E ∞ -cells and general linear groups of finite fields
aa r X i v : . [ m a t h . A T ] O c t E ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITEFIELDS SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
Abstract.
We prove new homological stability results for general linear groupsover finite fields. These results are obtained by constructing CW approxima-tions to the classifying spaces of these groups, in the category of E ∞ -algebras,guided by computations of homology with coefficients in the E -split Steinbergmodule. Introduction
The homology of general linear groups over finite fields was studied by Quillenin his seminal paper on the K -theory of finite fields [Qui72, Theorem 3]. One of hismain results is the complete determination of the homology groups H d (GL n ( F q ); F ℓ )where F q is a finite field with q = p r elements, GL n ( F q ) denotes the general lineargroup of the n -dimensional vector space F nq , and ℓ = p .When ℓ = p full information about these homology groups is only available ina stable range. Extending linear isomorphisms of F n − q to F nq = F n − q ⊕ F q by theidentity on the second subspace induces an injective homomorphism s : GL n − ( F q ) −→ GL n ( F q )called the stabilization map . Forming the colimit over such stabilisation maps,Quillen has proved [Qui72, Corollary 2] that(1.1) H d (GL ∞ ( F q ); F p ) = 0 for d > , which has implications for finite n using homological stability. The best homolo-gical stability result available in the literature is due to Maazen, who proved that H d (GL n ( F q ) , GL n − ( F q ); F p ) = 0 for d < n [Maa79]. It is an unpublished resultof Quillen that for q = 2 these groups vanish in the larger range q < n . (A proofappears in his unpublished notebooks [Qui, 1974-I, p. 10]. The first pages of thisnotebook were unfortunately bleached by sunlight, which makes it difficult to followthe argument. A forthcoming paper of Sprehn and Wahl [SW18] will include anexposition of Quillen’s argument.)1.1. Homological stability.
In this paper study the homology of general lineargroups of a finite fields of characteristic p , with coefficients in F p , using E k -cellularapproximations. We developed this technique in [GKRW18a], and in [GKRW18b]we applied it to studying homology of mapping class groups. For fields with morethan two elements we obtain the following strengthening of Quillen’s result. Theorem A. If q = p r = 2 , then H d (GL n ( F q ) , GL n − ( F q ); F p ) = 0 for d < n + r ( p − − . The bound n + r ( p − − r ( p − − q >
4. Our methods also apply to the field F ,where we obtain the following linear bound of slope 2 / Theorem B. H d (GL n ( F ) , GL n − ( F ); F ) = 0 for d < n − . Date : October 30, 2018.2010
Mathematics Subject Classification.
Combining this with Quillen’s calculation (1.1) (or using Corollary 6.2 when q = 2), we obtain the following vanishing theorem. Corollary C. If q = p r = 2 , then ˜ H d (GL n ( F q ); F p ) = 0 for d + 1 < n + r ( p − − .If q = 2 , then ˜ H d (GL n ( F ); F ) = 0 for d + 1 < n . A question of Milgram and Priddy.
In [MP87], Milgram and Priddy con-structed a cohomology class det n ∈ H n ( M n,n ( F ); F ), where M n,n ( F ) ⊂ GL n ( F )is the subgroup of (2 n × n )-matrices of the form (cid:20) id n ∗ n (cid:21) , which is invariant for the natural action of GL n ( F ) × GL n ( F ) as the subgroup ofblock-diagonal matrices. They suggested (see also [BHK +
07, Section 5]) that thisclass may be in the image under the restriction map H ∗ (GL n ( F ); F ) −→ H ∗ ( M n,n ( F ); F ) GL n ( F ) × GL n ( F ) . By Maazen’s result it would then be the lowest-degree such class. They showedthat this is indeed the case for det and det . However Corollary C implies that H n (GL n ( F ); F ) = 0 for n >
3, so that det n cannot be in the image of therestriction map for n >
3. Finally, in Section 6.3 we combine our techniques witha recent computation of Szymik to prove that det does lie in the image of therestriction map. This completely answers the question of Milgram and Priddy.1.3. Homology of Steinberg modules.
The reduced top integral homology ofthe Tits building associated to F nq is equal to the Steinberg module St( F nq ). It has anaction of GL n ( F q ), and in the modular representation theory of GL n ( F q ) a centralrole is played by the F p [GL n ( F q )]-module St( F nq ) ⊗ F p , as it is the only module whichis both irreducible and projective. This implies that H ∗ (GL n ( F q ); St( F nq ) ⊗ F p ) = 0.Combining the methods of this paper with Quillen’s calculation of the cohomologyof GL n ( F q ) away from the defining characteristic, we are also able to calculate thegroups H ∗ (GL n ( F q ); St( F nq ) ⊗ F ℓ ) for ℓ = p . We give this calculation in Section 7. Acknowledgments.
AK and SG were supported by the European Research Coun-cil (ERC) under the European Union’s Horizon 2020 research and innovation pro-gramme (grant agreement No. 682922). SG was also supported by NSF grant DMS-1405001. AK was also supported by the Danish National Research Foundationthrough the Centre for Symmetry and Deformation (DNRF92) and by NSF grantDMS-1803766. ORW was partially supported by EPSRC grant EP/M027783/1,and partially supported by the ERC under the European Union’s Horizon 2020research and innovation programme (grant agreement No. 756444).
Contents
1. Introduction 12. Recollections on a homology theory for E ∞ -algebras 33. General linear groups as an E ∞ -algebra 44. Computing E -homology of R
85. General linear groups of finite fields except F F ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 3 Recollections on a homology theory for E ∞ -algebras In this section we informally explain that part of the theory developed in [GKRW18a]which is necessary to prove Theorems A and B. We refer to that paper for amore formal discussion as well as for proofs; we shall refer to things labelled X in [GKRW18a] as E k . X throughout this paper. The reader may also find it help-ful to consult Sections 3 and 5 of [GKRW18b], which involves applications of thistheory to mapping class groups that are similar in technique to the applications inthis paper.The aforementioned theory concerns CW approximation for E ∞ -algebras. Be-cause we are interested mainly in the homology of certain E ∞ -algebras, we willwork in the category of simplicial k -modules . In this paper k shall always be afield, and we will also make this simplifying assumption in this section. In the casethat k = Q , an E ∞ -algebra in simplicial k -modules is equivalent to the data of acommutative algebra in non-negatively graded chain complexes over Q .In order to eventually keep track of the dimension n of the vector space F n weshall work with an additional N -grading (for us N always includes 0) which wecall rank , and so work in the category sMod N k of N -graded simplicial k -modules.Morphisms in this category are weak equivalences if they are weak equivalences ofunderlying N -graded simplicial sets, and the category is equipped with the symmet-ric monoidal structure given by Day convolution. That is, an object consists of asequence of simplicial k -modules M • ( n ) for n ∈ N , and the tensor product of twosuch objects is given by( M ⊗ N ) p ( n ) = M a + b = n M p ( a ) ⊗ k M p ( b ) . A non-unital E ∞ -operad in topological spaces is an operad whose space of n -ary operations is a contractible space with free S n -action, with the exception ofthe space of 0-ary operations, which is empty. An example of such an operad isprovided by taking the colimit as k → ∞ of the non-unital little k -cubes operads.Taking k [Sing( − )] we obtain an operad in simplicial k -modules, which we make N -graded by concentrating it in degree 0. We denote the resulting operad by C ∞ .An E ∞ -algebra in this paper shall usually mean an algebra over this operad in thecategory sMod N k .The module of indecomposables of an E ∞ -algebra R in sMod N k is defined usingthe exact sequence of N -graded simplicial k -modules M n ≥ C ∞ ( n ) ⊗ R ⊗ n −→ R −→ Q E ∞ ( R ) −→ R as an E ∞ -algebra. Thisconstruction is not homotopy invariant, and has a derived functor Q E ∞ L . (In thenotation of [GKRW18a] we have actually defined the relative indecomposables, butbecause C ∞ (1) ≃ ∗ at the level of derived functors there is no difference betweenthis and the absolute indecomposables, cf. equation (11.2) in Section E k .11.3. The E ∞ -homology groups are then defined as H E ∞ n,d ( R ) := H d ( Q E ∞ L ( R )( n )); in simplicial k -modules, homology means taking the homology of the associated chain complex,or equivalently taking homotopy groups.Its role is explained by Theorem E k .11.21, which says that E ∞ -homology deter-mines the cells needed to build a CW approximation to R in Alg E ∞ ( sMod N k ). Letus explain what this means. We write ∂D n,d ∈ sMod N k for the object which is 0when evaluated at any k = n , and is the simplicial k -module k [ ∂ ∆ d ] when evalu-ated at n ; we write D n,d for the analogous construction using k [∆ d ], so there isan inclusion ∂D n,d ⊂ D n,d . A cellular E ∞ -algebra is one obtained by iterated cell SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS attachments, the data for which is a map of simplicial k -modules k [ ∂ ∆ d ] → R ( n ),which by adjunction gives a morphism ∂D n,d k → R in sMod N k . Letting E ∞ ( − ) de-note the free E ∞ -algebra construction, this in turn gives a map E ∞ ( ∂D n,d k ) → R of E ∞ -algebras. The inclusion ∂D n,d ⊂ D n,d induces the left map in the followingdiagram, and to attach a cell we form the pushout in Alg E ∞ ( sMod N k ) E ∞ ( ∂D n,d k ) RE ∞ ( D n,d k ) R ∪ E ∞ D g,d . A cellular approximation to R is a weak equivalence C ∼ → R from a cellular E ∞ -algebra. If R (0) ≃ H E ∞ n,d ( R ) = 0 for d < f ( n ), then it follows that R admits a cellular approximation built only using cells of bidegrees ( n, d ) such that d ≥ f ( n ). This type of statement shall be called a vanishing line for E ∞ homology.In fact rather than just cellular, C can be taken to be CW, meaning that it comeswith a filtration where the d th stage is obtained from the ( d − d -cells. Such an approximation can be used for homology computations by takingthe spectral sequence associated to this skeletal filtration, or using iterated cellattachment spectral sequences as in Section E k .11.To prove Theorems A and B, we shall apply these ideas to a certain E ∞ -algebra R F p with R F p ( n ) ≃ F p [ B GL n ( F q )].We first consider R F p as an E -algebra, and prove a vanishing result for the E -homology groups H E n,d ( R F p ), defined analogously to the E ∞ -homology groups. Thisis done by first studying GL n ( F q )-equivariant semi-simplicial sets S E ( F nq ), first in-troduced by Charney, who called them “split buildings” and also proved they hadthe homotopy type of a wedge of ( n − Q E L ( R F p ) may be calculatedas the homotopy orbit space ( S E ( F nq )) // GL n ( F q ), and the vanishing line for E -homology is established by proving the vanishing of H ∗ (GL n ( F q ); H n − ( S E ( F nq ); F p ))in a range of degrees. The resulting vanishing line for E -homology can then betransferred from E -homology to E ∞ -homology using bar spectral sequences as inTheorem E k .14.2.By combining such vanishing results with calculations in low degrees and lowrank, we can produce an E ∞ -algebra with the same homological stability behavioras R F p in a range. For q = p r = 2 this E ∞ -algebra is the commutative algebra N and R F p maps to it. For q = 2 this is a custom-built E ∞ -algebra A which maps to R F . 3. General linear groups as an E ∞ -algebra Notation 3.1.
In the paper we shall use GL( V ) to denote the linear isomorphismsof a vector space V , and shall when appropriate write GL( F n ) for GL n ( F ).3.1. Construction of R.
Fix a field F , which need not be finite yet. We define agroupoid V F with objects the non-negative integers, and morphisms given by V F ( n, m ) = ( GL n ( F ) if n = m , ∅ otherwise.This is a skeleton of the category P F with objects finitely dimensional F -vectorspaces and morphisms linear isomorphisms, which has a symmetric monoidal struc-ture given by direct sum. Let us describe the corresponding symmetric monoidalstructure on this skeleton. On objects the monoidal product is given by n ⊕ m = n + m and on morphisms by block sum of matrices, as a homomorphism ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 5 GL n ( F ) × GL m ( F ) → GL n + m ( F ). This admits a symmetry given by the identity onobjects, and on morphisms by conjugation with the block ( n + m ) × ( n + m )-matrix T n,m := (cid:20) n id m (cid:21) , that is, A T − n,m AT n,m . This category has several useful properties. Lemma 3.2.
The symmetric monoidal groupoid ( V F , ⊕ , has the following prop-erties:(i) for the monoidal unit we have GL ( F ) = { e } ,(ii) the homomorphism − ⊕ − : GL n ( F ) × GL m ( F ) → GL n + m ( F ) induced by themonoidal structure is injective,(iii) there is a single object in each isomorphism class,(iv) the monoidal structure is strictly associative. Consider the category sSet V F of functors V F → sSet . This is simplicially en-riched and inherits a symmetric monoidal structure via Day convolution. As conse-quence, for any simplicial operad O there is a category Alg O ( sSet V F ) of O -algebrasin sSet V F . Furthermore, sSet V F admits a projective model structure, making it intoa simplicially enriched symmetric monoidal model category. If O is Σ-cofibrant, wemay transfer this to obtain a projective model structure on Alg O ( sSet V F ) with theproperty that the forgetful functor Alg O ( sSet V F ) → sSet V F preserves cofibrations inaddition to fibrations and weak equivalences.In particular, we may take O = C ∞ := colim k →∞ C k , where C k denotes the non-unital little k -cubes operad. There is a non-unital commutative monoid ∗ > in sSet V F given by ∗ > ( n ) = ( ∅ if n = 0, ∗ if n > E ∞ -algebra. As in Section E k .17.1, by cofibrantlyreplacing ∗ > ∈ Alg E ∞ ( sSet V F ) we construct a cofibrant non-unital E ∞ -algebra T such that T ( n ) ≃ ( ∅ if n = 0, ∗ if n > F -vector space its dimension gives a symmetric monoidal functor r : V F → N . Left Kan extension provides a left adjoint r ∗ : sSet V F → sSet N to therestriction functor r ∗ : sSet N → sSet V F , and this sends E ∞ -algebras to E ∞ -algebrasso may also be considered as functor r ∗ : Alg E ∞ ( sSet V F ) → Alg E ∞ ( sSet N ). As suchit is the left adjoint in a Quillen adjunction, and hence r ∗ ( T ) ≃ L r ∗ ( ∗ > ). Weshall denote this E ∞ -algebra by R . Since the forgetful functor preserves cofibrantobjects, we get R ( n ) ≃ ( ∅ if n = 0, B GL n ( F ) if n > E ∞ -algebra structure on an N -graded simplicial set weaklyequivalent to F n ≥ B GL( F n ).For any commutative ring k , we have that H ∗ ( R ( n ); k ) ∼ = H ∗ (GL n ( F ); k ) for n >
0. Since we are interested only in these homology groups, there is no loss inpassing from simplicial sets to simplicial k -modules and focusing our attention on R k := k [ R ] ∈ Alg E ∞ ( sMod N k ). SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
Tits buildings and relative Tits buildings.
Though R k is an E ∞ -algebra,the map C → C ∞ allows us to consider it as an E -algebra instead. As such ithas E -homology, which may be computed in terms of the E -splitting complex of R k as in Section E k .17.2. We will study these E -splitting complexes using Titsbuildings and their relative versions, whose definitions we recall in this section. Definition 3.3.
Let P be a finite-dimensional F -vector space. The Tits building T ( P ) is the poset with objects given by subspaces 0 ( V ( P and partial ordergiven by V ≤ V ′ if V ⊆ V ′ .We let T ( P ) denote the topological space obtained by taking the thin geometricrealization of the nerve of T ( P ).The Solomon–Tits theorem says that T ( P ) is homotopy equivalent to a wedgeof (dim( P ) − H dim( P ) − , a Z [GL( P )]-module that is free as an abelian group. Definition 3.4.
The
Steinberg module of P is the Z [GL( P )]-moduleSt( P ) := ˜ H dim( P ) − ( T ( P )) . In order to make certain inductive arguments later, we will also need relativeversions of the Tits building. These are subposets of the Tits building consisting ofsubspaces complementary to a fixed subspace W . Definition 3.5.
Let P be a finite-dimensional F -vector space and W ⊂ P be asubspace. Then the relative Tits building T ( P rel W ) is the poset with objectsgiven by non-zero proper subspaces V of P such that V ∩ W = { } and partialorder given by V ≤ V ′ if V ⊆ V ′ .We let T ( P rel W ) denote the topological space obtained by taking the thingeometric realization of the nerve of T ( P rel W ).In Lemma 3.9, we will prove that T ( P rel W ) is homotopy equivalent to a wedgeof (dim( P ) − dim W − P, pres W ) ≤ GL( P )consisting of linear isomorphisms of P which preserve W as a subspace (but do notnecessarily fix it pointwise). Definition 3.6.
