APPROXIMATING K_{∗}(ℤ) THROUGH DEGREE FIVE

John Rognes

Abstract. We approximate the K-theory spectrum of the integers using a spectrum level rank filtration. By means of a certain poset spectral sequence we explicitly compute the first three subquotients of this filtration. Assuming a conjecture about the filtration’s rate of convergence, we conclude that K_{4}(ℤ) = 0 and K_{5}(ℤ) is a copy of ℤ (the Borel summand) plus two–torsion of order at most eight.

1. Introduction.

This paper will show that, assuming a connectivity Conjecture, the higher algebraic K-group K_{4}(ℤ) is zero, while K_{5}(ℤ) is the free Borel summand plus two– torsion of order at most eight. The proof is by computation, utilizing constructions made in the author’s paper [14]. We first present the main computational result and explain how to determine K_{4}(ℤ) and the bound on K_{5}(ℤ) from it. Next we review the requisite constructions. Thereafter we give more detailed statements of our results, and finally present the calculations.

This work continues the fundamental computations from [1], [8] and [9] of the higher algebraic K-groups of the rational integers.

Let Kℤ denote the K-theory spectrum of the integers, and let F_{k}Kℤ denote the kth stage of the spectrum level rank filtration of Kℤ as defined in Section 2.

Theorem 1.1. H_{∗}(F_{3}Kℤ) ≅ (ℤ, 0, 0, ℤ/2, 0, ℤ + (#|4), …)

Here #|4 denotes an elementary abelian two–group of rank one or two.

We have made the following connectivity Conjecture, which these computations are compatible with :

Conjecture 1.2. F_{k}Kℤ → F_{k+1}Kℤ is (2k − 1)-connected for k ≥ 0.

From this follows our main Theorem :

Theorem 1.3. Assuming the connectivity Conjecture above, K_{4}(ℤ) = 0 and K_{5}(ℤ) ≅ ℤ + (#|8).

Here #|8 denotes a two–group of order at most eight.

We can also describe the order forty–eight elements in K_{3}(ℤ) [8], and the integral Borel generator in K_{5}(ℤ) [1]. See 2.1, 5.2 and 6.3 for notation. Duality refers to the bases given there.

Typeset by AMS-TEX

Theorem 1.4. The homology class in H_{2}(P_{1}) dual to p_{5} ∈ H^{3}(P_{1}) in the size one subquotient in the second subquotient of the rank filtration lifts to the nonzero class in H_{3}(Kℤ), which lifts further to an order forty–eight class in K_{3}(ℤ).

Assuming the connectivity Conjecture, twenty–four times the homology Thom class U ∈ H_{3}(SP_{1,12}) in the size two subquotient of the third subquotient of the rank filtration lifts to twice the generator of the free summand in H_{5}(Kℤ), which lifts further to an integral generator of the Borel summand in K_{5}(ℤ).

We will review the rank filtration and poset spectral sequences in Section 2, list the spectrum homology results in Section 3, prove Theorems 1.1, 1.3 and 1.4 in Section 4, and give the group homological calculations in the last three Sections.

The results are compatible with the Lichtenbaum–Quillen conjectures, also at the prime two [5], which predict that K_{4}(ℤ) = 0 and K_{5}(ℤ) ≅ ℤ.

Our computations prove Conjecture 1.2 for k ≤ 2. In [14] we more generally constructed a spectrum level rank filtration {F_{k}KR}_{k} of the free K-theory spectrum KR for all rings satisfying the strong invariant dimension property [10], i.e. there only exists split injections R^{i} ↪ R^{j} when i ≤ j. This includes all commutative rings. We conjectured in [14, 12.3] that F_{k}KR → F_{k+1}KR would be (2k − 1)-connected for local rings or Euclidean domains R, when k ≥ 0. Analogous computations to those of this paper, but easier, prove this form of the Conjecture for the cases R = 𝔽_{2}, 𝔽_{3}, ℝ and ℂ, when k ≤ 2.

In principle these methods extend to higher subquotients of the rank filtration, but it might be desirable to further structure the work using the tensor product on Kℤ before proceeding, and to work with Kℚ to benefit from better homological stability results for fields.

I thank Marcel Bökstedt and Jørgen Tornehave for helpful conversations on étale K-theory and crystallographic methods. This work was done during an exciting year at Århus University.

2. The poset spectral sequence.

This Section reviews the construction of the spectrum level rank filtration, the poset filtration on each subquotient of this rank filtration, and its associated poset spectral sequence. We introduce notation for parabolic groups and other matrix groups which we will need. Finally we consider some useful generalizations of the poset spectral sequence.

Recall that the nth space of the K-theory spectrum Kℤ can be defined as the diagonal of the nerve of the n-multisimplicial category wSn_{•}P(ℤ). Here P(ℤ) is the category of finitely generated projective ℤ-modules and Sn_{•} is Waldhausen’s S_{•}-construction iterated n times [15]. The kth stage F_{k}Kℤ of the spectrum level rank filtration of Kℤ is the (pre–)spectrum with nth space F_{k}wSn_{•}P(ℤ) given as the subcomplex of the nerve of wSn_{•}P(ℤ) consisting of the simplices which are diagrams in P(ℤ) involving only ℤ-modules of rank k or less.

The poset spectral sequence computes, for a given rank k, the spectrum homology of the subquotient spectrum F_{k}Kℤ/F_{k−1}Kℤ. The nth space of this (pre–)spectrum has simplices which are chains of isomorphic diagrams on some n-dimensional cubical indexing categories. To each simplex we can associate k distinguished unordered sites, or objects in the cubical indexing category ([14, §5]). These inherit a partial ordering from the product ordering on this n-dimensional cube, in the weak sense that two distinct sites may be mutually less than each other. Hence to each simplex we may associate an isomorphism class of partial orderings on the k-element set {1, …, k}. Such partial orderings may in turn be ordered by their sizes, where the size of a poset is defined as its number of components plus its number of equivalence classes, less two ([14, §7]). This gives rise to a filtration on the nth space of F_{k}Kℤ/F_{k−1}Kℤ, and as n tends to infinity the associated homology spectral sequence stabilizes. The colimit is the poset spectral sequence, which converges to H_{∗}(F_{k}Kℤ/F_{k−1}Kℤ). It is located in the first quadrant, and is of homological type.

We shall describe the poset spectral sequence’s E^{1}-term shortly, after making two further remarks on the geometry of the situation, and introducing some notation.

Embedded inside the nth space of the kth subquotient of the rank filtration is a complex called the n-dimensional building of ℤ^{k}, denoted D^{n}(ℤ^{k}). It consists of the simplices which are isomorphism chains of length zero, i.e. single cubical diagrams. We may assume that the final module occurring in such a diagram is precisely ℤ^{k}, and that all the other modules are included in it as submodules. D^{n}(ℤ^{k}) is a based GL_{k}ℤ-space, and its (reduced) homotopy orbit space, or Borel construction, is precisely the nth space of the kth subquotient of the rank filtration ([14, §3]). The buildings assemble into a spectrum, D(ℤ^{k}).

Call the submodules of ℤ^{k} generated by a subset of the standard basis for ℤ^{k} the axial submodules. The standard apartment is the subcomplex of the n-dimensional building with simplices the diagrams consisting of axial submodules. It is homeomorphic to an nk-sphere ([14, §4]), so for k ≥ 2 it is stably contractible as n tends to infinity.

Notation 2.1. For I ⊆ {1, …, k}, let P_{I} ⊆ GL_{k}ℤ denote the parabolic subgroup consisting of matrices (g_{ij}) such that g_{ij} = 0 if j ∈ I and i ∉ I. Similarly let P_{I,J} = P_{I} ∩ P_{J} etc. We omit curly parentheses in these subscripts. If we wish to emphasize the size k of the matrices in question we will write P_{I}(k). SP_{I} will denote the subgroup of P_{I} consisting of matrices with determinant one. For example, P_{1}(2) and P_{1,12}(3) are the upper triangular 2 × 2 and 3 × 3 matrices.

View the symmetric group Σ_{k} as embedded in GL_{k}ℤ as the permutation matrices, and let T_{k} ⊂ GL_{k}ℤ denote the diagonal torus consisting of matrices which are zero off the diagonal. Σ_{k} normalizes T_{k} and we set D_{k} = Σ_{k} · T_{k}. ST_{k} and SD_{k} are the determinant one subgroups. D_{2} is the dihedral group of order eight and SD_{3} the symmetric group on four letters.

Let C_{k} ⊆ GL_{k}ℤ denote the center. It consists of plus and minus the identity matrix.

Let W_{k} denote the integral rank (k − 1)! representation of Σ_{k} dual to the free abelian group generated by the iterated Lie brackets of weight one in each generator in the free Lie algebra on k generators, with Σ_{k} permuting these generators. Explicitly, W_{2} is the rank one sign representation of Σ_{2}, while W_{3} is a Σ_{3}-representation of rank two with ϕ = (12) acting by ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and ψ = (123) acting by ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ). We also view the W_{k} as D_{k}- and SD_{k}-representations through the natural homomorphisms SD_{k} → D_{k} → Σ_{k}.

Then the poset spectral sequence for the second subquotient of K-theory has E^{1}-term :

(GL_{2}ℤ PSS)  H_{∗}(GL_{2}ℤ) ⟵^{d^{1}} H_{∗}(P_{1}) ⟵^{d^{1}} H_{∗}(D_{2}; W_{2})

(group homology), and converges to the spectrum homology H_{∗}(F_{2}Kℤ/F_{1}Kℤ). The d^{1}-differential H_{∗}(P_{1}) → H_{∗}(GL_{2}ℤ) is induced by the inclusion of groups. The other d^{1}-differential factors as H_{∗}(D_{2}; W_{2}) → H_{∗}(T_{2}) → H_{∗}(P_{1}), where the right hand map is induced by the group inclusion and the left hand map occurs in the long exact sequence of D_{2}-homology coming from the short exact coefficient sequence W_{2} → ℤΣ_{2} → ℤ, using Shapiro’s lemma. See [14, 12.4 and 15.2] for a proof.

Similarly, the poset spectral sequence for the third subquotient of K-theory has E^{1}-term : (GL_{3}ℤ PSS)

H_{∗}GL_{3}ℤ ⟵^{d^{1}} H_{∗}P_{1} ⊕ H_{∗}P_{12} ⟵^{d^{1}} H_{∗}P_{1,12} ⊕ H_{∗}P_{12,3} ⟵^{d^{1}} H_{∗}P_{1,12,3} ⟵^{d^{1}} H_{∗}(D_{3}; W_{3})

converging to H_{∗}(F_{3}Kℤ/F_{2}Kℤ). Up to sign the two leftmost differentials are induced by inclusions of groups, together with (right) conjugation by ψ^{2} = (132) in the case of P_{12,3} ↪ P_{1}. The third differential d^{1} : E_{3∗}^{1} → E_{2∗}^{1} maps to a direct sum. We describe its two summands separately : The homomorphism to H_{∗}(P_{12,3}) is induced by inclusion. The homomorphism to H_{∗}(P_{1,12}) is the alternating signed sum of three maps induced by inclusion followed by (right) conjugation by respectively e = (), ϕψ = (23) and ψ^{2} = (132). The rightmost differential is harder to describe, and we shall use naturality to handle it. These descriptions follow easily from [14, 15.3].

