The cohomology of SL(3; Z[1=2])
HansWerner Henn
Abstract
We compute the cohomology of SL(3; Z[1=2]) with coefficients in
the prime fields and in the integers. On the way we obtain the co
homology of certain mod  2 congruence subgroups of SL(3; Z) with
coefficients in Fp for p > 2. Finally we compute the cohomology of
GL(3; Z[1=2]).
Contents
1 Introduction 2
2 Contractible spaces with actions of SL(3; Z) and SL(3; Z[1=2]) 8
2.1 The symmetric space and the BruhatTitsbuilding . . . . . . 8
2.2 Wellrounded lattices and the deformation retractions . . . . . 10
2.3 The space W0=SO(3) . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 i  equivariant cell structures on Z . . . . . . . . . . . . . . . 17
2.4.1 The case i = 0 . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 The cases i = 1 and i = 2 . . . . . . . . . . . . . . . . 19
2.5 Symmetries of wellrounded quadratic forms . . . . . . . . . . 20
2.6 The equivalence relations ~i on the spaces ix Di . . . . . . 23
2.6.1 3  cells . . . . . . . . . . . . . . . . . . . . . . . . . . *
*23
2.6.2 2  cells . . . . . . . . . . . . . . . . . . . . . . . . . . *
*23
2.6.3 1  cells . . . . . . . . . . . . . . . . . . . . . . . . . . *
*26
2.6.4 0  cells . . . . . . . . . . . . . . . . . . . . . . . . . . *
*31
3 The homology of the quotient spaces 34
3.1 Quotients of (X1 ; X1;s(i)) by i . . . . . . . . . . . . . . . . . 35
3.2 Quotients of X1;s(i) by i . . . . . . . . . . . . . . . . . . . . 41
3.3 Quotients of X1 by i . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Quotients by SL(3; Z[1=2]) . . . . . . . . . . . . . . . . . . . . 45
1
2 HansWerner Henn
4 The cohomology of SL(3; Z[1=2]) 49
4.1 Mod  2 cohomology . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Mod  3 cohomology . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Higher torsion in the integral cohomology . . . . . . . . . . . . 65
5 The cohomology of GL(3; Z[1=2]) 67
1 Introduction
So far there exist only very few complete computations of integral or mod
 p cohomology rings of arithmetic or more generally S  arithmetic groups.
Among the known results we mention the calculations for SL(2; Z) (which
is straightforward from the wellknown amalgamated product decomposi
tion SL(2; Z) ~= Z=6 *Z=2 Z=4), of SL(2; Z[1=2]) [Mi ] and that of SL(3; Z)
[So ]. Soule's computation is already fairly involved; e.g. he obtains that
the integral cohomology ring of SL(3; Z), after localization at the prime 2, is
generated by 7 elements which are subject to 22 relations. His result suggests
that the answer for SL(n; Z) would not be easily digestable (one should add
that it also seems to be completely out of reach at this point).
From a conceptual point of view the complexity of the answer in Soule's
calculation can also be explained by Quillen's work [Q ] which says among
other things that the minimal prime ideals in the mod  p cohomology ring
H*(; Fp) of an S  arithmetic group are in one to one correspondence with
the conjugacy classes of maximal elementary abelian p  subgroups of . (We
recall that an elementary abelian p  group is a group isomorphic to (Z=p)k for
some natural number k.) From this point of view those cases in which there
exists a unique conjugacy class of maximal elementary abelian p  subgroups
look more favourable than others. In the case of SL(n; Z) or GL(n; Z) it is
very difficult to determine the precise number of conjugacy classes of maximal
elementary abelian p  subgroups (this is essentially a problem of the integral
representation theory of elementary abelian p  groups) and thus the mod 
p cohomology of these groups must be complicated. The situation improves
if one inverts p and adjoins p  th roots of unity. In particular in the case
of SL(n; Z[1=2]) and GL(n; Z[1=2]) every elementary abelian 2  subgroup is
diagonalizable and there is a unique maximal one up to conjugacy.
This observation was presumably the basis of Quillen's conjecture (p. 591
of [Q ]), which in the case of H*(GL(n; Z[1=2]); F2) claims that the inclu
sion of rings Z[1=2] R (and identifying H*(GL(n; Z[1=2]); F2) as usual
with the mod 2  cohomology of the classifying space BGL(n; Z[1=2])) makes
H*(GL(n; Z[1=2]); F2) into a free, in particular a torsion free module over
The cohomology of SL(3; Z[1=2]) 3
the polynomial ring F2[w1; :::; wn] ~=H*(BGL(n; R); F2) with wi denoting as
usual the i  th universal Stiefel  Whitney class. In [HLS ] it was shown that
torsionfreeness implies thatQthe restrictionQmap aen : H*(GL(n; Z[1=2]); F2)
! H*(Dn; F2) (with Dn ~= ni=1(Z[1=2])x ~= ni=1(Z x Z=2) denoting the
subgroup of diagonal matrices of GL(n; Z[1=2])) is injective. Quillen also re
marked that with his conjecture a calculation of H*(GL(n; Z[1=2]); F2) should
be within reach. In fact, the image Im aen has been computed by Mitchell.
In order to state his result we identify the classes wi with their image under
restriction in H*(Dn; F2) ~=F[x1; :::; xn] E(a1; :::; an) (with E as usual de
noting an exterior algebra, and with all generators of dimension 1), namely
with the i  th elementary symmetric polynomial in the variables xi. We
also need classes ei 2 H2i1(Dn; F2): they are the symmetrizations of the
elements x21:::x2i1ai with respect to the canonical action of the symmetric
group Sn on n letters.
Now Mitchell's result reads as follows.
Theorem 1.1 [Mi ] Im aen ~=F2[w1; :::; wn] E(e1; :::; e2n1) . 2
Note that with this result Quillen's conjecture would imply an isomorphism
H*(GL(1; Z[1=2]); F2) ~= F2[w1; w2; :::] E(e1; e3; :::) and hence the Dwyer
 Friedlander version [DF ] of the Lichtenbaum  Quillen conjecture at p = 2.
Unfortunately Quillen's conjecture was too optimistic. Dwyer has recently
shown.
Theorem 1.2 [D ] The restriction map aen is not injective for all n. 2
The only previous complete computation of H*(GL(n; Z[1=2]); F2) was that
of [Mi ] for n = 2, and in this case aen turned out to be injective. Some
qualitative information on the size of the kernel of aen as n grows is provided
in [H2 ]. Dwyer shows, in fact, that ae32 is not injective, so that the case n *
*= 3
becomes an interesting test case in which one also has a nice candidate,
namely Im ae3, for the answer.
In fact, one of the main results of this paper shows that this candidate is
correct.
Theorem 1.3 The restriction homomorphism maps H*(GL(3; Z[1=2]); F2)
isomorphically onto the subalgebra F2[w1; w2; w3]E(e1; e3; e5) of H*(D3; F2).
This result is really an easy consequence of the following companion result
for SL(3; Z[1=2]). We denote its subgroup of diagonal matrices by SD3. Note
that the restriction map from H*(D3; F2) to H*(SD3; F2) kills the elements
w1 and e1. Let vi be the image of wi and d2i1 the image of e2i1, i = 2; 3.
4 HansWerner Henn
Theorem 1.4 The restriction homomorphism maps H*(SL(3; Z[1=2]); F2)
isomorphically onto the subalgebra F2[v2; v3] E(d3; d5) of H*(SD3; F2).
We remark that the corresponding result does not hold in the same way for
n = 2, i.e. the restriction map is not an isomorphism in this case, although
there is an abstract isomorphism H*(SL(2; Z[1=2]); F2) ~=F2[v2]E(d3) [Mi ].
How can Theorem 1.4 be proved? The standard approach would be to take a
suitable finite dimensional contractible space X on which := SL(3; Z[1=2])
acts properly and with finite isotropy groups (there is a canonical such can
didate, namely the product of the symmetric space SL(3; R)=SO(3) and the
BruhatTitsbuilding for SL(3; Q2), see Section 2.1 below). Then one would
take the Borel construction E x X as model for the classifying space B
and study its mod  2 cohomology H*(X; F2) via the cohomology spectral
sequence of the map E x X ! \X. If X has the structure of a
 CW Lcomplex then the E1  term of this spectral sequence is given as
Es;t1= oeHt(oe; F2) where oe runs through a set of representatives of the
 orbits of s  dimensional cells of X and oedenotes the isotropy group of
oe. This is how Soule studied the cohomology of SL(3; Z) [So ]. However, in
our case the space X looks too complicated to make this spectral sequence
manageable: in Section 2.6 we actually analyze the canonical X above and
we essentially produce a  equivariant deformation retract with finitely
many  orbits of cells; however, finite means 474 (!) orbits (see the table at
the beginning of Section 3) and so this standard approach looks unfeasible.
Instead we use a more manageable "centralizer spectral sequence"
Es;t2~=limsE2A*()Ht(C (E); F2) =) Hs+t(Xs; F2)
converging to the mod  2 cohomology of the Borel  construction of the 2
 singular locus Xs, i.e. the subspace of X consisting of all points whose
isotropy group contains an element of order 2. Here A*() is the category of
elementary abelian 2  subgroups of , lims is the s  th derived functor of
the inverse limit functor and C (E) is the centralizer in of the elementary
abelian 2  subgroup E . This spectral sequence is based on a homotopy
colimit decomposition of E x Xs and was introduced in [H1 ]. In this paper
we also evaluated this spectral sequence and obtained the following result
in which denotes as usual the suspension functor, e.g. 4Fp denotes the
graded Fp  vectorspace which is trivial in all dimensions except in dimension
4 where it is Fp.
Theorem 1.5 [H1 ] Let = SL(3; Z[1=2]) and let X be any mod  2 acyclic
finite dimensional  CW  complex for which the stabilizers of all cells are
The cohomology of SL(3; Z[1=2]) 5
finite. Then there is a short exact sequence
ae
0 ! 4F2 ! H*(Xs; F2) ! F2[v2; v3] E(d3; d5) ! 0
in which ae is an algebra homomorphism. Furthermore, if ss denotes the pro
jection map from E x Xs to the classifying space B then the composition
* * ae *
H*(; F2) ss!H (Xs; F2) ! F2[v2; v3] E(d3; d5) H (SD3; F2)
agrees with the restriction homomorphism of 1.4. 2
Now for * exceeding dim X, the dimension of X, we have isomorphisms
H*(Xs; F2) ~=H*(X; F2) ~=H*(; F2) and hence Theorem 1.5 is also a com
putation of H*(; F2) in large dimensions. In fact, X can be chosen to be of
dimension 5 (see [BS ] or Section 2 below) and Theorem 1.5 gives encouraging
evidence for Theorem 1.4.
In this paper we complete the proof of Theorem 1.4 by computing for
the canonical space X mentioned above, the relative groups H*(X; Xs; F2)
and the boundary homomorphism of the appropriate long exact cohomology
sequence. Note that, because the isotropy groups outside of Xs are finite of
order prime to 2, we have the following isomorphisms for the relative groups:
H*(X; Xs; F2) ~=H*(\(X; Xs); F2).
As a byproduct of our investigations we obtain the following results which
are of independent interest. In these results we abbreviate SL(3; Z[1=2]) by
, SL(3; Z) by 0, and we denote the subgroup of SL(3; Z) consisting of
all matrices whose first column agrees with the first standard basis vector
modulo 2 by 1, and the subgroup of all matrices which are upper triangular
modulo 2 by 2.
Theorem 1.6 Let X1 denote the symmetric space SL(3; R)=SO(3), X2 the
BruhatTitsbuilding of SL(3; Q2), X = X1 x X2 and let p be any prime.
Then the reduced cohomology of the quotient spaces by the obvious action of
the respective groups is given as follows:
a) He*(0\X1 ; Fp) = 0
b) He*(1 \X1 ; Fp) = 0
c) He*(2\X1 ; Fp) = 3Fp
d) He*(\X ; Fp) = 5Fp :
For p > 3 there are no elements of order p in these groups (because there
are obviously no elements of order p in SL(3; Q)) and hence we obtain the
following Corollary. For SL(3; Z) this was already known by [So ] and for
SL(3; Z)[1=2] by [Mo ]. The results for 1 and 2 are compatible with the
Euler chararacteristic computations in [Mo ].
6 HansWerner Henn
Corollary 1.7 Assume p > 3. Then
a) He*(0; Fp) = 0
b) He*(1 ; Fp) = 0
c) He*(2; Fp) = 3Fp
d) He*(; Fp) = 5Fp : 2
Theorem 1.8 Let X1 , X2, X and p be as in the previous theorem. Then we
get the following relative cohomology groups (where (X1;s(i) denotes the 2 
singular locus of X1 with respect to the action of i, and Xs the 2  singular
locus of X with respect to the action of ):
a) H*(0\(X1 ; X1;s(0)); Fp) = 0
b) H*(1\(X1 ; X1;s(1)); Fp) = 2(Fp)2
c) H*(2\(X1 ; X1;s(2)); Fp) = 3Fp 2(Fp)6
d) H*(\(X ; Xs); F2) = 5F2 :
(Observe that we restrict to the case p = 2 for the last part of the Theorem.)
The next result together with Theorem 1.5 and the last part of Theorem
1.8 finishes the proof of Theorem 1.4.
Proposition 1.9 The boundary homomorphism
H4(Xs; F2) ! H5(X ; Xs; F2)
is an epimorphism.
With the help of Theorem 1.6 we are also able to compute the mod  3
cohomology. Again this was known for SL(3; Z) by [So ].
Theorem 1.10 There are isomorphisms of F3  algebras (without unit) which
in the case of a), b) and d) are induced by restrictions to appropriate sub
groups:
Q 2
a) He*(0; F3) ~= i=1 eH*(S3; F3)
Q 2
b) He*(1 ; F3) ~= i=1 eH*(S3; F3)
c) He*(2; F3) ~=3F3
Q 2
d) He*(; F3) is isomorphic to the subalgebra of i=1He*(S3 x Z; F3) which
Q 2
can be characterized as follows: it is all of i=1 eH*(S3 x Z; F3) except in
degrees 1 and 4; in degree 1 it is trivial, and in degree 4 it is of dimension 3
and is generated by the image of the Bockstein of H3 and one further element
which restricts nontrivially to both factors.
The cohomology of SL(3; Z[1=2]) 7
The paper is organized as follows: In Section 2 we recall the symmetric
space X1 and the Bruhat  Tits building X2. We discuss the Soule  Lannes
method of replacing the symmetric space by a smaller space Z for which the
quotients by 0 = SL(3; Z) and the congruence subgroups 1 and 2 are
compact. The bulk of this long section is then devoted to patiently working
out an explicit cell structure of the quotients i\Z, i = 0; 1; 2, in fact even *
*a i
 equivariant cell structure on Z. This is straightforward but it is crucial for
the remainder of the paper; for i = 0 it is a variation of Soule's investigatio*
*ns
[So ]. In Section 3 we use these cell structures to prove Theorem 1.6 and
Theorem 1.8 as well as the corresponding results for the cohomology of the
quotients of the singular locus X1;s(i) resp. Xs. This is quite an elaborate
calculation but apart from the last part of Theorem 1.8 it is straightforward
given the results in Section 2. The last part of Theorem 1.8 is more tricky and
to settle it we use low dimensional information on H*(Xs; F2) as provided by
Theorem 1.5. In Section 4 we apply the results of Section 2 and Section 3 and
derive the remaining results listed in this introduction. We also determine
the height of torsion in H*(SL(3; Z[1=2]); Z) (Proposition 4.15). In Section
5 we compute H*(GL(3; Z[1=2]); Fp) for primes p > 2 (Proposition 5.1, 5.2)
and for p = 2, i.e. we derive Theorem 1.3.
Acknowledgements:___During the research presented in this paper the author
was supported by a Heisenberg fellowship of the DFG. The author is happy
to acknowledge numerous discussions with Jean Lannes which stimulated his
interest in the cohomology of SL(3; Z[1=2]). He also thanks Bob Oliver for
helpful discussions in connection with the proof of Proposition 1.9.
8 HansWerner Henn
2 Contractible spaces with actions of SL(3; Z)
and SL(3; Z[1=2] )
2.1 The symmetric space and the BruhatTitsbuilding
We start by recalling the contractible spaces on which our groups act with
finite stabilizer groups.
The symmetric space. The space Q(n) of positive definite quadratic
forms on Rn is equipped with an action of the multiplicative group R+ of
positive real numbers, given by (rq)(x) = rq(x) for r 2 R+ , q 2 Q(n) and
x 2 Rn. The quotient will be denoted by X1 (n), or simply by X1 if n is
clear from the context. The space X1 (n) is contractible because Q(n) is a
2
convex open cone in Rn . Furthermore, X1 (n) can be identified with the
symmetric space of SL(n; R), i.e. the space of left cosets SL(n; R)=SO(n),
via the map which sends a matrix A to the equivalence class of the positive
definite quadratic form q, given by q(x) = A1x where   denotes the
euclidean norm in Rn. The group SL(n; Z) acts on this coset space from the
left, and this action is proper, i.e. if C X1 (n) is compact then there are
only finitely many g 2 SL(n; Z) for which gC \ C 6= ;; in particular the
isotropy groups of the action are all finite.
The BruhatTitsbuilding. The group SL(n; Z[1=2]) acts on the coset
space X1 (n) as well. However, in this case the action is not proper. In order
to get a contractible space with proper action, the space X1 (n) has to be
enlarged by the appropriate BruhatTitsbuilding X2(n) (or simply X2 if n
is clear from the context) for the group SL(n; Q2). As reference for more on
this bulding we recommend [B2 ]. We recall here only some basic properties.
The space X2(n) is an (n  1)dimensional simplicial complex which can be
described as follows: an n  dimensional 2  adic lattice L is a Z2  submodule
of Qn2which is free of rank n. The group Qx2of units in Q2 acts on the set of
all such lattices via scalar multiplication, and the set of equivalence classes*
* is
the set of vertices in X2(n). A finite subset {l0; l1; :::; ln} of vertices spa*
*ns an n
 dimensional simplex in X2(n) if and only if there are representative lattices
Li in the class of li for i = 0; :::; n such that L0 ( L1 ( ::: ( Ln ( 1_2L0.
The space X2(n) is contractible (see Section V.8 and Theorem VI.3 in [B2 ]).
Furthermore the set of all 2  adic lattices can be identified with the set
of left cosets GL(n; Q2)=GL(n; Z2) via the map which sends a matrix A to
the lattice A(Zn2). The natural left action of SL(n; Q2) on this coset space
induces a simplicial left action of SL(n; Q2) on X2(n) and the quotient of
X2(n) by the action of SL(n; Q2) is an (n  1)  dimensional simplex n1.
The cohomology of SL(3; Z[1=2]) 9
Furthermore SL(n; Z[1=2]) is dense in SL(n; Q2) and therefore the quotient
of X2(n) by the action of SL(n; Z[1=2]) agrees with the quotient by the group
SL(n; Q2).
The group SL(n; Z[1=2]) embedds diagonally as a discrete subgroup into
SL(3; R)xSL(n; Q2) and acts properly on the contractible space X := X1 x
X2.
The projection maps and congruence subgroups. From now on we
concentrate on the case n = 3. We will be interested in the SL(3; Z[1=2]) 
equivariant projection map p : X  ! X2.
With respect to the action of GL(3; Z[1=2]) all vertices in X2 fall into a sin*
*gle
orbit and hence their isotropy groups (in SL(3; Z[1=2])) are conjugate in
the larger group GL(3; Z[1=2]), in particular they are abstractly isomorphic;
similarly with simplices of dimension one. For the vertex l0 corresponding
to the standard lattice L0 (which is spanned over Z32by the standard basis
vectors e1,e2 and e3, i.e. L0 = ), the isotropy group is SL(3; Z2) \
SL(3; Z[1=2]) = SL(3; Z) =: 0. For the edge consisting of the set {l0; l1}
with l0 the class of L0 and l1 the class of the lattice L1 = <1_2e1; e2; e3>, t*
*he
isotropy group is the subgroup 1 of 0 consisting of matrices whose first
column is equal to e1 modulo 2; for the twodimensional simplex spanned
by the set {l0; l1; l2} with l2 the class of the lattice L2 = <1_2e1; 1_2e2; e3*
*>, the
isotropy group is the subgroup 2 of 0 consisting of all matrices which are
upper triangular modulo 2. For simplicity of notation we will write instead
of SL(3; Z[1=2]).
The "fibres" of the map (which is induced by p)
ep: E x X  ! \X2 ~=2
over the 0 , 1  resp. 2  dimensional simplices respectively are homotopy 
equivalent to the classifying spaces B0, B1 and B2 respectively. We will
have to study the mod  2 (co)homology spectral sequence of epas well as
that of the map (which is also induced by p)
__p: \X  ! \X ~ 2
2 = :
In particular we need to understand the "fibres" of __p, i.e. the quotients
i\X1 , i = 0; 1; 2. These quotients are not compact and in the next section
we recall the SouleLannes method of finding a deformation retract of i\X1
which is compact, even a finite 3  dimensional complex (see [A ]).
10 HansWerner Henn
2.2 Wellrounded lattices and the deformation retrac
tions
Wellrounded lattices. We note that i\X1 ~=i\(SL(3; R)=SO(3)) may
also be obtained as quotient of i\SL(3; R) by the right action of SO(3).
Now the right SO(3)  space GL(3; Z) \GL(3; R) can be identified with the
space of all integral lattices in R3 (via the correspondence which sends a
matrix g to the lattice g1 (Zn)), and the space 0\SL(3; R) can be identified
with the space of equivalence classes (with respect to scalar multiplication)
of integral lattices L in R3, or equivalently with the space of integral lattic*
*es
whose minimal vectors are of length 1, i.e. for which m(L) := min {kxkx 2
L  {0}} = 1. We will denote this latter space by L0. Note that, in terms of
lattices, the right action of SO(3) on 0\SL(3; R) is given by L . g := g1 L
for L 2 L0 and g 2 SO(3).
Similarly the space 1\SL(3; R) can be identified with the space L1 of pairs
(L0; L1) of lattices such that m(L0) = 1 and L0 ( L1 ( 1_2L0, and the space
2\SL(3; R) can be identified with the space L2 of triples (L0; L1; L2) of
lattices such that m(L0) = 1 and L0 ( L1 ( L2 ( 1_2L0.
We recall that a lattice L in R3 is called wellrounded if its set of minimal
vectors, i.e. {x 2 L  {0}kxk = m(L)} spans R3. For i = 0; 1; 2 let Wi
denote the subspace of Li consisting of all tuples (L0; :::; Li) for which L0 is
wellrounded.
The deformation retractions. There is a beautiful geometric argument
which shows that Wi is an SO(3)  equivariant deformation retract of Li,
hence Wi=SO(3) is a deformation retract of Li=SO(3) ~= i\X1 . We recall
the construction ([A ]).
For i = 0; 1; 2 and 1 p 3 let Wpibe the set of tuples (L0; :::; Li) of
lattices such that the dimension of the subspace of R3 spanned by the set of
minimal vectors in L0 is at least p. Then W1i= Li, W3i= Wi and therefore
it suffices to show that Wp+1iis an SO(3)  equivariant deformation retract
of Wpifor p = 1; 2. So assume that the set of minimal vectors in L0 spans a
subspace U of dimension q p. If q > p then nothing happens to our tuple
in the next step of the deformation. Otherwise, consider a radial contracting
homotopy in the subspace U? of R3 perpendicular to U, and extend linearly
to a deformation of R3 by leaving U fixed. This defines a deformation Lj(t),
0 < t 1 of lattices (for 0 j i) with Lj(1) = Lj and there will be a
maximal t0 with 0 < t0 < 1 for which L0(t0) has a new vector of minimal
length 1. The corresponding tuple (L0(t0); :::; Li(t0)) of lattices lies in Wp+*
*1i
and is the image under the next step in the deformation. It is easy to see
that these constructions describe continuous SO(3)  equivariant maps which
The cohomology of SL(3; Z[1=2]) 11
combine to give an SO(3)  equivariant deformation retraction from Li to Wi
and induce a deformation retraction from i\X1 ~=Li=SO(3) to Wi=SO(3).
We will see in the next section that the spaces Wi=SO(3) are compact and
of dimension 3.
We can do even a bit better: the SO(3)  equivariant deformation re
traction of L0 can be lifted to give a left SL(3; Z)  equivariant and right
SO(3)  equivariant deformation retraction of SL(3; R) onto the subspace
Y := {g 2 SL(3; R)g1 (Zn) is a wellrounded lattice}. Dividing out by the
SO(3)  action gives a left SL(3; Z)  space Z and an SL(3; Z)  equivariant
deformation retraction from X1 to Z. The space Z will also be called the
space of (equivalence classes of) well  rounded quadratic forms.
The remainder of Section 2 is devoted to a detailed analysis of the spaces
i\Z ~=Wi=SO(3), in particular we will exhibit explicit finite cell structures
on them.
2.3 The space W0=SO(3)
Our first task is to understand the space 0\Z ~= W0=SO(3). This space
agrees with Soule's deformation retract of the space 0\SL(3; R)=SO(3) [So ];
however, our point of view is a bit different in so far as we emphasize lattices
rather than quadratic forms, i.e. we prefer to think in terms of W0=SO(3),
the space of wellrounded 3  dimensional lattices L with m(L) = 1, modulo
the action of SO(3).
We will see in a moment that in dimension 3 (unlike in higher dimensions)
the sublattice spanned by any set of 3 linearly independent vectors of minimal
length in a wellrounded lattice L is all of L, and therefore L is (up to the
action of SO(3)) determined by m(L) and the 3 scalar products between
these vectors. We will analyze which of these 3  tuples occur in this way
and which tuples give the same lattice, up to the action of SO(3). This
analysis will lead to an explicit description of the spaces Wi=SO(3). In this
section we will first concentrate on the case i = 0. Our first step is given by
the following Lemma.
Lemma 2.1 Suppose L R3 is a wellrounded lattice and let v1, v2 and
v3 be linearly independent vectors of minimal length m(L) in L. Then the
sublattice L0 spanned by these vectors is all of L.
Proof. By scaling and rotating L we may assume that m(L) = 1, v1 =
(1; 0; 0) and v2, v3 have the form: v2 = (a; x; 0) and v3 = (b; y; z). Assume
there exists w = (w1; w2; w3) 2 L  L0. By adding a suitable vector in L0 we
12 HansWerner Henn
may assume that w3 1_2z 1_2, w2 1_2x 1_2and w1 1_2. But then
w < 1 and we obtain a contradiction to the assumption that m(L) = 1. 2
The next two results will enable us to give an explicit description of the
space W0=SO(3). They will be proved together.
Proposition 2.2 Suppose v1, v2 and v3 are linearly independent vectors of
length 1 in R3 with scalar products a = , b = and c = .
Assume that a 0 and b 0. Then the lattice L spanned by v1, v2 and v3 is
wellrounded with m(L) = 1 if and only if
1. c 0 and a; b; c 1_2, or
2. c 0, a; b; c 1_2and a + b  c 1.
Clearly, the assumption on a and b can be assured by replacing, if necessary,
one of the vectors vi by its negative.
Proposition 2.3 Suppose v1, v2 and v3, a, b, c and L are as in Proposition
2.2. Furthermore assume a b c. Then the set of minimal vectors in L
contains v1; v2; v3 and in addition only the following vectors:
1. (v1  v2) if a = 1_2, b 6= 1_2and a + b  c 6= 1.
2. (v1  v2  v3) if a 6= 1_2, b 6= 1_2and a + b  c = 1.
3. (v1  v2) and (v1  v3) if a = b = 1_2and c 6= 0; 1_2.
4. (v1  v2) and (v1  v2  v3) if a = 1_2, b 6= 1_2and a + b  c = 1.
5. (v1  v2), (v1  v3) and (v2  v3) if a = b = c = 1_2.
6. (v1  v2), (v1  v3) and (v1  v2  v3) if a = b = 1_2and c = 0.
Again the assumption on a, b and c can always be assured by permuting
the vectors vi and passing to negatives if necessary.
Proof. 1. Let us first consider the case c 0.
Consider a vector w in L and write
w = n1v1 + n2v2 + n3v3; ni 2 Z; i = 1; 2; 3 :
Then
w2 = n21+ n22+ n23+ 2an1n2 + 2bn1n3 + 2cn2n3 ; (2.1)
or equivalently
w2 = a(n1 + n2)2 + b(n1 + n3)2 + c(n2 + n3)2
+(1  a  b)n21+ (1  a  c)n22+ (1  b  c)n23: (2.2)
If a > 1_2then n1 = n2 = 1, n3 = 0 gives a vector w with w = 2  2a < 1.
The same argument for b and c shows that, if m(L) = 1, then b; c 1_2. Now
assume that a; b; c 1_2. We distinguish different cases.
The cohomology of SL(3; Z[1=2]) 13
1.1. At least one ni = 0, w.l.o.g. n3 = 0. Then we obtain
w2 = n21+ n22+ 2an1n2 = a(n1 + n2)2 + (1  a)n21+ (1  a)n22: (2.3)
Because 1  a 1_2and 1  b 1_2it is clear from (2.3) that w2 1 unless
w = 0.
We also observe that the only vectors of length 1 in L with n3 = 0 are
the vectors v1, v2, and if a = 1_2, the vector (v1  v2). Similarly, the
only vectors with n2 = 0 are the vectors v1, v3, and if b = 1_2, the vector
(v1  v3). The only vectors with n1 = 0 are the vectors v2, v3, and if
c = 1_2, the vector (v2  v3).
1.2. We may now assume that all ni 6= 0. Then at least one of the sums
n1+n2, n1+n3, n2+n3 must be different from 0. If precisely one of the sums
is nonzero, say n2 + n3, then n1 = n3, n1 = n2 and n2 + n3 2 and
(2.2) yields w2 3  2a  2b + 2c 1; equality holds iff c = 0, a = b = 1_2,
n1 = n2 = n3 = 1, i.e. w = (v1  v2  v3). If at least two of the sums
are nonzero, say n1 + n3 and n2 + n3, then n1 + n3 2 and n2 + n3 2
and (2.2) yields w2 3  2a + 2b + 2c > 1, in particular there are no such
vectors of length 1.
2. Now consider the case c 0. Then we write
w2 = a(n1 + n2)2 + b(n1 + n3)2  c(n2  n3)2
+(1  a  b)n21+ (1  a + c)n22+ (1  b + c)n23: (2.4)
As before we see that a; b; c 1_2is necessary for L to satisfy m(L) = 1.
Now assume these inequalities hold. Again we distinguish different cases.
2.1. If at least one ni = 0 and w 6= 0, then we see as above that w2 1
and we only obtain additional vectors of length 1 iff a = 1_2resp. b = 1_2resp.
c = 1_2, namely the vectors (v1  v2) resp. (v1  v3) resp. (v2 + v3).
2.2. We may now assume that all ni 6= 0. Consider the sums n1+n2, n1+n3,
n2  n3. We subdivide into further cases. In case all sums are zero we have
n1 = n2 = n3 and from (2.4) we obtain again w2 3  2a  2b + 2c.
By taking n1 = n2 = n3 = 1 we see that the condition a + b  c 1 is
necessary for L to satisfy m(L) = 1, and there are further vectors of length
1 iff a + b  c = 1, namely the vectors (v1  v2  v3).
If two sums are zero, then the third one is as well, hence we may next assume
that at most one sum is zero, hence at least two of the terms n1 + n2 and
n1 + n3, n2  n3 are 2. In case n1 + n2 and n1 + n3 are 2, (2.4)
yields kwk2 3 + 2a + 2b + 2c > 1, in particular there are no such vectors of
length 1. The other two cases are analogous. 2
14 HansWerner Henn
After these preparations we can now describe the space W0=SO(3). Con
sider the following subspace D0 of R3 (see figure 1):
1
D0 := {(a; b; c) 2 R3 c b a __; and a + b  c 1 ifc 0} :
2
We define a map
fl : D0 ! Y = {g 2 SL(3; R)g1 (Zn) is a wellrounded lattice}
by sending the triple (a; b; c) to the unique matrix fl(a; b; c) with the follo*
*w
ing properties: fl(a; b; c) is (up to a scalar multiple guaranteeing fl(a; b; c*
*) 2
SL(3; R)) the inverse of the matrix whose i  th column is the basis vec
tor vi, where v1 = (1; 0; 0), v2 = (a; x; 0), v3 = (b; y; z) and x, y and z are
uniquely determined by the requirements x 0, ab + xy = c, z 0 and
vi = 1 for i = 1; 2; 3. By construction and Proposition 2.2 the lattice
fl(a; b;c)1(Zn) is well  rounded, hence fl(a; b; c) 2 Y. Let : D0 ! W0
denote the composition of fl with the canonical projection Y  ! W0; then
(a; b; c) is the well  rounded lattice spanned by the vectors v1, v2 and v3.
Note that by construction a = , b = and c = . Fi
nally let ' : D0 ! W0=SO(3) be the composition of with the canonical
projection W0 ! W0=SO(3). Clearly all these maps are continuous.
Finally we define an equivalence relation ~ on D0 by declaring the points
(1_2; b; c) with 0 c 1_2b equivalent to (1_2; b; b  c) and equivalent to (1_*
*2; b 
c; c) (cf. figure 1).
Theorem 2.4 The map ' : D0 ! W0=SO(3) is onto and induces a home
omorphism e': D0=~ ! W0=SO(3).
In the proof we will make repeated use of the following elementary fact.
Lemma 2.5 Assume v1, v2, v3 and v10, v20, v30are two sets of linearly
independent vectors of length 1 in R3 such that = for all
1 i < j 3. Then there exist unique rotations R; S 2 O(3) such that
Rvi = vi0and Svi = vi0for i = 1; 2; 3, and either R or S is in SO(3). 2
Proof of Theorem. That ' is onto can be seen as follows. Assume we are
given a wellrounded lattice L with minimal vectors of length 1. By Lemma
2.1 we can find spanning vectors w1, w2 and w3 in L of length 1, and after a
suitable permutation (and passing to additive inverses, if necessary) we may
assume that the scalar products a, b and c are as in 2.2 and 2.3. Then Lemma
2.5 implies '(a; b; c) = [L] where [L] denotes the image of L in W0=SO(3).
The cohomology of SL(3; Z[1=2]) 15
c


