he induced map Hi(Qm+1;Q)→Hi(Qm;Q) is an iso-
mo phism o m>3i/2. I is he e o e impo an o keep in mind ha
he abo e wo co olla ies only hold wi h espec o in eg al cohomology.
In eg al Cohomology o Moduli Spaces o G aphs 99
A quick no e abou ou no a ion is app op ia e he e. In gene al,
g oups wi hou any addi ional s uc u e will be w i en using mul iplica-
i e no a ion (e.g., Z/p ×Z/p ∼
=(Z/p)2) bu modules like cohomology
g oups will be w i en using addi i e no a ion (e.g., Z/p⊕Z/p ∼
=2(Z/p)).
In Sec ion 2 we e iew he basics abou ou e and au e space, and in
Sec ion 3 we p o e Theo em 1. Symme y g oups o g aphs wi h 2p−1
holes a e discussed in Sec ion 4, which enables us o p o e Theo em 2 in
Sec ion 5.
This pape is based on a disse a ion (see [8], [9]) w i en while he
au ho was a s uden o Ka en Vog mann a Co nell, and he au ho
would like o hank P o . Vog mann o he help and ad ice. The au ho
would also like o hank Hen y Glo e o his help ul commen s on his
pape .
2. Basics abou spec al sequences and Au (F
n
)
Le Gbe a g oup ac ing cellula ly on a fini e dimensional CW-com-
plex Xsuch ha he s abilize s abG(δ) o e e y cell δis fini e and
such ha he quo ien o Xby Gis fini e. Fu he suppose ha o
e e y cell δo X, he g oup s abG(δ) fixes δpoin wise. Le Mbe a
G-module. Recall (see [2]) ha he equi a ian cohomology g oups o
he G-complex Xwi h coefficien s in Ma e defined by
H∗
G(X;M)=H∗(G;C∗(X;M))
and ha i in addi ion Xis con ac ible (which will usually, bu no
always, be he case in his pape ) hen
H∗
G(X;M)=H∗(G;M).
In [2] a spec al sequence
˜
E ,s
1=
[δ]∈∆
n
Hs(s ab(δ); M)⇒H +s
G(X;M)(5)
is defined, whe e [δ] anges o e he se ∆
no o bi s o -simplices δ
in X.
I Mis Z/p o Z(p) hen a nice p ope y should be no ed abou he
spec al sequence (5). This p ope y will g ea ly educe he calcula ions
we need o go h ough, and in gene al will make conc e e compu a-
ions possible. Since each g oup s ab(δ) is fini e, a s anda d es ic ion-
ans e a gumen in g oup cohomology yields ha |s ab(δ)|annihila es
Hs(s ab(δ); M) o all s>0. (Fo examples o hese so s o a gumen s
see [2].) Since all p imes no equal o pa e di isible in Z/p o Z(p), his
in u n shows ha he p-pa o |s ab(δ)|annihila es Hs(s ab(δ); M)
100 C. A. Jensen
o s>0. In pa icula , i pdoes no di ide some |s ab(δ)|, hen his
[δ] does no con ibu e any hing o he spec al sequence (5) excep in
he ho izon al ow s= 0. I ollows ha i ou coefficien s a e Z/p o
Z(p) hen we a e mainly jus conce ned wi h he simplices δwhich ha e
“p-symme y”.
We now specialize o he cases whe e Gis Ou (Fn) o Au (Fn) and X
is ei he he spine ˆ
Xno “ou e space” o he spine Xno “au e space”.
Ha che and Vog mann’s defini ion o au e space closely ollows Culle
and Vog mann’s (p io ) defini ion o ou e space, excep ha he g aphs
a ising ha e basepoin s. We e iew some basic p ope ies and defini ions
o au e space below, whe e we concen a e on au e space because ha
is whe e mos o he calcula ions in his pape will ake place. Mos o
hese ac s can be ound in [3], [6], [12], and [13].