By alluding to naturality we now get to various variant poset spectral sequences. For subgroups of GL_{k}ℤ, such as SL_{k}ℤ, which act transitively on all submodule configurations ([14, §8]) we may consider the corresponding homotopy orbit space of D^{n}(ℤ^{k}) as a subcomplex of the GL_{k}ℤ-homotopy orbit space of D^{n}(ℤ^{k}). This subcomplex inherits a poset filtration and a spectral sequence as indicated above, which naturally maps to the GL_{k}ℤ poset spectral sequence. Our interest is in the case of SL_{3}ℤ. In view of the splitting GL_{3}ℤ ≅ SL_{3}ℤ×C_{3}, with C_{3} ≅ ℤ/2 generated by minus the identity matrix, we obtain the following SL_{3}ℤ poset spectral sequence which splits off the GL_{3}ℤ poset spectral sequence :

\begin{aligned}H_{∗}(SL_{3}ℤ) ⟵^{d^{1}} H_{∗}(SP_{1}) ⊕ H_{∗}(SP_{12}) & ⟵^{d^{1}} H_{∗}(SP_{1,12}) ⊕ H_{∗}(SP_{12,3}) \\ & ⟵^{d^{1}} H_{∗}(SP_{1,12,3}) ⟵^{d^{1}} H_{∗}(SD_{3}; W_{3}) \end{aligned} (SL_{3}ℤ PSS)

It converges to the spectrum homology of D(ℤ^{3})/hSL_{3}ℤ = {D^{n}(ℤ^{3})/hSL_{3}ℤ}. As the center acts trivially by conjugation on the n-dimensional buildings, we can recover F_{3}Kℤ/F_{2}Kℤ as D(ℤ^{3})/hSL_{3}ℤ ∧ BC_{3+}. Here X/hG = X ∧_{G} EG_{+} denotes the homotopy orbit space.

In a different direction, we may consider subgroups of GL_{k}ℤ containing Σ_{k} which leave the axial submodules invariant. The largest such is D_{k}, but we also consider SD_{k}. These groups act on the standard apartment, and the corresponding homotopy orbit spaces inherit a poset filtration and spectral sequence. As the standard apartment is stably contractible for k ≥ 2, so is its homotopy orbit space, and these spectral sequences will abut to zero. We will use :

(D_{2} PSS)  H_{∗}(D_{2}) ⟵^{d_{1}} H_{∗}(T_{2}) ⟵^{d_{1}} H_{∗}(D_{2}; W_{2})

and

\begin{aligned}H_{∗}SD_{3} ⟵^{d_{1}} H_{∗}S(ℤ/2 × D_{2}) ⊕ H_{∗}S(D_{2} × ℤ/2) & ⟵^{d_{1}} H_{∗}ST_{3} ⊕ H_{∗}S(D_{2} × ℤ/2) \\ & ⟵^{d_{1}} H_{∗}ST_{3} ⟵^{d_{1}} H_{∗}(SD_{3}; W_{3}) \end{aligned} (SD_{3} PSS)

As is clear from their construction, the groups occurring in these spectral sequences are the intersections of the parabolic subgroups of the GL_{k}ℤ poset spectral sequence with the group in question, i.e. D_{2} or SD_{3}. For instance, P_{1}(3) ∩ SD_{3} = S(ℤ/2 × D_{2}).

3. The results.

Proposition 3.1.

F_{1}Kℤ ≅ Σ^{∞}Bℤ/2_{+}, so H_{∗}(F_{1}Kℤ) ≅ (ℤ, ℤ/2, 0, ℤ/2, 0, ℤ/2, …).

Proof. This is [14, 15.1].

Theorem 3.2. (1) Localized at the prime two :

H_{∗}(F_{2}Kℤ/F_{1}Kℤ) ≅ (0, 0, ℤ/2, ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/2)^{3}, …)

There are nonzero classes in degrees three and four dual to p_{5} ∈ H^{3}(P_{1}) and the sum of the duals of p_{1}p_{3} and p_{2}p_{3} in H^{4}(P_{1}) respectively.

(2) Modulo two–torsion :

H_{∗}(F_{2}Kℤ/F_{1}Kℤ) ≅ (0, 0, 0, ℤ/3, 0, 0, 0, ℤ/3, 0, …)

Proposition 3.3. The connecting maps ∂: H_{2i}(F_{2}Kℤ/F_{1}Kℤ) → H_{2i−1}(F_{1}Kℤ) are onto for all i ≥ 1.

Let #n denote a group with n elements.

Theorem 3.4. (1) Localized at the prime two, F_{3}Kℤ/F_{2}Kℤ is four–connected, H_{5}(F_{3}Kℤ/F_{2}Kℤ) ≅ ℤ is generated by 2U + V ∈ H_{3}(SP_{1,12}), where U is a Thom class of infinite order and V a class of order two, and H_{6}(F_{3}Kℤ/F_{2}Kℤ) ≅ ℤ/2.

(2) Modulo two–torsion

H_{∗}(F_{3}Kℤ/F_{2}Kℤ) ≅ (0, 0, 0, 0, ℤ/3, ℤ, 0, 0, #9, 0, …)

and H_{5}(F_{3}Kℤ/F_{2}Kℤ) ≅ ℤ is generated by 3U ∈ H_{3}(SP_{1,12}), where U is as above.

Proposition 3.5. (1) The connecting map

∂: H_{5}(F_{3}Kℤ/F_{2}Kℤ) → H_{4}(F_{2}Kℤ/F_{1}Kℤ)

is nonzero, hitting the sum of the classes dual to p_{1}p_{3} and p_{2}p_{3} in E_{1,3}^{∞} in the GL_{2}ℤ PSS.

(2) Modulo two–torsion, the connecting maps

∂: H_{4i}(F_{3}Kℤ/F_{2}Kℤ) → H_{4i−1}(F_{2}Kℤ/F_{1}Kℤ)

are onto for all i ≥ 1.

4. The proofs.

Proof of Theorems 1.1 and 1.4. Let us assemble the beginnings of the spectrum level rank filtration of Kℤ into two spectrum homology spectral sequences as below; one localized at two, the other away from two :

(H_{∗}(Kℤ; ℤ_{(2)}))  \begin{matrix}0 & & (ℤ/2)^{3} & & ? \\ ℤ/2 & ⟵ & (ℤ/2)^{3} & & ? \\ 0 & & (ℤ/2)^{2} & & ℤ/2 \\ ℤ/2 & ⟵ & (ℤ/2)^{2} & ⟵ & ℤ_{(2)} \\ 0 & & ℤ/2 & & 0 \\ ℤ/2 & ⟵ & ℤ/2 & & 0 \\ ℤ_{(2)} & & 0 & & 0 \end{matrix}

and

(H_{∗}(Kℤ; ℤ[\frac{1}{2}]))  \begin{matrix}0 & ℤ/3 & ⟵ & #9 \\ 0 & 0 & & 0 \\ 0 & 0 & & 0 \\ 0 & 0 & & ℤ[\frac{1}{2}] \\ 0 & ℤ/3 & ⟵ & ℤ/3 \\ 0 & 0 & & 0 \\ ℤ[\frac{1}{2}] & 0 & & 0 \end{matrix}

The connectivity Conjecture can be seen as a potential vanishing line in these spectral sequences.

There may be a connecting map, i.e. a d^{1}-differential, E_{2,4}^{1} → E_{1,4}^{1} in the two– primary case, so the integral E^{∞}-term begins as follows :

\begin{matrix}0 & ? & ? \\ 0 & #|4 & ? \\ 0 & 0 & ℤ \\ 0 & ℤ/2 & 0 \\ 0 & 0 & 0 \\ ℤ & 0 & 0 \end{matrix}

This proves Theorem 1.1. Twice the ℤ-generator in E_{2,3}^{1} survives to E^{∞}, so the second part of Theorem 1.4 follows from Theorem 3.4. The first part of Theorem 1.4 is contained in Theorem 3.2. □

Recall Bökstedt’s construction of a spectrum JKℤ_{2}^{∧} related to two–primary étale K-theory, receiving a spectrum map f from Kℤ_{2}^{∧} [3]. Here X_{2}^{∧} denotes the Bousfield–Kan completion of X at two. We can define JKℤ_{2}^{∧} as the homotopy fiber of the composite map of spectra

[figure: ko; \overline{ψ^{3} − 1}; bspin; \overline{c}; bsu]

from connective real K-theory to three–connected complex K-theory, with ψ^{3} the Adams operation, and c complexification. Then JKℤ_{2}^{∧} is a covering of the pullback in the diagram of two–completed algebraic K-theory spectra induced by a square of ring homomorphisms :

ℤ  \overline{}ℝ

[figure]

ℤ_{3}  \overline{}ℂ

and f is the canonical map from Kℤ. Bökstedt proves, using étale methods, that ΩΩ^{∞}JKℤ_{2}^{∧} splits off ΩΩ^{∞}Kℤ_{2}^{∧} as a space, with ΩΩ^{∞}f as the retraction. Hence the spherical classes of H_{∗}(JKℤ_{2}^{∧}) split off those of H_{∗}(Kℤ).

Proof of Theorem 1.3. We will compare the Atiyah–Hirzebruch spectral sequences (AH–SS) E^{2} = H_{∗}(X; π_{∗}S) ⟹ π_{∗}(X) for the spectra F_{3}Kℤ, Kℤ and JKℤ_{2}^{∧}.

From [13] and [8] we know that the spectrum homology of Kℤ begins H_{∗}(Kℤ) ≅ (ℤ, 0, 0, ℤ/2, …), and there are no differentials in the AH–SS for Kℤ originating in total degree four or below. Let λ ∈ H_{3}(Kℤ) and η ∈ π_{1}(S) denote the nonzero elements.

By the assumed connectivity Conjecture 1.2, the inclusion map F_{3}Kℤ → Kℤ induces a five–connected map H_{∗}(F_{3}Kℤ) → H_{∗}(Kℤ). So H_{4}(Kℤ) = 0 and the free summand in H_{5}(F_{3}Kℤ) must map isomorphically to a free summand in H_{5}(Kℤ) (cf. [1]).

From the description above, it follows that

H_{∗}(JKℤ_{2}^{∧}) ≅ (ℤ_{2}^{∧}, 0, 0, ℤ/2, 0, ℤ_{2}^{∧}, …),

and

π_{∗}(JKℤ_{2}^{∧}) ≅ (ℤ_{2}^{∧}, ℤ/2, ℤ/2, ℤ/16, 0, ℤ_{2}^{∧}, …).

In particular, there are d^{2}-differentials hitting the nonzero classes in bidegrees (3, 1), (3, 2) and (4, 1) in the AH–SS for JKℤ_{2}^{∧}.