6




 
 jC= (1_2; 1_2; 1_2)
 j j 
 j 
 j 
 j 
 j 
 j 
 j j 
 j 
 j 
 j 
O = (0; 0; 0) ____________________________________________bjA
A D = (1_2; 1_2; 1_4)
A 
A 
A 
A 
A 
A 
A 
A = (1_2; 0; 0)________________________________B_=A(1_2; 1_2; 0)
@ A
@ A
@ A
@ A
@ A
@ A
@ A
@ A
ff @@_________A
a E = (1_2; 1_4; 1_4) F = (1_3; 1_3; 1_3)
Figure 1: The space D0 and the equivalence relation ~ . The equivalence
relation is given by identifying the triangle ABD with the triangles ACD
and ABE via reflections at the edges AD and AB.
16 HansWerner Henn
Next we show that equivalent triples have the same image under ' so that
' induces a continuous map 'e. So assume 0 c 1_2b and consider the
lattice (1_2; b; c). This has (at least) 4 pairs of minimal vectors, namely
v1 = (1; 0; 0), v2 = (a; x; 0), v3 = (b; y; z) and (v1  v2). (Here
x, y and z are as before.) Then it is straightforward to check that the
scalar products between the vectors v01:= v1, v02:= v1  v2 and v03:= v3
are (1_2; b; b  c) and those between the vectors v001:= v2  v1, v002:= v2 and
v003:= v3 are (1_2; b  c; c) and Lemma 2.5 implies again that the image
under ' of these triples agree.
Now we turn to injectivity of e'. As D0= ~ is compact and W0=SO(3) is
Hausdorff, this will show that e'is a homeomorphism and finish the proof.
So assume '(a; b; c) = '(a0; b0; c0). By assumption the corresponding lattices
L := (a; b; c) and L0 := (a0; b0; c0) agree up to a rotation R 2 SO(3), i.e.
L = RL0. In particular, L and L0 have the same number of minimal vectors,
the vectors Rv01, Rv02, Rv03form a set of linearly independent vectors of length
1 in L and the triple (a0; b0; c0) occurs as a triple of scalar products between
three linearly independent vectors of length 1 of L. We have to show that
this happens only if (a; b; c) and (a0; b0; c0) are equivalent under the relati*
*on
~.
In the "generic" case, i.e. if a 6= 1_2and a+bc 6= 1, L has only the minimal
vectors v1, v2 and v3 (cf. Proposition 2.3), and in this case it is obvious
that the triple of scalar products is uniquely determined by L and by the
condition a b c.
Now assume (a; b; c) 6= (a0; b0; c0) and we have 6 pairs of minimal vectors.
By Proposition 2.3 this can only happen if w.l.o.g. (a; b; c) = (1_2; 1_2; 1_2*
*) and
(a0; b0; c0) = (1_2; 1_2; 0). However, these points are clearly equivalent unde*
*r ~.
Next assume we have precisely 5 pairs of minimal vectors in L. By Proposi
tion 2.3 and because ' is constant on ~  equivalence classes we may assume
that our triples are of the form (1_2; 1_2; c) and (1_2; 1_2; c0) with 0 < c; c*
*0 1_4and
we have to show c = c0. The lattice L = (1_2; 1_2; c) has the following pairs
of minimal vectors of length 1: v1, v2, v3, (v1  v2) and (v1  v3),
and the triple (1_2; 1_2; c0) must occur as a triple of scalar products of 3 li*
*nearly
independent vectors taken from those 5 pairs. It is now straightforward to
check that this can happen only if c = c0.
Finally assume we have exactly 4 pairs of linearly independent vectors of
length 1 in L = (a; b; c). Again by Proposition 2.3 and because ' is constant
on ~  equivalence classes we may assume that the triple (a; b; c) satisfies
either a = 1_2and 0 c 1_2b < 1_4, or a 6= 1_2, c 0 and a + b  c = 1. We
have to show that L determines uniquely a triple of this form. In the first
case the set of minimal vectors consists of v1, v2, v3, (v1  v2), in the
The cohomology of SL(3; Z[1=2]) 17
second case of v1, v2, v3, (v1  v2  v3). Again it is straightforward
to see that all triples of linearly independent vectors taken from those sets
which lead to scalar products of the required form, lead indeed to the same
scalar products, and thus the proof is complete. 2
2.4 i  equivariant cell structures on Z
2.4.1 The case i = 0
We recall the map fl : D0  ! Y which sends d = (a; b; c) 2 D0, up to a
scalar multiple, to the inverse of the matrix whose i  th colum is the vector
vi specified in the last section (cf. the discussion before Theorem 2.4). The
composition of the map fl : D0  ! Y with the quotient map ssZ : Y  !
Y=SO(3) ~= Z will be denoted by 0. Note that 0(d) is the equivalence
class of the positive definite quadratic form for which the scalar products
between the standard basis vectors e1; e2; e3 are given by =
for i i; j 3, i.e. = 1 for i = 1; 2; 3, = a, = b
and = c. In particular this map is injective and a homeomorphism
from D0 to 0(D0). The 0  equivariant extension 0 x D0 ! Z which
sends (g; d) to g 0(d) will still be denoted by 0. Let ~0 be the equivalence
relation on 0 x D0 induced by the map 0, i.e. defined by (g; d) ~0 (g0; d0)
iff 0(g; d) = 0(g0; d0). Then ~0 is 0  equivariant, i.e. if (g; d) ~0 (g0; d*
*0)
then (hg; d) ~0 (hg0; d0) for every h 2 0.
Let gAD 2 0 be given by gAD (e1) = e1, gAD (e2) = e1+e2 and gAD (e3) =
e3, and let gAB 2 0 be given by gAB (e1) = e2  e1, gAB (e2) = e2 and
gAB (e3) = e3. Then we have the following result which is a refinement of
Theorem 2.4.
Theorem 2.6 The equivalence relation ~0 on 0 x D0 induced by the map
0 : 0 x D0 ! Z is the smallest 0  equivariant equivalence relation gen
erated by the following elementary relations: (g; d) and (g0; d0) are elementary
equivalent if either
1. g0 = 1, d = d0 and g belongs to the isotropy group Hd 0 of the (class
of the) quadratic form 0(d).
2. g0 = 1, d = (1_2; b; c), 0 c 1_2b, and either
1 AD
d0= (__; b; b  c); g = g ; or
2
1 AB
d0= (__; b  c; c); g = g :
2
18 HansWerner Henn
Furthermore the induced map 0 : 0 x D0= ~0 ! Z is a homeomorphism
of 0  spaces.
Proof. First we observe that points of 0 x D0 which are elementary equiv
alent are mapped to the same point in Z under 0. This is trivial for the
first elementary relation. For the second one it follows because by defini
tion of gAD and Lemma 2.5 we have gAD fl(1_2; b; c) 2 fl(1_2; b; b  c)SO(3),
i.e. gAD 0(1_2; b; c) = 0(1_2; b; b  c). Similarly, gAB is defined such*
* that
gAB fl(1_2; b; c) 2 fl(1_2; b  c; c)SO(3), i.e. gAB 0(1_2; b; c) = 0(1_2; *
*b  c; c).
It follows that 0 induces a map 0 as claimed.
Furthermore 0 induces (on passing to the quotients with respect to the
actions of 0) the surjection ' of Theorem 2.4. In particular, if follows that
0 and hence 0 is surjective. Next assume that 0(g; d) = 0(g0; d0). Then
Theorem 2.4 shows that d ~ d0 and by definition of ~0 we may therefore
assume that d = d0. But then we clearly have g1 g0 2 Hd and by 0 
equivariance of ~0 we see that (g; d) ~0 (g0; d0) and injectivity of 0 follows.
Finally it is easy to see that the map 0 is an open map and hence a home
omorphism (e.g. by using that the actions of 0 on 0 x D0= ~0 and Z are
proper, and that the induced map on the quotient spaces is a homeomorphism
by Theorem 2.4). 2
Cell structures on D0, D0= ~ and a 0  equivariant cell structure
on Z. Theorem 2.6 allows us to establish a 0  equivariant cell structure
on Z in terms of a cell structure on D0 resp. on D0= ~. We start with cell
structures on D0 and D0= ~ (see figure 1).
0. The 0  dimensional cells of D0 are the vertices O = (0; 0; 0), A = (1_2; 0*
*; 0),
B = (1_2; 1_2; 0), C = (1_2; 1_2; 1_2), D = (1_2; 1_2; 1_4), E = (1_2; 1_4; 1_*
*4) and F = (1_3; 1_3; 1_3).
On D0= ~ this gives 5 cells which will still be labelled O, A, B ~ C, D ~ E
and F .
1. The 1  dimensional cells of D0 are the edges OC, OF , OA, EF , BF ,
AB, AC, AD, AE, BD, CD and BE. On D0= ~ this gives 8 cells labelled
OC, OF , OA, EF , BF , AB ~ AC, AD ~ AE and BD ~ CD ~ BE.
2. The 2  dimensional cells of D0 are the quadrangles OAEF (characterized
by b = c) and OCBF (a = b), and the triangles OAC (b = c), BEF
(a + b  c = 1) and ABD, ACD and ABE. On D0= ~ this gives 5 cells
labelled OAEF , OCBF , OAC, BEF and ABD ~ ACD ~ ABE.
3. D0 has one cell of dimension 3, namely the interior of D0, and this gives
also one cell for D0= ~.
It follows easily from Proposition 2.3 (see also Section 2.5 below) that the
isotropy groups Hd of the action of 0 on Z at 0(d) are constant within the
The cohomology of SL(3; Z[1=2]) 19
interior of each cell e of D0 and this is the reason for the choice of our cell
structure on D0. If we denote this isotropy group by He then Theorem 2.6
shows that Z has an equivariant cell structure with one orbit (0=He) x e of
cells for each equivalence class of cells in D0= ~. The attaching maps can be
read off from figure 1.
2.4.2 The cases i = 1 and i = 2
We consider the right 0  spaces i\0. In case i = 1 this coset space
can be identified with the set S1 of nonzero vectors in (F32 {0}) and in
case i = 2 with the set S2 of pairs consisting of a line in F32and a plane
in F32containing the line, i.e. with the set of complete flags in F32. In
fact, there is a canonical left action of 0 on the sets Si, and if we con
vert this into a right action in the usual way via s . g := g1 s, then the
map 0  ! S1; g 7! g1 (e1) mod 2 induces an isomorphism of right 0 
spaces 1\0 ! S1; similarly the map 0 ! S2 which sends g to the flag
( mod 2) induces an isomorphism of right
0  spaces 2\0 ! S2. (Here < > denotes the subgroup generated by the
elements within the brackets and 100, 010 are standard basis vectors in F32.)
The sets D0 x Si will be denoted by Di.
Now we choose representatives for the right cosets of i in 0. Such a
choice of a representative gs for each s 2 Si gives an explicit i  equivariant
homeomorphism
ix D0 x Si ! 0 x D0; (g; d; s) 7! (ggs; d) :
In order to obtain a i  equivariant cell structure on Z we will carry over
the 0  equivariant equivalence relation ~0 on 0 x D0 to a i  equivariant
equivalence relation ~i on ix Di. For this we note that the isotropy groups
Hd act from the right on the coset spaces Si. Likewise the matrices gAB and
gAD act from the right on Si. The following result is now a straightforward
consequence of Theorem 2.6.
Theorem 2.7 The equivalence relation ~i on ix Di induced by the map
i : ix Di ! Z; (g; d; s) 7! ggs 0(d)
is the smallest i  equivariant equivalence relation generated by the follow
ing elementary relations: (g; d; s) and (g0; d0; s0) are elementary equivalent *
*if
either
1. g0 = 1, d = d0, there exists an element h 2 Hd with s = s0h (in particular
s and s0 belong to the same Hd  orbit with respect to the right action of Hd
on the set Si) and g is determined by ggs = gs0h.
20 HansWerner Henn
2. g0 = 1, d = (1_2; b; c), 0 c 1_2b and either
1 0 AD AD
d0= (__; b; b  c); s = s g ; ggs = gs0g ; or
2
1 0 AB AB
d0= (__; b  c; c); s = s g ; ggs = gs0g :
2
Furthermore the induced map i : ixDi= ~i! Z is a homeomorphism of
i  spaces, the i  equivariant equivalence relation ~i induces an equivalence
relation (denoted by ~(i)) on the quotient Di of ix Di such that the induced
map e : Di= ~(i)! i\Z is a homeomorphism. 2
i  equivariant cell structures on Z and cell structures on i\Z.
Theorem 2.7 yields i  equivariant cell strucures on the space Z (and then
ordinary cell structures on the quotients i\Z). The indexing set for the i
 orbits of cells on Z (resp. the cells on the quotients i\Z) are equivalence
classes of pairs (e; s) with e a cell in D0 and s 2 Si, with the equivalence
relation generated by the following elementary relations: (e; s) ~i (e0; s0) iff
either
1. e = e0 and s and s0 are in the same He  orbit, or
2a. e is ABD or a face of it, e0 is ACD or the corresponding face of it and
s = s0gAD , or
2b. e is ABD or a face of it, e0 is ABE or the corresponding face of it and
s = s0gAB .
The i  orbits of cells of Z are then of the form i=H(e;s)x (e; s) where
(e; s) runs through a set of representatives of equivalence classes of such pai*
*rs
and the isotropy group H(e;s)of the cell (e; s) is given by i\ gsHegs1, i.e
agrees up to conjugation by gs with gs1igs\He which is the isotropy group
of s with respect to the right action of He on Si. The attaching maps can
again be read off from figure 1.
In the next two sections we will make this concrete, i.e. we will describe in
explicit form the isotropy groups He, their actions on the sets Si, and also
the effect of the action of gAB and gAD on Si.
2.5 Symmetries of wellrounded quadratic forms
In order to make the equivariant cell structure of the spaces i\Z concrete
we need to determine the isotropy groups H(a;b;c)of the action of 0 on Z at
0(a; b; c). Of course, H(a;b;c)preserves the length of vectors and the scalar
The cohomology of SL(3; Z[1=2]) 21
products between them (both taken, of course, with respect to a representa
tive quadratic form of (the equivalence class of quadratic forms) 0(a; b; c)),
and hence H(a;b;c)acts on the set of minimal vectors in the standard lattice.
These sets have been determined in Theorem 2.4 (we just have to replace
the letter v by e everywhere). The standard basis vectors are always mini
mal vectors and so H(a;b;c)is determined by this action. It is clear that the
groups H(a;b;c)are constant in the interior of each cell of D0 and this gives
the justification for the choice of our cell structure on D0.
The case of the 3  dimensional cell is particularly simple. If (a; b; c) is i*
*n its
interior then we have only the 3 standard basis vectors and their negatives
in the set of minimal vectors and it is easy to check that H(a;b;c)= {1}.
Tables 1, 2 and 3 below give the isotropy groups on the open cells of D0 of
dimension 2, 1 and 0. In fact it will be enough for us to take one cell from
each ~  equivalence class of cells. The first column lists the name of the cel*
*l,
the second one the set of minimal vectors on the standard lattice with respect
to (a representative quadratic form of) 0(a; b; c) if (a; b; c) is an interior*
* point
of the appropriate cell and the third column gives the isotropy group. The
last column describes the action of the isotropy group on the tuple (e1; e2; e3)
of minimal vectors explicitly; the 3  tuples in this column are the images of
the tuple (e1; e2; e3) under the action of appropriate generators. The proofs
are straightforward and are left to the reader.
We use the following notation in these tables: for the symmetric group on
n  letters we write Sn, o denotes the wreath product construction, and the
dihedral group with n elements is denoted by Dn.
Table 1: Symmetries on the 2dimensional cells
______________________________________________________
_Cell__Minimal_vectorsIsotropy____Generators_______
 e1; e2; e3;   
 ABD   trivial 
________(e1__e2)_________________________________
_OAC___e1;_e2;_e3______Z=2______(e2;_e1;_e3)___ 
_OAEF__e1;_e2;_e3______Z=2________(e2;_e1;_e3)___ 
_OCBF__e1;_e2;_e3______Z=2______(e1;_e3;_e2)___ 
 e1; e2; e3  (e2; e1; e1  e2  e3) 
 BEF   Z=2 x Z=2 
_______(e1__e2__e3)________(e3;_e1__e2__e3;_e1)_
22 HansWerner Henn
Table 2: Symmetries on the 1dimensional cells
_______________________________________________________
_Cell_Minimal_vectors___Isotropy____Generators______
    (e1; e3; e2) 
 OC  e1; e2; e3  S3  
__________________________________(e2;_e1;_e3)__ 
    (e1; e3; e2) 
 OF  e1; e2; e3  S3  
____________________________________(e2;_e1;_e3)__ 
    (e2; e1; e3) 
 OA  e1; e2; e3 Z=2 x Z=2 
____________________________________(e2;_e1;_e3)__ 
  e1; e2; e3;   
 AB   Z=2  (e2  e1; e2; e3)
_________(e1__e2)_________________________________
  e1; e2; e3;   
 AD   Z=2  (e1; e2  e1; e3)
_________(e1__e2)_________________________________
  e1; e2; e3;   (e1; e3; e2) 
 BD   Z=2 x Z=2 
_____(e1__e2);_(e1__e3)_______(e1;_e3__e1;_e2__e1) 
  e1; e2; e3;   BEF symmetries, 
 BF   D8  
_______(e1__e2__e3)_____________(e1;_e3;_e2)__ 
  e1; e2; e3   BEF symmetries, 
 EF   D8  
_______(e1__e2__e3)_______________(e2;_e1;_e3)__ 
Table 3: Symmetries on the 0dimensional cells
________________________________________________________________
_CellMinimal_vectors_Isotropy____Description_________________
2
 O e1; e2; e3 S4 ~=(Z=2) o S3symmetry of a cube;3 
   index 2 in (Z=2) o S3 
   permuting the set of pairs
_________________________________{e1};_{e2};_{e3}___________~2
 C e1; e2; e3 S4 = (Z=2) o S3Z=2 x Z=2 generated by: 
  (e1  e2)  (e2  e3; e1  e3; e3) 
  (e1  e3)  (e3  e2; e2; e1  e2); 
______(e2__e3)__________________S3_symmetry_as_on_OC_______~2
 F e1; e2; e3 S4 = (Z=2) o S3Z=2 x Z=2 action as on;BEF
_____(e1__e2__e3)______________S3_symmetry_as_on_OF_______
 A e1; e2; e3  D12 {e3} is invariant. 
  (e1  e2)  Standard action on the regular
   planar hexagon formed by 
_________________________________e1;_e2;_(e1__e2)__________
 D e1; e2; e3  D8 Z=2 x Z=2 action as on BD;
  (e1  e2)  additional generator: 
______(e1__e3)__________________(e1;_e2__e1;_e3)________
The cohomology of SL(3; Z[1=2]) 23
2.6 The equivalence relations ~i on the spaces i x Di
By Theorem 2.7 the equivalence relations ~i, as well as i  equivariant cell
structures on Z and ordinary cell structures on i\Z, are determined by the
right actions of the isotropy groups Hd 0, d 2 D, on the sets Si together
with the right action of gAB and gAD on these sets. Clearly, the associated
left action of Hd on the sets Si has identical orbits and isotropy groups as the
right action; in the case of gAB and gAD the left and right actions are even
identical because both elements agree with their own inverses; we prefer to
work with the left actions.
As remarked above the isotropy groups and hence their actions are constant
on the cells of D and we consider cell by cell separately. In our analysis the
elements in S2 will be labelled by pairs consisting of a nonzero vector in F32
and a plane in F32containing this vector, e.g. 010x = z, 011x = 0, : :.:The
plane with equation x + y + z = 0 will be abbreviated by = 0. We will
also abbreviate (ABD; 100y = z) by ABD100y = z and so on. The proofs
in this section are all straightforward and are left to the reader.
2.6.1 3  cells
By Theorem 2.7 and by Section 2.5 there are no identifications in the interior
of the 3  cells. As there is only one 3  cell in D0, the 3  cells in 1\Z will
be labelled just by the nonzero vectors F32. So there are seven 3  cells in
1\Z, labelled:
100; 010; 001; 110; 101; 011; 111 :
Similarly there are 21 cells of dimension 3 in 1\Z which will be labelled:
100y = 0; 100z = 0; 100y = z
010x = 0; 010z = 0; 010x = z;
001x = 0; 001y = 0; 001x = y;
110z = 0; 110x = y; 110 = 0;
101y = 0; 101x = z; 101 = 0;
011x = 0; 011y = z; 011 = 0;
111x = y; 111x = z; 111y = z:
The isotropy group of the 3  cell is trivial, so there is nothing else to do in
this case.
2.6.2 2  cells
1. ABD By Section 2.5 and Theorem 2.7 all the relations involving these 2
 cells are of type 2, i.e. the following cells become equivalent.
24 HansWerner Henn
ABDs~i ACDgAD s~i ABEgAB s : (2.5)
Clearly the isotropy groups are trivial for all these cells.
We will now make the maps gAD and gAB explicit. We recall that by
definition these matrices induce the linear maps on F32given by
gAD (100) = 100; gAD (010) = 110; gAD (001) = 001 ;
gAB (100) = 110; gAB (010) = 010; gAB (001) = 001 :
Explicit knowledge of the action of these maps on the sets Si will be used
repeatedly later on and therefore these maps are explicitly described in tables
4 and 5 below.
Table 4: Action of gAD and gAB on S1
_______________________________________
 s 100 010 001 110 101 011 111 
_____________________________________
 gAD s100 110 001 010 101 111 011 
_____________________________________
 gAB s110 010 001 100 111 011 101 
_____________________________________
Table 5: Action of gAD and gAB on S2
__________________________________________________________________
 s 100y = 0 100z = 0 100y = z 010x = 0 010z = 0 010x = z 
_______________________________________________________________
 gAD s100y = 0 100z = 0 100y = z 110x = y 110z = 0 110 = 0 
______________________________________________________________
 gAB s110x = y 110z = 0 110 = 0  010x = 0 010z = 0 010x = z 
________________________________________________________________
 s 001x = 0 001y = 0 001x = y 110z = 0 110x = y 110 = 0 
______________________________________________________________
 gAD s001x = y 001y = 0 001x = 0 010z = 0 010x = 0 010x = z 
_______________________________________________________________
 gAB s001x = 0 001x = y 001y = 0 100z = 0 100y = 0 100y = z 
________________________________________________________________
 s 101y = 0 101x = z 101 = 0  011x = 0 011y = z 011 = 0 
______________________________________________________________
 gAD s101y = 0 101 = 0 101x = z 111x = y 111y = z 111x = z 
_______________________________________________________________
 gAB s111x = y 111x = z 111y = z 011x = 0 011 = 0 011y = z 
________________________________________________________________
 s 111x = y 111x = z 111y = z  
__________________________________ 
 gAD s011x = 0 011 = 0 011y = z  
__________________________________ 
 gAB s101y = 0 101x = z 101 = 0  
_______________________________________________________________
The cohomology of SL(3; Z[1=2]) 25
2. OAC In the interior of these cells all relations are of type 1. In other
words we have to determine the action of the group HOAC ~= Z=2 on the
vector space F32. By table 1 the action of the nontrivial element h 2 HOAC
on F32is given by
h100 = 010; h010 = 100; h001 = 001 :
Hence we get the following orbits and isotropy groups for the action on S1:
__________________________________________________
_Isotropy_groups_{1}_____{1}___Z=2_Z=2_Z=2__
_Orbits_________100;_010101;_011_001_110_111_
For the action on S2 we obtain:
_________________________________________________________________
_Isotropy_groups_{1}______{1}______{1}______{1}______{1}___
 Orbits 100y = 0 100z = 0 100y = z101y = 0 101x = z 
________________010x_=_0_010z_=_0010x_=_z_011x_=_0_011y_=_z__
_Isotropy_groups_{1}______{1}______{1}______Z=2_____Z=2____