Conside he au omo phism g oup Au (Fn) o a ee g oup Fno
ank n(whe e nwill be 2p−1 o mos o ou wo k). Le (Rn,
0)be he
n-lea ed ose, a wedge o nci cles. We say a basepoin ed g aph (G, x0)
is admissible i i has no ee edges, all e ices excep he basepoin
ha e alence a leas h ee, and he e is a basepoin -p ese ing con-
inuous map φ:Rn→Gwhich induces an isomo phism on π1. The
iple (φ, G, x0) is called a ma ked g aph. Two ma ked g aphs (φi,G
i,x
i)
o i=0,1 a e equi alen i he e is a homeomo phism α:(G0,x
0)→
(G1,x
1) such ha (α◦φ0)#=(φ1)#:π1(Rn,
0)→π1(G1,x
1). Define
a pa ial o de on he se o all equi alence classes o ma ked g aphs
by se ing (φ0,G
0,x
0)≤(φ1,G
1,x
1)i G1con ains a o es (a disjoin
union o ees in G1which con ains all o he e ices o G1) such ha
collapsing each ee in he o es o a poin yields G0, whe e he collapse
is compa ible wi h he maps φ0and φ1.
F om [5] and [6] we ha e ha Au (Fn) ac s wi h fini e s abilize s on a
con ac ible space Xn. The space Xnis he geome ic ealiza ion o he
pose o ma ked g aphs ha we defined abo e. Le Qnbe he quo ien
o Xnby Au (Fn). No e ha he CW-complex Qnis no necessa ily a
simplicial complex. Since Au (Fn) has a o sion ee subg oup o fini e
index [5] and i ac s on he con ac ible, fini e dimensional space Xn
wi h fini e s abilize s and fini e quo ien , Au (Fn) has fini e cd.
Le pbe an odd p ime numbe , and le Z(p)be he localiza ion o Z
a he p ime ideal (p). Then we can apply he spec al sequence (5) o
ge
˜
E ,s
1=
[δ]∈∆
n
Hs(s ab(δ); Z(p))⇒H +s(Au (Fn); Z(p))(6)
whe e [δ] anges o e he se ∆
no o bi s o -simplices δin Xn.
In eg al Cohomology o Moduli Spaces o G aphs 101
The spec al sequence (6) equi es as inpu he s abilize s s abAu (Fn)
(δ)
o simplices δin Xn. Smillie and Vog mann [12] examined he s uc u e
o hese s abilize s in de ail, and we lis hei esul s he e. Conside a
gi en -simplex
(φ ,G
,x
)>···>(φ1,G
1,x
1)>(φ0,G
0,x
0)
wi h co esponding o es collapses
(H ⊆G ),...,(H2⊆G2),(H1⊆G1).
Fo each i∈0,1,..., , le Fibe he in e se image unde he map
G →···→Gi+1 →Gi
o o es collapses, o he o es Hi. Tha is, we ha e
F ⊆···⊆F2⊆F1⊆G .
I is shown in [12] ha he s abilize o he simplex unde conside a ion is
isomo phic o he g oup Au (G ,F
1,...,F
,x
) o basepoin ed au omo -
phisms o he g aph G ha espec each o he o es s Fi. Fo example,
he s abilize o a poin (φ, G, x0)inXnis isomo phic o Au (G, x0).
3. G aphs wi hou basepoin s
Theo em 1 is a di ec consequence o he s abili y heo ems in [5] and
he spec al sequence calcula ions in [4].
P oo o Theo em 1: F om [5],
H4k+i(Ou (F∞); Q)=H4k+i(Au (F∞); Q)
and i m≥4k2+10k+1+i2/4+2ik +5i/2, hen he s anda d map
H4k+i(Au (Fm); Z)→H4k+i(Ou (Fm); Z)
is an isomo phism. Obse e ha H4k+i(Ou (F4k2+10k+1+i2/4+2ik+5i/2);
Z)=H4k+i(Ou (F∞); Z) is a fini ely gene a ed abelian g oup. I i
con ains a o sion ee summand isomo phic o Z, hen we a e done
and H4k+i(Ou (F∞); Q)= 0. O he wise, choose a p ime qsuch ha
q+1≥4k2+10k+1+i2/4+2ik +5i/2 and so ha o all p imes
p≥q he e is no p- o sion in H4k+i(Ou (F∞); Z).We will show ha
H4k+i+1(ˆ
Qp+1;Z) has p- o sion o infini ely many p imes p, which will
p o e he heo em.