Since the ℤ/2 in H_{3}(JKℤ_{2}^{∧}) = E_{3,0}^{2} survives to E^{∞}, it is a spherical class, and thus splits off H_{3}(Kℤ). It follows that f is five–connected, and by the relative Hurewicz theorem H_{5}(Kℤ) → H_{5}(JKℤ_{2}^{∧}) ≅ ℤ is onto. The latter observation is due to Marcel Bökstedt.

Then there is a nonzero differential d^{2} : E_{5,0}^{2} → E_{3,1}^{2} in the AH–SS for Kℤ, factoring through H_{5}(f), the d^{2} for JKℤ_{2}^{∧}, and H_{3}(f)^{−1}. It is necessarily zero on torsion, and takes a generator of a free summand to ηλ.

Using the π_{∗}(S)-module structure on the AH–SS, η times this differential must hit η^{2}λ which generates E_{3,2}^{2} ≅ ℤ/2. However, we do not know if the differential ending in bidegree (4, 1) in the AH–SS for JKℤ_{2}^{∧} lifts to one for Kℤ.

In conclusion, the contributions to π_{∗}(F_{3}Kℤ) through degree five are: π_{∗}(S), the nontrivial extension π_{3}(S) → K_{3}(ℤ) → H_{3}(Kℤ), some possibly surviving two– torsion in E_{4,1}^{2} = H_{3}(Kℤ) ∗ π_{1}(S) ≅ ℤ/2, and twice the ℤ-generator plus some possible two–torsion remaining in E_{5,0}^{2} = H_{5}(F_{3}Kℤ). Theorem 1.3 follows. □

5. Computations at the second stage.

Let us settle on the following notation :

Notation 5.1. In GL_{2}ℤ let s = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and r = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) generate D_{2}, x = sr =

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) and y = rs = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) generate T_{2}, s and t = ( \begin{matrix}0 & 1 \\ −1 & 1 \end{matrix} ) generate SL_{2}ℤ, and

x, y and c = t^{−1}s^{−1} = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) generate P_{1}. s and t are of order four and six, with

s^{2} = t^{3}.

Let σ and ρ in H^{1}(D_{2}; ℤ/2) be dual to s and r, ξ and η in H^{1}(T_{2}; ℤ/2) be dual to x and y, and ξ, η and γ in H^{1}(P_{1}; ℤ/2) be dual to x, y and c.

Let ω ∈ H^{2}(D_{2}; ℤ/2) be the restriction of the second Stiefel–Whitney class from H^{2}(GL_{2}ℝ; ℤ/2).

The following cohomology algebras will be useful :

Lemma 5.2.

(1) H^{∗}(T_{2}) ≅ ℤ[t_{1}, t_{2}, t_{3}]/t_{3}^{2} = t_{1}t_{2}(t_{1} + t_{2}) where the generators (t_{1}, t_{2}, t_{3}) are of degrees (2, 2, 3) and of orders (2, 2, 2) respectively. Reduced modulo two, t_{1} = ξ^{2}, t_{2} = η^{2} and t_{3} = ξη(ξ + η).

(2) H^{∗}(GL_{2}ℤ)_{(2)} ≅ H^{∗}(D_{2}) ≅ ℤ[x_{1}, x_{2}, x_{3}, x_{4}]/x_{1}^{2} = x_{1}x_{2}, x_{3}^{2} = x_{2}x_{4} where the generators (x_{1}, x_{2}, x_{3}, x_{4}) are of degrees (2, 2, 3, 4) and of orders (2, 2, 2, 4) respectively. Reduced modulo two, x_{1} = σ^{2}, x_{2} = ρ^{2}, x_{3} = ρω and x_{4} = ω^{2}.

(3) H^{∗}(P_{1}) ≅ ℤ[p_{1}, p_{2}, p_{3}, p_{4}, p_{5}]/p_{3}^{2} = (p_{1} + p_{2})p_{3}, p_{3}p_{4} = p_{3}p_{5} = (p_{1} + p_{2})p_{5}, p_{4}^{2} = p_{1}p_{2}(p_{1} + p_{2}), p_{4}p_{5} = p_{5}^{2} = p_{1}p_{2}p_{3} where the generators (p_{1}, …, p_{5}) are of degrees (2, 2, 2, 3, 3) and of orders (2, 2, 2, 2, 2) respectively. Reduced modulo two, p_{1} = ξ^{2}, p_{2} = η^{2}, p_{3} = γ^{2}, p_{4} = ξη(ξ + η) and p_{5} = ξηγ.

Proof. (Note the implicit relations 2·t_{1} = 0 etc.) First compute the ℤ/2-cohomology algebras. The case T_{2} is easy, the case D_{2} is given in [6, p. 322] (σ^{2} = σρ), and for P_{1} use the Lyndon–Hochschild–Serre spectral sequence (LHS–SS) for the split extension ⟨c⟩ = ℤ → P_{1} → T_{2} which collapses at E_{2} = ℤ/2[ξ, η] ⊗ E(γ). The algebra structure (γ^{2} = ξγ + ηγ) will be forced by the result for D_{2} and the form of the algebra homomorphism induced by P_{1} ⊂ GL_{2}ℤ (see Lemma 5.3).

Next extract the integral information using a Bockstein spectral sequence [2]. Let β_{2^{n}} denote the Bockstein differentials killing elements of integral order 2^{n} (n ≥ 1). Then the β_{2} differentials square all degree one generators. The remaining nonzero differentials can be determined from the computations in [7], and are β_{2}(ω) = ρω and β_{4}(σω) = ω^{2}, both in the case of H^{∗}(D_{2}).

The two–primary identification of H^{∗}(D_{2}) and H^{∗}(GL_{2}ℤ) comes about by comparing the extensions ⟨s⟩ = ℤ/4 → D_{2} → ℤ/2 = ⟨r⟩ and ⟨s, t⟩ = SL_{2}ℤ → GL_{2}ℤ → ℤ/2, where the inclusion ℤ/4 ↪ SL_{2}ℤ ≅ ℤ/4 ∗_{ℤ/2} ℤ/6 induces a homology isomorphism at two.

The algebra relations given follow from those in ℤ/2-cohomology. An induction argument on the growth of the order of H^{∗}(P_{1}) shows that they form a complete set. It is easy to check that H^{∗}(T_{2}), H^{∗}(D_{2}) and H^{∗}(P_{1}) contain no odd torsion. □

Lemma 5.3. The inclusion P_{1} ⊂ GL_{2}ℤ induces the algebra homomorphism given by x_{1} ↦ p_{1} +p_{2} +p_{3}, x_{2} ↦ p_{1} +p_{2}, x_{3} ↦ p_{4} and x_{4} ↦ p_{1}p_{2} on integral cohomology. With ℤ/2-coefficients, the homomorphism is σ ↦ ξ + η + γ, ρ ↦ ξ + η and ω ↦ ξη.

Proof. Most of this follows from expressing the generators x, y and c in s, r and t. ω ∈ H^{2}(D_{2}; ℤ/2) comes from w_{2} ∈ H^{2}(GL_{2}ℝ; ℤ/2), and its restriction to P_{1} factors through the real parabolic Borel subgroup P_{1}ℝ ⊂ GL_{2}ℝ consisting of upper– triangular real matrices. The inclusion T_{2} ⊂ P_{1}ℝ is a homotopy equivalence, so the restriction of ω to P_{1} is the same as that coming from P_{1}ℝ ≃ T_{2}, namely ξη. Jørgen Tornehave supplied this argument. □

Staying with induced homomorphisms, we gather together :

Lemma 5.4. The inclusion k_{1} : T_{2} ↪ D_{2} induces x_{1} ↦ t_{1} + t_{2}, x_{2} ↦ t_{1} + t_{2}, x_{3} ↦ t_{3} and x_{4} ↦ t_{1}t_{2} on cohomology.

The injection k_{2} : T_{2} ↪ D_{2} taking x to s^{2} and y to r induces x_{1} ↦ 0, x_{2} ↦ t_{2}, x_{3} ↦ t_{3} and x_{4} ↦ t_{1}^{2} + t_{1}t_{2}.

Proof. This is easiest with ℤ/2-coefficients, where the classes in degree one are determined by what group elements they detect. The image of the second Stiefel– Whitney class is determined by its restrictions to the three order two subgroups of T_{2}, which can be found by the Cartan formula. □

The last groups in the GL_{2}ℤ PSS are found as follows :

Lemma 5.5. The LHS–SS for H_{∗}(D_{2}; W_{2}) using the split extension ℤ/4 → D_{2} → ℤ/2 collapses at E^{2}, and splits, i.e. there are no extensions in the E^{∞}-term. Thus :

H_{∗}(D_{2}; W_{2}) ≅ (ℤ/2, ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/2)^{3}, …)

Proof. Actually, we can compute H_{∗}(D_{2}; W_{2}) directly using Hamada’s resolution [7]. Alternatively, we can determine it up to extensions from the long exact sequence linking this group to H_{∗}(D_{2}) and H_{∗}(T_{2}), which appears in another guise as the D_{2} PSS abutting to zero. Comparing with the computation of H_{∗}(D_{2}; W_{2} ⊗ ℤ/2) ≅ H_{∗}(D_{2}; ℤ/2) above, it follows that there are no extensions, for any cyclic summands of order four or more in H_{∗}(D_{2}; W_{2}) would give too small a rank for H_{∗}(D_{2}; W_{2} ⊗ ℤ/2). Lastly, we can find the E^{2}-term via computing H_{∗}(ℤ/4; W_{2}) with the periodic resolution, and the collapsing and splitting results follow in view of the preceding remarks. □

We may now describe the complete two–primary behavior of the GL_{2}ℤ PSS.

Proof of Theorem 3.2 (1). We will compare the D_{2} PSS with the GL_{2}ℤ PSS :

H_{∗}(D_{2})  ⟵^{d^{1}} H_{∗}(T_{2}) ⟵^{d^{1}} H_{∗}(D_{2}; W_{2})
↓≅  ↓  ∥
H_{∗}(GL_{2}ℤ)_{(2)} ⟵^{d^{1}} H_{∗}(P_{1}) ⟵^{d^{1}} H_{∗}(D_{2}; W_{2})

Lemmas 5.2 and 5.5 give the E^{1}-term of the GL_{2}ℤ PSS. The left d^{1}-differential is given by Lemma 5.3, while the (graded) rank of the right d^{1}-differential agrees with that for the D_{2} PSS as H_{∗}(T_{2}) → H_{∗}(P_{1}) is a split injection. This in turn agrees with the rank of the kernel of H_{∗}(T_{2}) → H_{∗}(D_{2}) by exactness in the D_{2} PSS, which is found using Lemma 5.4 :

\begin{matrix}(ℤ/2)^{4} & ⟵^{4} & (ℤ/2)^{7} & ⟵^{2} & (ℤ/2)^{3} \\ (ℤ/2)^{2} & ⟵^{2} & (ℤ/2)^{4} & ⟵^{1} & (ℤ/2)^{3} \\ (ℤ/2)^{2} ⊕ ℤ/4 & ⟵^{3} & (ℤ/2)^{5} & ⟵^{1} & (ℤ/2)^{2} \\ ℤ/2 & ⟵^{1} & (ℤ/2)^{2} & ⟵^{0} & (ℤ/2)^{2} \\ (ℤ/2)^{2} & ⟵^{2} & (ℤ/2)^{3} & ⟵^{1} & ℤ/2 \\ ℤ_{(2)} & ⟵^{1} & ℤ_{(2)} & ⟵^{0} & ℤ/2 \end{matrix}

Here each morphism is superscripted by its rank, i.e. the rank of its image.