 Orbits 101 = 0 001x = 0 111x = z 001x = y 111x = y 
________________011_=_0_001y_=_0_111y_=_z___________________
_Isotropy_groups_Z=2______Z=2______Z=2____  
_Orbits_________110x_=_y_110z_=_0110_=_0___________________
3. OAEF Again all relations are of type 1. Furthermore the action of
HOAEF on F32is the same as in the case of OAC. Therefore we obtain the
same list of orbits and isotropy groups.
4. OCBF Once again all relations are of type 1. By table 1 the action of
the nontrivial element h 2 HOCBF on F32is given by
h100 = 100; h010 = 001; h001 = 010
and we get the following orbits and isotropy groups for the action on S1 resp.
S2:
__________________________________________________
_Isotropy_groups_{1}_____{1}___Z=2_Z=2_Z=2__
_Orbits_________010;_001110;_101_100_011_111_
26 HansWerner Henn
_________________________________________________________________
_Isotropy_groups_{1}______{1}______{1}______{1}_____{1}____
 Orbits 010x = 0 010z = 0010x = z 110z = 0 110x = y 
________________001x_=_0_001y_=_0001x_=_y_101y_=_0_101x_=_z__
_Isotropy_groups_{1}______{1}______{1}______Z=2_____Z=2____
 Orbits 110 = 0 100y = 0 111x = y 100y = z 111y = z 
________________101_=_0_100z_=_0_111x_=_z___________________
_Isotropy_groups_Z=2______Z=2______Z=2____  
_Orbits_________011x_=_0_011y_=_z011_=_0___________________
5. BEF All relations are of type 1. By table 1 the action of two generators
h1 and h2 of HBEF ~=Z=2 x Z=2 on F32is given by
h1100 = 010; h1010 = 100; h1001 = 111 ;
h2100 = 001; h2010 = 111; h2001 = 100 :
Hence we get the following orbits and (types of) isotropy groups for the action
on S1 resp. S2:
__________________________________________________
_Isotropy_groups______{1}________Z=2_Z=2_Z=2_
_Orbits_________100;_111;_010;_001110_101_011_
_____________________________________________________________________
_Isotropy_groups_{1}_______{1}_______{1}______Z=2_______Z=2____
 Orbits 100y = 0 100z = 0 100y = z 110x = y 101x = z 
 111x = z 111x = y 111y = z 110z = 0 101y = 0 
 010x = z 010z = 0 010x = 0   
________________001y_=_0_001x_=_y__001x_=_0_____________________
_Isotropy_groups_Z=2___Z=2_x_Z=2_Z=2_x_Z=2_Z=2_x_Z=2___________
 Orbits 011y = z 110 = 0  101 = 0  011 = 0  
________________011x_=_0_______________________________________
2.6.3 1  cells
1. OC There are only relations of type 1. By table 1 the action of two
generators h1 and h2 of HOC ~=S3 on F32is given by
h1100 = 100; h1010 = 001; h1001 = 010;
h2100 = 010; h2010 = 100; h2001 = 001;
The cohomology of SL(3; Z[1=2]) 27
and we get the following orbits and isotropy groups for the actions on S1
resp. S2:
_________________________________________________
_Isotropy_groups___Z=2__________Z=2______S3__
_Orbits_________100;_010;_001110;_101;_011111_
_______________________________________________________
_Isotropy_groups______{1}__________Z=2______Z=2____
 Orbits 100y = 0; 100z = 0100y = z 110x = y 
 010x = 0; 010z = 0010x = z 101x = z 
________________001x_=_0;_001y_=_0001x_=_y_011y_=_z__
_Isotropy_groups______Z=2__________Z=2______Z=2____
 Orbits  110z = 0 110 = 0 111x = y 
  101y = 0 101 = 0 111x = z 
____________________011x_=_0______011_=_0_111y_=_z_
2. OF Again there are only relations of type 1, and furthermore the action
of HOF on F32is the same as in the case of OC. Therefore we obtain the same
list of orbits and isotropy groups.
3. OA There are only relations of type 1. By table 1 the action of two
generators h1 and h2 of HOA ~= Z=2 x Z=2 on F32is given by
h1100 = 100; h1010 = 010; h1001 = 001 ;
h2100 = 010; h2010 = 100; h2001 = 001 :
Hence we get the same orbits as in the case of the 2  cells OAC resp. OAEF .
However, the isotropy groups are now larger: the trivial ones get replaced by
Z=2 generated by h1h2, the Z=2 gets replaced by Z=2 x Z=2.
4. AB and AC There are relations of both types. Those of type 2 lead
to ABs ~i ACgAD s and are described by tables 4 and 5. As far as relations
of type 1 are concerned we can concentrate on the edge AB. By table 2 the
action of the nontrivial element h 2 HAB on F32is here given by
h100 = 110; h010 = 010; h001 = 001 :
Hence we get the following orbits and isotropy groups for the action on S1
resp. S2 (the orbits for AB and those for AC in the same column correspond
to each other via gAD ; the same conventions will hold in later tables of this
section): __________________________________________________
_Isotropy_groups_{1}_____{1}___Z=2_Z=2_Z=2__
_Orbits_for_AB__100;_110101;_111_010_001_011_
_Orbits_for_AC__100;_010101;_011_110_001_111_
28 HansWerner Henn
__________________________________________________________________
_Isotropy_groups_{1}______{1}______{1}______{1}______{1}____
 Orbits for AB 100y = 0 100z = 0 100y = z101y = 0 101 = 0 
________________110x_=_y_110z_=_0110_=_0_111x_=_y_111y_=_z__
 Orbits for AC 100y = 0 100z = 0 100y = z101y = 0 101x = z 
________________010x_=_0_010z_=_0_010x_=_z011x_=_0_011y_=_z__
_Isotropy_groups_{1}______{1}______{1}______Z=2______Z=2____
 Orbits for AB 101x = z 001x = y011 = 0 010x = 0 010z = 0 
________________111x_=_z_001y_=_0_011y_=_z__________________
 Orbits for AC 101 = 0 001x = 0 111x = z 110x = y 110z = 0 
________________011_=_0_001y_=_0_111y_=_z____________________
_Isotropy_groups_Z=2______Z=2______Z=2____  
_Orbits_for_AB__010x_=_z_001x_=_0_011x_=_0_  
_Orbits_for_AC__110_=_0_001x_=_y_111x_=_y___________________
5. AD and AE There are again relations of both types. Those of type
2 lead to ADs ~i AEgAB s and are described by tables 4 and 5. As far as
relations of type 1 are concerned we can concentrate on the edge AD. By
table 2 the action of the nontrivial element h 2 HAD on F32is here given by
h100 = 100; h010 = 110; h001 = 001 :
Hence we get the following orbits and isotropy groups for the action on S1
resp. S2:
__________________________________________________
_Isotropy_groups_{1}_____{1}___Z=2_Z=2_Z=2__
_Orbits_for_AD__110;_010111;_011_001_100_101_
_Orbits_for_AE__100;_010101;_011_001_110_111_
__________________________________________________________________
_Isotropy_groups_{1}______{1}______{1}______{1}______{1}____
 Orbits for AD 110x = y 110z = 0110 = 0 111x = y 111x = z 
________________010x_=_0_010z_=_0_010x_=_z011x_=_0_011_=_0__
 Orbits for AE 100y = 0 100z = 0 100y = z101y = 0 101x = z 
________________010x_=_0_010z_=_0_010x_=_z011x_=_0_011y_=_z__
_Isotropy_groups_{1}______{1}______{1}______Z=2______Z=2____
 Orbits for AD 111y = z 001x = 0 101x = z100y = 0 100z = 0 
________________011y_=_z_001x_=_y101_=_0____________________
 Orbits for AE 101 = 0 001x = 0 111x = z 110x = y 110z = 0 
________________011_=_0_001y_=_0_111y_=_z____________________
_Isotropy_groups_Z=2______Z=2______Z=2____  
_Orbits_for_AD__100y_=_z_001y_=_0_101y_=_0_  
_Orbits_for_AE__110_=_0_001x_=_y_111x_=_y___________________
The cohomology of SL(3; Z[1=2]) 29
6. BD, CD and BE There are again relations of both types. Those of
type 2 lead to BDs ~i BEgAB s resp. BDs ~i CDgAD s and are described by
tables 4 and 5. As far as relations of type 1 are concerned we can concentrate
on the edge BD. By table 2 the action of two generators h1 and h2 of
HBD ~=Z=2 x Z=2 on F32is here given by
h1100 = 100; h1010 = 001; h1001 = 010 ;
h2100 = 100; h2010 = 101; h2001 = 110 :
Hence we get the following orbits and isotropy groups for the action on S1
resp. S2:
___________________________________________________________
_Isotropy_groups_{1}___Z=2_x_Z=2_Z=2_x_Z=2_Z=2_x_Z=2__
 Orbits for BD 010; 001  100  011  111 
________________101;_110______________________________
 Orbits for CD 110; 001  100  111  011 
________________101;_010______________________________
 Orbits for BE 010; 001  110  011  101 
________________111;_100______________________________
_____________________________________________________________________
_Isotropy_groups_{1}_______{1}_______{1}_______Z=2______Z=2____
 Orbits for BD 010x = 0 010z = 0 010x = z 100y = 0 011x = 0 
 001x = 0 001y = 0 001x = y 100z = 0 011 = 0 
 101 = 0 101y = 0 101x = z   
________________110_=_0_110z_=_0__110x_=_y_____________________
 Orbits for CD 110x = y 110z = 0 110 = 0  100y = 0 111x = y 
 001x = y 001y = 0 001x = 0 100z = 0 111x = z 
 101x = z 101y = 0 101 = 0   
________________010x_=_z_010z_=_0__010x_=_0____________________
 Orbits for BE 010x = 0 010z = 0 010x = z 110x = y 011x = 0 
 001x = 0 001x = y 001y = 0 110z = 0 011y = z 
 111y = z 111x = y 111x = z   
________________100y_=_z_100z_=_0__100y_=_0_____________________
_Isotropy_groups_Z=2____Z=2_x_Z=2_Z=2_x_Z=2_Z=2_x_Z=2__________
 Orbits for BD 111x = y 100y = z 011y = z 111y = z  
________________111x_=_z_______________________________________
 Orbits for CD 011x = 0 100y = z 111y = z 011y = z  
________________011_=_0________________________________________
 Orbits for BE 101y = 0 110 = 0  011 = 0  101 = 0  
________________101x_=_z_______________________________________
30 HansWerner Henn
7. BF There are only relations of type 1. By table 1 and table 2 we know
the action of three generating involutions of HBF ~=D8 on F32
h1100 = 010; h1010 = 100; h1001 = 111 ;
h2100 = 001; h2010 = 111; h2001 = 100 ;
h3100 = 100; h2010 = 001; h2001 = 010 :
Hence we get the following orbits and isotropy groups for the action on S1
resp S2: ___________________________________________________
_Isotropy_groups______Z=2________Z=2_x_Z=2_D8__
_Orbits_________100;_111;_010;_001110;_101__011_
_________________________________________________________
_Isotropy_groups______{1}___________Z=2______Z=2_____
 Orbits 100y = 0; 111x = z 100y = z 110x = y 
 010x = z; 001y = 0 111y = z 110z = 0 
 100z = 0; 111x = y 010x = 0 101x = z 
________________001x_=_y;_010z_=_0_001x_=_0_101y_=_0__
_Isotropy_groups___Z=2_x_Z=2_____Z=2_x_Z=2____D8_____
 Orbits  110 = 0  011y = z 011 = 0 
___________________101_=_0________011x_=_0___________
8. EF Again there are only relations of type 1 and by table 1 and table 2
we know the action of three generating involutions of HEF ~=D8 on F32
h1100 = 010; h1010 = 100; h1001 = 111 ;
h2100 = 001; h2010 = 111; h2001 = 100 ;
h3100 = 010; h3010 = 100; h3001 = 001 :
Hence we get the following orbits and isotropy groups for the action on S1
resp. S2:
___________________________________________________
_Isotropy_groups______Z=2________Z=2_x_Z=2_D8__
_Orbits_________100;_111;_010;_001101;_011__110_
_________________________________________________________
_Isotropy_groups______{1}___________Z=2______Z=2_____
 Orbits 100y = 0; 111x = z 100z = 0 101x = z 
 010x = z; 001y = 0 111x = y 101y = 0 
 010x = 0; 111y = z 010z = 0 011y = z 
________________100y_=_z;_001x_=_0_001x_=_y_011x_=_0__
_Isotropy_groups___Z=2_x_Z=2_____Z=2_x_Z=2____D8_____
 Orbits  101 = 0  110x = y 110 = 0 
___________________011_=_0________110z_=_0___________
The cohomology of SL(3; Z[1=2]) 31
2.6.4 0  cells
1. O There are only relations of type 1. By table 3 the action of HO ~= S4
factors through an action of S3, and this action agrees with that in the
case of the edge OC. Therefore we get the same orbits; the isotropy groups
"grow" by Z=2 x Z=2, more precisely the trivial isotropy group gets replaced
by Z=2 x Z=2, Z=2 gets replaced by D8 and S3 by S4.
2. B and C There are relations of both types. Those of type 2 lead to
Bs ~i CgAD s and are described by table 4 and 5. As far as relations of
type 1 are concerned we can concentrate on C. By table 3 the action of
HC ~= S4 ~=Z=2 x Z=2 o S3 is described as follows: for the generators h1 and
h2 of Z=2 x Z=2 we have
h1100 = 011; h1010 = 101; h1001 = 001 ;
h2100 = 011; h2010 = 010; h2001 = 110 :
The action of S3 is again as in the case of the edge OC. Hence we get the
following orbits and isotropy groups for the action on S1 resp. S2:
_________________________________________________
_Isotropy_groups_______Z=2_x_Z=2__________S4__
_Orbits_for_C___100;_010;_001;_110;_101;_011111_
_Orbits_for_B___100;_110;_001;_010;_101;_111011_
_________________________________________________________
_Isotropy_groups______Z=2________Z=2_x_Z=2____D8_____
 Orbits for C 100y = 0; 100z = 0 100y = z 111x = y 
 010x = 0; 010z = 0 010x = z 111x = z 
 001x = 0; 001y = 0 001x = y 111y = z 
 110z = 0; 110 = 0 110x = y  
 101y = 0; 101 = 0 101x = z  
________________011x_=_0;_011_=_0_011y_=_z___________
 Orbits for B 100y = 0; 100z = 0 100y = z 011x = 0 
 010x = z; 010z = 0 010x = 0 011y = z 
 001y = 0; 001x = y 001x = 0 011 = 0 
 110z = 0; 110x = y 110 = 0  
 101y = 0; 101x = z 101 = 0  
________________111x_=_y;_111x_=_z_111y_=_z__________
32 HansWerner Henn
3. F Here there are only relations of type 1. By table 3 the action of
HF ~= S4 ~=Z=2 x Z=2 o S3 is described as follows: for the generators h1 and
h2 of Z=2 x Z=2 we have
h1100 = 010; h1010 = 100; h1001 = 111 ;
h2100 = 001; h2010 = 111; h2001 = 100 :
The action of S3 is again as in the case of the edge OC resp. OF . Hence we
get the following orbits and isotropy groups for the action on S1 resp. S2:
_________________________________________________
_Isotropy_groups______S3______________D8______
_Orbits_________100;_010;_001;_111110;_101;_011_
_________________________________________________________
_Isotropy_groups______Z=2________Z=2_x_Z=2____D8_____
 Orbits 100y = 0; 010x = 0 110x = y 110 = 0 
 100z = 0; 010z = 0 110z = 0 101 = 0 
 100y = z; 010x = z 101x = z 011 = 0 
 001x = 0; 111x = y 101y = 0  
 001y = 0; 111x = z 011x = 0  
________________001x_=_y;_111y_=_z_011y_=_z__________
4. A Again there are only relations of type 1. By table 3 the action
of HA ~=D12 factors through an action of S3 and permutes the elements
100, 010 and 110 while 001 is fixed under the action. Therefore we get the
following orbits and isotropy groups for the action on S1 resp. S2:
_________________________________________________
_Isotropy_groupsZ=2_x_Z=2____Z=2_x_Z=2___D12_
_Orbits_________100;_110;_010101;_111;_011001_
__________________________________________________________
_Isotropy_groups______Z=2_________Z=2_x_Z=2_Z=2_x_Z=2_
 Orbits 101x = z; 101 = 0 100y = z 100y = 0 
 011y = z; 011 = 0 110 = 0  110x = y 
________________111x_=_z;_111y_=_z_010x_=_z__010x_=_0_ 
_Isotropy_groups___Z=2_x_Z=2______Z=2_x_Z=2_Z=2_x_Z=2__
 Orbits  100z = 0 101y = 0 001x = 0 
  110z = 0 011x = 0 001x = y 
____________________010z_=_0______111x_=_y__001y_=_0__
The cohomology of SL(3; Z[1=2]) 33
5. D and E There are relations of both types. Those of type 2 lead to
Ds ~i EgAB s and are described by table 4 and 5. As far as relations of type
1 are concerned we can concentrate on D. By table 3 we know the action of
three generating involutions of HD ~= D8
h1100 = 100; h1010 = 001; h1001 = 010 ;
h2100 = 100; h2010 = 101; h2001 = 110 ;
h3100 = 100; h3010 = 110; h3001 = 001 :
Hence we get the following orbits and isotropy groups for the action on S1
resp. S2:
___________________________________________________
_Isotropy_groups______Z=2________Z=2_x_Z=2_D8__
_Orbits_for_D___110;_101;_010;_001111;_011__100_
_Orbits_for_E___100;_111;_010;_001101;_011__110_
_________________________________________________________
_Isotropy_groups______{1}____________Z=2______Z=2____
 Orbits for D 010x = 0; 010x = z 010z = 0 011x = 0 
 001x = 0; 001x = y 001y = 0 011 = 0 
 101x = z; 101 = 0 110z = 0 111x = y 
________________110x_=_y;_110_=_0_101y_=_0__111x_=_z_
 Orbits for E 100y = 0; 100y = z 100z = 0 101y = 0 
 010x = 0; 010x = z 010z = 0 101x = z 
 001x = 0; 001y = 0 001x = y 011x = 0 
________________111x_=_z;_111y_=_z_111x_=_y__011y_=_z__
_Isotropy_groups___Z=2_x_Z=2______Z=2_x_Z=2___D8_____
 Orbits for D  100y = 0 011y = z 100y = z 
____________________100z_=_0______111y_=_z___________
 Orbits for E  110z = 0 101 = 0  110 = 0 
____________________110x_=_y______011_=_0____________
34 HansWerner Henn
3 The homology of the quotient spaces
Let p be any prime. In this section we will compute the mod  p cohomology
resp. homology of the quotients of X1 , X1;s(i) and the pair (X1 ; X1;s(i)) by
the groups i, and also the cohomology of the quotients of X , Xs and (X ; Xs)
by := SL(3; Z[1=2]); in particular we prove Theorem 1.6, Corollary 1.7 and
Theorem 1.8.
In Sections 2.4 and 2.6 we described cell structures on the spaces i\Z '
i\X1 . Let Zs(i) be the 2  singular locus of Z with respect to the action
of i, i = 0; 1; 2, so that i\Zs(i) ' i\X1;s(i). We will use the results
of Section 2 to give the boundary homomorphisms of the chain complexes
C*(i\(Z; Zs(i))) and C*(i\Zs(i)) (with integral coefficients) in an explicit
form. Then we compute the homology groups of interest from these com
plexes. As these complexes are quite big our computations will be simplified
by Euler characteristic considerations. We summarize the discussion of Sec
tion 2 relevant for the Euler characteristic O in the following table.
____________________________________________________________________
_______________0cells1cells2cells3cellsnumber_of_all_cellsO__
_0\Z____________5_____8______5_____1___________19_________1__
_0\Zs(0)________5_____8______4_____0___________17_________1__
_0\(Z;_Zs(0))___0_____0______1_____1____________2_________0___
_1\Z___________13_____31____26_____7___________77_________1__
_1\Zs(1)_______13_____26____12_____0___________51_________1_
_1\(Z;_Zs(1))___0_____5_____14_____7___________26_________2___
_2\Z___________24_____72____69_____21_________186_________0__
_2\Zs(2)_______23_____49____21_____0___________93_________5_
_2\(Z;_Zs(2))___1_____23____48_____21__________93_________5__
In order to determine the incidence matrices, i.e. the boundary homo
morphisms in the relevant cellular chain complexes, we will have to choose
orientations for our cells. We will choose the orientation of the edges and
triangles in D0 in accordance with the ordering of the vertices in their names
so that for example [ACD] = [ADC] and for the boundary of [ABD] we
obtain [AB] + [BD]  [AD]. (Here [ACD]; [ABD] etc. denote the basis
elements in the chain complex given by the cells ACD; ABD etc.; similar
notation will be used below.) The 2  dimensional cell OAEF is oriented
such that its boundary is [OA] + [AE] + [EF ]  [OF ]; likewise with OCBF .
The 3  dimensional cell in D can then be oriented such that its boundary is
given by [OAEF ]  [ABD]  [ADC]  [AEB]  [BEF ]  [OCBF ]  [OAC].
Then we get an orientation of the cells (e; s) in D0 x Si (by choosing the
The cohomology of SL(3; Z[1=2]) 35
orientation of e) and finally we get induced orientations for the cells in i\Z.
For example in C*(0\Z) we obtain [ABD] = [ACD] = [ADC].
3.1 Quotients of (X1 ; X1;s (i)) by i
We will compute the homology of the homotopyequivalent quotients of the
pairs (Z; Zs(i)) by i.
1. 0 : There is only one 2  and one 3  dimensional cell in 0\(Z; Zs(0)) and
it is clear that the boundary map @3 : C3 ! C2 in the cellular chain complex
C*(0\(Z; Zs(0)) is an isomorphism. This implies part a) of Theorem 1.8.
2. 1 : Using the description of 1\Z that we gave in Section 2.4 and 2.6 it
is straightforward to check that the boundary maps @2 and @3 in the cellular
complex C*(1\(Z; Zs(1)) are given by the matrices in tables 6 and 7 below.
In these matrices the columns and rows are labelled by cells in 1\(Z; Zs(1)),
i.e. by equivalence classes of "nonsingular" cells in D1 (cf. Section 2.4.2),
and we have chosen representatives from equivalence classes where necessary.
Furthermore all zero entries in these matrices have been omitted.
One sees at once that @3 has trivial kernel, i.e. H3(1\(Z; Zs(1)); Fp) = 0
and that @2 is onto, i.e. H1(1\(Z; Zs(1)); Fp) = 0. Then the Euler charac
teristic argument implies that H2(1\(Z; Zs(1)); Fp) ~=(Fp)2 and we obtain
part b) of Theorem 1.8. For later use we specify two 2  dimensional cycles
which form a basis of H2. We can take the cycles
[ABD100]  [OAC100] and [ABD011] + [OAEF 101] : (3.1)
3. 2 : Now consider the complex C*(2\(Z; Zs(2))). First of all it is clear
that @1 : C1 ! C0 is onto and hence we obtain H0(2\(Z; Zs(2)); Fp) = 0.
Furthermore, using our description in Section 2.6 again, it is straightforward
to check that @2 and @3 are given by the matrices in tables 8  11 below.
These matrices show that the kernel of @3 is of dimension 1 and is generated
by the cycle:
[100y = 0][100z = 0][010x = 0]+[010z = 0]+[001x = 0][001y = 0] :
(3.2)
In particular we get H3(2\(Z; Zs(2)); Fp) ~= Fp. (Here [100y = 0] etc.
denote the 3  dimensional cells in 2\Z corresponding to the elements
100y = 0 etc. in S2.) Furthermore, the image of @2 is of dimension 22,
i.e. H1(2\(Z; Zs(2)); Fp) = 0, and then the Euler characteristic argument
implies H3(2\(Z; Zs(2)); Fp) ~=(Fp)6 and hence part c) of Theorem 1.8.
Again for later use we specify six 2  dimensional cycles whose homology
classes form a basis of H2. We can take the cycles
c1 : = [OAC100y = z]  [ABD100y = z] (3.3)
36 HansWerner Henn
c2 : = [OAC100z = 0] + [ABD100y = 0]  [ABD100z = 0] 
[OAC100y = 0] (3.4)
c3 : = [ABD001y = 0] + [BEF 100z = 0] + [OAC001x = 0] +
+[ABD100y = 0]  [OAC100y = 0] + [OCBF 111x = y] (3.5)
c4 : = [ABD011x = 0] + [OAEF 101y = 0] (3.6)
c5 : = [ABD011y = z]  [OAC111x = z] + [OAEF 101 = 0] (3.7)
c6 : = [ABD011 = 0]  [OAC111x = z] + [OAEF 101x = z] : (3.8)
Table 6: The boundary homomorphism C2 ! C1 for 1\(Z; Zs(1))
_______________________________________________________________________*
*
 ____________ABD______________BEF___OAC___OAEF__OCBF__*
*
_________100010_001_110_101_011_111_100_100_101_100101_010_110_
 BD 010  1 1  1 1   1    
______________________________________________________________
 AB 100 1  1   1   
         