102 C. A. Jensen
Le p≥max{q,25}wi h p≡3 (mod 4). (No e ha he e a e in-
fini ely many possibili ies o p, as he e a e infini ely many p imes
ha a e g ea e han a gi en numbe and cong uen o 3 modulo 4.)
Because H4k+i(Ou (Fp+1); Z) has no p- o sion, he e is also no p- o -
sion in H4k+i(Ou (Fp+1); Z(p)). F om he calcula ion o Glo e and
Mislin in [4]o heE2-page o he equi a ian spec al sequence used
o calcula e H∗(Ou (Fp+1); Z(p)), we know ha his E2-page, in he
ows 0 ≤s<2(p−1), is gi en by
E ,s
2=
H (ˆ
Qp+1;Z(p))s=0
Z/p = 0 and s=4k>0,k∈Z+
(np)Z/p = 1 and s=4k>0,k∈Z+
0 o he wise
whe e np=(p−1)/12 −pand p∈{0,1}. Since p≥25, no e ha
np≥1.
Hence a class ˆα∈Ei,4k
2in he E2-page su i es a leas un il he
E4k+1-page. The class ˆα∈Ei,4k
2canno su i e o he E∞page, how-
e e , because he e is no p- o sion in he fini e (since H4k+i(Ou (Fp+1);
Q) = 0) addi i e g oup H4k+i(Ou (Fp+1); Z(p)).
I ollows ha he e is p- o sion in
E4k+i+1,0
4k+1 =H4k+i+1(ˆ
Qp+1;Z(p)).
Thus H4k+i+1(ˆ
Qp+1;Z) has p- o sion.
4. Symme y g oups o g aphs
We will use spec al sequence (6) o compu e a po ion o he coho-
mology o Au (Fn). Since ou coefficien ing is Z(p), we ha e al eady
ema ked ha o he e ms in he spec al sequence abo e he ho izon al
axis, we a e conce ned only wi h simplices whose s abilize s a e di isible
by p. In addi ion, he s abilize o a simplex consis s o g aph au o-
mo phisms ha espec he o es collapses in he simplex. We will find
which simplices a ise in he case n=2p−1. In o he wo ds, we wan o
calcula e which g aphs Gwi h a Z/p ac ion on hem ha e π1(G)∼
=Fn.
Recall ha a Z/p-g aph Gis educed i i con ains no Z/p-in a ian
sub o es s.
In eg al Cohomology o Moduli Spaces o G aphs 103
We now examine he cohomology o he quo ien Qno he spine Xn
o au e space. The e a e na u al inclusions QmQm+1, and i is
known [6] ha he induced map Hi(Qm+1;Q)→Hi(Qm;Q)isan
isomo phism o m>3i/2. Ou goal is o show ha , in con as ,
H5(Qm;Z) ne e s abilizes as m→∞. This is done by showing ha
as minc eases, o sion om inc easingly highe p imes is in oduced in
H5(Qm;Z). To his end, we do specific calcula ions in he spec al se-
quence (5) applied o he ac ion o Au (Fn)onXn o n=2p−1. The
E ,0
2- e m o his spec al sequence is H (Qn;Z(p)), and he sequence
con e ges o H (Au (Fn); Z(p)). Resul s om Ha che and Vog mann [6]
on he cohomology o Au (Fn) a e hen used o ob ain he esul .
In his sec ion, we do he g ound wo k necessa y o compu e he
E1-page o he spec al sequence: we find all simplices o Xnwi h p-sym-
me y and compu e he cohomology o he s abilize s o hese simplices
wi h coefficien s in Z(p). In Sec ion 5 we will compu e he E2-page o
he spec al sequence, and use his calcula ion o ob ain he esul .