This gives the following E^{2}-term, where the classes in the zeroth column correspond to the cohomology classes 2x_{4}, 2x_{4}^{2}, …:

\begin{matrix}0 & ℤ/2 & ℤ/2 \\ 0 & ℤ/2 & (ℤ/2)^{2} \\ ℤ/2 & ℤ/2 & ℤ/2 \\ 0 & ℤ/2 & (ℤ/2)^{2} \\ 0 & 0 & 0 \\ 0 & 0 & ℤ/2 \end{matrix}

By exactness, these classes are hit by d^{2}-differentials in the D_{2} PSS, and by naturality, the same happens in the GL_{2}ℤ PSS.

This leaves us with the E^{∞}-term :

\begin{matrix}0 & ℤ/2 & ℤ/2 \\ 0 & ℤ/2 & (ℤ/2)^{2} \\ 0 & ℤ/2 & ℤ/2 \\ 0 & ℤ/2 & ℤ/2 \\ 0 & 0 & 0 \\ 0 & 0 & ℤ/2 \end{matrix}

giving H_{∗}(F_{2}Kℤ/F_{1}Kℤ; ℤ_{(2)}) ≅ (0, 0, ℤ/2, ℤ/2, #4, #4, …).

The classes surviving in E_{1,2}^{∞} and E_{1,3}^{∞} are easily seen to be dual to p_{5} ∈ H^{3}(P_{1}) and the sum of the duals of p_{1}p_{3} and p_{2}p_{3}, respectively. All the groups and ranks involved grow linearly with period dividing four, so this also applies to the abutment. A similar computation with ℤ/2-coefficients gives H_{n}(F_{2}Kℤ/F_{1}Kℤ; ℤ/2) ≅ (ℤ/2)^{n−1} for n ≥ 1, so all homology groups occurring are actually elementary abelian. □

The odd primary case is much easier :

Proof of Theorem 3.2 (2). Away from the prime two H_{∗}(P_{1}) ≅ H_{∗}(1), as is seen from the LHS–SS for ℤ → P_{1} → T_{2}, and H_{∗}(D_{2}; W_{2}) = 0 since D_{2} is a two–group. As for H_{∗}(GL_{2}ℤ), the inclusion ⟨t^{2}, b⟩ = Σ_{3} ⊂ GL_{2}ℤ induces a homology isomorphism away from two. This follows from the map of extensions from ℤ/3 → Σ_{3} → ℤ/2 to SL_{2}ℤ → GL_{2}ℤ → ℤ/2 taking ⟨t^{2}⟩ = ℤ/3 ↪ SL_{2}ℤ ≅ ℤ/4 ∗_{ℤ/2} ℤ/6, which induces an isomorphism away from two. Hence the E^{1}-term for H_{∗}(F_{2}Kℤ/F_{1}Kℤ) modulo two–torsion looks like :

H_{∗}(Σ_{3}) ⟵^{d_{1}} H_{∗}(1) ⟵^{d_{1}} 0

Thus H_{∗}(F_{2}Kℤ/F_{1}Kℤ; ℤ[\frac{1}{2}]) ≅ H̃_{∗}(Σ_{3}; ℤ[\frac{1}{2}]) ≅ (0, 0, 0, ℤ/3, 0, 0, 0, ℤ/3, 0, …). □

Proof of Proposition 3.3. First prove the case i = 1. This would follow from the connectivity Conjecture, but to make Theorem 1.1 independent of the Conjecture we provide a chain–level computation below. In the GL_{2}ℤ PSS,

d^{2} : ker d^{1} ≅ H_{∗}(D_{2})/ im H_{∗}(T_{2}) → coker d^{1} ≅ H_{∗}(GL_{2}ℤ)/ im H_{∗}(P_{1})

has kernel E_{2,∗−1}^{∞} = ker d^{2} ≅ H_{∗}(P_{1})/ im H_{∗}(T_{2}). The connecting map ∂ extends to H_{∗}(T_{2}) → H_{∗}(P_{1}) → H_{∗}(ℤ/2) where the composite is zero. Dually, by inspection of the S_{•}-construction, ∂^{∗}: H^{∗}(ℤ/2) ≅ ℤ[u]/2u → H^{∗}(P_{1}) is a sum of algebra homomorphisms. The case i = 1 shows that ∂^{∗}(u) = p_{3}; whence ∂^{∗}(u^{i}) = p_{3}^{i} ≠ 0 in general.

Let E denote the category (with cofibrations and weak equivalences) of finite based sets. Its K-theory spectrum KE is mapped to Kℤ by the functor taking a finite set to the free ℤ-module generated by its non-base point elements. The image in D^{n}(ℤ^{k}) is precisely the standard apartment ([14, §4]). KE admits a rank filtration and poset filtrations on the subquotients compatible with the map to Kℤ.

Let F_{1} and F_{2} denote the nth spaces of F_{1}Kℤ and F_{2}Kℤ, for n suitably large. We express the degrees of chains and homology groups relative to n. Let Φ_{0} ⊂ Φ_{1} ⊂ Φ_{2} = F_{2}/F_{1} denote the poset filtration, and let Φ̃_{i} be the preimage of Φ_{i} in F_{2}. Simplices in F_{1} involve a single line (free rank one ℤ-module), Φ_{0} includes simplices involving a single plane (free rank two ℤ-module), Φ_{1} adjoins configurations of a line contained in a plane, and Φ_{2} adjoins pairs of lines. Similarly write F_{i}E, Φ_{i}E and Φ̃_{i}E for the corresponding constructions in KE. There are splittings Φ̃_{0} ≃ F_{1} ∨ Φ_{0} and Φ̃_{0}E ≃ F_{1}E ∨ Φ_{0}E, and stably F_{2}E/F_{1}E ≃ ∗.

Start with the one–cycle b: ℤ^{2} ⟶^{r} ℤ^{2} generating H_{1}Φ_{0}E which lifts to b̃ ∈ H_{1}Φ̃_{0}E by the splitting. As F_{1}E → F_{2}E is a stable equivalence, the image of b̃ in F_{2}E is null–homologous; say ∂ã = b̃, and let ã ↦ a under F_{2}E → F_{2}E/F_{1}E. Then a is a two–chain in F_{2}E/F_{1}E with ∂a = b. The image of a in Φ_{2}E/Φ_{1}E generates, and its image in the K-theory of ℤ represents the nonzero class in E_{2,0}^{∞} of the GL_{2}ℤ PSS. Our aim is to compute the connecting map on this class.

We should consider a lifting of a to F_{2} with boundary contained in F_{1} (this is possible as the class of a survives to E^{∞}). Modulo odd torsion, b is exact when included into Φ_{0}, as r = t · cx, so the desired lifting of a is obtained by adjoining to ã a lifting of the class of cx in Φ_{1}/Φ_{0}. Call the resulting two–chain a′. ∂a′ is a cycle in F_{1}. By the splitting, ∂ã = b̃ makes no contribution. The chain cx = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} )

represents an automorphism of the flag ℤ^{1} ↪ ℤ^{2}, and its boundary contributes to F_{1} by the restrictions of this automorphism to the lines ℤ^{1} and ℤ^{2}/ℤ^{1}, i.e. the diagonal entries. Hence one maps to the generator, and the other to zero. Added together, the connecting map ∂ takes the generator a onto the generator of H_{1}(F_{1}Kℤ). □

6. Two–primary computations at the third stage.

The computations at the third stage are somewhat more involved, especially in the two–primary case. We present the calculation at two of the spectrum homology of the third subquotient of the rank filtration, using the SL_{3}ℤ poset spectral sequence. Thereafter we compute the connecting map from the third to the second subquotient. In the next Section we will we do the same at odd primes.

In some more detail, the computation of the E^{1}-term of the SL_{3}ℤ PSS at the prime two is accomplished by using Soulé’s computation [11] of H^{∗}(SL_{3}ℤ) and LHS–SS techniques for the various parabolic subgroups occurring. These methods suffice to describe most of the d^{1}-differentials needed, except d^{1} : E_{2,∗}^{1} → E_{1,∗}^{1}. For these differentials we use a description of the relevant parabolic subgroups as isometry groups of a tessellation of the plane (crystallographic groups) to compute their ℤ/2-cohomology algebras, from which the ranks of the differentials can be estimated. This suffices to describe the E^{2} = E^{∞}-term in the range we are studying.

We need further notation for 3×3 matrices, which is the default size for matrices in this Section :

Notation 6.1. We carry over the notation from 5.1 for matrices in GL_{2}ℤ to SL_{3}ℤ by means of a fixed embedding g ↦ ( \begin{matrix}det(g) & 0 \\ 0 & g \end{matrix} ). In addition we shall use the

matrices a = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and b = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ). We also use an involution on SL_{3}ℤ given by

g ↦ ḡ = (c_{(13)}g^{T})^{−1} which first reflects the matrix about a line perpendicular to the main diagonal, and then inverts the matrix. Here c_{g} denotes (left) conjugation h ↦ ghg^{−1}, while we use c^{g} to denote (right) conjugation h ↦ g^{−1}hg. The involution g ↦ ḡ takes SP_{1} to SP_{12}, and SP_{1,2} to SP_{1,12,13}. For instance x = (s̄)^{2} and y = r̄s̄.

Let Di_{n} denote the dihedral group with 2n elements. We view Di_{2} as embedded in Di_{4} = D_{2} as ⟨s^{2}, r⟩, and Di_{6} as embedded in GL_{2}ℤ as ⟨t, r⟩. Thus Di_{2} ⊂ Di_{6}.

We let ℤ_{det} and ℤ_{std}^{2} denote the determinant and standard representations of GL_{2}ℤ, or any of its subgroups. For a group on given generators let ℤ_{sgn(x,...,y)} denote the representation on ℤ where the subset {x, …, y} of generators act by reversing sign.

Let us begin with extending some of the group homology computations from the preceding Section to twisted coefficients.