____101_________________1_______1___________1_________________
 AD 110  1 1     1  
         
____111_____________________1_1__________________1__________
Table 7: The boundary homomorphism C3 ! C2 for 1\(Z; Zs(1))
______________________________________________
____________100_010__001_110_101__011_111_
 ABD 100   1  
     
 010 1  
 _001__________1____________________
 1101 1 1  
 101  1 
 _011___________________________1___
________111__________________1____1__1__
 BEF 100 1 1 1  1 
_________________________________________
 OAC 100 1 1   
     
________101__________________1__1______
 OAEF 100 1 1   
     
________101__________________1____1______
 OCBF 010  1 1   
     
________110_____________1___1__________
The cohomology of SL(3; Z[1=2]) 37
*
* Table 8: The boundary homomorphism C2 ! C1 for 2\(Z;@
_____*
*___________________________________________________________________@
 *
* ___________________________________________ABD_____________@
 *
* ____100________010________001________110________101@
____*
*______y=_0z_=y0=_zx=_0z_=x0=_zx=_0y_=x0=_yz=x0=_y_=_0y=_0x_=_z@
 BD*
*010x = 0   1  1  1  @
 *
*010z = 0   1  1  1  1 @
___*
*010x_=_z___________________1__________1_______1___________1___@
 AB*
*001y = 0    1 1   @
 _*
*011y_=_z______________________________________________________@
 *
*100y = 01    1  @
 *
*100z = 0  1    1  @
 _*
*100y_=_z_______1__________________________________1___________@
 *
*101y = 0      1 @
 *
*101 = 0      @
___*
*101x_=_z__________________________________________________1___@
 AD*
*001x = 0   1 1   @
 _*
*101x_=_z_________________________________________________1___@
 *
*110x = y   1   1  @
 *
*110z = 0   1  1  @
 _*
*110_=_0___________________1______________________1__________@
 *
*111x = y      @
 *
*111x = z      @
___*
*111y_=_z______________________________________________________@
_OC*
*100y_=_0______________________________________________________@
_OF*
*100y_=_0______________________________________________________@
_EF*
*100y_=_0______________________________________________________@
_BF*
*100y_=_0______________________________________________________@
38 HansWerner Henn
*
* Table 9: The boundary homomorphism C2 ! C1 for 2\(Z;@
_____*
*___________________________________________________________________@
 *
* _______________OAC____________________________OAEF_______@
 *
* ____100________101____001_111____100________101_____@
____*
*______y=_0z_=y0=_zy=_0x_=_z=_0x=_0x_=yz=0z_=_0y_=yz=0x_=_z=_0x@
 BD*
*010x = 0       @
 *
*010z = 0       @
___*
*010x_=_z_____________________________________________________@
 AB*
*001y = 0    1    @
 _*
*011y_=_z__________________________1__________________________@
 *
*100y = 01      @
 *
*100z = 0  1      @
 _*
*100y_=_z_______1_____________________________________________@
 *
*101y = 0   1     @
 *
*101 = 0   1     @
___*
*101x_=_z___________________1_________________________________@
 AD*
*001x = 0       @
 _*
*101x_=_z_____________________________________________________@
 *
*110x = y     1   @
 *
*110z = 0     1   @
 _*
*110_=_0_______________________________________1______________@
 *
*111x = y      1  @
 *
*111x = z      1  @
___*
*111y_=_z__________________________________________________1__@
_OC*
*100y_=_011_________________1______________________________@
_OF*
*100y_=_0_____________________________1__1__________________@
_EF*
*100y_=_0______________________________1_______1______________@
_BF*
*100y_=_0_____________________________________________________@
The cohomology of SL(3; Z[1=2]) 39
*
* Table 10: The boundary homomorphism C3 ! C2 for 2\(Z@
*
* __________________________________________________________________@
*
*  ____100________010________001________110_____@
*
* ___________y=_0z_=y0=_zx=_0z_=x0=_zx=_0y_=x0=_yz=x0=_y_=_0y=@
*
*  ABD100y = 0     1  @
*
*  100z = 0     1  @
*
*  _100y_=_z__________________________________________1____@
*
*  010x = 0     1  @
*
*  010z = 0     1  @
*
*  _010x_=_z__________________________________________1____@
*
*  001x = 0    1   @
*
*  001y = 0    1   @
*
*  _001x_=_y______________________1___1___1_______________@
*
*  110z = 0  1  1  1  @
*
*  110x = y1  1   1  @
*
*  _110_=_0________1___________1______________________1___@
*
*  101y = 0      @
*
*  101x = z      @
*
*  _101_=_0________________________________________________@
*
*  011x = 0      @
*
*  011y = z      @
*
*  _011_=_0________________________________________________@
*
*  111x = y      1@
*
*  111x = z      @
*
* ____111y_=_z_______________________________________________@
*
*  BEF100y = 01  1  1   @
*
*  100z = 0  1  1  1   @
*
* ____100y_=_z______1_1__________1________________________@
40 HansWerner Henn
*
* Table 11: The boundary homomorphism C3 ! C2 for 2\(Z;@
*
* __________________________________________________________________@
*
*  ____100_________010________001________110____@
*
* ____________y_=z0=_0y_=_zx=z0=_0x_=xz=0y_=_0x_=zy=0x_=_y_=_0y@
*
*  OAC 100y = 01 1    @
*
*  100z = 0  1  1    @
*
*  _100y_=_z______1__________1__________________________@
*
*  101y = 0     @
*
*  101x = z      @
*
*  _101_=_0_______________________________________________@
*
*  001x = 0   1 1   @
*
* _____111x_=_z______________________________________________@
*
*  OAEF100y = 01  1    @
*
*  100z = 0  1  1    @
*
*  _100y_=_z_______1__________1___________________________@
*
*  101y = 0     1@
*
*  101x = z      @
*
*  _101_=_0_______________________________________________@
*
*  001x = 0    1 1   @
*
* _____111x_=_z______________________________________________@
*
*  OCBF010x = 0  1 1   @
*
*  010z = 0   1  1   @
*
*  _010x_=_z__________________1__________1______________@
*
*  110z = 0    1 @
*
*  110x = y     1  @
*
*  _110_=_0___________________________________________1__@
*
*  100y = 011     @
*
* _____111x_=_y______________________________________________@
The cohomology of SL(3; Z[1=2]) 41
3.2 Quotients of X1;s (i) by i
We will derive Theorem 1.6 from Theorem 1.8 and from the following result.
Theorem 3.1 Let p be any prime. Then the reduced cohomology of the
quotients of X1;s(i) by the action of the respective groups is given as follows.
a) He*(0\X1;s(0); Fp) = 0
b) He*(1 \X1;s(1); Fp) = (Fp)2
c) He*(2\X1;s(2); Fp) = (Fp)6
Proof. We will compute the mod  p (co)homology from the cell complexes
of the homotopy equivalent spaces i\Zs(i). First of all we note that in all
cases we have eH0(i\Zs(i); Fp) = 0 (e.g. because Z is connected and because
of Theorem 1.8).
a) Because of Euler characteristic considerations it suffices in the case of
0 to show that the boundary map @2 : C2 ! C1 in the cellular complex
C*(0\Zs(0)) is a monomorphism. This can be easily seen from figure 1.
b) In the case of 1 the Euler characteristic argument shows that is enough
to verify eH2(1\Zs(1); Fp) = 0. For this we need to show that the boundary
homomorphism @2 is injective. This boundary homomorphism can be easily
determined by the information provided in Section 2.6 and is described in
table 12 below. Injectivity is now easily checked.
c) Finally we consider the case of 2. Again by the Euler characteristic
argument it suffices to show eH2(2\Zs(2); Fp) = 0. The boundary map @2 is
now described in tables 13 and 14 and again injectivity is easily checked. 2
3.3 Quotients of X1 by i
In order to determine eH*(i\X1 ; Fp) ~=He*(i\Z1 ; Fp) it remains to compute
the relevant connecting homomorphismsm in the long exact sequence for the
homology of the pair i\(Z; Zs(i)).
In case i = 0 we obtain clearly eH*(0\Z; Fp) = 0 and in the other two cases
one checks easily that under @2 : C2(i\Z) ! C1(i\Z) the images of the
relative cycles in (3.1) resp. (3.3) ff. are linearly independent in the quotie*
*nt
of C1(i\Zs) by the image of @2 : C2(i\Zs(i))  ! C1(i\Zs(i)); in fact,
to see this it is enough to determine the "OA"  part of the total boundary
of these relative cycles and compare with tables 12 resp. 13 and 14. In
other words, the corresponding connecting homomorphism in the long exact
sequence is injective and then even an isomorphism because of dimension
reasons. Part a), b) and c) of Theorem 1.6 follow.
42 HansWerner Henn
Table 12: The boundary homomorphism C2 ! C1 for 1\Zs(1)
_________________________________________________________________
  OAC  OAEF  OCBF  BEF 
 ________________________________________________ 
 001 110 111 001 110 111 100 011 111 110 101 011 
____________________________________________________________
 BD 100     1 
      