Unless o he wise s a ed, p≥5 will be p ime and n=2p−1. The
assump ion ha p≥5 is o con enience mo e han any o he eason, as
he main esul s will only conside a bi a ily la ge p imes pand so we
should no de o e ex a ime o he ( ai ly easy o esol e) complica ions
in oduced by conside ing he p ime p= 3. These complica ions a ise
om he ac ha he dihed al g oup D6is he same as he symme ic
g oup S3, so ha we canno dis inguish be ween dihed al and symme ic
symme y in ha case.
We now define some g aphs ha we will need o his sec ion. (Re e
o Figu es 1 and 2 o illus a ions o mos o hese g aphs.) Le Θp−1
be he g aph wi h wo e ices and pedges, each o which goes om one
e ex o he o he (see Figu e 1). Say he “le mos e ex” o Θp−1is
he basepoin . Hence when we w i e Θp−1∨Rp−1 hen we a e s ipula ing
ha he ose Rp−1is a ached o he non-basepoin ed e ex o Θp−1,
while when we w i e Rp−1∨Θp−1 hen we a e saying ha he ose is
a ached o he basepoin o Θp−1. Le Φ2(p−1) be a g aph wi h 3pedges
a1,...,a
p,b1,...,b
p,c1,...,c
p, and p+ 3 e ices 1,...,
p,x,y,z. The
basepoin is xand each o he edges aibegin a xand end a i. The
edges biand cibegin a yand z, espec i ely, and end a i. No e ha
he e a e ob ious ac ions o Z/p on Θp−1and Φ2(p−1), gi en by o a ion,
and ha hese ac ions a e unique up o conjugacy. Le Ψ2(p−1) be he
g aph ob ained om Φ2(p−1) by collapsing all o he edges ai o a poin .
104 C. A. Jensen
Le Ω2(p−1) be he g aph ob ained om Φ2(p−1) by collapsing ei he he
edges bio he edges ci( he esul ing g aphs a e isomo phic) o a poin .
No e ha he only diffe ence be ween Ψ2(p−1) and Ω2(p−1) is whe e he
basepoin is loca ed.
Rp−1Θp−1Θp−1∨Rp−1
Rp−1∨Θp−1Φ2(p−1)
Ψ2(p−1) Ω2(p−1)
Figu e 1. Some g aphs wi h p-symme y
Gi en a fini e subg oup Go Au (Fn) o some in ege n, we say ha
a ma ked g aph
η1:R →Γ1
is a G-equi a ian blowup in he fixed poin space XG
o a ma ked g aph
η2:R →Γ2
i he e is a 1-simplex η1>η
2in XG
.
In eg al Cohomology o Moduli Spaces o G aphs 105
ΞpΥ2p−1
Υ1
2p−1
Υ2
2p−1
Figu e 2. Some g aphs wi h D2p-symme y
Le Υ1
2p−1and Υ2
2p−1be he wo possible g aphs ha can be ob ained
om Υ2p−1by equi a ian ly blowing up he p alence 4 e ices in o
2p alence 3 e ices. Tha is, Υ1
2p−1can be ob ained by fi s aking
ap-gon and hen a aching p ee edges o he p e ices o he p-gon.
Say each o hese new edges eibegins a he e ex xiand ends a he
e ex yi, and suppose ha he e ices xia e he ones ha a e a ached
o he p-gon. Now o m he 1-skele on o he double cone o suspension
o e he p e ices yi. This gi es he g aph Υ1
2p−1. The g aph Υ2
2p−1
can be hough o as ollows: Fi s ake a p-gon and cone off o e he
p e ices o he p-gon. Now also cone off o e he pmidpoin s o he
pedges o he p-gon. No e ha he e is an ob ious Zp-ac ion on each o
Θp−1,Ξ
p,Υ
2p−1,Υ
1
2p−1, and Υ2
2p−1.
Le Ξpbe he 1-skele on o he cone o e a p-gon, so ha Ξphas p+1
e ices and 2pedges, one e ex has alence pand he o he p e ices
all ha e alence 3. Le Υ2p−1be he 1-skele on o he suspension o a
p-gon. Hence Υ2p−1has p+ 2 e ices and 3pedges; wo o he e ices
ha e alence pand he o he pha e alence 4.