Lemma 6.2. The following homology groups eventually grow linearly with period dividing four :

H_{∗}(T_{2}; ℤ_{det}) ≅ (ℤ/2, ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/2)^{3}, …)
H_{∗}(P_{1}(2); ℤ_{det}) ≅ (ℤ/2, ℤ + ℤ/2, (ℤ/2)^{4}, (ℤ/2)^{3}, (ℤ/2)^{6}, (ℤ/2)^{5}, (ℤ/2)^{8}, …)
H_{∗}(Di_{4}; ℤ_{det}) ≅ (ℤ/2, ℤ/4, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, ℤ/4 + (ℤ/2)^{2}, …)
H_{∗}(P_{1}(2); ℤ_{std}^{2}) ≅ (ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/2)^{4}, (ℤ/2)^{5}, (ℤ/2)^{6}, …)
H_{∗}(Di_{2}; ℤ_{std}^{2}) ≅ (ℤ/2, 0, ℤ/2, 0, ℤ/2, 0, …)
H_{∗}(Di_{4}; ℤ_{std}^{2}) ≅ (ℤ/2, ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/2)^{3}, …)
H_{∗}(Di_{6}; ℤ_{std}^{2})_{(2)} ≅ (0, 0, 0, 0, 0, 0, …)

and

H_{∗}(GL_{2}ℤ; ℤ_{det})_{(2)} ≅ H_{∗}(D_{2}; ℤ_{det}) = H_{∗}(Di_{4}; ℤ_{det})
H_{∗}(GL_{2}ℤ; ℤ_{std}^{2})_{(2)} ≅ (0, ℤ/2, ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{3}, …)

Proof. By ‘eventually growing linearly with period four’ we mean that, beginning at the end of the list given, H_{n+4} is abstractly the sum of H_{n} and some fixed group.

In all but the last two cases, these groups are found using the LHS–SS for one of the extensions C_{2} → T_{2} → ℤ/2, ℤ → P_{1}(2) → T_{2} or ℤ/n → Di_{n} → ℤ/2. The E^{2}- terms are found using the periodic resolutions for C_{2} or ℤ/n, or the short resolution for ℤ. Each spectral sequence can be seen to collapse at E^{2} = E^{∞} using that these extensions are split, and Hamada’s resolution [7] in the case of Di_{4} = D_{2}.

The last two cases follow from the amalgamation GL_{2}ℤ ≅ Di_{4}∗_{Di_{2}}Di_{6}, by noting that the inclusions induce an isomorphism H_{∗}(Di_{2}; ℤ_{det}) → H_{∗}(Di_{6}; ℤ_{det})_{(2)} and an injection H_{∗}(Di_{2}; ℤ_{std}^{2}) ↪ H_{∗}(Di_{4}; ℤ_{std}^{2}). □

We will study SP_{1,12} using an extension ℤ̌^{3} → SP_{1,12} → ST_{3}. Here ℤ̌^{3} = ⟨a, b, c⟩ fits into the extensions ⟨b⟩ = ℤ → ℤ̌^{3} → ℤ^{2} = ⟨a, c⟩, ⟨a, b⟩ = ℤ^{2} → ℤ̌^{3} → ℤ = ⟨c⟩, and similarly with a and c interchanged. ℤ̌^{3} acts freely and cocompactly on the contractible three–manifold ℝ̌^{3} consisting of upper–triangular real matrices with ones on the diagonal. The orbit space Bℤ̌^{3} is compact and orientable (admitting the seventh geometry in Thurston’s list [12]). By Poincaré duality and using b = [a, c], the one possible d^{2}-differential in the LHS–SS for the first extension is an isomorphism. Let a and c generate ℤ_{a} and ℤ_{c} and let their Poincaré duals generate ℤ_{bc} and ℤ_{ab} (when dualized back to homology).

Lemma 6.3. H_{∗}(ℤ̌^{3}) ≅ (ℤ, ℤ_{a} + ℤ_{c}, ℤ_{ab} + ℤ_{bc}, ℤ_{abc}). As ST_{3}-modules from the split extension

ℤ̌^{3} → SP_{1,12} → ST_{3}

ℤ_{a} ≅ ℤ_{bc} ≅ ℤ_{sgn(y)}, ℤ_{c} ≅ ℤ_{ab} ≅ ℤ_{sgn(x,y)} and ℤ ≅ ℤ_{abc} is the trivial module. In particular the orientation class U = [Bℤ̌^{3}] ∈ H_{3}(ℤ̌^{3}) is a generator, and is invariant under the ST_{3}-action. □

Recall from [11, Theorem 4] :

Theorem 6.4. (Soulé)

H_{∗}(SL_{3}ℤ) ≅ (ℤ, 0, (ℤ/2)^{2}, (ℤ/4)^{2} + (ℤ/3)^{2}, ℤ/2, (ℤ/2)^{4}, (ℤ/2)^{3}, …).

Soulé gives the whole cohomology algebra; we have merely listed these groups for concreteness.

Some other groups in the SL_{3}ℤ PSS have already been computed. Clearly GL_{2}ℤ ≅ SP_{12,3} and P_{1}(2) ≅ SP_{1,12,3} through the fixed embedding followed by the involution.

From [11, Proposition 2(iii)] we also recall :

Proposition 6.5. H^{∗}(SD_{3})_{(2)} ≅ ℤ[y_{1}, y_{2}, y_{3}]/y_{1}^{4} + y_{1}y_{2}^{2} + y_{1}^{2}y_{3} = 0 where the generators (y_{1}, y_{2}, y_{3}) are of degrees (2, 3, 4) and of orders (2, 2, 4) respectively.

Here we have used SD_{3} ≅ Σ_{4}. From this we may set up the E^{1}-term of the SD_{3} PSS, except that we do not know the fourth column H_{∗}(SD_{3}; W_{3}). These groups may presumably be found using a stable element argument as in [4, XII Theorem 10.1]. Instead we have used that the SD_{3} PSS abuts to zero and computed its d^{1}-differentials by methods similar to those we will use for the SL_{3}ℤ PSS below, to conclude :

Lemma 6.6.

H_{∗}(SD_{3}; W_{3})_{(2)} ≅ (0, ℤ/2, 0, ℤ/2, ℤ/2, ℤ/2, ℤ/2, (ℤ/2)^{2}, …)

and d^{1} : E_{4,∗}^{1} = H_{∗}(SD_{3}; W_{3})_{(2)} → E_{3,∗}^{1} = H_{∗}(ST_{3}) in the SD_{3} PSS is an injection in this range.

Proof. We omit the calculation, as it is rather similar to what follows, but simpler. We only point out that D_{2} ≅ S(ℤ/2 × D_{2}) and T_{2} ≅ ST_{3} by the fixed embedding. It turns out that E^{2} = 0 in this range. □

View ℤ^{2} as embedded in SL_{3}ℤ as ⟨a, b⟩ or ⟨b, c⟩ as appropriate.

Lemma 6.7. The LHS–SS for the split extension ℤ^{2} → SP_{1,12,13} → ST_{3} ≅ T_{2} collapses at E^{2} and splits. Similarly for ℤ^{2} → SP_{1,2} → ST_{3} :

(E^{∞})  \begin{matrix}ℤ/2 & ℤ/2 & (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{3} & (ℤ/2)^{3} \\ (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{4} & (ℤ/2)^{4} & (ℤ/2)^{6} & (ℤ/2)^{6} \\ ℤ & (ℤ/2)^{2} & ℤ/2 & (ℤ/2)^{3} & (ℤ/2)^{2} & (ℤ/2)^{4} \end{matrix}

So H_{∗}(SP_{1,12,13}) ≅ (ℤ, (ℤ/2)^{4}, (ℤ/2)^{4}, (ℤ/2)^{8}, (ℤ/2)^{8}, (ℤ/2)^{12}, …).

Proof. SP_{1,12,3} and SP_{1,2,13} split off SP_{1,12,13}, and force the collapse and splitting. □

Lemma 6.8. The LHS–SS for the split extension ℤ^{2} → SP_{1,12} → SP_{1,2,23} ≅ P_{1}(2) collapses at E^{2} and splits :

(E^{∞})  \begin{matrix}ℤ/2 & ℤ + ℤ/2 & (ℤ/2)^{4} & (ℤ/2)^{3} & (ℤ/2)^{6} & (ℤ/2)^{5} \\ ℤ/2 & (ℤ/2)^{2} & (ℤ/2)^{3} & (ℤ/2)^{4} & (ℤ/2)^{5} & (ℤ/2)^{6} \\ ℤ & (ℤ/2)^{3} & (ℤ/2)^{2} & (ℤ/2)^{5} & (ℤ/2)^{4} & (ℤ/2)^{7} \end{matrix}

Similarly for ℤ̌^{3} → SP_{1,12} → ST_{3} :

(E^{∞})  \begin{matrix}ℤ & (ℤ/2)^{2} & ℤ/2 & (ℤ/2)^{3} & (ℤ/2)^{2} & (ℤ/2)^{4} \\ (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{4} & (ℤ/2)^{4} & (ℤ/2)^{6} & (ℤ/2)^{6} \\ (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{4} & (ℤ/2)^{4} & (ℤ/2)^{6} & (ℤ/2)^{6} \\ ℤ & (ℤ/2)^{2} & ℤ/2 & (ℤ/2)^{3} & (ℤ/2)^{2} & (ℤ/2)^{4} \end{matrix}

So H_{∗}(SP_{1,12}) ≅ (ℤ, (ℤ/2)^{4}, (ℤ/2)^{5}, ℤ + (ℤ/2)^{9}, (ℤ/2)^{12}, (ℤ/2)^{15}, …).

Proof. To find the E^{2}-terms use Lemma 6.2. The conjugation action of SP_{1,2,23} makes H_{1}(ℤ^{2}) abstractly isomorphic to ℤ_{std}^{2} and H_{2}(ℤ^{2}) isomorphic to ℤ_{det}.

The inclusions SP_{1,12,13} ↪ SP_{1,12} and SP_{1,2} ↪ SP_{1,12} map onto the zeroth through second rows of the latter spectral sequence, so by Lemma 6.7 no differentials may originate in these rows. We claim that the orientation class U generating E_{0,3}^{2} ≅ ℤ survives to E^{∞}, from which it follows (by considering the H^{∗}(ST_{3})- module structure in the cohomological case) that no differentials originate in the third row either. Thus E^{2} = E^{∞}. To prove the claim we compare BSP_{1,12} with the total space of an orientable sphere bundle over BST_{3}. View the unit cube [0, 1]^{3} as a fundamental domain in ℝ̌^{3} and construct a ST_{3}-equivariant map Bℤ̌^{3} → S^{3} by collapsing its boundary to a point. By inspection, ST_{3} acts orientably on S^{3}, and U maps to the orientation class of S^{3}, which by the Thom isomorphism survives to the homology of the total space. Thus U survives in H_{∗}(SP_{1,12}).

For the splitting results, there can be no extensions in the zeroth through second row in the latter spectral sequence by the same comparisons we used to show collapsing, and Lemma 6.7. Also there can be no extensions from the third to zeroth or first row by the splittings used in the proof of Lemma 6.7. Lastly, there can be no extensions from the third to second row as these groups appear without extensions in the second row of the former spectral sequence (and for ℤ^{2} → SP_{1,12} → SP_{1,12,3}). □

Note. For the remainder of this Section, all homology groups are given modulo odd torsion.

