The cohomology of SL(3; Z[1=2]) Hans-Werner 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 Bruhat-Tits-building . . . . . . 8 2.2 Well-rounded 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 well-rounded 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 Hans-Werner 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 well-known 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 torsion-freeness 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 H2i-1(Dn; F2): they are the symmetrizations of the elements x21:::x2i-1ai 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; :::; e2n-1) . 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 d2i-1 the image of e2i-1, i = 2; 3. 4 Hans-Werner 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 Bruhat-Tits-building 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 Bruhat-Tits-building 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 Hans-Werner 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 non-trivially 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 Hans-Werner Henn 2 Contractible spaces with actions of SL(3; Z) and SL(3; Z[1=2] ) 2.1 The symmetric space and the Bruhat-Tits-building 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) = ||A-1x|| 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 Bruhat-Tits-building. 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 Bruhat-Tits-building 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 n-1. 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 two-dimensional 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 Soule-Lannes method of finding a deformation retract of i\X1 which is compact, even a finite 3 - dimensional complex (see [A ]). 10 Hans-Werner Henn 2.2 Well-rounded lattices and the deformation retrac- tions Well-rounded 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 g-1 (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 {kxk|x 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 := g-1 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 well-rounded 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 well-rounded. 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)|g-1 (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 well-rounded 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 well-rounded 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 Hans-Werner Henn may assume that |w3| 1_2|z| 1_2, |w2| 1_2|x| 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 well-rounded 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 ||w||2 = n21+ n22+ n23+ 2an1n2 + 2bn1n3 + 2cn2n3 ; (2.1) or equivalently ||w||2 = 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 ||w||2 = 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 ||w||2 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 non-zero, say n2 + n3, then n1 = -n3, n1 = -n2 and |n2 + n3| 2 and (2.2) yields ||w||2 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 non-zero, say n1 + n3 and n2 + n3, then |n1 + n3| 2 and |n2 + n3| 2 and (2.2) yields ||w||2 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 ||w||2 = 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 ||w||2 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 ||w||2 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 Hans-Werner 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)|g-1 (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 well-rounded 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 | | | | | | || j|C|= (1_2; 1_2; 1_2) | j j | | j | | j | | j | | j | | j | | j j | | j | | j | | j | O = (0; 0; 0) __________________________|__________________-b||jA 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 Hans-Werner 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+b-c 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 Hans-Werner 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 g-1 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 non-zero 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 := g-1 s, then the map 0 - ! S1; g 7! g-1 (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 Hans-Werner 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\ gsHegs-1, i.e agrees up to conjugation by gs with gs-1igs\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 well-rounded 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 2-dimensional cells ______________________________________________________ |_Cell__|Minimal_vectors|Isotropy|____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 Hans-Werner Henn Table 2: Symmetries on the 1-dimensional 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 0-dimensional cells ________________________________________________________________ |_Cell|Minimal_vectors|_Isotropy____|Description_______________|__ 2 | O |e1; e2; e3 |S4 ~=(Z=2) o S3|symmetry 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 S3|Z=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 S3|Z=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 non-zero 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 non-zero 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 Hans-Werner 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 1|00 010 001 110 101 011 111 | |_______|______________________________| | gAD s1|00 110 001 010 101 111 011 | |_______|______________________________| | gAB s1|10 010 001 100 111 011 101 | |_______|______________________________| Table 5: Action of gAD and gAB on S2 __________________________________________________________________ | s 1|00y = 0 100z = 0 100y = z |010x = 0 010z = 0 010x = z | |_______|___________________________|____________________________|_ | gAD s1|00y = 0 100z = 0 100y = z |110x = y 110z = 0 110 = 0 | |_______|___________________________|___________________________|_ | gAB s1|10x = y 110z = 0 110 = 0 | 010x = 0 010z = 0 010x = z | |_______|__________________________|_____________________________|__ | s 0|01x = 0 001y = 0 001x = y |110z = 0 110x = y 110 = 0 | |_______|___________________________|___________________________|_ | gAD s0|01x = y 001y = 0 001x = 0 |010z = 0 010x = 0 010x = z | |_______|___________________________|____________________________|_ | gAB s0|01x = 0 001x = y 001y = 0 |100z = 0 100y = 0 100y = z | |_______|___________________________|____________________________|__ | s 1|01y = 0 101x = z 101 = 0 | 011x = 0 011y = z 011 = 0 | |_______|__________________________|____________________________|_ | gAD s1|01y = 0 101 = 0 101x = z |111x = y 111y = z 111x = z | |_______|___________________________|____________________________|_ | gAB s1|11x = y 111x = z 111y = z |011x = 0 011 = 0 011y = z | |_______|___________________________|____________________________|__ | s 1|11x = y 111x = z 111y = z | | |_______|___________________________| | | gAD s0|11x = 0 011 = 0 011y = z | | |_______|___________________________| | | gAB s1|01y = 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 non-trivial 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;_0101|01;_011_|001_|110_|111_| For the action on S2 we obtain: _________________________________________________________________ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__{1}___|___{1}___| | Orbits |100y = 0 |100z = 0 |100y = z1|01y = 0 |101x = z | |________________|010x_=_0_|010z_=_00|10x_=_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_=_01|10_=_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 non-trivial 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;_0011|10;_101_|100_|011_|111_| 26 Hans-Werner Henn _________________________________________________________________ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__{1}___|__{1}___|_ | Orbits |010x = 0 |010z = 00|10x = z |110z = 0 |110x = y | |________________|001x_=_0_|001y_=_00|01x_=_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_=_z0|11_=_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;_0011|10_|101_|011_| _____________________________________________________________________ |_Isotropy_groups|_{1}___|____{1}____|___{1}____|__Z=2____|___Z=2___|_ | Orbits |100y = 0 |100z = 0 |100y = z |110x = y 1|01x = z | | |111x = z |111x = y |111y = z |110z = 0 1|01y = 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;_0011|10;_101;_0111|11_| _______________________________________________________ |_Isotropy_groups|______{1}________|__Z=2____|__Z=2____| | Orbits |100y = 0; 100z = 01|00y = z |110x = y | | |010x = 0; 010z = 00|10x = z |101x = z | |________________|001x_=_0;_001y_=_00|01x_=_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 non-trivial 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;_1101|01;_111_|010_|001_|011_| |_Orbits_for_AC__|100;_0101|01;_011_|110_|001_|111_| 28 Hans-Werner Henn __________________________________________________________________ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__{1}___|___{1}____| | Orbits for AB |100y = 0 |100z = 0 |100y = z1|01y = 0 |101 = 0 | |________________|110x_=_y_|110z_=_01|10_=_0_|111x_=_y_|111y_=_z_|_ | Orbits for AC |100y = 0 |100z = 0 |100y = z1|01y = 0 |101x = z | |________________|010x_=_0_|010z_=_0_|010x_=_z0|11x_=_0_|011y_=_z_|_ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__Z=2___|___Z=2___|_ | Orbits for AB |101x = z |001x = y0|11 = 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 non-trivial 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;_0101|11;_011_|001_|100_|101_| |_Orbits_for_AE__|100;_0101|01;_011_|001_|110_|111_| __________________________________________________________________ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__{1}___|___{1}____| | Orbits for AD |110x = y |110z = 01|10 = 0 |111x = y |111x = z | |________________|010x_=_0_|010z_=_0_|010x_=_z0|11x_=_0_|011_=_0_|_ | Orbits for AE |100y = 0 |100z = 0 |100y = z1|01y = 0 |101x = z | |________________|010x_=_0_|010z_=_0_|010x_=_z0|11x_=_0_|011y_=_z_|_ |_Isotropy_groups|_{1}____|__{1}___|___{1}____|__Z=2___|___Z=2___|_ | Orbits for AD |111y = z |001x = 0 |101x = z1|00y = 0 |100z = 0 | |________________|011y_=_z_|001x_=_y1|01_=_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 Hans-Werner 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;_0011|10;_101__|011_| _________________________________________________________ |_Isotropy_groups|______{1}________|___Z=2____|__Z=2____|_ | Orbits |100y = 0; 111x = z |100y = z 1|10x = y | | |010x = z; 001y = 0 |111y = z 1|10z = 0 | | |100z = 0; 111x = y |010x = 0 1|01x = z | |________________|001x_=_y;_010z_=_0_|001x_=_0_1|01y_=_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;_0011|01;_011__|110_| _________________________________________________________ |_Isotropy_groups|______{1}________|___Z=2____|__Z=2____|_ | Orbits |100y = 0; 111x = z |100z = 0 1|01x = z | | |010x = z; 001y = 0 |111x = y 1|01y = 0 | | |010x = 0; 111y = z |010z = 0 