112 C. A. Jensen
he symme ic g oup Σpac s on he collec ions o edges defined abo e,
and so he cohomology o he g oup o g aph au omo phisms o he g aph
is he same as ha o he symme ic g oup. I |{y0,z
1,z
2}| = 2 hen he
only edges in he g aph a e he ei, i, and giand he g aph is ei he
Υ2a
2p−1o Υ2b
2p−1. On he o he hand, i |{y0,z
1,z
2}| = 3, hen he g aph
has one addi ional edge besides he ei, i,o gi. Acco dingly, he g aph
looks like a Φ2p−1(see Figu e 1) wi h one addi ional edge added. This
addi ional edge can go om any o he {y0,z
1,z
2} o any o he one,
including possibly he same one. In any case, i is defini ely ue ha
he g aph has au omo phism g oup wi h he same cohomology as Σp.
The lemma ollows.
Fo an example o wha we we e ying o a oid in he p oo o he
abo e lemma, e e o he h ee examples gi en in he Figu e 4. The
g aphs pic u ed ha e an ob ious Z/p-symme y gi en by o a ion abou
he basepoin , which is indica ed by a solid do . Bu hey ha e no
dihed al flip, and hei basepoin -p ese ing au omo phism g oups a e
all exac ly Z/p, whe e p= 5 in he examples pic u ed and whe e ob ious
analogues exis o o he odd p imes. The anks o he undamen al
g oups o he g aphs pic u ed a e 3p−1, 3p, and 2p, espec i ely. The
las ank, 2p, is he lowes ank possible whe e one can ha e a g aph
wi h exac ly Z/p symme y.
Co olla y 9. A e ex in he p-singula locus o Xnhas a mos dihed al
symme y i i s cohomology is he same as ha o D2po D2p×Σp, and
e ices in he p-singula locus will ne e ha e exac ly Z/p symme y.
In he figu es below, a do ed line o a hollow do indica es ha he
gi en edge o e ex, espec i ely, does no ha e he indica ed p ope y.
A solid do , a solid line, o a 2-simplex wi h an X in i , means ha he
gi en e ex, edge, o 2-simplex, espec i ely, does ha e he indica ed
p ope y.
By analyzing he Z/p-in a ian sub o es s o all o he g aphs explic-
i ly lis ed in he p oo o Lemma 7, we can see wha ypes o s abilize s
highe dimensional simplices ( a he han jus e ices) ha e.
We will show ha he simplices wi h a mos dihed al symme y will
all in o wo (exhaus i e bu no disjoin ) ca ego ies. The fi s ca ego y
consis s o hose ha a e lis ed in Figu e 5. The second ca ego y consis s
o simplices whose maximal e ex ( ecall ha Xnis he ealiza ion o
a pose ) has he o m Ξp∨Γp−1whe e Γp−1is some basepoin ed g aph
wi h undamen al g oup o ank p−1, he wedge does no necessa ily
In eg al Cohomology o Moduli Spaces o G aphs 113
ake place a he basepoin , and whe e he o es collapses o he simplex
espec he Z/p ac ion on Ξp.
Ξp∨Θp−1Υ1
2p−1Θp−1∨Ξp
Rp∨Θp−1Θp−1∨Rp
Υ2p−1
Θ2p−1
Υ2
2p−1
Υ2a
2p−1Υ2b
2p−1
Figu e 5. Some simplices wi h a mos dihed al symme y
Rp∨Θp−1Υ2p−1Θp−1∨Rp
Υ2
2p−1
Υ2a
2p−1Υ2b
2p−1
Figu e 6. Simplices wi h exac ly Z/p symme y
114 C. A. Jensen
We will also show ha he simplices lis ed in Figu e 6 a e he only
ones wi h exac ly Z/p symme y.
Co olla y 10. Le p≥5be p ime, n=2p−1, and conside he p-sin-
gula locus o he spine Xno au e space.
•The only simplices wi h a mos dihed al symme y a e ei he :
(i) lis ed in Figu e 5; o (ii) ha e maximal e ex o he o m Ξp∨
Γp−1.
•The only simplices wi h exac ly Z/p symme y a e hose lis ed in
Figu e 6.