Lemma 6.9. The LHS–SS for the split extension ℤ^{2} → SP_{1} → SP_{1,23} ≅ GL_{2}ℤ collapses at E^{2} and splits :

(E^{∞})  \begin{matrix}ℤ/2 & ℤ/4 & (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{3} & ℤ/4 + (ℤ/2)^{2} \\ 0 & ℤ/2 & ℤ/2 & (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{3} \\ ℤ & (ℤ/2)^{2} & ℤ/2 & ℤ/4 + (ℤ/2)^{2} & (ℤ/2)^{2} & (ℤ/2)^{4} \end{matrix}

So H_{∗}(SP_{1}) ≅ (ℤ, (ℤ/2)^{2}, (ℤ/2)^{3}, (ℤ/4)^{2} + (ℤ/2)^{3}, (ℤ/2)^{6}, (ℤ/2)^{8}, …).

Proof using Lemma 6.16. The E^{2}-term is computed using Lemma 6.2. To prove collapsing and splitting we will compare with the crystallographic computation of H^{∗}(SP_{1}; ℤ/2) in Lemma 6.16. The E^{2}-term above poses an upper bound on the (graded) rank of the group H_{∗}(SP_{1}), which agrees with the lower bound given there. Any differentials or extensions would have reduced this rank, and can therefore not occur. □

We are now able to list the E^{1}-term for the SL_{3}ℤ PSS :

2^{3} ⟵^{3}  2^{9} ⊕ 2^{9}
2^{4} ⟵^{4}  2^{8} ⊕ 2^{8}  ⟵^{12}  2^{15} ⊕ 2^{4}  ⟵^{6} 2^{7} ⟵^{1} 2
2 ⟵^{1}  2^{6} ⊕ 2^{6}  ⟵^{11}  2^{12} ⊕ 2^{2}  ⟵^{3} 2^{4} ⟵^{1} 2
4^{2} ⟵^{2.0} 4^{2} + 2^{3} ⊕ 4^{2} + 2^{3} ⟵^{2.6} ℤ + 2^{9} ⊕ 4 + 2^{2} ⟵^{4} 2^{5} ⟵^{1} 2
2^{2} ⟵^{2}  2^{3} ⊕ 2^{3}  ⟵^{4}  2^{5} ⊕ 2  ⟵^{2} 2^{2} ⟵^{0} 0
0 ⟵^{0}  2^{2} ⊕ 2^{2}  ⟵^{4}  2^{4} ⊕ 2^{2}  ⟵^{2} 2^{3} ⟵^{1} 2
ℤ ⟵^{1}  ℤ ⊕ ℤ  ⟵^{1}  ℤ ⊕ ℤ  ⟵^{1} ℤ ⟵^{0} 0

We have not yet computed the d^{1}-differentials given here, but include them for convenience. Here ‘4^{n} + 2^{m}’ is short for a group (ℤ/4)^{n} + (ℤ/2)^{m}, a differential superscript ‘n’ means the image has rank n, while a superscript ‘n.m’ means the image is isomorphic to (ℤ/4)^{n} + (ℤ/2)^{m}. We use ‘+’ to denote direct sum within a homology group, and ‘⊕’ to separate summands of the PSS.

Let d_{s}^{1} denote the differential d_{s}^{1} : E_{s,∗}^{1} → E_{s−1,∗}^{1}.

Lemma 6.10. (1) d_{1}^{1} is surjective. (2) d_{4}^{1} is injective (through homological degree seven).

Proof. In view of [11, Theorem 4(ii) and Proposition 3(i)] the cohomology of SL_{3}ℤ is detected on the subgroups S(ℤ/2×D_{2}) ⊂ SD_{3} = [O] and [M′P] ≅ Di_{4}, adapting Soulé’s notation somewhat. These are subconjugate in SL_{3}ℤ to SP_{1} (c^{a^{T}} [M′P] ⊂ SP_{1}), so H^{∗}(SL_{3}ℤ) ↪ H^{∗}(SP_{1}) injects and part (1) of the Lemma follows. Part (2) is immediate from Lemma 6.6 and the splitting of ST_{3} off SP_{1,12,3}. □

Lemma 6.11. The image of the differential d_{3}^{1} is isomorphic to

(ℤ, (ℤ/2)^{2}, (ℤ/2)^{2}, (ℤ/2)^{4}, (ℤ/2)^{3}, (ℤ/2)^{6}, …).

Proof. Compute the dual differential in cohomology :

d_{1}^{3,∗}: H^{∗}(SP_{1,12}) ⊕ H^{∗}(SP_{12,3}) → H^{∗}(SP_{1,12,3})

On the second summand, the map is given by Lemma 5.3. On the first summand, the map is the signed sum (c^{e})^{∗} − (c^{ϕψ})^{∗} + (c^{ψ^{2}})^{∗} followed by restriction over i: SP_{1,12,3} → SP_{1,12}. We choose a (possibly redundant) set of algebra generators t_{1}, t_{2}, t_{3}, p_{3}, p̄_{3}, p_{5}, p̄_{5}, q_{3}, q̄_{3}, q_{5}, q̄_{5}, and U for H^{∗}(SP_{1,12}) in bidegrees (2, 0), (2, 0), (3, 0), (1, 1), (1, 1), (2, 1), (2, 1), (1, 2), (1, 2), (2, 2), (2, 2) and (0, 3) in the LHS–SS for the extension ℤ̌^{3} → SP_{1,12} → ST_{3}; compare Lemma 6.8. We may assume that the t’s, p’s and p̄’s restrict to the generators of Lemma 5.2 by the obvious restrictions. Also let (x̄_{1}, …, x̄_{4}) denote the generators for H^{∗}(SP_{12,3}), and (p̄_{1}, …, p̄_{5}) the generators for H^{∗}(SP_{1,12,3}). In particular, p̄_{1}, p̄_{2} and p̄_{3} are the Bockstein images of classes detecting x̄ = xy, ȳ = y and c̄ = a^{−1}.

Then by Lemma 5.3 x̄_{1} ↦ p̄_{1} + p̄_{2} + p̄_{3}, x̄_{2} ↦ p̄_{1} + p̄_{2}, x̄_{3} ↦ p̄_{4} and x̄_{4} ↦ p̄_{1}p̄_{2}. By explicit calculation with ℤ/2–coefficients we find :

(c^{e}i)^{∗}: (t_{1}, t_{2}, t_{3}, p_{3}, p̄_{3}, p_{5}, p̄_{5}) ↦ (p̄_{1}, p̄_{1} + p̄_{2}, p̄_{4}, 0, p̄_{3}, 0, p̄_{5})
(c^{ϕψ}i)^{∗}: (t_{1}, t_{2}, t_{3}, p_{3}, p̄_{3}, p_{5}, p̄_{5}) ↦ (p̄_{1} + p̄_{2}, p̄_{1}, p̄_{4}, 0, 0, 0, 0)
(c^{ψ^{2}}i)^{∗}: (t_{1}, t_{2}, t_{3}, p_{3}, p̄_{3}, p_{5}, p̄_{5}) ↦ (p̄_{2}, p̄_{1}, p̄_{4}, p̄_{3}, 0, p̄_{5}, 0)

The other generators map to zero.

Granted this, it is straightforward to list generators and maps for d_{1}^{3} in low degrees, giving the image stated above. □

We shall now consider cocompact group actions of our parabolic groups on the plane. We obtain a lower bound for the rank of d_{2}^{1} by computing the composite H_{∗}(SP_{1,12,13}; ℤ/2) ⊕ H_{∗}(SP_{1,2}; ℤ/2) → H_{∗}(SP_{1,12}; ℤ/2) → H_{∗}(SP_{1}; ℤ/2). The corresponding map into H_{∗}(SP_{12}; ℤ/2) will be obtained by applying the involution.

Notation 6.12. Let X = (0, 0), Y = (1/2, 0), Z = (1/2, 1/2), U = (0, 1/2), V = (1/3, 1/3) and W = (1, 0) be points in the xy-plane. There is an action of SP_{1} on this plane defined as follows : a and b translate by unit vectors in the xand y-directions, r reflects in the line x + y = 0, s rotates counterclockwise by π/2 radians, and t applies the matrix ( \begin{matrix}1 & 1 \\ −1 & 0 \end{matrix} ). Let ℤ^{2} ⋉ Di_{n} denote the subgroup of SP_{1} corresponding to our embeddings Di_{n} ↪ GL_{2}ℤ for n = 2, 4, 6.

Let [X] denote the isotropy group at X for the ℤ^{2} ⋉ Di_{4} action on the plane at X, and similarly for the other letters. Let [XY] denote the isotropy of the edge from X to Y, etc. Let [X]^{∗}, [X]^{∨} and [X]^{∘} denote the isotropy groups for the ℤ^{2} ⋉ Di_{2}, ℤ^{2} ⋉ Di_{6} and SP_{1,12,13} actions, respectively. Further, let [X]′ etc. denote the subgroups of SP_{1,2} corresponding to [X]^{∘} under the involution. We shall use these accents throughout the following calculations to separate the cases.

The top square of the following diagram is a pullback square due to the amalgamation GL_{2}ℤ ≅ Di_{4} ∗_{Di_{2}} Di_{6} :

(∗)  \begin{matrix}H^{∗}(ℤ^{2} ⋉ Di_{6}; ℤ/2)^{∨} & ⟶ & H^{∗}(ℤ^{2} ⋉ Di_{2}; ℤ/2)^{∗} \\ ↑ & & ↑ \\ H^{∗}(SP_{1}; ℤ/2) & ⟶ & H^{∗}(ℤ^{2} ⋉ Di_{4}; ℤ/2) \\ ↓ & & ↓ \\ H^{∗}(SP_{1,2}; ℤ/2)′ & & H^{∗}(SP_{1,12,13}; ℤ/2)^{∘} \end{matrix}

Lemma 6.13. (1) △XY Z is a fundamental domain for the action of ℤ^{2} ⋉ Di_{4} on the plane. [X] = ⟨s, r⟩ ≅ D_{2}. [Y] = ⟨sr, ars⟩ ≅ T_{2}. [Z] = ⟨as, abr⟩ ≅ D_{2}.

[XY] = ⟨sr⟩ ≅ ℤ/2. [XZ] = ⟨rs⟩ ≅ ℤ/2. [Y Z] = ⟨ars⟩ ≅ ℤ/2.

(2) □XY ZU is a fundamental domain for the action of SP_{1,12,13} on the plane. [X]^{∘} = ⟨sr, rs⟩ ≅ T_{2}. [Y]^{∘} = ⟨sr, ars⟩ ≅ T_{2}. [Z]^{∘} = ⟨bsr, ars⟩ ≅ T_{2}. [U]^{∘} = ⟨rs, bsr⟩ ≅ T_{2}. [XY]^{∘} = ⟨sr⟩ ≅ ℤ/2. [Y Z]^{∘} = ⟨ars⟩ ≅ ℤ/2. [UZ]^{∘} = ⟨bsr⟩ ≅ ℤ/2. [XU]^{∘} = ⟨rs⟩ ≅ ℤ/2.