0|11y = z | |________________|100y_=_z;_001x_=_0_|001x_=_y_0|11x_=_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;_0111|11_| |_Orbits_for_B___|100;_110;_001;_010;_101;_1110|11_| _________________________________________________________ |_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 Hans-Werner 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;_1111|10;_101;_011_| _________________________________________________________ |_Isotropy_groups|______Z=2________|Z=2_x_Z=2_|___D8____|_ | Orbits |100y = 0; 010x = 0 |110x = y 1|10 = 0 | | |100z = 0; 010z = 0 |110z = 0 1|01 = 0 | | |100y = z; 010x = z |101x = z 0|11 = 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_groups|Z=2_x_Z=2___|_Z=2_x_Z=2___|D12_| |_Orbits_________|100;_110;_0101|01;_111;_0110|01_| __________________________________________________________ |_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;_0011|11;_011__|100_| |_Orbits_for_E___|100;_111;_010;_0011|01;_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 Hans-Werner 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. ____________________________________________________________________ |_______________|0-cells|1-cells|2-cells|3-cells|number_of_all_cells|O__| |_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 homotopy-equivalent 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 Hans-Werner 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______________B|EF_||__OAC_|__|OAEF_||_OCBF__||||||* *||| |_________|100010_001_|110_101_011_|111_|100_|100_101_|100101_|010_110_| | BD 0|10 | 1 1 | 1 1 | | 1 | | | | |_____|___|___________|____________|___|_____|_______|________|________| | AB 1|00 |1 | 1 | | |1 | | | || || || || || || || || || || |____1|01_|___________|_____1______|_1__|____|_____1__|_______|________| | AD 1|10 | -1 |-1 | | | | 1 | | || || || || || || || || || || |____1|11_|___________|_________-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__|_____________|____|_||||| || |110||1| 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 Hans-Werner Henn * * Table 9: The boundary homomorphism C2 ! C1 for 2\(Z;@ _____* *___________________________________________________________________@ | * * ||_______________OAC______________||______________OAEF_______@ | * * ||____100___|_|____101____00|1_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_=_0-|1-1_____|____________|-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|=@ * * | ABD|100y = 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_|________|____________|____________|____________|__@ * * | BEF|100y = 0-|1 | -1 | -1 | | @ * * | |100z = 0 | -1 | -1 | -1 | | @ * * |____|100y_=_z_|_____-1_|-1__________|-1__________|____________|__@ 40 Hans-Werner 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 = 0-|1 |-1 | | | @ * * | |100z = 0 | -1 | -1 | | | @ * * | _|100y_=_z_|_____-1_|_________-1_|____________|____________|_@ * * | |101y = 0 | | | | |-@ * * | |101x = z | | | | | @ * * | _|101_=_0_|_________|____________|____________|____________|_@ * * | |001x = 0 | | |-1 -1 | | @ * * |_____|111x_=_z_|________|____________|____________|____________|_@ * * | OAEF|100y = 01| | 1 | | | @ * * | |100z = 0 | 1 | 1 | | | @ * * | _|100y_=_z_|______1__|________1__|____________|____________|_@ * * | |101y = 0 | | | | |1@ * * | |101x = z | | | | | @ * * | _|101_=_0_|_________|____________|____________|____________|_@ * * | |001x = 0 | | | 1 1 | | @ * * |_____|111x_=_z_|________|____________|____________|____________|_@ * * | OCBF|010x = 0 | |-1 |-1 | | @ * * | |010z = 0 | | -1 | -1 | | @ * * | _|010x_=_z_|________|_________-1_|_________-1_|____________|_@ * * | |110z = 0 | | | |-1 |-@ * * | |110x = y | | | | -1 | @ * * | _|110_=_0_|_________|____________|____________|_________-1__|@ * * | |100y = 0-|1-1 | | | | @ * * |_____|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 Hans-Werner 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 |100-|1 | | 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____00|1_111||_____110____10|0111||__@ * * |__________x|=_yx_=_yx|=zy=_0_=_0x|=_yx_=xy=|yz_=_0_=_0y|=_zy_=yz=@ * * | BD|100y = 0 | | | | | | @ * * | _|100y_=_z_|____|____________|________|____________|_______|___@ * * | |011x = 0 | | | | | | @ * * | _|011y_=_z_|____|____________|________|____________|_____1__|-1@ * * | |111x = y | | | | | | @ * * |___|111y_=_z_|____|____________|________|____________|____-1_|_1_@ * * | AC|001x = y1| | | | | | @ * * | _|111x_=_y_|__1__|___________|________|____________|_______|___@ * * | |110x = y | | 1 | | | | @ * * | |110z = 0 | | 1 | | | | @ * * |___|110_=_0_|_____|__________1__|_______|____________|_______|___@ * * | AE|001x = y | | | 1 | | | @ * * | _|111x_=_y_|____|____________|_____1__|____________|_______|___@ * * | |110x = y | | | | 1 | | @ * * | |110z = 0 | | | | 1 | | @ * * |___|110_=_0_|_____|____________|________|_________1__|_______|___@ * * | OC|100y = z-|1 | | | | 1 | @ * * | _|111x_=_y_|_-1_|____________|________|____________|_____1__|__@ * * | |110x = y | | -1 | | | | 1 @ * * | |110z = 0 | | -1 | | | | @ * * |___|110_=_0_|_____|_________-1__|_______|____________|_______|___@ * * | OF|100y = z | | | -1 | |-1 | @ * * | _|111x_=_y_|____|____________|_____-1_|____________|____-1_|___@ * * | |110x = y | | | |-1 | | -1@ * * | |110z = 0 | | | | -1 | | @ * * |___|110_=_0_|_____|____________|________|_________-1__|______|___@ 44 Hans-Werner Henn * * Table 14: The