The
Steinberg module of P relative to W is the Z [GL( P, pres W )]-module St( P rel W ) := ˜ H dim( P ) − dim W − ( T ( P rel W )) . The following is the “linear dual” of the poset T ( P rel W ) as in Definition 3.5. Definition 3.7.
Let P be a finite-dimensional F -vector space and W ⊂ P be asubspace. Then the dual relative Tits building T ( P rel + W ) is the poset with objectsgiven by non-zero proper subspaces V of P such that V + W = P and partial ordergiven by V ≤ V ′ if V ⊆ V ′ .We let T ( P rel + W ) denote the topological space obtained by taking the thingeometric realization of the nerve of T ( P rel + W ).If W = P this is equal to the Tits building T ( P ) and has a GL( P )-action. Forproper non-trivial W , it has an action of the subgroup GL( P, pres W ) of GL( P )consisting of linear isomorphisms of P which preserve W .By dualizing we can identify it with a relative Tits building. Letting W ◦ ⊂ V ∨ denote the linear subspace of linear forms which are identically 0 on W , there are ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 7 mutually inverse functors T ( P rel W ) ∼ = ←→ T ( P ∨ rel + W ◦ ) V V ∨ U ∨ ←− [ U, where the bottom map uses the canonical identification of the double dual ( P ∨ ) ∨ with P . These identifications are equivariant for the action of GL( P, pres W ) onthe left hand side and the action of GL( P ∨ , pres W ◦ ) on the right side, uponidentifying these groups by dualizing.One can reduce the study of relative Tits buildings to the study of ordinary Titsbuildings and those relative Tits buildings where W ⊂ P is a line. This is due toQuillen and appears in his unpublished notebooks. We give his proof below, whichcan also be found in [SW18]: Lemma 3.8 (Quillen) . Let P be a F -vector space of finite dimension ≥ and ( W ( P be a non-zero proper subspace. Let L be a one-dimensional subspace of W . Then there is a homotopy equivalence (3.1) T ( P rel + W ) ≃ T ( P/L rel + W/L ) ∗ T ( P rel + L ) , equivariant for the action of the subgroup GL( P, pres L , pres W ) ≤ GL( P, pres W ) of those linear maps which also preserve the subspace L .Proof. To prove this homotopy equivalence, we define a pair of subposets: (i) L ⊂T ( P rel + W ) is the full subposet consisting of those V such that L ⊂ V , (ii) L ⊂T ( P rel + W ) is the full subposet consisting of those V such that V + L = P . Wewrite L i for the thin geometric realisation of the nerve of L i .The poset L is a subposet of L and there is deformation retraction L → L induced by the relation V ≤ V + L , regarded as a natural transformation from theidentity functor on the poset L regarded as a category, to the functor V V + L .Hence L ≃ L . Similarly, there are mutually inverse functors L ∼ = ←→ T ( P/L rel + W/L ) V V /Lp − ( V /L ) ←− [ V /L, where p : P → P/L is the projection map. These exhibit L and T ( P/L rel + W/L )as posets which are isomorphic, equivariantly for the action of GL( P, pres L , pres W ).The poset L is a full subposet of T ( P | W ) and the only missing elements arethe rank ( m −
1) subspaces H satisfying H + L = P , i.e. the complements to L . Thus T ( P rel + W ) is obtained from L by attaching cones on the various links | Link L ( H ) | , that is, there is a pushout diagram F H | H + L = P | Link L ( H ) | L F H | H + L = P Cone( | Link L ( H ) | ) T ( P rel + W ) . The subposet Link L ( H ) ⊂ L is given by those 0 ( V ( H such that V + L = P ,and we claim that the inclusion Link L ( H ) ֒ → L is a homotopy equivalence. Itshomotopy inverse is given by V ( V + L ) ∩ H . To see this is well-defined, we needto check that (a) ( V + L ) ∩ H can be neither 0 nor H , and (b) ( V + L ) ∩ H + L = P .For (a), if ( V + L ) ∩ H = 0, then V + L must have had rank 1 and hence V = L ,but then V + W = P can not hold, as W ( P . Similarly, if ( V + L ) ∩ H = H , then SØREN GALATIUS, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS L ⊂ V + L ⊂ H and H + L = P . For (b), if ( V + L ) ∩ H + L has dimension dim( P ),then ( V + L ) ∩ H has dimension dim( P ) −
1, hence must equal H and we saw abovethis can not happen. The homotopy is given by the zigzag V ≤ V + L ≥ ( V + L ) ∩ H .Because the attaching maps are homotopic to the identity, T ( P rel + W ) is homo-topy equivalent to the join of L with the set { H | H + L = P } , which is homotopyequivalent to T ( P/L rel + W/L ) and isomorphic to T ( P rel + L ) respectively. (cid:3) By induction on dim W , using the fact that T ( P rel + L ) is non-empty as long asdim( P ) ≥
2, we obtain:
Lemma 3.9. T ( P rel + W ) is (dim W − -spherical. E -splitting complexes. Using Remark E k .17.2.10 and the fact that thesymmetric monoidal groupoid V F satisfies the conditions in Lemma 3.2, the E -splitting complex of R k can be described in terms of Young-type subgroups. Thespaces S E ( P ) obtained this way were originally invented by Charney in analogywith the Tits building; in the Tits building one considers proper subspaces ratherthan proper splittings of a vector space. Definition 3.10.
Let P be a finite-dimensional F -vector space. The E -splittingcomplex S E • ( P ) is the semisimplicial set with p -simplices given by ordered collection( V , . . . , V p +1 ) of non-zero proper subspaces of P , such that the natural map V ⊕ . . . ⊕ V p +1 → P induced by the inclusions is an isomorphism. The face maps d i takes the sum of the i th and ( i + 1)st term.We let S E ( P ) denote the topological space obtained by taking the thick geo-metric realization of S E • ( P ).Charney proved that S E ( P ) is a wedge of (dim( P ) − H dim( P ) − , which is a Z [GL( P )]-module that is free as an abelian group. Definition 3.11.
The E -Steinberg module of P is the Z [GL( P )]-moduleSt E ( P ) := ˜ H dim( P ) − ( S E ( P )) . Corollary E k .11.5 implies that the homology groups of GL n ( F ) with coefficientsin St E ( F n ) coincide with E -homology groups of R k up to a shift:(3.2) H E n,d ( R k ) ∼ = H d − n +1 (GL n ( F ); St E ( F n ) ⊗ k ) . It is useful to think of the semi-simplicial set S E • ( P ) in terms of the following poset. Definition 3.12.
Let P be a finite-dimensional F -vector space. The split building S E ( P ) is the poset with objects given by ordered pairs ( V , V ) of non-zero propersubspaces of P such that V ∩ V = { } and V ⊕ V → P is an isomorphism. Thepartial order is given by ( V , V ) ≤ ( V ′ , V ′ ) if V ′ is a subspace of V and V is asubspace of V ′ .We recognise the semi-simplicial set S E • ( P ) as the non-degenerate simplices ofthe nerve N • S E ( P ), so the thick geometric realisation of S E • ( P ) is homeomorphicto the thin geometric realisation of N • S E ( P ).4. Computing E -homology of R In this section we describe a method to prove that the E -splitting complexof a finite-dimensional vector space is highly-connected and that homology withcoefficients in the E -Steinberg module vanishes in a range. The connectivity resultis due to Charney [Cha80], who proves it more generally for Dedekind domains. Infact, we follow her proof strategy but additionally keep track of various homologygroups with coefficients. ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 9 Poset techniques.
Recall that the E -splitting complex S E ( F n ) can be de-scribed as the geometric realization of the nerve of the poset S E ( F n ). There arewell-developed techniques to prove the connectivity of posets and maps betweenthem (that is, upon taking geometric realization of the nerves).Recall that if ( Y , ≤ ) is a poset and y ∈ Y then Y
Let f : X → Y be a map of posets, n ∈ Z , and t Y : Y → Z be afunction. Assume that(i) Y is n -spherical,(ii) for every y ∈ Y , f ≤ y is ( n − t Y ( y )) -spherical, and(iii) Y >y is ( t Y ( y ) − -spherical.Then X is n -spherical and there is a canonical filtration F n +1 ⊂ F n ⊂ · · · ⊂ F − = ˜ H n ( X ) such that F − /F ∼ = ˜ H n ( Y ) ,F q /F q +1 ∼ = M t Y ( y )= n − q ˜ H n − q − ( Y >y ) ⊗ ˜ H q ( f ≤ y ) . The statement.