 011    1 1  1 
      
 111    1 1  1 
___________________________________________________________
 AC 001 1    
      
 110  1    
      
 111  1    
___________________________________________________________
 AE 001   1   
      
 110   1   
      
 111   1   
___________________________________________________________
 OC 1001   1  
      
 110  1   1  
      
 111  1   1  
___________________________________________________________
 OF 100  1 1  
      
 110   1  1  
      
 111   1  1  
___________________________________________________________
 EF 110   1   1 
      
 101     1 1 
      
 100   1 1   
___________________________________________________________
 BF 011    1  1 
      
 110    1 1 
      
 100    1 1  
___________________________________________________________
 OA 001 1  1   
      
 110  1  1   
      
 111  1  1   
      
 100     
      
 101     
___________________________________________________________
The cohomology of SL(3; Z[1=2]) 43
*
* Table 13: The boundary homomorphism C2 ! C1 for 2\@
*
* __________________________________________________________________@
*
*  _________OAC_______________OAEF________________OC@
*
*  _001111_____110____001_111_____110____100111__@
*
* __________x=_yx_=_yx=zy=_0_=_0x=_yx_=xy=yz_=_0_=_0y=_zy_=yz=@
*
*  BD100y = 0       @
*
*  _100y_=_z_______________________________________________@
*
*  011x = 0       @
*
*  _011y_=_z__________________________________________1__1@
*
*  111x = y       @
*
* ___111y_=_z_________________________________________1__1_@
*
*  AC001x = y1      @
*
*  _111x_=_y___1___________________________________________@
*
*  110x = y   1     @
*
*  110z = 0   1     @
*
* ___110_=_0________________1_______________________________@
*
*  AE001x = y    1    @
*
*  _111x_=_y______________________1________________________@
*
*  110x = y     1   @
*
*  110z = 0     1   @
*
* ___110_=_0___________________________________1____________@
*
*  OC100y = z1     1  @
*
*  _111x_=_y__1______________________________________1____@
*
*  110x = y   1     1 @
*
*  110z = 0   1     @
*
* ___110_=_0_______________1_______________________________@
*
*  OF100y = z    1  1  @
*
*  _111x_=_y______________________1_________________1____@
*
*  110x = y    1   1@
*
*  110z = 0     1   @
*
* ___110_=_0___________________________________1___________@
44 HansWerner Henn
*
* Table 14: The boundary homomorphism C2 ! C1 for 2\@
*
* __________________________________________________________________@
*
*  _________OAC_______________OAEF________________OC@
*
*  _001111_____110____001_111_____110____100111__@
*
* __________x=_yx_=_yx=zy=_0_=_0x=_yx_=xy=yz_=_0_=_0y=_zy_=yz=@
*
*  EF100z_=_0__________________1___1________________________@
*
*  110x = y     1 1   @
*
*  _110_=_0___________________________________1____________@
*
*  101x = z       @
*
* ___101_=_0________________________________________________@
*
*  BF100y_=_z______________________________________1___1____@
*
*  110x = y       @
*
*  _110_=_0________________________________________________@
*
*  011y = z       1 @
*
* ___011_=_0________________________________________________@
*
*  OA100y = 0       @
*
*  100z = 0       @
*
*  _100y_=_z_______________________________________________@
*
*  001x = 0       @
*
*  _001x_=_y1_________________1____________________________@
*
*  101y = 0       @
*
*  101x = z       @
*
*  _101_=_0________________________________________________@
*
*  110x = y   1   1   @
*
*  110z = 0   1   1   @
*
*  _110_=_0________________1__________________1____________@
*
*  111x = y  1   1    @
*
* ___111x_=_z_______________________________________________@
The cohomology of SL(3; Z[1=2]) 45
3.4 Quotients by SL(3; Z[1=2])
As before we abbreviate SL(3; Z[1=2]) by . In this section we are concerned
with the proof of part d) of Theorem 1.6 and Theorem 1.8, i.e. with the
computation of the mod  p cohomology of \X and the mod  2 cohomology
of the pair (\X ; \Xs). For this we consider the  equivariant projection
map p : X  ! X2 and the spectral sequences (which arise from the skeletal
filtrations of the bases) of the following associated maps
__p ~ 2
X : \X  ! \X2 = ;
__p ~ 2
(X;Xs): \(X ; Xs) ! \X2 = ;
and also that of
__p ~ 2
Xs : \Xs ! \X2 = :
The fibres of these maps over the i  simplices in 2 are homeomorphic
to the spaces i\X1 resp. to i\(X1 ; X1;s(i)) resp. to i\X1;s(i) (cf.
Section 2.1). Therefore Theorem 3.1 and the already proven parts a), b)
and c) of Theorem 1.6 and Theorem 1.8 immediately give the following E1
 terms for the cohomology spectral sequences converging to H*(\X ; Fp),
H*(\(X ; Xs); Fp) resp. H*(\Xs; Fp).
t t t
6 6 6
__________ __________ __________
           