P oo : We examine each o he g aphs lis ed in Lemma 7 sepa a ely. By
enume a ing he Z/p in a ian sub o es s o each o hese g aphs, one can
lis all o he simplices in he p-singula locus o Xn. We can igno e he
g aphs in Lemma 7 ha do no ha e dihed al symme y, as all o hei
symme y comes om symme ic g oups. When you collapse in a ian
sub o es s o hese g aphs, you s ill ge g aphs wi h symme y coming
om he symme ic g oup.
So we a e le wi h analyzing he g aphs om Lemma 7 wi h dihed al
symme y, which we e:
•Θp−1∨Ξp. (The e a e ac ually wo possibili ies he e as he enume -
a ion in Lemma 7 did no speci y basepoin s. The cen al e ex
o Ξpcould be a ached o ei he he basepoin o Θp−1o he
o he e ex o Θp−1.)
•Γp−1∨Ξp, whe e Γp−1is a basepoin ed g aph wi h undamen al
g oup o ank p−1 which has no p-symme y (o whe e Γp−1is
Θp−1bu he cen al e ex o Ξpis a ached o he midpoin o
an edge o Γp−1).
•Υ2p−1.
•Υ1
2p−1.
•Υ2
2p−1.
Fo he fi s wo ypes o g aphs, you can ob ain simplices wi h dihe-
d al symme y by collapsing all o he spokes o Ξpand/o any o es in
he o he g aph o he wedge sum (ei he Θp−1o Γp−1). The esul ing
simplex wi h maximal e ex Θp−1∨Ξpo Γp−1∨Ξpwill clea ly ha e
a mos dihed al symme y, and will also jus as clea ly no gi e you a
g aph wi h exac ly Z/p symme y.
In a simila manne , simplices in he p-singula locus o Xnwi h
maximal e ex Υ2p−1o Υ1
2p−1a e (exhaus i ely) lis ed in Figu e 5.
No e ha Υ2p−1can only be blown up (while s ill p ese ing he Z/p
In eg al Cohomology o Moduli Spaces o G aphs 115
ac ion so ha we s ay in he p-singula locus o Xn) in wo ways, o
ei he Υ1
2p−1o Υ2
2p−1. The la e wo g aphs canno be blown up a all.
Finally, he simplices wi h maximal e ex Υ2
2p−1a e lis ed in Fig-
u e 5 o Figu e 6. No e ha we can ob ain edges and 2-simplices wi h
exac ly Z/p symme y, e en hough no ac ual e ex o Xnhas ex-
ac ly Z/p symme y. This is because you can choose sub o es s o Υ2
2p−1
ha do espec he dihed al “flip” o Υ2
2p−1. In o he wo ds, his flip
will no ake he sub o es o i sel again. Hence he esul ing simplex
will jus ha e symme y g oup Z/p. Las o all, no e ha you can also
choose sub o es s o Υ2
2p−1which do espec he dihed al flip, and hese
gi e simplices wi h dihed al symme y.
5. The in eg al cohomology o he quo ien ne e
s abilizes
We will p o e Theo em 2 in his sec ion. As in Sec ion 4, all p imes p
conside ed a e assumed o be g ea e han o equal o 5.
Lemma 11. Fo he ows 0≤s<2(p−1), he E2page o he spec al
sequence (5) applied o calcula e H∗(Au (F2p−1); Z(p))is gi en by
E ,s
2=
H (Q2p−1;Z(p))s=0
Z/p =2and s=4k−2>0,k ∈Z+
0o he wise.
P oo : As E ,0
1is he cochain complex C (Q2p−1;Z(p)), i ollows ha
E ,0
2=H (Q2p−1;Z(p)) as claimed abo e.
None o he simplices in X2p−1con ibu e any hing o he odd ows
be ween 0 and 2(p−1) o he abo e spec al sequence, om Co olla y 10.
Also om Co olla y 10, he ones ha con ibu e o ows o he o m 4k−2,
k∈Z+, a e all lis ed in Figu e 6. Le Abe he subcomplex o Qn
gene a ed by all o he simplices pic u ed in Figu e 6 and le Bbe he
subcomplex gene a ed by jus he simplices co esponding o do ed lines
o hollow do s in Figu e 6. Then he ow s=4k−2on heE1page o
he spec al sequence is C (A, B;Z/p). Examining Figu e 6 we see ha
H (A, B;Z/p)=Z/p =2
0 o he wise.