boundary homomorphism C2 ! C1 for 2\@ * * __________________________________________________________________@ * * | ||_________OAC_______|_|_______OAEF________||________OC@ * * | _|001111||_____110____00|1_111||_____110____10|0111||__@ * * |__________x|=_yx_=_yx|=zy=_0_=_0x|=_yx_=xy=|yz_=_0_=_0y|=_zy_=yz=@ * * | EF|100z_=_0_|____|____________|_1___1__|____________|_______|___@ * * | |110x = y | | | | 1 1 | | @ * * | _|110_=_0_|_____|____________|________|_________1__|_______|___@ * * | |101x = z | | | | | | @ * * |___|101_=_0_|_____|____________|________|____________|_______|___@ * * | BF|100y_=_z_|____|____________|________|____________|_1___1__|__@ * * | |110x = y | | | | | | @ * * | _|110_=_0_|_____|____________|________|____________|_______|___@ * * | |011y = z | | | | | | 1 @ * * |___|011_=_0_|_____|____________|________|____________|_______|___@ * * | OA|100y = 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 | | | | 0|3 |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 Hans-Werner 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 Xs-ss!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 non-trivially 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 Hans-Werner Henn is induced by the map which associates to the triple (L0; L1; L2) of lattices (with L0 being well-rounded and m(L0) = 1) the pair (L1; L2). Because L1 need not be well-rounded 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- X2k-1with 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 1-skeleton @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 Hans-Werner 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 non-trivial 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 i-th 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 non-trivial. 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 non-trivially by the map H1(2; C2) d12;1 ss1 -! H1(2; C1) -! H1(2; F2[2=o1]) then y is also mapped non-trivially 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 non-trivially 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 Cs-1 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 -! Cs-1. 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 non-trivial 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 "non-singular" 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 Hans-Werner Henn of o2. Furthermore oe1 and oe2 are the only cells in their 2 - orbit for which the incidence numbers are non-trivial, 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 gs-1) 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 gs-1 2 2, and in fact, by Section 2.4.2, the element gsgAB gs-1 is the unique non-trivial 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! g-1 o1 induces an isomorphism F2[2]=F2[2] ~= F2[2=g-1o1] 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=g-1o1] -! 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=g-1o1] where ess1denotes the composition of ss1 with left multiplication by g-1 . 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 non-trivial 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 Hans-Werner 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) Diederichsen-Reiner 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 Hans-Werner 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 Diederichsen-Reiner 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 non-trivial 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 non-trivial 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 Hans-Werner 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 non-trivial 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 non-trivially 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 Hans-Werner 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 i-th 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 Hans-Werner 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 non-trivial 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 Hans-Werner 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 non-trivial one-dimensional 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: 2-t 1+t+t2 t2-t 1+t+t2 t2-t . .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 non-trivial in H2(Q0*F3[D12]K), i.e. is not in the image of 2-t 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 Hans-Werner 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 Hans-Werner 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 References [A] A.Ash, Small-dimensional classifying spaces for arithmetic subgroups of gen* *eral linear groups, Duke Math. Journal 51 (1984), 459-468 [BS]A. Borel et J. 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Lannes and L. Schwartz. Localization of unstable A - modul* *es and equivariant mod - p cohomology, Math. Ann. 301 (1995), 23-68 [Mi]S. Mitchell, On the plus construction for BGLZ[1_2], Math.Zeit. 209 (1992),* * 205-222 [Mo] K. Moss, Homology of SL(n; Z[1=p]), Duke Math. Journal 47 (1980), 803-818 [Q] D. Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. of M* *ath. 94 (1971), 549-572, 573-602 [Se]J. P. Serre, Trees, Springer Verlag, 1980 [So]C. Soule, The cohomology of SL(3; Z), Topology 17 (1978), 1-22 Hans-Werner Henn Mathematisches Institut der Universit"at Im Neuenheimer Feld 288 D-69120 Heidelberg Germany Current address: Departement de Mathematique Universite Louis Pasteur 7, rue Rene Descartes F-67084 Strasbourg France