(3) △XY V is a fundamental domain for the action of ℤ^{2} ⋉ Di_{6} on the plane. [X]^{∨} = ⟨t, r⟩ ≅ Di_{6}. [Y]^{∨} = ⟨rt^{2}, at^{3}⟩ ≅ T_{2}. [V]^{∨} = ⟨brt, bt^{2}⟩ ≅ Σ_{3}. [XY]^{∨} = ⟨rt^{2}⟩ ≅ ℤ/2. [XV]^{∨} = ⟨rt^{3}⟩ ≅ ℤ/2. [Y V]^{∨} = ⟨art^{5}⟩ ≅ ℤ/2.

(4) △XW Z is a fundamental domain for the action of ℤ^{2} ⋉ Di_{2} on the plane. [X]^{∗} = ⟨s^{2}, r⟩ ≅ T_{2}. [Y]^{∗} = ⟨as^{2}⟩ ≅ ℤ/2. [Z]^{∗} = ⟨abr, rs^{2}⟩ ≅ T_{2}. ([W]^{∗} = c_{a}[X]^{∗}.) [XY]^{∗} = 1. [XZ]^{∗} = ⟨rs^{2}⟩ ≅ ℤ/2. [ZW]^{∗} = ⟨abr⟩ ≅ ℤ/2.

(5) Under the involution, the isotropy subgroups of SP_{1,12,13} map to the following subgroups of SP_{1,2} : [X]′ = ⟨s^{2}, rs⟩ ⊂ [X]. [Y]′ = ⟨s^{2}, strs⟩ ∼ ⟨s^{2}, rs^{2}⟩ ⊂ [X]. [Z]′ = ⟨b^{−1}s^{2}, strs⟩ ∼ ⟨abs^{2}, rs^{2}⟩ ⊂ [Z]. [U]′ = ⟨rs, b^{−1}s^{2}⟩ ∼ ⟨sr, as^{2}⟩ = [Y]. Here ‘∼’ denotes (right) conjugation in SP_{1} by srtrs^{−1}, b^{−1}srtrs^{−1} and b^{−1}s respectively in the three cases. □

In the Lemma below we have named the generators of the ℤ/2-cohomology algebras of the groups above as follows : For a dihedral group of order eight like [X], with given group generators ⟨s, r⟩ with s of order four, take cohomology generators x_{1}, x_{2}, x_{3} corresponding to σ, ρ and ω in Notation 5.1. H^{∗}([X]; ℤ/2) = ℤ/2[x_{1}, x_{2}, x_{3}]/x_{1}^{2} = x_{1}x_{2}. For a Klein four group (or dihedral group of order twelve) like [Y], with given group generators ⟨x, y⟩, take cohomology generators y_{1} and y_{2} corresponding to ξ and η. H^{∗}([Y]; ℤ/2) = ℤ/2[y_{1}, y_{2}]. For a symmetric group on three letters like [V]^{∨} call the cohomology generator v_{1}^{∨}. Use analogous notation for the other letters and accents.

Lemma 6.14. (1) H^{∗}(ℤ^{2} ⋉ Di_{4}; ℤ/2) ≅ ℤ/2[g_{1}, g_{2}, g_{3}, g_{4}, g_{5}]/g_{1}^{2} = g_{1}g_{2}, g_{1}g_{3} = g_{2}g_{3}, g_{3}g_{4} = 0, g_{1}g_{5} = g_{3}g_{5}, g_{4}g_{5} = 0 with generators g_{1} = x_{1} + y_{1} + y_{2} + z_{1}, g_{2} = x_{2} + y_{1} + y_{2} + z_{2}, g_{3} = y_{2} + z_{1}, g_{4} = x_{3} and g_{5} = z_{3} in degrees (1, 1, 1, 2, 2). g_{1}, g_{2} and g_{3} detect s, r and a respectively.

(2) H^{∗}(SP_{1,12,13}; ℤ/2) ≅ ℤ/2[g_{1}^{∘}, g_{2}^{∘}, g_{3}^{∘}, g_{4}^{∘}]/g_{1}^{∘}g_{3}^{∘} = g_{2}^{∘}g_{4}^{∘} = 0 with generators g_{1}^{∘} = x_{1}^{∘} + y_{1}^{∘}, g_{2}^{∘} = y_{2}^{∘} + z_{2}^{∘}, g_{3}^{∘} = z_{1}^{∘} + u_{2}^{∘} and g_{4}^{∘} = x_{2}^{∘} + u_{1}^{∘} all in degree one.

(3) H^{∗}(ℤ^{2} ⋉ Di_{6}; ℤ/2) ≅ ℤ/2[g_{1}^{∨}, g_{2}^{∨}, g_{3}^{∨}] (no relations) with generators g_{1}^{∨} = x_{1}^{∨} + y_{2}^{∨} + v_{1}^{∨}, g_{2}^{∨} = x_{2}^{∨} + y_{1}^{∨} + v_{1}^{∨} and g_{3}^{∨} = y_{1}^{∨}y_{2}^{∨} + (y_{2}^{∨})^{2} in degrees (1, 1, 2). g_{1}^{∨} and g_{2}^{∨} detect t and r respectively.

(4) H^{∗}(ℤ^{2} ⋉ Di_{2}; ℤ/2) ≅ ℤ/2[g_{1}^{∗}, g_{2}^{∗}, g_{3}^{∗}, g_{4}^{∗}]/g_{1}^{∗}g_{3}^{∗} = (g_{3}^{∗})^{2}, g_{2}^{∗}g_{3}^{∗} = 0, g_{3}^{∗}g_{4}^{∗} = 0, g_{1}^{∗}(g_{1}^{∗} + g_{2}^{∗})g_{4}^{∗} = (g_{4}^{∗})^{2} with generators g_{1}^{∗} = x_{1}^{∗} + y_{1}^{∗} + z_{2}^{∗}, g_{2}^{∗} = x_{2}^{∗} + z_{1}^{∗} + z_{2}^{∗}, g_{3}^{∗} = y_{1}^{∗} and g_{4}^{∗} = z_{1}^{∗}z_{2}^{∗} in degrees (1, 1, 1, 2). g_{1}^{∗}, g_{2}^{∗} and g_{3}^{∗} detect s^{2}, r and a respectively.

(5) H^{∗}(SP_{1,2}; ℤ/2) ≅ ℤ/2[g′_{1}, g′_{2}, g′_{3}, g′_{4}]/g′_{1}g′_{3} = g′_{2}g′_{4} = 0 with generators g′_{1} = x′_{1} + y′_{1}, g′_{2} = y′_{2} + z′_{2}, g′_{3} = z′_{1} + u′_{2} and g′_{4} = x′_{2} + u′_{1} all in degree one.

Proof. We do case (1). By comparing with Lemma 6.2, all differentials in the spectral sequence associated with the cellular filtration on △XY Z have maximal rank, so there is an exact sequence

0 → H^{∗}(ℤ^{2} ⋉ Di_{4}; ℤ/2) → H^{∗}([X]; ℤ/2) ⊕ H^{∗}([Y]; ℤ/2) ⊕ H^{∗}([Z]; ℤ/2) → H^{∗}([XY]; ℤ/2) ⊕ H^{∗}([XZ]; ℤ/2) ⊕ H^{∗}([Y Z]; ℤ/2) → H^{∗}(1; ℤ/2) → 0 where the maps are easily computed. The structure of H^{∗}(ℤ^{2} ⋉ Di_{4}; ℤ/2) follows. □

Lemma 6.15. (1) SP_{1,12,13} → ℤ^{2} ⋉ Di_{4} induces g_{1} ↦ g_{1}^{∘} + g_{2}^{∘} + g_{3}^{∘} + g_{4}^{∘}, g_{2} ↦ g_{1}^{∘} + g_{2}^{∘} + g_{3}^{∘} + g_{4}^{∘}, g_{3} ↦ g_{2}^{∘} + g_{3}^{∘}, g_{4} ↦ g_{1}^{∘}g_{4}^{∘} and g_{5} ↦ g_{2}^{∘}g_{3}^{∘}.

(2) ℤ^{2} ⋉ Di_{2} → ℤ^{2} ⋉ Di_{6} induces an injection g_{1}^{∨} ↦ g_{1}^{∗}, g_{2}^{∨} ↦ g_{2}^{∗} and g_{3}^{∨} ↦ (g_{3}^{∗})^{2} + g_{4}^{∗} with cokernel of rank one in every odd degree.

(3) ℤ^{2} ⋉ Di_{2} → ℤ^{2} ⋉ Di_{4} induces g_{1} ↦ 0, g_{2} ↦ g_{2}^{∗}, g_{3} ↦ g_{3}^{∗}, g_{4} ↦ (g_{1}^{∗})^{2} + g_{1}^{∗}g_{2}^{∗} + (g_{3}^{∗})^{2} + g_{4}^{∗} and g_{5} ↦ g_{4}^{∗}.

Proof. The homomorphisms are found by detecting what happens on the isotropy subgroups, using the exact sequence(s) of the previous proof. Case (2) makes use of the observation that c_{bt}[Y]^{∨} = [Z]^{∗}. Lemma 5.4 becomes useful at this stage. We omit the details. □

We may now use the pullback square in (∗) to determine the ℤ/2-cohomology of SP_{1}. As the map in Lemma 6.15 (2) is an injection with cokernel of rank one in odd degrees, so is H^{∗}(SP_{1}; ℤ/2) ↪ H^{∗}(ℤ^{2} ⋉ Di_{4}; ℤ/2). The Bockstein operations on H^{∗}(SP_{1}; ℤ/2) are determined by those on the isotropy subgroups, and we recover the order two elements in H^{∗}(SP_{1}) as the kernel of β_{2}.

Lemma 6.16. H^{∗}(SP_{1}; ℤ/2) ≅ (ℤ/2, (ℤ/2)^{2}, (ℤ/2)^{5}, (ℤ/2)^{8}, (ℤ/2)^{11}, (ℤ/2)^{14}, …) (growing linearly) is generated by g_{1}, g_{2}, g_{4}, g_{5} + g_{3}^{2} and g_{1}(g_{3})^{i} for i ≥ 1. □

Lemma 6.17. SP_{1,2} → SP_{1} induces g_{1} ↦ g′_{4}, g_{2} ↦ g′_{2} + g′_{4}, g_{4} ↦ (g′_{1})^{2} + g′_{1}g′_{2} + g′_{1}g′_{4}, g_{5} + g_{3}^{2} ↦ g′_{2}g′_{3} + (g′_{3})^{2} and g_{1}(g_{3})^{i} ↦ (g′_{3})^{i}g′_{4} for i ≥ 1.