We find it clarifying to use the notation S E ( · , · | F n ) := S E ( F n )for the split building, as this notation allows us to easily denote several variants. Definition 4.2.
Let W ⊂ F n be a non-zero subspace. The relative split building S E ( · , W ⊂ · | F n ) is the subposet of S E ( F n ) with objects those splittings ( V , V )such that W ⊂ V .We let S E ( · , W ⊂ · | F n ) denote the thin geometric realization of its nerve.In the following theorem we prove that this is homotopy equivalent to a wedgeof ( n − dim W − E ( F n rel W ) := ˜ H n − dim W − ( S E ( · , W ⊂ · | F n )) . This notation is in analogy with St( F n rel W ), as the objects of S E ( · , W ⊂· | F n ) are pairs ( A, B ) with A ∩ W = { } . This has an action of the subgroupGL( P, pres W ) ≤ GL( P ) consisting of linear isomorphisms of P which preserve W as a subspace, and hence also of the subgroupGL( P, fix W ) ≤ GL( P, pres W )consisting of linear isomorphisms of P which fix W pointwise. Theorem 4.3.
Let F be a field, let W ⊂ F n be a non-zero subspace and suppose n ≥ . Then we have that: S E ( · , · | F n ) is ( n − -spherical, S E ( · , W ⊂ · | F n ) is ( n − dim W − -spherical.Suppose further that k and c ∈ N are such that H i (GL( F n ); St( F n ) ⊗ k ) = 0 ,H i (GL( F n , fix W ); St( F n rel W ) ⊗ k ) = 0 , for all n ≥ and all i ≤ c . Then for all n ≥ and i ≤ c it is true that: H i (GL( F n ); St E ( F n ) ⊗ k ) = 0 ,H i (GL( F n , fix W ); St E ( F n rel W )) ⊗ k ) = 0 . Remark . A Serre spectral argument shows H i (GL( F n , fix W ); St E ( F n rel W ) ⊗ k ) = 0 for i ≤ c implies H i (GL( F n , pres W ); St E ( F n rel W ) ⊗ k ) = 0 for i ≤ c .Hence the former is a stronger statement.It shall be helpful to name two facts that are used in the proof of Theorem 4.3,as well as the two hypotheses in the second part of Theorem 4.3:(1) The Tits building T ( F n ) is ( n − T ( F n rel W ) is ( n − dim W − St For n ≥ i ≤ c , H i (GL( F n ); St( F n ) ⊗ k ) = 0.(2) St For n ≥ i ≤ c , H i (GL( F n , fix W ); St( F n rel W ) ⊗ k ).Without loss of generality we can change basis of F n so that W = F w ×{ } , the spanof the first w basis vectors. The proof of Theorem 4.3 will be an upwards inductionover ( n, w ) in lexicographic order, where we interpret w = 0 as the absolute version.4.2.1. The base cases.
The base cases are those with n ≤
1, in which case allstatements are empty.4.2.2.
The first reduction.
Proposition 4.5.
The case ( n, is implied by the cases { ( n, w ) | w > } .Proof. Consider the map of posets X := S E ( · , · | F n ) f −→ Y := T ( F n ) op ( A, B ) B, and for U ∈ Y let t Y ( U ) = dim( U ) −
1. We verify the assumptions of Theorem 4.1:(i) Y = T ( F n ) op is ( n − Y >U is given by T ( U ) so is (dim( U ) −
2) = ( t Y ( U ) − X f ≤ U is given by those ( A, B ) such that U ⊂ B , i.e. S E ( · , U ⊂ · | F n ). Bythe hypothesis this is ( n − dim( U ) −
1) = ( n − − t Y ( U ))-spherical.Theorem 4.1 then implies that X = S E ( · , · | F n ) is ( n − k [GL( F n )]-module St E ( F n ) with F − /F = St( F n ) ⊗ k andthe other filtration quotients F q /F q +1 given by M
1) basis vectors.For any filtered k [ G ]-module F − = M ⊃ F ⊃ · · · ⊃ F n ⊃ E p,q = H p + q ( G ; F q /F q +1 ) = ⇒ H p + q ( G ; M ) , which we apply to the above filtration of the k [GL( F n )]-module St E ( F n ). We candistinguish two different types of columns on the E -page. Firstly, for q = − E p, − = H p − (GL m ( F ); St( F m ) ⊗ k ) which vanishes for p − ≤ c and n ≥ St . Secondly, for q ≥ E p,q ∼ = H p + q (cid:16) GL( F n , pres F n − q − ); St( F n − q − ) ⊗ St E ( F n rel F n − q − ) ⊗ k (cid:17) . ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 11 To prove that these vanish for p + q ≤ c we consider the extensionGL( F n , fix F n − q − ) −→ GL( F n , pres F n − q − ) −→ GL( F n − q − )and apply the Serre spectral sequence F s,t = H s GL( F n − q − ); St( F n − q − ) ⊗ H t h GL( F n , fix F n − q − ); St E ( F n rel F n − q − ) ⊗ k i ! H s + t (cid:16) GL( F n , pres F n − q − ); St( F n − q − ) ⊗ St E ( F n rel F n − q − ) ⊗ k (cid:17) . Now F s,t = 0 for t ≤ c by inductive assumption as n ≥
2, so the target vanishes for s + t ≤ c , and the claim follows. (cid:3) Cutting down.
The following technique is due to Charney [Cha80, p. 4].Given V and W subspaces of P , let S E ( · ⊂ V, W ⊂ · | P ) be the subposet of S E ( · , · | P ) consisting of those splittings ( A, B ) such that A ⊂ V and W ⊂ B . Lemma 4.6.
Let V and W be subspaces of P such that V ∩ W = 0 and V ⊕ W = P .Let C be a complement to W in P containing V . Then there is an isomorphism S E ( · ⊂ V, W ⊂ · | P ) ∼ = S E ( · ⊂ V, · | C ) , equivariantly for the subgroup of GL( P ) preserving the subspaces V , W and C .Proof. Mutually inverse functors are given by S E ( · ⊂ V, W ⊂ · | P ) ∼ = ←→ S E ( · ⊂ V, · | C )( A, B ) ( A, B ∩ C )( A ′ , B ′ ⊕ W ) ←− [ ( A ′ , B ′ )This is well-defined, as it can not happen that B ∩ C = { } , because V ⊕ W = P implies that B ∩ C has dimension at least 1. (cid:3) The second reduction.
Proposition 4.7.
The case ( n, w ) is implied by the cases { ( n ′ , w ′ ) | n ′ < n } .Proof. When w = n − S E ( · , F w ⊂ · | F n ) is isomorphic to T ( F n rel F w ) and wealready know the result. So let us assume that w ≤ n − X := S E ( · , F w ⊂ · | F n ) f −→ Y := T ( F n rel F w )( A, B ) A, and take t Y ( V ) = n − dim V − w . We verify the assumptions of Theorem 4.1:(i) T ( F n rel F w ) is ( n − w −
1) = n -spherical by (2).(ii) Y >V is isomorphic to T ( F n /V rel F w /V ), which is ( n − dim V − w −
1) =( t Y ( V ) − X f ≤ V is S E ( · ⊂ V, F w ⊂ · | F n ), which we claim is (dim V −
1) = ( n − w − − t Y ( V ))-spherical.There are two cases: the first case is that V ⊕ F w = F n , and then thecomplex is contractible (the splitting ( V, F w ) is terminal) and hence is a wedgeof no ( n − w − − t Y ( V ))-spheres. The second case is that V ⊕ F w = F n .Taking a complement C to F w , we can apply the cutting down lemma tosee it is isomorphic to S E ( · ⊂ V, · | C ). Dualizing makes this isomorphic to S E ( · , V ◦ ⊂ · | C ∨ ), which is ( n − w − ( w − dim V ) −
1) = (dim( V ) −
1) =( n − t Y ( V ))-spherical by induction. ∗ ∗ ∗ ∗ ∗ ∗ Figure 1.