Consequen ly he E2page is as claimed o he ows s=4k−2.
116 C. A. Jensen
Ou final ask is o calcula e he E2page o he ows s=4k. Sim-
plices in he p-singula locus o he spine wi h “a mos dihed al sym-
me y” con ibu e o hese ows. F om Co olla y 10, we ha e a cha ac-
e iza ion o such simplices. Define he subcomplex Mo he p-singula
locus o he spine X2p−1o au e space o be he subcomplex gene a ed
by simplices wi h “a mos dihed al symme y”. Mo e p ecisely, om
Co olla y 10, we know i is gene a ed by he simplices co esponding o
hose in Figu e 5 (i.e., co esponding in he sense ha we a e aking M
o be a subcomplex o he spine a he han i s quo ien and Figu e 5 is
a pic u e in he quo ien ) in addi ion o simplices whose maximal e ex
has unde lying g aph o he o m Ξp∨Γp−1(whe e he o es collapses
in he simplices espec he Z/p ac ion on Ξp). Recall ha an -sim-
plex wi h a mos dihed al symme y con ibu es exac ly one Z/p o
E ,4k
1, while all o he simplices ( hose wi hou dihed al o exac ly Z/p
symme y) con ibu e no hing o his ow.
Le Nbe he subcomplex o Mgene a ed by simplices in Mwhich do
no ha e a mos dihed al symme y. Obse e ha none o he simplices
in Nha e a mos dihed al symme y. Also no e ha he ow E∗,4k
1is
he ela i e cochain complex
C∗(M/Au (F2p−1),N/Au (F2p−1); Z/p).
Le Mbe he subcomplex o Mgene a ed by Nand by simplices
whose maximal e ex is Υ2
2p−1. Hence Mis he subcomplex consis ing
o Nand he bo om wo hi ds o Figu e 5. The e is an Au (F2p−1)-equi-
a ian de o ma ion e ac ion o Mon o M, gi en on he e ices o
he pose by:
•Con ac ing he spokes o he g aph Ξpin Θp−1∨Ξp.
•Con ac ing he spokes o he g aph Ξpin Γp−1∨Ξp, whe e Γp−1
has no p-symme y.
•Con ac ing he pou wa d adia ing edges a ached o he p-gon
in he cen e o he g aph Υ1
2p−1. In he e minology used a he
beginning o Sec ion 4 while defining Υ1
2p−1, we a e con ac ing he
edges ei.
Tha i is a de o ma ion e ac ion ollows om he Pose Lemma in [10]
a ibu ed o Quillen.
As he homo opy e ac ing M o Mis Au (F2p−1)-in a ian , i de-
scends o a de o ma ion e ac ion o M/Au (F2p−1) oM/Au (F2p−1).
Hence he ela i e cohomology g oups
H∗(M/Au (F2p−1),N/Au (F2p−1); Z/p)
In eg al Cohomology o Moduli Spaces o G aphs 117
and
H∗(M/Au (F2p−1),N/Au (F2p−1); Z/p)
a e isomo phic. Now e e ing o Figu e 5, we see ha
H (M/Au (F2p−1),N/Au (F2p−1); Z/p)=0
o all because we can con ac all o he simplices in M/Au (F2p−1)
uni o mly in o N/Au (F2p−1).
An immedia e consequence is
P oo o Theo em 2: F om [6], i m≥8k+3, hen he s anda d map
H4k(Au (Fm+1); Z)→H4k(Au (Fm); Z)
is an isomo phism. Obse e ha H4k(Au (F8k+3);Z)=H4k(Au (F∞);Z)
is a fini ely gene a ed abelian g oup. I i con ains a o sion ee sum-
mand isomo phic o Z, hen we a e done and H4k(Au (F∞); Q)=0.
