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 straightf