3 _______1 3 _______1 3_______
           
2     2  6 6  2   
_______ _______ _______
1     1     1 6 6 
_______ _______ _______
0 3 3 1  0     03 3 1 
___________ __________ __________
0 1 2 s 0 1 2 s 0 1 2 s
Es;t1(\X ) Es;t1(\(X ; Xs)) Es;t1(\Xs)
The numbers in these diagramms give the dimension of Es;t1as an Fp  vector
space. Missing numbers are to be interpreted as 0. The differential d1 on the
line t = 0 (in the first and the third case) is as in the case of the simplicial
chains on 2, in particular we get in these cases E20;0~=Fp and E2s;0= 0
if s > 0 . In particular, we immediately obtain He*(\X ; Fp) ~= 5Fp, i.e.
46 HansWerner Henn
part d) of Theorem 1.6. We also see that Hi(\(X ; Xs); Fp) = 0 for i 2,
independent of the precise behaviour of the spectral sequences.
What remains to be calculated is the differential d1;21: E1;21! E2;21in the
second case and the differential d1;11: E1;11! E2;11in the third case. The con
necting homomorphisms of the long exact sequences associated to the pairs
i\(X1 ; X1;s(i)) induce (by Section 3.3) isomorphisms between Ei;11(\Xs)
and Ei;21(\(X ; Xs)) for i = 1; 2, hence it suffices to do the calculation in o*
*ne
case. We will show that in the third case d11;1is an isomorphism if p = 2 and
this will finish the proof of Theorem 1.8.
For this consider the mod  2 cohomology spectral sequences (arising from
a skeletal filtration of the base) of the maps
E x Xs ! \Xs
and
E x X  ! \X :
(For a discussion of the existence of a cellular structure on these bases we
refer to the remark at the end of this section.) The E1  terms of both spectral
sequences agree except on the line t = 0 because the mod  2 cohomology
of a fibre outside of \Xs vanishes in positive dimensions. Consequently the
E2  terms of both spectral sequences also agree except on the line t = 0 and
there we get Es;02~=Hs(\Xs; F2) resp. Es;02~=Hs(\X ; F2).
Claim 1: E0;12= 0 in both spectral sequences.
Proof. We consider the spectral sequence converging to H*(E x X ; F2).
As we have seen above the groups Ei;02~=Hi(\X ; F2) are trivial for i = 1; 2.
Therefore we have E0;12= 0 if and only if H1(E x X ; F2) = 0. From
Theorem 1.5 we know that H1(E x Xs; F2) = 0 and from the discussion
above we know that H1(E x (X ; Xs); F2) ~=H1(\(X ; Xs); F2) = 0. Then
the long exact sequence of the pair Ex (X ; Xs) shows H1(Ex X ; F2) = 0
and we are done. 2
Now consider the class v2 2 H2(E x Xs; F2) which is pulled back from
the second universal Stiefel Whitney class in H*(BSL(3; R); F2) under the
induced map of the composition E x Xsss!B i!BSL(3; R) (where ss
is given by sending Xs to a point and i by the canonical inclusion ,!
SL(3; R)).
Claim 2: In the spectral sequence converging to H*(E x Xs; F2) the class
v2 is detected on E0;21.
The cohomology of SL(3; Z[1=2]) 47
L
Proof. We have E0;21~= eH2(e; F2) where e runs through a set of 
orbits of 0  dimensional cells in Xs and e denotes the isotropy group of
the cell e. We may write e = (e2; e1 ); if the X2  component e2 is given
by the vertex l0 defined by the standard Z2  lattice in Q2 then we have
e = e1 , the isotropy group of the cell e1 X1;s with respect to the
action of SL(3; Z). For any such cell the class v2 restricts to the Stiefel
Whitney class of the representation of e arising from the embedding e ,!
SL(3; Z[1=2]) ,! SL(3; R). Now e contains at least one element of order 2
and all such elements are conjugate in SL(3; R). It follows that v2 restricts
nontrivially to any subgroup of order 2 in e and hence the claim is proved.
2
Now assume that the differential d1;11(\Xs) is not an isomorphism. Then
H2(\Xs; F2) 6= 0 and in the spectral sequence converging to H*(E x
Xs; F2) we have E2;02~=H2(\Xs; F2) 6= 0. Because of Claim 1 we conclude
that all of E2;02survives to E1 , and because of Claim 2 we see that the
assumption implies that H2(E x Xs; F2) is a vector space of dimension
bigger than 1 in contradiction to Theorem 1.5. This finishes the proof of
part d) of Theorem 1.8. 2
Remark. a) Our approach to the computation of H*(\(X ; Xs); F2) is rather
indirect and one might wonder why we did not analyze the differential
M3
E1;21~= H2(1\(X1 ; X1;s(1)); F2) ! H2(2\(X1 ; X1;s(2)); F2) ~=E2;21
i=1
directly? The reason is that the three summands in the source (correspond
ing to the three  orbits of 1  dimensional cells in X2 resp. the three edges
in 2) are mapped differently under this differential; only on one summand
is the map induced by the inclusion 2 1, on the other two summands it
is induced by the inclusion of 2 into the isotropy groups of the edges {l0; l2}
resp. {l1; l2} where as in Section 2.1 l0; l1; l2 are the classes of the Z2  l*
*attices
L0 = , L1 = <1_2e1; e2; e3> and L2 = <1_2e1; 1_2e2; e3> respectivel*
*y. The
component of the differential corresponding to the inclusion 2 1 (corre
sponding to the edge {l0; l1}) is straightforward to determine: with respect to
our cell structures on the spaces i\Z it is induced by a cellular map which is
determined by the forgetful map S2 ! S1. The component corresponding
to the edge {l0; l2} can also be worked out on the level of the spaces i\Z. In
contrast the isotropy group H{l1;l2}of the edge {l1; l2} is not contained in 0,
hence the deformation retraction X1 ! Z is not H{l1;l2} equivariant and
this makes this component of the differential much harder to evaluate. In
terms of integral lattices in R3 and the spaces Wi=SO(3) this last component
48 HansWerner Henn
is induced by the map which associates to the triple (L0; L1; L2) of lattices
(with L0 being wellrounded and m(L0) = 1) the pair (L1; L2). Because L1
need not be wellrounded one has to work out the effect of the deformation
retraction L1=SO(3) ! W1=SO(3) of Section 2.2 explicitly. Although one
would not expect that this could cause unsurmountable problems it would
be at the very least very laborious and the author found his initial attempts
to carry this out very frustrating.
b) We have tacitly used above that X is a  equivariant CW  complex
and we will use it again, namely in the final step of the proof of 1.4 which
combines Proposition 1.9 and Theorem 1.5. It is quite likely that there is
such a stucture but we do not know of a suitable reference. However, it is easy
to show that X has the equivariant homotopy type of a  CW complex,
and this will be enough: for example, we can do induction on the skeleta
X2kof the evident  equivariant cell structure of the simplicial complex X2
using that the preimages of the space X2kand X2k X2k1with respect to the
projection map X  ! X2 are understood by Section 2.4.
The cohomology of SL(3; Z[1=2]) 49
4 The cohomology of SL(3; Z[1=2])
4.1 Mod  2 cohomology
In this section we will complete the proof of Theorem 1.4. Because of Theo
rem 1.5 and Theorem 1.8 it is enough to prove Proposition 1.9, i.e. that the
connecting homomorphism
H4(Xs; F2) ! H5(X ; Xs; F2) ~=F2
in the long exact sequence of the pair E x (X ; Xs) is an epimorphism, or
equivalently that the natural map
H5(X ; Xs; F2) ! H5(X ; F2)
is trivial.
For this consider the following commutative diagramm in which the hori
zontal maps are induced by inclusions and the vertical maps by projections:
H5(X ; Xs; F2) ! H5(X ; F2)
x x
~=?? q*??
~=
H5(\(X ; Xs); F2) ! H5(\X ; F2)
The indicated arrows are isomorphisms because of Section 3.4 resp. because
the isotropy groups in X  Xs are of odd order. Therefore we have to show
that the map q* : H5(\X ; F2) ! H5(X ; F2) is trivial.
This will be a consequence of the following two results.
Lemma 4.1 If the map q2* : H3(2\X1 ; F2) ! H32(X1 ; F2) (induced by
projection) is trivial then so is q* : H5(\X ; F2) ! H5(X ; F2).
Lemma 4.2 The map q2* : H3(2\X1 ; F2) ! H32(X1 ; F2) is trivial.
Proof of Lemma 4.1 This follows immediately from naturality applied to
the following situation. If X 1denotes the preimage of the 1skeleton @2 of
2 under the projection map X  ! X2 ! \X2 ~= 2 then consider the
following commutative diagram in which the vertical maps are induced by
projections and the horizontal maps by inclusions:
H5(X ; X 1; F2)  ! H5(X ; F2)
x x
? ?
? ?
H5(\(X ; X 1); F2)  ! H5(\X ; F2) :
50 HansWerner Henn
It is clear from Section 3.4 that the horizontal arrow on the bottom line of
the diagram is an isomorphism. By excision we see
H5(\(X ; X 1); F2) ~=H5((2; @2) x (2\X1 ); F2) ~=2H3(2\X1 ; F2)
and
H5(X ; X 1; F2) ~=H52((2; @2) x X1 ; F2) ~=2H32(X1 ; F2)
and the claim follows. 2
Proof of Lemma 4.2 This is more involved. We prefer to work in ho
mology and there we have to show that the map q2* : H23(X1 ; F2)  !
H3(2\X1 ; F2) is trivial, or equivalently that the nontrivial element (cf.
Theorem 1.6) of H3(2\X1 ; F2) is not a permanent cycle in the homology
spectral sequence of the projection map. For this consider the short exact
sequence of F22  modules
0 ! C3 @3!C2 @2!I ! 0 (4.1)
in which Ci denotes the ith cellular chain group (with coefficients in F2)
of the contractible 2  space Z and I is the image ofLthe boundary map
@2 : C2 ! C1. Note that Ci can be identified with oF2[2=o] where o
runs through a set of representatives of 2  orbits of i  cells in Z and o is
the isotropy subgroup of the chosen representative o .
The E1  term of the spectral sequence of the projection map is given as
E1s;t~=Ht(2; Cs) and the differential d23;0: E23;0! E21;1can be described as
follows: The group E23;0is given as
E23;0~=H3(2\Z) ~=Ker (H0(2; C3) ! H0(2; C2))
while E21;1is given as quotient
Ker (H1(2; C1) ! H1(2; C0))
E21;1~=________________________________:
Im (H1(2; C2) ! H1(2; C1))
All maps are, of course, induced by the differentials in the chain complex
C*. If z is an element in E23;0 H0(2; C3) then z = @y for some y 2
H1(2; I) (with @ denoting the connecting homomorphism associated to the
exact sequence (4.1)) and d23;0z is represented by i*y 2 H1(2; C1) (with i
denoting the inclusion of I into C1. In particular we see that the following
two conditions are equivalent:
The cohomology of SL(3; Z[1=2]) 51
1. d23;0: E23;0~=F2 ! E21;1is nontrivial.
d12;1
2. Im (H1(2; I) i*!H1(2; C1)) is strictly larger than Im (H1(2; C2) !
H1(2; C1)).
We will verify the second condition and this will complete the proof of
Lemma 4.2. In fact, it suffices to verify theLsecond condition after projecting
off to a suitable summand in H1(2; C1) ~= o H1(2; F2[2=o]) where as
above o runs through a set of representatives of 2  orbits of 1  cells in
Z. We choose the 1  dimensional cells o1 resp. o2 given by 1 . (AC; 001x =
y) 2 x D2= ~2 and 1 . (OA; 001x = y) 2 x D2= ~2 (where 1 2
and our conventions for labelling the cells in Z ~= 2 x D2= ~2 are those of
Section 2.4.2). We will denote the corresponding projections by ss1 and ss2
respectively. Lemma 4.2 will then follow from the following two results. 2
Lemma 4.3 If y 2 H1(2; C2) is mapped nontrivially by the map H1(2; C2)
d12;1 ss1
! H1(2; C1) ! H1(2; F2[2=o1]) then y is also mapped nontrivially by
d12;1 ss2
H1(2; C2) ! H1(2; C1) ! H1(2; F2[2=o2]).
Lemma 4.4 There is an element u 2 H1(2; I) which maps nontrivially
by the map H1(2; I) i*!H1(2; C1) ss1!H1(2; F2[2=o1]) and trivially by
H1(2; I) i*!H1(2; C1) ss2!H1(2; F2[2=o2]).
Proof of Lemma 4.3. The differential d1s;*can be described as followsL
(cf. chapter VII.8Lof [B1 ]): via the identifications of Cs with oeF2[2=oe]
and of Cs1 with oF2[2=o] the oeo component of d1s;*is induced by the
@oe;o
corresponding component F2[2=oe]  ! F2[2=o] of the boundary map
Cs ! Cs1. By equivariance this component is determined by the image
of the coset 1 . oe2 F2[2=oe], i.e. by understanding the incidence numbers
[oe : go ] betweenPthe cell oe and all cells in the 2  orbit of o . We obtain
@oe;o(1 . oe) = go [oe : go ]go where the sum is over the 2  orbit of o .
Because we project off via ss1 and ss2 and we are interested in homology in
degree 1 only it suffices to consider "singular" 2  dimensional cells oe Zs(2)
for which [oe : goi] is nontrivial for some cell in the orbit of oi; in partic*
*ular
all cells of the form g . (ACD; s), g . (ABD; s) and g . (ABE; s) in 2\Z ~=
2 x D2= ~2 are "nonsingular" and can be ignored. By going through the
discussion of the relevant 2  cells in Section 2.6 and using Theorem 2.7 we see
that we only get contributions to @oe;oi(1.oe) for oe = oe1 := 1.(OAC001; x = y)
in the case of o1, and oe = oe1 or oe = oe2 := 1 . (OAEF; 001x = y) in the case
52 HansWerner Henn
of o2. Furthermore oe1 and oe2 are the only cells in their 2  orbit for which
the incidence numbers are nontrivial, namely equal to 1.
Therefore it suffices to consider the following situation in which we identify
H1(2; F2[2=oe]) with H1(oe; F2) for oe 2 {oe1; oe2; o1; o2} and drop the coef
ficients from the notation; the maps i1 resp. i2 denote the inclusions of oe1
resp. oe2into o2 (cf. table 1 and table 2 for the isotropy groups and their
inclusions).
H1(oe1) H1(oe2) ~=H1(Z=2) H1(Z=2) (a; b)
? ?
? ?
y y
H1(o1) H1(o2) ~=H1(Z=2) H1(Z=2 Z=2) (a; i1*a + i2*b) :
The proof of the Lemma is now reduced to showing that a 6= 0 implies
i1*a + i2*b 6= 0, and this is clear. 2
Proof of Lemma 4.4. Of course, the connecting homomorphism associated
to the exact sequence (4.1) has to send the element u in question to the
element in H3(2\Z; F2) H0(2; C3) given by the cycle of (3.2):
[100y = 0] + [100z = 0] + [010x = 0] + [010z = 0] + [001x = 0] + [001y = 0] :
Consider now the following chain in C3 whose class in H0(2; C3) agrees with
this cycle:
z : = [1 . (100y = 0)] + [1 . (100z = 0)] + [1 . (010x = 0)] +
+[1 . (010z = 0)] + [1 . (001x = 0)] + [1 . (001y = 0)] :
Let oe denote the 2  dimensional cell 1.(ABD; 001x = 0) in 2xD2= ~2~= Z.
This cell generates a free F2[2]  submodule that we denote by F2[2].
Let ss3 : C2  ! F2[2] denote the projection map. Then Section 2.4.2
and inspection of table 5 in Section 2.6 yield ss3@3z = oe where = (1 +
gsgAB gs1) 2 F2[2], s 2 S2 ~= 2\0 is the element 001x = 0, gs 2 0 is a
fixed chosen coset representative of s and gAB 2 0 is as in Section 2.4. Note
that because of sgAB = s we have gsgAB gs1 2 2, and in fact, by Section
2.4.2, the element gsgAB gs1 is the unique nontrivial element in the isotropy
group of the cell 1 . (AB; 001x = 0). In Z the cell g . (AB; 001x = 0) gets
identified with the cell 1 . (AC; 001x = y) = o1 if g is determined by the
equation ggs = gs0gAD , s02 S2 is the element 001x = y, gs0is a fixed chosen
coset representative of s0and gAD is again as in Section 2.4.2. It follows that
the assignment oe 7! g1 o1 induces an isomorphism F2[2]=F2[2] ~=
F2[2=g1o1] of F2[2]  modules.
The cohomology of SL(3; Z[1=2]) 53
Now let F2[2] be the F2[2]  submodule of C3 generated by the cycle z;
obviously this is a free F2[2]  module. Let C@z2be the F2[2]  submodule of
C2 which is generated by all cells appearing in @3z if this is written as linear
combination of cells; observe that C@z2is a direct summand of C2 as a F2[2]
 module. Next let I@z be the quotient of C@z2by the F2[2]  submodule
generated by @3z. Then we get the following diagramm of exact sequences
of F2[2]  modules:
0  ! C3 @3! C2 ! I ! 0
x x x
? ? ?
? i? j?
0  ! F2[2]  ! C@z2 ! I@z ! 0
? ? ?
? ? ?
y ss3y fss3y
0  ! F2[2]  ! F2[2] ! F2[2=g1o1] ! 0
where the left hand vertical arrow in the upper half of the diagram is an
inclusion, i is the inclusion of a direct summand, the upper left hand rectangle
commutes and induces j. The lower left hand vertical arrow is given by z 7!
oe, hence the lower left hand rectangle commutes by the formula for ss3@3(z)