An element of K as in the proof of Proposition 4.7 with n = 5, w = 2, and V = h e , e i . Theorem 4.1 then implies that X = S E ( · , F w ⊂ · | F n ) is ( n − w − k [GL( F n , fix F w )]-module St E ( F n rel F w ) with F − /F =St( F n rel F w ) ⊗ k and further filtration quotients F q /F q +1 given by M 2, St( F nq ) ⊗ F p is an irreducible projective F p [GL n ( F q )]-module [Hum87, Sections 2 and 3] and thus we have H ∗ (GL n ( F q ); St( F nq ) ⊗ F p ) = 0for all n ≥ 2. More precisely, the higher homology vanishes because it is projective,and the coinvariants vanish because, if not, they would provide a nontrivial quotientof an irreducible module.The similar looking groups H d (GL n ( F q ); St E ( F nq ) ⊗ F p ) cannot vanish for all d ≥ n ≥ 2: if they did, the E -homology of R F p would vanish for n ≥ R F p ≃ E ( S , ), which has the wrong homology. Consequently, byTheorem 4.3, the homology of the relative Steinberg modules St( F nq rel W ) ⊗ F p cannot vanish in all degrees for n ≥ does vanish in a range of degrees.To do so, we will use the dual relative Tits building T ( P rel + W ) of Definition3.7. Given a subspace W ⊂ P , this is the subposet of T ( P ) of non-zero propersubspaces V such that V + W = P . By taking duals, it is isomorphic to the ordinaryrelative Tits building T ( P ∨ rel W ◦ ). The group GL( P, pres W , fix P/W ) acts on T ( P rel + W ), and under dualization this goes to action of the isomorphic groupGL( P ∨ , fix W ◦ ). As before, without loss of generality W = F wq . Lemma 5.2. Let n ≥ and < w < n , then H d (cid:0) GL( F nq , pres F wq , fix F nq / F wq ); ˜ H w − ( T ( F nq rel + F wq ) ⊗ F p (cid:1) = 0 for d < r ( p − − .Proof. Let L ⊂ F wq be a 1-dimensional subspace, and let K be the subgroup ofGL( F nq , pres F wq , fix F nq / F wq ) given by K := GL( F nq , pres L , pres F wq , fix F nq / F wq ) . The group GL( F nq , pres F wq , fix F nq / F wq ) acts transitively on the set of lines P ( F wq ) in W with K the stabiliser of L . Thus this subgroup has index | P ( F wq ) | = q w − q − ≡ p. It follows by a transfer argument that the map H ∗ (cid:0) GL( F nq , pres L , pres F wq , fix F nq / F wq ); ˜ H w − ( T ( F wq rel + F nq ); F p ) (cid:1) H ∗ (cid:0) GL( F nq , pres F wq , fix F nq / F wq ); ˜ H w − ( T ( F wq rel + F nq ); F p ) (cid:1) i ∗ is surjective, so it is enough to show the vanishing of the source.By Lemma 3.8 we have a weak equivalence T ( F nq rel + F wq ) ≃ T ( F nq /L rel + F wq /L ) ∗ T ( F nq rel + L ) , ∗ ∗ ∗ ∗ ∗ Figure 2. An element of K ′ when n = 5 and L = h e i . equivariant for the subgroup K defined above. Thus we get an isomorphism˜ H w − ( T ( F nq rel + F wq ) F p ) ∼ = ˜ H w − ( T ( F nq /L rel + F wq /L ); F p ) ⊗ ˜ H ( T ( F nq rel + L ); F p )of F p [ K ]-modules. To deal with the source of i ∗ , we consider the subgroup of K which acts trivially on F nq /L , i.e. K ′ := GL( F nq , pres L , fix F nq /L ) ⊂ K. This is the kernel of the natural surjective homomorphism K → GL( F nq /L ) (seeFigure 2 for an example). It is isomorphic to the semidirect product F n − q ⋊ F × q .The group K ′ acts trivially on ˜ H w − ( T ( F nq /L rel + F wq /L ); F p ), so using the Serrespectral sequence it is enough to show vanishing of H i ( K ′ ; ˜ H ( T ( F nq rel + L ); F p )) inthe claimed range. Now T ( F nq rel + L ) is the set of hyperplanes in F nq complementaryto L , and K ′ acts transitively on these with stabiliser F × q . Because this stabiliseris F p -acyclic the exact sequence0 −→ ˜ H ( T ( F nq rel + L ); F p ) −→ F p {T ( F nq rel + L ) } −→ F p −→ H d +1 ( F n − q ; F p ) F × q = H i +1 ( K ′ ; F p ) ∼ −→ H d ( K ′ ; ˜ H ( T ( F nq rel + L ); F p ))is an isomorphism for d ≥ 0. By [Qui72, Lemma 16], this vanishes for d + 1 In Section 3, we defined a non-unital E ∞ -algebra R F p ∈ Alg E ∞ ( sMod NF p ) satisfying H ∗ ( R F p ( n )) ∼ = ( n = 0, H ∗ (GL n ( F q ); F p ) if n > , and by (3.2) we have H d − n +1 (GL n ( F q ); St E ( F nq ) ⊗ F p ) ∼ = H E n,d ( R F p ). Thus The-orem 5.1 implies that as long as n ≥ r ( p − − ≥ q = 2) we have H E n,d ( R F p ) = 0 in degrees d < n + r ( p − − N F p ∈ sMod NF p be given by N F p ( n ) := ( n = 0, F p if n > sMod NF p , so in par-ticular of a non-unital E ∞ -algebra. The augmentations ε : H (GL n ( F q ); F p ) → F p assemble to a map of E ∞ -algebras f : R F p −→ N F p . Lemma 5.4. H E ∞ n,d ( f ) = 0 for d < max( r (2 p − 3) + 1 , n + r ( p − − . ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 15 Proof. On the one hand, N F p is weakly equivalent to E ( S , F p σ ) as an E -algebra,so its E -homology vanishes in all bidegrees except (1 , H E n,d ( R F p ) = 0 when d < n + r ( p − − n ≥ H n,d ( R F p ) = 0 for 0 < d < r (2 p − H n,d ( f ) = 0for d < r (2 p − 3) + 1. Using the Hurewicz theorem for E -homology (Corollary E k .11.12) and the long exact sequence on E -homology, we conclude H E n,d ( f ) = 0for d < max( r (2 p − 3) + 1 , n + r ( p − − ρ ( n ) = max( r (2 p − 3) + 1 , n + r ( p − − 1) satisfies ρ ∗ ρ ≥ ρ , so is lax monoidal, and by applying spectral sequence comparison to the barspectral sequences in the vanishing line transfer theorem E k .14.2.1, we concludethat H E ∞ n,d ( f ) = 0 for d < max( r (2 p − 3) + 1 , n + r ( p − − (cid:3) From R F p we may construct a unital strictly associative algebra R F p , which isequivalent to the unitalization R + F p . Picking a representative σ : S , F p → R F p , weget a corresponding map σ · − : S , F p ⊗ R F p −→ R F p and we may form its homotopy cofibre R F p /σ . The operation S , F p ⊗ − shifts therank grading by 1, and the homology groups of this homotopy cofibre may beidentified with the relative homology groups of the stabilization map H n,d ( R F p /σ ) ∼ = H d (GL n ( F q ) , GL n − ( F q ); F p ) . To prove Theorem A it thus suffices to prove the following vanishing range for thegroups H n,d ( R F p /σ ). Theorem 5.5. H n,d ( R F p /σ ) = 0 for d < max( r (2 p − , n + r ( p − − .Proof. By Theorem E k .11.5.3 we can extend the map f to a diagram of E ∞ -algebras R F p N F p C , fg ≃ where g is a relative CW object obtained by attaching E ∞ -cells of bidegrees ( n, d )satisfying d ≥ max( r (2 p − 3) + 1 , n + r ( p − − E k .10.3.5 constructs a spectral sequence E n,p,q = ˜ H n,p + q,q ∗ R F p ∨ E ∞ E ∞ _ i ∈ I S n i ,d i ,d i F p x i = ⇒ H n,p + q ( N F p ) , where ( n i , d i ) satisfy n i ≥ d i ≥ max( r (2 p − 3) + 1 , n i + r ( p − − E -page has homology given by the tensor product of H ∗ , ∗ ( R F p ), put in filtrationdegree 0, with W ∞ -algebras W ∞ ( F p { x i } ). Each of these W ∞ -algebras is a freegraded-commutative algebra with generators Q I x i obtained by applying iteratedDyer–Lashof operations to x i . Let us investigate how these affect bidegrees. The case p odd . Applying a Dyer–Lashof operation Q s changes bidegrees by( n, d ) ( pn, d + 2 s ( p − s ≥ d , and βQ s by ( n, d ) ( pn, d + 2 s ( p − − s > d . This