Proof. The image of each class in H^{∗}(SP_{1}; ℤ/2) in H^{∗}(SP_{1,2}; ℤ/2) is determined by the restrictions to [X]′, [Y]′, [Z]′ and [U]′. These can be found using the subconjugacies listed in Lemma 6.13 (5). □

This information suffices to write down the composite H^{∗}(SP_{1}) ⊕ H^{∗}(SP_{12}) → H^{∗}(SP_{1,12}) → H^{∗}(SP_{1,12,13}) ⊕ H^{∗}(SP_{1,2}) on the order two elements. On the first summand, the map is determined by Lemma 6.15 (1) and Lemma 6.17, using the Bockstein structure. The involution takes the basis for H^{∗}(SP_{1}) to a basis for H^{∗}(SP_{12}) and interchanges the bases for H^{∗}(SP_{1,12,13}) and H^{∗}(SP_{1,2}), while respecting the map. Therefore the matrix form of the map on the second summand follows directly from that of the first. The combined matrix will have the form

( \begin{matrix}A & B \\ B & A \end{matrix} ).

We have computed these matrices through cohomological degree six, finding :

Lemma 6.18. Restricted to the order two elements, the composite H^{∗}(SP_{1}) ⊕ H^{∗}(SP_{12}) → H^{∗}(SP_{1,12}) → H^{∗}(SP_{1,12,13}) ⊕ H^{∗}(SP_{1,2}) has ranks (4, 4, 8, 11, 12) in cohomological degrees two through six. □

Proposition 6.19. In the SL_{3}ℤ PSS :

(1) Through homological degree five, d_{2}^{1} maps onto the order two elements in ker d_{1}^{1}.

(2) In homological degree three, the Thom class U ∈ H_{3}(SP_{1,12}) and a ℤ/4generator in H_{3}(SP_{12,3}) map to independent elements of order four in E_{1,3}^{1}.

(3) There exists an order two element V ∈ H_{3}(SP_{1,12}) such that d_{2}^{1}(2U) = d_{2}^{1}(V). It is the image under inclusion of the dual of (g_{2}^{∘})^{2}(g_{3}^{∘})^{2} in H_{3}(SP_{1,12,13}).

Proof. Part (1) is the outcome of the preceding discussion.

Part (2) is proved by considering the map from the first spectral sequence in Lemma 6.8 to that in Lemma 6.9. The map induced on the second row is H_{∗}(P_{1}(2); ℤ_{det}) → H_{∗}(GL_{2}ℤ; ℤ_{det}), which in degree one takes the class of c which generates a free summand in H_{1}(P_{1}(2); ℤ_{det}) to the class of t^{−1}s^{−1}, which modulo three–torsion equals the class of s^{−1} in H_{1}(D_{2}; ℤ_{det}), of order four. On the other hand, the subconjugation SP_{12,3} ↪ SP_{1} maps isomorphically to the zeroth row of the spectral sequence in Lemma 6.9, converging to H_{∗}(SP_{1}).

The existence in part (3) follows from part (1), but we need the more precise definition of the class V. Twice the order four classes in H_{3}(SP_{1}) are dual to g_{4}^{2} and g_{5}^{2} + g_{3}^{4}, as is seen from the Bockstein structure. The latter class is the one hit from H_{1}(GL_{2}ℤ; ℤ_{det}) (cf. the description of g_{5} in Lemma 6.14 (1)). From the matrix form of the map H^{4}(SP_{1}; ℤ/2)⊕H^{4}(SP_{12}; ℤ/2) → H^{4}(SP_{1,12,13}; ℤ/2)⊕H^{4}(SP_{1,2}; ℤ/2), we may see that the dual of (g_{2}^{∘})^{2}(g_{3}^{∘})^{2} is mapped diagonally to the duals of g_{5}^{2} + g_{3}^{4} and ḡ_{5}^{2} + ḡ_{3}^{4} respectively. Thus V is precisely the image in H_{3}(SP_{1,12}) of the dual of (g_{2}^{∘})^{2}(g_{3}^{∘})^{2}. □

Proof of Theorem 3.4 (1). The outcome of the computations above is that through total degree six in the SL_{3}ℤ PSS, the only group surviving to E^{2} (and also to E^{∞}) is the ℤ-summand generated by 2U+V in bidegree (2, 3). Hence H_{∗}(D(ℤ^{3})/hSL_{3}ℤ) ≅ (0, 0, 0, 0, 0, ℤ, 0, …), and using

F_{3}Kℤ/F_{2}Kℤ ≅ D(ℤ^{3})/hSL_{3}ℤ ∧ BC_{3+}

from Section 2, the Theorem follows. □

Proof of Proposition 3.5 (1). By inspection of the S_{•}-construction, the connecting map ∂: H_{5}(F_{3}Kℤ/F_{2}Kℤ) → H_{4}(F_{2}Kℤ/F_{1}Kℤ) is a restriction of a homomorphism (π_{12} + π_{23})_{∗}: H_{3}(SP_{1,12}) → H_{3}(P_{1}(2)). Here π_{ij} projects SP_{1,12} → P_{1}(2) to the elements in the ith or jth rows and columns. We are interested in the image of the class 2U + V ∈ H_{3}(SP_{1,12}), where U is the Thom class and V is the image of the dual to (g_{2}^{∘})^{2}(g_{3}^{∘})^{2}.

π_{12} and π_{23} take U to zero by Lemma 6.8.

(g_{2}^{∘})^{2}(g_{3}^{∘})^{2} is detected on [Z]^{∘}, and by computing the maps π_{ij}: [Z]^{∘} → P_{1}(2) we find that the dual of (g_{2}^{∘})^{2}(g_{3}^{∘})^{2} maps respectively to the sum of the duals of p_{1}p_{2}, p_{1}p_{3} and p_{2}p_{3}, and to the dual of p_{1}p_{2} in H_{3}(P_{1}(2)). Thus ∂(2U + V) is the sum of the duals of p_{1}p_{3} and p_{2}p_{3} in H_{3}(P_{1}(2)), which is in ker d^{1} : H_{3}(P_{1}(2)) → H_{3}(GL_{2}ℤ) but not in im d^{1} : H_{3}(D_{2}; W_{2}) → H_{3}(P_{1}(2)). In short, this class is nonzero in E_{1,3}^{∞} and thus in H_{4}(F_{2}Kℤ/F_{1}Kℤ). □

7. Odd primary computations at the third stage.

We compute the SL_{3}ℤ PSS at odd primes. In view of the splitting GL_{3}ℤ = SL_{3}ℤ × C_{3}, this is the same as the GL_{3}ℤ PSS. In this Section all homology groups are given modulo two–torsion.

Lemma 7.1.

H_{∗}(SL_{3}ℤ) ≅ (ℤ, 0, 0, (ℤ/3)^{2}, 0, 0, 0, (ℤ/3)^{2}, …).

H_{∗}(SP_{1}) ≅ H_{∗}(SP_{12}) ≅ (ℤ, 0, 0, (ℤ/3)^{2}, 0, 0, 0, (ℤ/3)^{2}, …).

H_{∗}(SP_{12,3}) ≅ (ℤ, 0, 0, ℤ/3, 0, 0, 0, ℤ/3, …). The subconjugation SP_{12,3} ↪ SP_{1} induces an injection.

H_{∗}(SP_{1,12}) ≅ (ℤ, 0, 0, ℤ, 0, 0, 0, 0, …). H_{3}(SP_{1,12}) is generated by the Thom class U, which is mapped to a class of order three in H_{3}(SP_{1}), independent of the image of H_{∗}(SP_{12,3}).

H_{∗}(SP_{1,12,3}) ≅ H_{∗}(1).

H_{∗}(SD_{3}; W_{3}) ≅ (ℤ/3, 0, 0, 0, ℤ/3, 0, 0, 0, …).

All these groups are eventually repeating with period four.

Proof. The result for H_{∗}(SL_{3}ℤ) is from [11].

SP_{1} and SP_{12} are isomorphic under the involution. The LHS–SS for the split extension ⟨a, b⟩ = ℤ^{2} → SP_{1} → GL_{2}ℤ collapses at E^{2} and splits, as the first row is zero. The zeroth row is isomorphic to H_{∗}(GL_{2}ℤ) and the second row is isomorphic to H_{∗}(GL_{2}ℤ; ℤ_{det}), both of which are easily computed.

For H_{∗}(SP_{1,12}) we use the split extension ⟨a, b⟩ = ℤ^{2} → SP_{1,12} → P_{1}(2) which maps to the one used for SP_{1}. The LHS–SS collapses at E^{2} and splits, in view of the P_{1}(2)-homology computations in Lemmas 5.2 and 6.2. In bidegree (1, 2) the inclusion SP_{1,12} ⊂ SP_{1} induces a surjection H_{1}(P_{1}(2); ℤ_{det}) ↠ H_{1}(GL_{2}ℤ; ℤ_{det}) as the class of c generating ℤ maps to the class of t^{−1}s^{−1}, which is of order precisely three.

The results for P_{12,3} and P_{1,12,3} are obvious.

To compute H_{∗}(SD_{3}; W_{3}) we use the SD_{3} PSS abutting to zero. Clearly its only differentials d^{4} : E_{4,4i}^{4} → E_{0,4i+3}^{4} are isomorphisms. □

Lemma 7.2. In the SL_{3}ℤ PSS d_{1}^{1} : E_{1,∗}^{1} → E_{0,∗}^{1} is surjective.

Proof. By [11, Proposition 1] the odd primary cohomology of SL_{3}ℤ is detected on certain subgroups [O] = SD_{3} and [Q] ≅ Di_{6}. [Q] is contained in SP_{1}, and the cohomology of [O] is detected on ⟨rs^{2} = −ϕ, ψ⟩ ≅ Σ_{3}, which is subconjugate to SP_{1} (c^{(ab)^{T}} ⟨−ϕ, ψ⟩ ⊂ SP_{1}). Thus H^{∗}(SL_{3}ℤ) → H^{∗}(SP_{1}) is an injection and the Lemma follows. □

Proof of Theorem 3.4 (2). The SL_{3}ℤ PSS appears as :

(ℤ/3)^{2} ⟵^{2} (ℤ/3)^{4} ⟵^{1} ℤ/3  0  0
0  0  0  0  0
0  0  0  0  0
0  0  0  0  ℤ/3
(ℤ/3)^{2} ⟵^{2} (ℤ/3)^{4} ⟵^{2} ℤ/3 ⊕ ℤ  0  0
0  0  0  0  0
0  0  0  0  0
ℤ  ⟵^{1} ℤ^{2}  ⟵^{1}  ℤ^{2}  ⟵^{1} ℤ  ℤ/3

Hence H_{∗}(F_{3}Kℤ/F_{2}Kℤ) ≅ (0, 0, 0, 0, ℤ/3, ℤ, 0, 0, #9, 0, …). □

Proof of Proposition 3.5 (2). This is similar to the proof of Proposition 3.3. Analogous splittings and KE-theoretic equivalences appear, and a three–chain in the bar resolution for SP_{1} replaces the class cx. We omit the details. □

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Matematisk Institutt, Universitetet i Oslo