O he wise, choose a p ime qsuch ha 2q−1≥8k+ 3 and so ha o
all p imes p≥q he e is no p- o sion in H4k(Au (F8k+3); Z). We will
show ha H4k+1(Q2p−1;Z) has p- o sion o all p imes p≥q, which will
p o e he heo em.
Le p≥q. F om he lemma abo e, i we use he s anda d equi a ian
spec al sequence o calcula e H∗(Au (F2p−1); Z(p)), hen a class α∈
E2,4k−2
1in he E1-page su i es a leas un il he E4k−1-page.
Because H4k(Au (F2p−1); Z) has no p- o sion and H4k(Au (F2p−1);
Q) = 0, we ha e H4k(Au (F2p−1); Z(p)) = 0. Hence he class α∈E2,4k−2
1
canno su i e o he E∞page. I ollows ha he e is p- o sion in
E4k+1,0
4k−1. Recall ha E ,0
1co esponds o he cellula chain complex wi h
Z(p)coefficien s o Q2p−1. The p- o sion in E4k+1,0
4k−1, he e o e, would
ha e o ha e been c ea ed when going om he E1 o E2pages, because
any o he o sion abo e he ho izon al axis o he spec al sequence
could no map on o a o sion ee elemen on he ho izon al axis. So
H4k+1(Q2p−1;Z(p)) has p- o sion, and hus H4k+1(Q2p−1;Z) has p- o -
sion.
Re e ences
[1] G. Baumslag and T. Taylo , The cen e o g oups wi h one
defining ela o , Ma h. Ann. 175 (1968), 315–319.
[2] K. S. B own,“Cohomology o g oups”, G adua e Tex s in Ma h-
ema ics 87, Sp inge -Ve lag, New Yo k-Be lin, 1982.
[3] M. Culle and K. Vog mann, Moduli o g aphs and au omo -
phisms o ee g oups, In en . Ma h. 84(1) (1986), 91–119.
118 C. A. Jensen
[4] H. H. Glo e and G. Mislin, On he p-p ima y cohomology
o Ou (Fn) in he p- ank one case, J. Pu e Appl. Algeb a 153(1)
(2000), 45–63.
[5] A. Ha che , Homological s abili y o au omo phism g oups o
ee g oups, Commen . Ma h. Hel . 70(1) (1995), 39–62.
[6] A. Ha che and K. Vog mann, Ce heo y o g aphs, J. Lon-
don Ma h. Soc. (2) 58(3) (1998), 633–655.
[7] A. Ha che and K. Vog mann, Ra ional homology o Au (Fn),
Ma h. Res. Le . 5(6) (1998), 759–780.
[8] C. A. Jensen, Cohomology o Au (Fn), Co nell Uni e si y
Ph. D. disse a ion, I haca, New Yo k (1998).
[9] C. A. Jensen, Cohomology o Au (Fn) in he p- ank wo case,
J. Pu e Appl. Algeb a 158(1) (2001), 41–81.
[10] S. K s i´
c and K. Vog mann, Equi a ian ou e space and au-
omo phisms o ee-by-fini e g oups, Commen . Ma h. Hel . 68(2)
(1993), 216–262.
[11] H. Minkowski, Zu Theo ie de posi i en quad a ischen Fo men,
C elles J. 101 (1887), 196–202.
[12] J. Smillie and K. Vog mann, A gene a ing unc ion o he Eule
cha ac e is ic o Ou (Fn), in: “P oceedings o he No hwes e n con-
e ence on cohomology o g oups” (E ans on, Ill., 1985), J. Pu e
Appl. Algeb a 44(1–3) (1987), 329–348.
[13] J. Smillie and K. Vog mann, Au omo phisms o g aphs, p-sub-
g oups o Ou (Fn) and he Eule cha ac e is ic o Ou (Fn), J. Pu e
Appl. Algeb a 49(1–2) (1987), 187–200.
[14] R. G. Swan, The p-pe iod o a fini e g oup, Illinois J. Ma h. 4
(1960), 341–346.
Depa men o Ma hema ics
Uni e si y o New O leans
New O leans, LA 70148
U.S.A.
E-mail add ess:[email p o ec ed]
Rebu el 7 de ma ¸c de 2001.