does not decrease the degree. We claim it also preserves theproperty of satisfying the inequality d i ≥ n i + r ( p − − 1. It suffices to verify thisin the following cases, as these are the smallest s such that Q s ( x ) can be non-zero: (i) the element Q k ( x ) if | x | = 2 k − ε with ε = 0 , 1: if 2 k − ε = | x | ≥ n + r ( p − − | x | + 2 k ( p − 1) = | x | + ( | x | + ε )( p − ≥ p | x | ≥ p ( n + r ( p − − ≥ pn + r ( p − − r ( p − − > βQ k +1 ( x ) if | x | = 2 k + 2 − ε with ε = 0 , 1: if 2 k + 2 − ε = | x | ≥ n + r ( p − − | x | + 2( k + 1)( p − − | x | + ( | x | + ε )( p − − ≥ p | x | − ≥ p ( n + r ( p − − − ≥ pn + r ( p − − . The case p = 2. The operation Q s changes bidegrees by ( n, d ) (2 n, d + s ) for s > d . Again it does not decrease the degree and we claim it preserves the propertyof satisfying the inequality d i ≥ n i + r − 1. As above it suffices to verify this forthe smallest s which can be non-zero, Q d ( x ) for | x | = d (which equals squaring).Assuming | x | ≥ n + r − 1, we get | x | + d = 2 d ≥ n + r − ≥ n + r − . Hence all of the F p -algebra generators of W ∞ ( F p { x i } ) lie in bidegrees ( d, n )satisfying d ≥ max( r (2 p − 3) + 1 , n + r ( p − − F n,p,q = ˜ H n,p + q,q ∗ R F p ∨ E ∞ E ∞ _ i ∈ I S n i ,d i ,d i F p x i /σ = ⇒ H n,p + q ( N F p /σ ) , which converges to 0 for ( n, p + q ) = (0 , 0) and is a module over the previous spectralsequence. We may express its F -page as H ∗ , ∗ , ( R F p /σ ) ⊗ O i ∈ I W ∞ ( F p { x i } [ n i , d i , d i ]) . Our goal is to show that H n,d ( R F p /σ ) = 0 for d < max( r (2 p − , n + r ( p − − H , ( R F p /σ ) = F p . This follows for d < r (2 p − 3) by thetheorem of Friedlander–Parshall [FP83, Lemma A.1].For d ≥ r (2 p − 3) we give a proof by contradiction. Suppose that d is the first non-zero degree in which there is an n such that H n,d ( R F p /σ ) = 0 for d < n + r ( p − − F n,d, = 0, and since the spectral sequence { F rn,p,q } has to converge to 0with the exception of tridegree (0 , , 0) and the d r -differential changes tridegree by(0 , r − , − r ), it must be that F n,d +1 − q,q = 0 for some q ≥ 1. The group F n,d +1 − q,q isspanned by products of an element of H n ′ ,d ′ , ( R F p /σ ) with an element of the tensorproduct of W ∞ -algebras of tridegree ( n ′′ , d ′′ , q ), which must satisfy d ′′ ≥ q ≥ n ′ , d ′ ) = (0 , 0) then we have ( n ′′ , d ′′ , q ) = ( n, d + 1 , q ), which is impossiblebecause then d ′′ = d + 1 < n + r ( p − − n ′′ + r ( p − − 1, but all elements inthe tensor product of W ∞ -algebras satisfy d ′′ ≥ n ′′ + r ( p − − n ′ , d ′ ) = (0 , 0) then, as d ′′ ≥ d ′ < d and so as d was asumed to be minimal such that there is an n such that H n,d ( R F p /σ ) = 0 for d < n + r ( p − − 2, we must have that d ′ ≥ n ′ + r ( p − − 2. As d ′ + d ′′ < n ′ + n ′′ + r ( p − − d ′′ < n ′′ , but this contradicts d ′′ ≥ n ′′ + r ( p − − (cid:3) ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 17 Sharpness. In this section we explain in what sense Theorem A and its con-sequences in Corollary C are optimal.We have already pointed out that H n,d ( R F p ) = 0 for 0 < d < r (2 p − H n,r (2 p − ( R F p ) = 0 for 2 ≤ n ≤ p .When r = 1 these groups are one-dimensional and H n, p − ( R F p ) vanishes for n > p [Spr15b, Theorem 3.1]. Thus it is known that the last rank in which H n, p − ( R F p )does not vanish is p , and our Corollary C says that H n, p − ( R F p ) vanishes for n + p − > p − n > p . Thus for r = 1, in Theorem A neither theoffset nor the slope can be improved while keeping the other fixed.This also shows our vanishing results for the homology of St E ( F np ) ⊗ F p aresharp. For the stabilization of the non-zero class in H p, p − ( R F p ) to vanish in H p +1 , p − ( R F p ), there must be an E -cell in bidegree ( p + 1 , p − 2) and hence H E p +1 , p − ( R F p ) = 0. Thus H p − (GL p +1 ( F p ); St E ( F p +1 p ) ⊗ F p ) = 0, which is justoutside our vanishing line of d < n − H d (GL n ( F p ); St E ( F np ) ⊗ F p ). Remark . When r = 1, following Sprehn’s proof that H p, p − ( R F p ) = F p , one canshow that this group is generated by βQ ( σ ). This implies that H E ∞ p, p − ( R F p ) = 0,which is not a consequence of our vanishing line H E n,d ( R F p ) = 0 for d < n +( p − − Remark . Further unstable classes in bidegrees ( n, d ) = ( p N , r (2 p N − p N − − N ≥ r (2 p N − p N − − ≥ p N + r ( p − − r = 1, p = 3, and N = 2, their class is in bidegree (9 , 11) andour vanishing range for n = 9 is d < 9. In general their classes lie above a line ofslope rp − rp . 6. General linear group of the field F The proof of Theorem B for GL n ( F ) is substantially different. It only uses that S E ( P ) is (dim( P ) − E -Steinberg module.6.1. Low rank computations. We will do some computations in low rank and lowhomological degree of the homology groups H n,d ( R F ) ∼ = H d (GL n ( F ); F ), takingparticular care to describe these in terms of Dyer-Lashof operations. Before doingso we give a variation of a result of Quillen [Qui72, Corollary 1], where instead ofchanging the field in all entries of the matrices but we only do so in the first row. Lemma 6.1. The iterated stabilization maps σ k +1 : GL n − ( F ) −→ GL k + n ( F ) σ k : GL n ( F ) −→ GL k + n ( F ) have the same image on F -homology in degrees d < k + 1 .Proof. The 2-Sylow subgroup of GL n ( F ) is given by the group B n ( F ) ⊂ GL n ( F )of upper triangular matrices. We shall define a larger group H containing thisgroup. Choosing an inclusion F ֒ → F k +1 we may consider F as a subfield of F k +1 . Then H is defined to be the subgroup of B n ( F k +1 ) of matrices such thatall entries below the first row lie in F ⊂ F k +1 , that is H := (cid:20) F × k +1 ( F k +1 ) n − B n − ( F ) (cid:21) . This contains B n ( F ) as the subgroup of matrices all of whose entries lie in F ⊂ F k +1 .By considering F k +1 as a ( k + 1)-dimensional F -vector space, once we pick abasis we get a homomorphism φ : H → GL k + n ( F ). If we pick this basis such that1 ∈ F k +1 is the last basis vector, then because 1 is the only unit in F the followingdiagram commutes: B n − ( F ) B n ( F ) H GL n − ( F ) GL n ( F ) GL n + k ( F ) . φσ σ k We claim that the inclusion B n − ( F ) ֒ → B n ( F ) ֒ → H is an isomorphism on F -homology in degrees d < k + 1. To do so, we use the commutative diagram { e } F × k +1 ⋉ F n − k +1 B n − ( F ) HB n − ( F ) B n − ( F ) . Thus the claim follows by comparing the Serre spectral sequences for the two verticalsequences, as soon as we recall that the F -homology of the group F × k +1 ⋉ F n − k +1 vanishes for d < k + 1 by [Qui72, Theorem 6].Thus in (6.1), the first two vertical maps are surjective on homology with F -coefficients, while the composite of the top two arrows is an F -homology isomor-phism in degrees d < k + 1. The result follows by a diagram chase. (cid:3) Corollary 6.2. σ k · − : H n,d ( R F ) → H n + k,d ( R F ) vanishes for < d < k + 1 .Proof. This is a proof by contradiction. Let x be an element of degree 0 < d < k + 1and of minimal rank n such that σ k x = 0. As d > n ≥ 1, as H ,d ( R F ) = 0. By the previous lemma there is an element y in rank n − ≥ σ k x = σ k +1 y , but σ k y = 0 as n was minimal. (cid:3) 012 1 2 3 40 σ σ σ σ Q ( σ ) Q ( σ ) σQ ( σ ) ( Q ( σ )) d / g Figure 3. The additive generators of the F -homology of GL n ( F ) forlow degree and low rank. Lemma 6.3. See Figure 3 for a depiction of the following results:(i) H ,d ( R F ) = 0 for all d > .