and induces the map ess3. By going through table 5 in Section 2.6 one checks
j i ess1
that ess3agrees with the composition I@z  ! I  ! C1  ! F2[2=g1o1]
where ess1denotes the composition of ss1 with left multiplication by g1 .
Now the upper half of the diagramm shows that there is an element u0 2
H1(2; I@z) such that @j*u0 = z. (Use that z 2 H0(2; C3) comes from an
element, still denoted by z, in H0(2; F2[2]) whose image in H0(2; C@z2)
vanishes because i is split injective.) Pick any such u0 and let u := j*u0.
Then the lower half of the diagramm shows that the connecting homomor
phism maps ess3*(u0) to the nontrivial element in H0(2; F2[2]) ~=F2, in
particular ess3*(u0) 6= 0 and hence ss1*i*u 6= 0.
Finally, using Section 2.6 once more, it is straightforward to check that
ss2@2 : C2 ! F2[2=o2] vanishes on C@z2and hence ss2ij : I@z ! F2[2=o2]
is the zero map and the second statement of the Lemma follows. 2
4.2 Mod  3 cohomology
In this section we prove Theorem 1.10. We take advantage of our investi
gations in Sections 2.4, 2.5 and 2.6. In particular we will make use of the
description of the j  space Z given by Theorem 2.6 resp. Theorem 2.7, i.e.
54 HansWerner Henn
we will identify Z with jx Dj= ~j and write A, B, : :f:or the points of this
space given by the class of (1; A), (1; B), : :.:
We break the proof into two parts.
Proof of Theorem 1.10 (a)(c). Let Zs;3(j) denote the 3  singular locus of
Z with respect to the action of j.
Part c) of the Theorem follows immediately from Section 2.6 because in
this case Zs;3(2) = ;, and therefore we get He*(2; F3) ~= eH*2(Z; F3) ~=
eH*(2\Z; F3) ~=3F3 by Theorem 1.6.
In the cases of 0 and 1 we derive from Section 2.6 that the space j\Zs;3(j)
consists of two components. In the case of 0 one of the components consists
of the image of the 0  orbit of the 0  cell A in Zs;3(0) (with isotropy group
isomorphic to D12). The other one consists of the image of the 0  orbit of
the subcomplex with the two 1  simplices OC and OF in Zs;3(0); the 0 
orbits of the 1  dimensional cells have isotropy isomorphic to S3 and the
0  orbits of the three 0  dimensional cells have isotropy group isomorphic
to S4. In the case of 1 one component in 1\Zs;3(1) comes from the 0 
cell A001 (where we use the convention of Section 2.6 for labelling the cells),
again with isotropy group isomorphic to D12; the other one comes from the
subcomplex with 1  cells OC111 and OF 111, with isotropy groups isomor
phic to S3 for the 1  dimensional cells and the 0  dimensional cell F 111,
and isomorphic to S4 for the 0  dimensional cells O111 and C111.
Now we consider the spectral sequences associated to the maps
Ej xj Zs(j) ! j\Zs;3(j); j = 0; 1 :
Because the inclusions of S3 into S4 and of S3 into D12 induce isomorphisms
in mod  3 cohomology we find in both cases an isomorphism
Y2
H*j(Zs;3(j); F3) ~= H*(S3; F3) :
i=1
Furthermore it is clear that H*(j\Zs(j); F3) = F3F3; from Theorem 1.6 we
know eH*(j\Z; F3) = 0 and therefore we conclude H*(j\(Z; Zs;3(j)); F3) =
F3. Finally the long exact sequence for the Borel cohomology of the pair
(Z; Zs;3(j)) gives part a) and b). 2
To prove part (d) of Theorem 1.10 one could now consider the spectral
sequence of the map E x X  ! 2 and we will actually make some use of
this spectral sequence. However, both for the final description of the result
The cohomology of SL(3; Z[1=2]) 55
as well as for the proofs centralizers of elementary abelian 3  subgroups turn
out to be helpful again. In fact, we will combine information derived from the
knowledge of these centralizers with information coming from the analysis of
the spectral sequence of the map Ex X  ! 2 . We start by analyzing the
elementary abelian 3  subgroups of GL(3; Z[1=2]). First it is clear that there
are no elementary abelian 3  subgroups of rank 2 (isomorphic to Z=3 x Z=3)
because there are none in GL(3; R).
Proposition 4.5 In GL(3; Z[1=2]) there are precisely two conjugacy classes
of subgroups isomorphic to Z=3.
Proof. These conjugacy classes are in one to one correspondence with the
isomorphism classes of modules M over the group algebra Z[1=2][Z=3] which
are free of rank 3 as Z[1=2]  modules and on which Z=3 acts faithfully. Such
modules are classified by the (obvious modification for the ring Z[1=2] of the)
DiederichsenReiner Theorem (cf. Theorem (74.3) of [CR ]); one of the two
classes corresponds to the free Z[1=2][Z=3]  module F on one generator, the
other one to T R, the direct sum of the trivial module T and the module
R := Z[1=2][i3] where a fixed chosen generator of Z=3 acts by multiplication
with i3, a fixed chosen third root of unity. 2
We pick a subgroup E1 corresponding to the module F and a subgroup E2
corresponding to T R. If E is a subgroup of a group G we write CG (E) for
the centralizer of E in G and NG (E) for the normalizer of E in G. The units
in a ring R will be denoted by Rx . We will now analyze the centralizers and
normalizers of Ei. We start with the case of E2.
Proposition 4.6 The centralizer CGL(3;Z[1=2])(E2) is isomorphic to Z=3 x
Z[1=2]x x Z[1=2]x .
Proof. The centralizer is isomorphic to the group of automorphisms of
the corresponding Z[1=2][Z=3]  module, i.e. to the group of units in its
endomorphism ring. After tensoring with Q both F and R T become
isomorphic; both R Q and T Q are irreducible, in particular there are no
Z[1=2][Z=3]  module maps between R and T .
Therefore we obtain CGL(3;Z[1=2])(E2) ~= Z[1=2]x x (Z[1=2][i3])x and it is
easy to check (say by using the norm map from the cyclotomic extension
Q[i3] down to Q) that the map
Z=3 x Z[1=2]x ! (Z[1=2][i3])x
(a; b) 7! ia3b
is an isomorphism. 2
56 HansWerner Henn
Remark. The norm can also be used to show that Z[i3] is a Euclidean
ring and therefore a principal ideal domain. This simplifies the proof and
statement of the DiederichsenReiner Theorem for modules over Z[Z=3] and
Z[1=2][Z=3].
Corollary 4.7 E2 is contained in SL(3; Z[1=2]) and there is a unique con
jugacy class of elementary abelian 3  subgroups in SL(3; Z[1=2]) which maps
to the GL(3; Z[1=2])  conjugacy class of E2. Furthermore
CSL(3;Z[1=2])(E2) ~=Z=3 x Z[1=2]x ;
NSL(3;Z[1=2])(E2) ~=CSL(3;Z[1=2])(E2) o Z=2
and the isomorphism can be chosen such that the conjugation action of Z=2
on Z=3 is nontrivial while it is trivial on Z[1=2]x .
Proof. It is clear that E2 is contained in SL(3; Z[1=2]) and also that
CSL(3;Z[1=2])(E2) is isomorphic to Z=3 x Z[1=2]x . Furthermore, if oe denotes
the Galois automorphism of Z[1=2][i3] then oe (id) normalizes E2 and this
shows NSL(3;Z[1=2])(E2) ~= CSL(3;Z[1=2](E2) o Z=2 with the conjugation action
as claimed.
Now assume E0 is another subgroup of SL(3; Z[1=2]) which becomes conju
gate in GL(3; Z[1=2]) to E2, say be an element g. Now the determinant from
GL(3; Z[1=2]) to Z[1=2]x remains onto when restricted to CGL(3;Z[1=2])(E2),
hence we can write g = g1g2 with g1 2 CGL(3;Z[1=2])(E2) and g2 2 SL(3; Z[1=2])
and this implies that E and E0 are already conjugate in SL(3; Z[1=2]). 2
Proposition 4.8 The centralizer CGL(3;Z[1=2])(E1) is isomorphic to Z=3 x
Z[1=2]x x Z.
Proof. The module F contains the direct sum of the submodules Ker (g  1)
(generated as abelian group by 1 + g + g2) and Ker (1 + g + g2) (generated
by 1  g and 1  g2) with quotient isomorphic to Z=3 (observe that 3 =
(1 + g + g2) + (1  g) + (1  g2)). These submodules are isomorphic to T
resp. R and are preserved by any automorphism of F . In other words we get
a homomorphism
Aut(F ) ! Aut (R T ) ~=Z=3 x Z[1=2]x x Z[1=2]x :
This is obviously injective and we claim that its image is isomorphic to Z=3x
Z[1=2]x x Z. To see this note that the subgroup Z=3 is clearly in the image;
scalar automorphisms show that the diagonal of Z[1=2]x x Z[1=2]x is also in
The cohomology of SL(3; Z[1=2]) 57
the image. Therefore it suffices to determine which of the automorphisms
ffffl;n: R T ! R T , (r; t) 7! (r; ffl2nt) (with ffl 2 {0; 1} and n 2 Z) exte*
*nds
to one of F . Because of 3 = (1 + g + g2) + (1  g) + (1  g2) an extension
exists iff ffl2n(1 + g + g2) + (1  g) + (1  g2) = (g + g2)(ffl2n  1) + (ffl2*
*n + 2)
is divisible by 3. This happens iff ffl2n  1 is divisible by 3. In other words*
*, n
may be chosen arbitrarily but ffl is then determined by n. 2
Corollary 4.9 E1 is contained in SL(3; Z[1=2]) and there is a unique con
jugacy class of elementary abelian 3  subgroups in SL(3; Z[1=2]) which maps
to the GL(3; Z[1=2])  conjugacy class of E1. Furthermore
CSL(3;Z[1=2])(E1) ~=Z=3 x Z ;
NSL(3;Z[1=2])(E1) ~=CSL(3;Z[1=2])(E1) o Z=2
and the isomorphism can be chosen such that the conjugation action of Z=2
on Z=3 is nontrivial while it is trivial on Z.
Proof. The proof is analogous to that of Proposition 4.7. One only has
to check that the restriction of the determinant to CGL(3;Z[1=2])(E1) remains
onto and that the automorphism oe (id) of R T (with oe the Galois
automorphism of Z[1=2][i3]) extends to an automorphism of F . 2
We can now use the "centralizer spectral sequence"
Es;t2~=limsEHt(C (E); F3) =) Hs+t(Xs;3; F3)
of [H1 ] to compute H*(Xs;3; F3) where as before = SL(3; Z[1=2]), X is the
space X1 x X2, but now Xs;3denotes the 3  singular locus of X with respect
to the action of ; the limit is here taken over the category of elementary
abelian 3  subgroups of . Because the 3  rank of is equal to 1, the
spectral sequence degenerates into an isomorphism
Y Y
H*(Xs;3; F3) ~= (H*(C (E); F3))N (E)=C (E)~= H*(N (E); F3)
(E) (E)
where the product is indexed by conjugacy classes of elementary abelian 3 
subgroups of (see 3.3.1 of [H1 ]). In our case there are two conjugacy classes
whose normalizers are isomorphic to S3 x Z x Z=2 resp. S3 x Z resp., so we
obtain the following result.
Proposition 4.10
Y2
H*(Xs;3; F3) ~= eH*(S3 x Z; F3)
i=1
58 HansWerner Henn
We now turn towards the proof of part (d) of Theorem 1.10. By Proposition
4.10 it suffices to prove the following result.
Proposition 4.11 a) H*(X ; Xs;3; F3) ~=F3 2(F3)2 5(F3).
b) The boundary map
H*(Xs;3; F3) ! H*+1(X ; Xs;3; F3)
is surjective. Its kernel in degree 4 is of dimension 3 and is generated by the
image of the Bockstein of H3 and one further element which has nontrivial
restriction to the two factors in Proposition 4.10.
The proof of Proposition 4.11 depends on another result whose proof we
give at the end of this section.
Proposition 4.12 The restriction map H*(1; F3) ! H*(2; F3) is onto,
and with respect to the isomorphism H3(1; F3) ~=H3(S3; F3) H3(S3; F3)
of part (b) of Theorem 1.10 the kernel in degree 3 restricts nontrivially to
both factors.
Proof of Proposition 4.11. We consider the E1  term of the spectral se
quence of the map
E x (X ; Xs;3) ! 2 ~=\X2 :
By the proof of part (a)  (c) of Theorem 1.10 we get E0;11~=E1;11~=(F3)3,
E2;01~=E2;31~=F3 and Es;t1= 0 in all other cases. In particular, we see that
H3(X ; Xs;3; F3) = H4(X ; Xs;3; F3) = 0.
Now one could try to directly compute the differentials in order to prove
(a). This can presumably be done directly, but we proceed in a different way
which at the same time turns out to be quite useful for the proof of part (b).
We use the spectral sequence of the map
E x X ! 2 ~=\X2 :
By Theorem 1.10 its E1  term is given by
Y3
Es;*1~= H*(s; F3) ifs = 0; 1 and E2;*1~=H*(2; F3) ~=F3 3F3 :
i=1
By Proposition 4.10 we already know H*(; F3) in large dimensions. This
together with the multiplicative structure of the spectral sequence forces the
The cohomology of SL(3; Z[1=2]) 59
behaviour of d1 : E0;*1! E1;*1and gives for all * > 0 with * 3; 4 mod 4
that the kernel and cokernel of the map
d1 : (F3)6 ~=E0;*1! E1;*1~=(F3)6
is of dimension 2. By Proposition 4.12 the restriction map H*(1; F3)  !
H*(2; F3) is onto and therefore d1 : E1;*1! E2;*1is onto as well. In par
ticular we find that the spectral sequence collapses at E2 and with some
extra effort one could probably also determine the multiplicative structure.
Here we need only the additive result in dimensions up to 5 where we find
H0(; F3) ~=F3, H1(; F3) = H2(; F3) = 0, H3(; F3) ~=H5(; F3) = (F3)2,
H4(; F3) ~=(F3)3.
Part (a) and the surjectivity in part (b) of the proposition follow now
immediately from the long exact sequence of the pair (X ; Xs;3) together with
the knowledge that H3(X ; Xs;3; F3) = H4(X ; Xs;3; F3) = 0. It is also clear
that the kernel in degree 4 is of dimension 3 and contains the image of the
Bockstein of H3. The following proposition finishes the proof. 2
Proposition 4.13 The restriction maps
H*(; F3) ! H*(N (Ei); F3) ~=H*(S3 x Z); F3)
are surjective for i = 1; 2 except in degree 1.
Proof of Proposition 4.13. We abbreviate N (Ei) by Ni. By Smith theory
the space X Eiis mod 3  acyclic, so we try to understand the Ni  space X Ei
and for this we consider the canonical Ni  equivariant map X Ei ! X2Ei
induced by the  equivariant projection X  ! X2. The quotient of X2 by
the action of is 2. It is an elementary exercise to verify that the quotient
of X2Eiby Ni is the 1  skeleton @2 of 2, and furthermore that the isotropy
group of the j  simplices in @2 are isomorphic to Ni\ j for j = 0; 1.
Now we compare the mod  p cohomology spectral sequences of the maps
E x X ! 2 ~=\X2
and
E xNi X Ei! @2 ~=Ni\X2Ei:
As observed before the first spectral spectral sequence has as E1  terms
Es;*1~=(H*(s; F3))3 ifs = 0; 1 and E2;*1~=H*(2; F3) ~=F3 3F3 ;
while the second has
fE1s;*~=(H*(s \ Ni; F3))3 ifs = 0; 1 and Ef12;*= 0 :
60 HansWerner Henn
The map on E1  terms is induced by the restriction maps H*(s; F3) !
H*(s \ Ni; F3) for s = 0; 1. The groups s \ Ni are easily identified with
S3 (for E1) resp. S3 x Z=2 (for E2). The map on Es;*1corresponds for
s = 0; 1 to the projection onto the ith factor, i = 1; 2 (with respect to the
product decomposition of the source, cf. Theorem 1.10(a+b)). Now we use
Proposition 4.12 to finish the proof. 2
Finally we turn towards the proof of Proposition 4.12.
Proof of Proposition 4.12. We dualize and work in homology. Furthermore
we use Theorem 1.10(a+b) to identify H*(0; F3) with H*(1; F3) via the
map induced by inclusion. Therefore we may consider the homomorphism
H*(2; F3) ! H*(0; F3) again induced by inclusion. By Shapiro's lemma
this homomorphism can be identified with the map
H*(0; F3[0=2]) ! H*(0; F3)
induced by the canonical map of 0  modules F3[0=2] ! F3. We denote
the kernel of this map by K. The following lemma is the main step in the
proof.
Lemma 4.14 H2(0; K) ~=F3.
We continue with the proof of Proposition 4.12. The lemma immediately
implies the first part of 4.12. For the second part we consider the two non
conjugate elementary abelian 3  subgroups E1 and E2 of 0; they are the
3  Sylow subgroups of the two S3's which detect H*(0; F3). The proof
of the proposition will be complete once we have seen that the inclusions
of both S3's into 0 induce isomorphisms H2(S3; K)  ! H2(0; K). (We
note that F3[0=2] is projective as F3[S3]  module, and hence H2(S3; K) ~=
H3(S3; F3) ~= F3.) In fact, this follows at once from the following observa
tion: via mod  2 reduction both S3's map monomorphically to SL(3; F2)
and there they agree with the normalizer of a 3  Sylow subgroup; therefore
the composition
H2(S3; K) ! H2(0; K) ! H2(SL(3; F2); K)
(the second arrow being induced by mod  2 reduction) is an isomorphism.
2
We turn towards the proof of Lemma 4.14. We could explicitly work out
a projective resolution of the trivial F3[0]  module F3 from the resolution
provided by the cellular chains of Z, and then compute H2(0; K) from this
projective resolution. As this would be quite involved we construct just as