(ii) H ,d ( R F ) = F for all d > , generated by Q d ( σ ) .(iii) H , ( R F ) = 0 , and H , ( R F ) = F , generated by σQ ( σ ) .(iv) H , ( R F ) = 0 , and H , ( R F ) = F , generated by ( Q ( σ )) . ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 19 (v) The map σ · − : H , ( R F ) → H , ( R F ) is zero.Proof. Item (i) follows from the observation that GL ( F ) is the trivial group.For (ii), we use that GL ( F ) is isomorphic to the symmetric group S , byacting on the set F \ { } of non-zero vectors, so in particular the inclusion S → GL ( F ) of permutation matrices is an isomorphism on F -homology. The inclusionof permutation matrices forms part of an E ∞ -map, and the F -homology of S iswell-known to be generated by Q d ( σ ) in degree d .For (iii), the cohomology ring of GL ( F ) with Z -coefficients was computed in[TY83], and from it the F -homology is easily deduced using the Universal Coeffi-cients Theorem.To see that the generator of H is σQ ( σ ), we use Stiefel–Whitney classes.The action of GL ( F ) on the set F \ { } gives a 7-dimensional real GL ( F )-representation ρ . Composing with the inclusion ι : S ֒ → GL ( F ) ֒ → GL ( F )gives a real S -representation, which is the sum of 2 copies of the sign represen-tation and 5 copies of the trivial representation. Thus its total Stiefel–Whitneyclass is w ( ρ ◦ ι ) = (1 + w (sign)) = 1 + w (sign) . In the usual identification H ∗ ( S ; F ) = F [ x ] the Stiefel–Whitney class w (sign) is x . As Q ( σ ) ∈ H ( S ; F )is non-zero, it must pair to 1 with x = w (sign) , from which it follows that σQ ( σ ) ∈ H (GL ( F ); F ) pairs to 1 with w ( ρ ), and is hence non-zero.For (iv), we use the exceptional isomorphism A ∼ = GL ( F ) and the com-putation of the cohomology ring of A with F -coefficients by Adem–Milgram[AM04, Theorem VI.6.5]. To describe the generator in degree 2, we again useStiefel–Whitney classes. Acting on the set of non-zero vectors F \ { } gives a15-dimensional real GL ( F )-representation ρ . By composing with the block per-mutation matrix inclusion ι × : S × S → GL ( F ) we obtain a 15-dimensionalreal S × S -representation ρ ◦ ι × . This splits as one copy of sign ⊠ sign, 3copies of triv ⊠ sign, 3 copies of sign ⊠ triv, and 8 copies of triv ⊠ triv, where ⊠ denotes the external tensor product of representations. Thus, under the iden-tification H ∗ ( S × S ; F ) = F [ x , x ] given by the K¨unneth theorem, its totalStiefel–Whitney class is w ( ρ ◦ ι × ) = (1 + x + x ) · (1 + x ) · (1 + x ) = 1 + x x + classes of degree ≥ x x pairs to 1 with the homology class Q ( σ ) × Q ( σ ) ∈ H ( S × S ; F ), which under ι × maps to Q ( σ ) .For item (v), it is a consequence of Corollary 6.2 that stabilizing twice vanishesin degrees < 4, so in particular σ Q ( σ ) = 0. An alternative proof may extractedfrom [vdK75]. (cid:3) The proof of Theorem B. The canonical generator σ of H (GL ( F ); F )may be represented by a map σ : S , → R F . We further pick a 0- and a 1-simplex in F [ C ∞ (2)] ∈ sMod F representing the product and the operation Q ( − ).These may be used to first produce a representative S , → R F for Q ( σ ), andthen one for σQ ( σ ). As we proved that σQ ( σ ) = 0, we may pick a null-homotopy D , → R F for the latter. Out of this data, we may build a map in Alg E ∞ ( sMod NF ). f : A −→ R F with A := E ∞ ( S , F σ ) ∪ E ∞ σQ ( σ ) D , F τ By Lemma 6.3, this satisfies H n,d ( f ) = 0 for n ≤ d ≤ 2. Using theHurewicz theorem in E ∞ -homology, Corollary E k .11.12, we see that H E ∞ n,d ( f ) = 0for n ≤ d ≤ 2. On the other hand, we know that H E ∞ n,d ( A ) is concentrated in bidegrees ( n, d ) = (1 , , (3 , H E ∞ n,d ( R F ) = 0 for d < n − 1. The conclusionis that H E ∞ n,d ( f ) = 0 for d < n if n ≤ 3, and for d < n − n > R F we may construct a unital strictly associative algebra R F ≃ R + F . From σ we get a corresponding map σ ·− : R F → R F whose homotopy cofibre R F /σ satisfies H n,d ( R F /σ ) ∼ = H d (GL n ( F ) , GL n − ( F ); F ) , and to prove Theorem B it thus suffices to prove a vanishing range for H n,d ( R F /σ ). Proposition 6.4. H n,d ( R F /σ ) = 0 for d < n − .Proof. We apply the CW approximation theorem, Theorem E k .11.21, to extendthe map f to a commutative diagram A R F C fg ≃ with g : A → C a relative CW approximation with only cells of bidegrees ( n, d )satisfying d ≥ n if n ≤ d ≥ n − n > 3. We get a corresponding skeletalspectral sequence(6.1) E n,p,q = ˜ H n,p + q,q ∗ A ∨ E ∞ E ∞ _ i ∈ I S n i ,d i ,d i F x i /σ = ⇒ H n,p + q ( R F /σ ) , with n i and d i satisfying the bounds given above. By Proposition E k .16.4.1 coprod-ucts of unital E ∞ -algebras coincide with tensor products up to weak equivalence.Thus on homology there is an isomorphism H ∗ , ∗ , ( A /σ ) ⊗ O i ∈ I W ∞ ( F { x i [ n i , d i , d i ] } ) , where W ∞ is the Dyer-Lashof algebra as described in Section E k .16.2. Since d i ≥ n i if n i ≤ d i ≥ n i − n i > 3, for all i ∈ I , in fact all non-zero elements of N W ∞ ( F { x i } ) satisfy these bounds. Thus it remains to show that H n,d ( A /σ ) = 0for d < n − .Computing the homology of A /σ is quite complicated, and we shall be contentwith computing enough of it to establish the required vanishing. To do so, wefirst investigate the spectral sequence for the cell-attachment filtration on A , asdescribed in Corollary E k .10.17. This spectral sequence interacts well with Dyer–Lashof operations as described in Section E k .16.6, converges to H d,p + q ( A ), and has E -page given by(6.2) E n,p,q ( A ) = H n,p + q,q (cid:18) E ∞ (cid:16) S , , F σ ∨ D , , F τ (cid:17)(cid:19) ∼ = W ∞ ( F { σ, τ } ) . This is the free graded commutative algebra on the tri-graded F -vector space L with basis given by iterated Q i ’s applied to σ and τ . See Figure 4 for a table ofthese generators in a range; we point out that the only generators of slope < / σ , Q ( σ ), and τ . Claim. The differentials of (6.2) have the following properties:(i) All differentials vanish on Q I ( σ ) for any admissible I .