much of this resolution as necessary.
The cohomology of SL(3; Z[1=2]) 61
Proof of Lemma 4.14. We consider the mod  3 cellular chain complex of Z
and break it apart into the following exact sequences of 0  modules where
i1ffi2 = @2 and i0ffi1 = @1:
0 ! C3 @3!C2 ffi2!I2 ! 0 (4.2)
0 ! I2 i1!C1 ffi1!I1 ! 0 (4.3)
0 ! I1 i0!C0 "! F3 ! 0 (4.4)
The lemma will follow from the long exact sequence in Tor0*(; K) associ
ated to the exact sequence (4.4) and the following claims. (Here and in the
sequel we abbreviate TorF3[0]*(; K) by Tor0*(; K).)
Claim 1: Tor 02(C0; K) ~=(F3)4.
Claim 2: Tor 01(I1; K) = 0 and Tor02(I1; K) ~=(F3)3.
Claim 3: The map Tor02(I1; K) ! Tor02(C0; K) induced by i0 is injec
tive.
L
Proof of Claim 1. The 0  modules Ciare direct sums oeF3[0=oe] where
oe runs through the set of 0  orbits of i  dimensional cells in Z and oeis
the isotropy group of a chosen representative of the orbit oe. By Sections 2.4
and 2.5 we have 5 orbits of 0  cells in Z corresponding to the vertices A,
O, C, F and D in 0\Z, with respective isotropy groups D12 (for A), S4
(for O, C and F ) and D8 (for D). By Shapiro's lemma we have therefore
isomorphisms
Tor02(C0; K) ~=H2(D12; K) (H2(S4; K))3 H2(D8; K) :
The contribution coming from D8 is trivial because the order of D8 is prime
to 3. Furthermore 2 has no 3  torsion, hence the 3  Sylow subgroup of all
the other finite subgroups acts freely on F3[0=2] and hence this module is
projective when restricted to the other finite subgroups. As a consequence we
obtain H2(D12; K) ~= H3(D12; F3) ~= F3 and H2(S4; K) ~= H3(S4; F3) ~= F3
and the claim follows.
2
Proof of Claim 2. Here we use the exact sequences (4.2) and (4.3). Just as
above we find M
Tor 0i(Cs; K) ~= Hi+1(oe; F3) (4.5)
oe
62 HansWerner Henn
if s = 0; 1; 2; 3 and all i > 0. If i = 0, we observe that for all oe we have
H1(oe; F3) = 0, and hence the functors Tor 00(Cs; ) carry the short exact
sequence
0 ! K ! F3[0=2] ! F3 ! 0
into short exact sequences. Hence we have a short exact sequence of com
plexes
0 ! Tor00(C*; K) ! Tor00(C*; F3[2=0]) ! Tor00(C*; F3) ! 0 (4.6)
for which the homology is known in the middle and on the right by Theorem
1.6.
From the exact sequence (4.2), formula (4.5) and our analysis of the cell
structures and their symmetries in Sections 2.4, 2.5 and 2.6 we deduce that
Tor0i(I2; K) = 0 if i > 1. For i = 1 it is isomorphic to the homology
in dimension 3 of the complex Tor 00(C*; K); this in turn is isomorphic to
the homology in dimension 3 of the complex Tor 00(C*; F3[2=0]), i.e. to
H3(2\X1 ; F3) ~=F3 by Theorem 1.6. For i = 0 we obtain
Tor00(I2; K) ~=Coker (Tor 00(C3; K) @3!Tor 00(C2; K)) :
Now we can compute Tor0i(I1; K) for i = 1; 2 from the long exact sequence
in Tor which is associated to the exact sequence (4.3). Using once more our
analysis in section 2.4, 2.5 and 2.6 we see that Tor 01(C1; K) = 0 and we
obtain a short exact sequence
0 ! (F3)2 ~=Tor 02(C1; K) ! Tor02(I1; K) ! Tor01(I2; K) ~=F3 ! 0
(4.7)
where the contribution to Tor02(C1; K) comes from the two 0  orbits of 1 
dimensional cells corresponding to OC and OF with symmetry group isomor
phic to S3 in both cases. For Tor01(I1; K) we use again that Tor01(C1; K) = 0
and that the map Tor 00(I2; K)  ! Tor 00(C1; K) is injective (because the
complex Tor00(C*; K) has no homology in degree 2 by Theorem 1.6). 2
Proof of Claim 3. We proceed in several steps. In a first step we reduce
the evaluation of
i0*: Tor02(I1; K) ! Tor02(C0; K) ~=H2(D12; K) (H2(S4; K))3
to the study of a particular chain map. In a second step we descroibe this
chain map and in a final step we finish the computation of i0*.
The cohomology of SL(3; Z[1=2]) 63
First_step._We consider the restriction of the map i0* to the subgroup
Tor02(C1; K) ~= (H2(S3; K))2 ~= (F3)2 (cf. (4.7)). This restriction is in
duced by injections of the isotropy groups (isomorphic to S3) of the edges
OC and OF into the isotropy groups (isomorphic to S4) of the vertices
O, C and F . Each of these injections induces an isomorphism of coho
mology in H2(; K) ~=H3(; F3), hence these injections map (H2(S3; K))2
monomorphically to the summand (H2(S4; K))3 of Tor02(C0; K). It suffices
therefore to show that the composition of i0 : I1 ! C0 with the projection
ss : C0  ! F3[0=D12] induces a nontrivial map in Tor 02(; K). To this
end we should construct F3[0]  projective resolutions P* of I1 and Q* of
F3[0=D12] and lift ssi0 to a chain map ff : P* ! Q*.
In the case of Q* we start with the minimal projective resolution Q0*for F3
as a F3[D12]  module and take Q* = F3[0] F3[D12]Q0*. In the case of I1 we
obtain a projective resolution P* by taking first projective resolutions R2 *of
I2 and R1 *of C1; then we lift i1 : I2 ! C1 to a chain map ei1: R2 *! R1 *
and obtain a double complex R** whose total complex gives the desired
projective resolution P*. More concretely, we can take the exact sequence
0 ! C3 ! C2 ! I2 ! 0 as a projective resolutionLR2 *of I2. For C1
we use the direct sum decomposition C1 ~= oeF3[0=oe] and, if Roe1*is a
minimalLprojective resolution of the trivial F3[oe]  module F3, then we take
R1* = oeF3[0] F3[oe]Roe1*. In these terms, the projective resolution P* of
I1 looks as follows:
@P3 @P2 @P1 @P0
. . .! R14 ! R13 ! R12 C3 ! R11 C2 ! R10
i @R1 j i @R1 fi j
with @P3= @R13, @P2= 2 , @P1= 1 R11 and @P0= (@R10 fi10).
0 0 @02
We denote the direct summands of C2 resp. R1 corresponding to the 2 
dimensional cells BEF and_OCBF_resp._the 1  dimensional faces of these
2  dimensional cells by C2 resp._R1 ._It_is clear that the lift ei1of i1 can be
chosen in such a way that fi10(C2 ) R10 so that
____ @P3____ @P2 ____ @P1____ ___ @P0____
. ..! R14 ! R13  ! R12 ! R11 C2 ! R10
is a subcomplex of P*. Furthermore the lift ff of ssi0 can be chosen such
that this subcomplex maps trivially to Q*. Therefore, if we denote the direct
summands of C2 resp. C1 correponding to the other 2  dimensional resp. 1 
dimensional cells by fC2resp. fC1and if we use that fC1is projective (because
all isotropy groups are of even order), i.e. gR10 = fC1and gR1*= 0 for * > 0,
we obtain a factorization of ff to a map effof complexes fP*! Q* as follows
64 HansWerner Henn
(with i10 being induced by @C1) :
@R20 if10
0 ! C3  ! fC2 ! fC1
? ? ?
ff2?y ff1?y ff0?y
@Q3 @Q2 @Q1
Q3 ! Q2  ! Q1 ! Q0
Second_step._Now we construct effin detail. First we observe that the com
plexes fP*resp. Q are induced from F3[D12]  complexes P*0resp. Q0*and that
effcan also be constructed as a map induced from a chain map ff0 of D12 
chain complexes. So it is enough to construct ff0.
We denote the natural S3  modules F3[S3=S2] by S and the tensor prod
uct of S with the nontrivial onedimensional S3  module by S(1). With
respect to the action of the 3  Sylow subgroup of S3 the two module struc
tures agree, so if we denote a fixed generator of this 3  Sylow subgroup by
t, we have well defined linear maps S(1)  ! S and S(1)  ! S which
deserve to be labelled t2  t. We consider all these modules as D12  modules
via the natural homomorphism D12 ! S3 and leave it to the reader to
verify that the minimal resolution Q0*of the trivial F3[D12]  module F3 is
periodic of order 4 and can be described as follows:
2t 1+t+t2 t2t 1+t+t2 t2t
. .S.(1) t! S ! S ! S(1) ! S(1) ! S :
Now we define maps ff0ifor i = 1; 2; 3 and leave it to the reader to verify that
they fit together to give a D12  equivariant chain map ff0 : P*0 ! Q0*as
desired.
As before we denote the isotropy groups of the cells OA, AB, : :b:y OA ,
AB , : :.:Then the map
ff00: F3[D12=OA ] F3[D12=AB ] F3[D12=AD ] ! S
is given as follows (we choose the letter e as a generic letter for the generat*
*ors
of the various modules while gAB and gAD are the elements in A = D12which
have been introduced in section 2.4):
ff00(eOA ) = eS; ff00(eAB ) = gAD eS; ff00(eAD ) = gAB eS :
This is D12  equivariant if the subgroup S2 which occurs in the definition of
the module S is chosen as the subgroup generated by the image of gAB gAD gAB
with respect to the projection D12 ! S3; for t we take the image of gAD gAB .
The map
ff01: F3[D12=OAC ] F3[D12=OAF ] F3[D12=ABD ] ! S(1)
The cohomology of SL(3; Z[1=2]) 65
may then be given by
ff01(eOAC ) = 0; ff01(eOAF ) = 0; ff01(eABD ) = eS(1) :
Finally we have
ff02: F3[D12] ! S(1); ff02(e) = eS(1) :
Third_step._We are now ready to finish the calculation of i0*. The following
element of K F3[0=2] (cf. formula (3.2) in section 3.1)
[100y = 0]  [100z = 0]  [010x = 0] + [010z = 0] + [001x = 0]  [001y = 0]
represents a class in H2(P*0F3[D12]K) and it suffices to show that its image
via
ff02 idK : K ~=F3[D12] F3[D12]K ! S(1) F3[D12]K
is nontrivial in H2(Q0*F3[D12]K), i.e. is not in the image of
2t
S F3[D12]K t! S(1) F3[D12]K :
We leave this verification to the patient reader with the hint that the calu
lation can be significantly simplified by making use of the decomposition of
F3[0=2] as a F3[D12]  module (cf. section 2.6). 2
4.3 Higher torsion in the integral cohomology
It is clear from Corollary 1.7 that the p  torsion in H*(SL(3; Z[1=2]); Z)
is trivial for primes p > 3. Furthermore the mod  3 Bockstein spectral
sequence and Theorem 1.10 shows that the 3  torsion is all of order 3 and is
easily understood from the results in the last section. Therefore we restrict
attention to higher 2  torsion.
For this consider the mod  2 Bockstein spectral sequence for SL(3; Z[1=2]):
We know from Theorem 1.4 that H*(SL(3; Z[1=2]); F2) maps injectively onto
the subalgebra of H*(SD3; F2) ~= F2[x; y] E(f; g) generated by v2 = x2 +
xy+y2, v3 = x2y+xy2, d3 = x2g+y2f and d5 = x4g+y4f (cf. [H1 ]). Therefore
we have Sq1v2 = v3 while Sq1 is zero on the other algebra generators and
thus we see that the E2  term of this spectral sequence is isomorphic to
F2[v22] E(d3; d5). The crucial point is now which order Bockstein of d3 kills
v22.
To settle this we consider the mod  2 Bockstein spectral sequence for
GL(2; Z[1=2]): In this case H*(GL(2; Z[1=2]); F2) maps injectively onto the
subalgebra of H*(D2; F2) ~= F2[x; y] E(f; g) generated by w1 = x + y,
66 HansWerner Henn
w2 = xy, e1 = e + f and e3 = x2g + y2f [H1 ] and the E2  term identifies
with F2[w22] E(e1; e3). The restriction map from H*(SL(3; Z[1=2]); F2) to
H*(GL(2; Z[1=2]); F2) maps d3 to e3 and v2 to w2 + w21, hence v23to w22+ w41
which in the E2  term is identified with w22. Therefore it suffices to determi*
*ne
which higher order Bockstein of e3 kills w22.
Now we compare the mod  2 Bockstein spectral sequence for GL(2; Z[1=2])
with that of SL(2; Z[1=2]); we recall that H*(SL(3; Z[1=2]); F2) ~= F2[w2]
E(e3) (cf. [Mi ]). The notation suggests the behaviour of the restriction map,
i.e. the elements w2 and e3 of H*(GL(2; Z[1=2]); F2) map to the elements
in H*(SL(2; Z[1=2]); F2) with the same name. Furthermore, the element
w2 comes from an integral class in H*(SL(2; Z[1=2]); Z) (namely the first
Chern class c1), hence Sq1 acts trivially on it. Therefore, the E2  term
in the case of SL(2; Z[1=2]) is isomorphic to F2[w2] E(e3) and hence it
is enough to determine which higher order Bockstein of e3 kills w22in the
Bockstein spectral sequence for SL(2; Z[1=2]), or equivalently which is the
additive order of the second power of the integral lift c1 of w2. This can
be checked to be of order 8, e.g. by playing off the mod  2 cohomology
computation [Mi ] against an integral cohomology computation based on the
amalgam description SL(2; Z[1=2]) ~= SL(2; Z) * SL(2; Z) [Se]. Here is
the subgroup of SL(2; Z) consisting of all matrices which are upper triangular
modulo 2.
We summarize our discussion in the following result.
Proposition 4.15 The higher 2  torsion in H*(SL(3; Z[1=2]); Z) is all of
order 8 and is represented in the mod  2 Bockstein spectral sequence by the
classes v22n and d5v22n (n > 0); the classes 1 and d5 represent classes of
infinite order. 2
The integral cohomology of SL(3; Z[1=2]) can now be easily written down
explicitly. We leave the details to the interested reader.
The cohomology of SL(3; Z[1=2]) 67
5 The cohomology of GL(3; Z[1=2])
Let GL (3; Z[1=2]) be the preimage of the subgroup {1} of (Z[1=2])x under
the determinant GL(3; Z[1=2])  ! (Z[1=2])x . The group GL (3; Z[1=2])
splits as SL(3; Z[1=2]) x Z=2, so we understand its mod p  cohomology by
Theorem 1.4, Corollary 1.7 and Theorem 1.10. We will work out the mod 
p cohomology spectral sequences of the group extension
1 ! GL (3; Z[1=2]) ! GL(3; Z[1=2]) ! Z ! 1 (5.1)
where the homomorphism from GL(3; Z[1=2]) to Z is the determinant fol
lowed by the quotient map (Z[1=2])x  ! (Z[1=2])x ={1} ~=Z.
Note that the matrix 2 . idis central in GL(3; Z[1=2]), hence it acts trivially
on GL (3; Z[1=2]) by conjugation. Its determinant is 8 = 23 which corre
sponds to the element 3 in Z under the determinant map. It follows that the
conjugation action of Z on H*(GL (3; Z[1=2]); Fp) factors through an action
of Z=3.
The case p > 3. The conjugation action of Z=3 on H5(GL (3; Z[1=2]); Fp)
~=Fp comes from one on integral cohomology, hence it is necessarily trivial.
Furthermore the spectral sequence necessarily collapses at E2 and we obtain
the following result.
Proposition 5.1 Assume p > 3. Then there is an isomorphism of algebras
H*(GL(3; Z[1=2]); Fp) ~=H*(SL(3; Z[1=2]); Fp) H*((Z[1=2])x ; Fp) :
We have chosen Z[1=2]x as second factor in order to get "symmetric state
ments" for the different primes.
The case p = 2. Again we look at the spectral sequence of the group exten
sion (5.1). As in the case of primes p > 3 we claim that the conjugation action
of Z=3 on H*(GL (3; Z[1=2]); F2) ~= H*(SL(3; Z[1=2]); F2) H*(Z=2; F2) is
trivial.
First we note that this action leaves the two factors H*(SL(3; Z[1=2]); F2)
and H*(Z=2; F2) invariant and is clearly trivial on the second factor. By
dimensional reasons it is clear that the action is trivial on v2, and because of
Sq1v2 = v3 it is also trivial on v3. Now we know that the action of Z=3 on
H3(SL(3; Z[1=2]; F2) ~=(F2)2 has an invariant subspace (namely the subspace
generated by v3) and this forces it to be also trivial on d3. Next the formula
Sq2d3 = d5 and multiplicativity of the action shows that Z=3 acts trivially
as claimed. We obtain E2 ~= H*(SL(3; Z[1=2]); F2) H*((Z[1=2])x ; F2) as
algebras. By Theorem 1.1 E2 consists of permanent cycles, i.e. the spectral
sequence collapses and we have finally proved Theorem 1.3.
68 HansWerner Henn
The case p = 3. Once more we look at the spectral sequence of the
group extension (5.1). Using the restriction map to the cohomology of the
centralizers CSL(3;Z[1=2])(Ei) together with the description of these groups
as provided by Section 4.2, it is easy to see that the action of Z=3 on
eH*(GL (3; Z[1=2]); F3) is trivial. So as before we obtain an isomorphism
E2 ~=H*(SL(3; Z[1=2]); F3) H*((Z[1=2])x ; F3) as algebras, i.e. there is no
room for differentials and the spectral sequence collapses. By using the re
striction maps to the centralizers CGL(3;Z[1=2])(Ei) we see that the E2  term
gives also the algebra structure. We state the result of our discussion in the
following result.
Proposition 5.2 There is an isomorphism of algebras
H*(GL(3; Z[1=2]); F3) ~=H*(SL(3; Z[1=2]); F3) H*((Z[1=2])x ; F3) :
The cohomology of SL(3; Z[1=2]) 69
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HansWerner Henn
Mathematisches Institut der Universit"at
Im Neuenheimer Feld 288
D69120 Heidelberg
Germany
Current address:
Departement de Mathematique
Universite Louis Pasteur
7, rue Rene Descartes
F67084 Strasbourg
France