(ii) d ( τ ) = σQ ( σ ). For i ≥ d j ( τ i ) = 0 for j < i and d i ( τ i ) = ( Q ( σ )) i Q i − , i − ,..., ( σ ) + σ i Q i , i − ,..., ( σ ) . (iii) d ( Q I ( τ )) = d ( Q I ( τ )) = 0 for admissible I = ∅ and e ( I ) > ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 21 σ Q ( σ ) Q ( σ ) Q ( σ ) Q ( σ ) Q ( σ ) Q , ( σ ) Q , ( σ ) Q , ( σ ), Q , ( σ ) τ Q ( τ ) d = n d / n Figure 4. The additive generators of L in the range n ≤ d ≤ 5, withfiltration degree suppressed. Proof of claim. Part (i) follows from the map E ∞ ( S , , F σ ) → A . We prove part (ii)by induction. As the cell τ , of filtration 1, is attached along σQ ( σ ), of filtration 0,we have d ( τ ) = σQ ( σ ). We can compute d ( τ ) = 2 d ( τ ) = 0, and then compute d ( τ ) using the fact that τ = Q ( τ ) so by Theorem E k .16.6.2 it survives to the E i +1 -page and its differential on this page is given by d ( τ ) = d ( Q ( τ )) = Q ( d ( τ )) = Q ( σQ ( σ ))= Q ( σ ) + σ Q , ( σ )using the Cartan formula.Suppose now that the result holds for τ i . Then on the E i -page the class τ i exists and so on this page τ i +1 is the square of τ i , so is Q i +1 ( τ i ) (recall that τ has degree 2). It then follows from Theorem E k .16.6.2 that τ i +1 survives until E i +1 , and the differential here satisfies d i +1 ( τ i +1 ) = d i +1 ( Q i +1 ( τ i ))= Q i +1 ( d i ( τ i ))= Q i +1 (( Q ( σ )) i Q i − , i − ,..., ( σ ) + σ i Q i , i − ,..., ( σ )) . Let us consider the terms separately: Q i +1 (( Q ( σ )) i Q i − , i − ,..., ( σ )) may becomputed using the Cartan formula. Since Q s vanishes on elements of degree d < s , the only options are (a) splitting Q i +1 into 2 i terms Q and a single Q i ,resulting in ( Q ( σ )) i +1 Q i , i − , i − ,..., ( σ ) because Q s acts as squaring on elementsof degree s , or (b) splitting Q i +1 into 2 i − Q , a single term Q and a single Q i − , resulting in 2 i contributions ( Q ( σ )) i +1 − Q , ( σ )( Q i − , i − ,..., ( σ )) .Similarly, for Q i +1 ( σ i Q i , i − ,..., ( σ )) the only non-zero contribution to theCartan formula splits Q i +1 into 2 i terms Q , and a single Q i +1 , resulting in σ i +1 Q i +1 , i ,..., ( σ ).For (iii), use Theorem E k .16.6.2. (cid:3) We compare this to the cell-attachment filtration spectral sequence on A /σ using q : A −→ A /σ. This spectral sequence converges to H d,p + q ( A /σ ) and has E -page(6.3) E n,p,q ( A /σ ) = H n,p + q,q (cid:18) E ∞ (cid:16) S , , F σ ∨ D , , F τ (cid:17) /σ (cid:19) . In terms of our description of E ∗ , ∗ , ∗ ( A ) this is given by dividing out by the ideal( σ ). That is, it is the free graded commutative algebra with generators given byiterated Q i ’s applied to σ and τ , except for the generator σ itself. Let us write thesegenerators as q ∗ ( Q I ( σ )) := { Q I ( σ ) } and so on. We shall compute differentials inthis spectral sequence using the fact that they commute with the map q ∗ of spectralsequences, and more generally using the module structure of (6.3) over (6.2), − · − : E n,p,q ( A ) ⊗ E n ′ ,p ′ ,q ′ ( A /σ ) −→ E n + n ′ ,p + p ′ ,q + q ′ ( A /σ ) , given by A /σ having the structure of a filtered module over the filtered ring A . Claim. E n,p,q ( A /σ ) = 0 for p + q < n − . Proof of claim. The first differentials we compute using naturality are d j ( { Q I ( σ ) } ) = 0 for j ≥ d ( { τ i } ) = 0 for i ≥ d ( { Q I ( τ ) } ) = d ( { Q I ( τ ) } ) = 0 for I = ∅ and e ( I ) > . Because τ k survives to E ( A ), we can determine d ( { τ k } ) using (ii): d ( { τ k } ) = d ( τ k · { } )= d ( τ k ) · { } + τ k · d ( { } )= ( kτ k − ( Q ( σ ) + σ Q , ( σ )) · { } = { kQ ( σ ) τ k − } . In particular, d ( { τ r } ) = 0 while d ( { τ r +2 } ) = { Q ( σ ) τ r } . Using the modulestructure we deduce from this that for all i and r (6.4) d ( { Q ( σ ) i τ r } ) = 0 ,d ( { Q ( σ ) i τ r +2 } ) = { Q ( σ ) i +3 τ r } . On the other hand, we cannot compute d ( { τ k +1 } ) by naturality, as τ k +1 doesnot survive to E ( A ). However, { τ } has internal degree 1, so d ( { τ } ) has internaldegree − d ( { τ k +1 } ) for k ≥ 1, we use themodule structure: d ( { τ k +1 } ) = d ( τ k · { τ } )= ( kτ k − ( Q ( σ ) + σ Q , ( σ ))) · { τ } + τ k · d ( { τ } )= { kQ ( σ ) τ k − } . In particular, d ( { τ r +1 } ) = 0, while d ( { τ r +3 } ) = { Q ( σ ) τ r +1 } . Using themodule structure we deduce from this that for all i and r (6.5) d ( { Q ( σ ) i τ r +1 } ) = 0 ,d ( { Q ( σ ) i τ r +3 } ) = { Q ( σ ) i +3 τ r +1 } . The conclusion of this discussion is as follows. Firstly E ∗ , ∗ , ∗ ( A /σ ) = E ∗ , ∗ , ∗ ( A /σ ).Secondly, E ∗ , ∗ , ∗ ( A /σ ) is a free module over the subalgebra W = ∅ of W ∞ ( F { σ, τ } )generated by those Q I ( σ ) and Q I ( τ ) with I = ∅ , and the d -differential is W = ∅ -linear. As a W = ∅ -module, E ∗ , ∗ , ∗ ( A /σ ) is given by W = ∅ ⊗ (cid:0) F [ Q ( σ ) , τ ] , d (cid:1) , ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 23 with differential d given by (6.4) and (6.5). Thus we have that E ∗ , ∗ , ∗ ( A /σ ) isa free W = ∅ -module on the generators Q ( σ ) i τ j with i ∈ { , , } and j ≡ , n, d ) satisfying d ≥ n − . As the additivegenerators of W = ∅ have slope at least , we conclude that E n,p,q ( A /σ ) = 0 for p + q < n − . (cid:3) Once E n,p,q ( A /σ ) vanishes in this range, the vanishing result for its E ∞ -pageand hence for H n,p + q ( A /σ ) follow. As explained earlier, this gives the desiredvanishing range for H n,p + q ( R F /σ ). (cid:3) Remark . One can not do better in the above proof. The class τ survivesto E , , ( A ) and satisfies d ( τ ) = Q ( σ ) Q , ( σ ) + σ Q , , ( σ ), so we have d ( { τ } ) = { Q ( σ ) Q , ( σ ) } . But on the other hand we have d ( { Q , ( σ ) Q ( σ ) τ } ) = { Q ( σ ) Q , ( σ ) } by (6.4) and the module structure, so d ( { τ } ) = 0 ∈ E , , ( A /σ ). Now all higherdifferentials on { τ } would land in negative internal degree, so { τ } is a permanentcycle. There is no reason for it to be a boundary.6.3. Sharpness. We saw in Section 6.1 the map H d ( S n , S n − ; F ) −→ H d (GL n ( F ) , GL n − ( F ); F )is an isomorphism for ( n, d ) = (2 , , (4 , Q ( σ ) and Q ( σ ) do not desta-bilize. The class in bidegree ( n, d ) = (4 , 2) exactly fails to satisfy the bound d < n − . Remark . This rules out some potential improvements of homological stabilityranges for other automorphism groups. Suppose we have a ring R with homo-morphism to F (e.g. Z ), then upon picking a point σ ∈ BGL ( R ) we get a map E + ∞ ( S , F σ ) → F n ≥ B GL n ( R ) of E ∞ -algebras. As a result of the above observation,the composition of the maps E + ∞ ( S , F σ ) −→ G n ≥ B GL n ( R ) −→ G n ≥ B GL n ( F )will induce an isomorphism on relative homology with F -coefficients of the stabiliza-tion map in bidegree ( n, d ) = (2 , , (4 , H d (GL n ( R ) , GL n − ( R ); F ) = 0for ( n, d ) = (2 , , (4 , E + ∞ ( S , F σ ) −→ G n ≥ B Aut( F n ) −→ G n ≥ B GL n ( F ) , the right map given by the action of a based homotopy automorphism of W n S on H ( W n S ; F ), implies H d (Aut( F n ) , Aut( F n − ); F ) = 0 for ( n, d ) = (2 , , (4 , H ( GL ( F ); F ) = 0, based oncomputer calculations. Combining this with Theorem G of [Hep16] it follows thatthis group must be one-dimensional and generated by Q ( σ ) . In the followinglemma we give our own proof of the latter result. Lemma 6.7. H , ( R F ) is at most 1-dimensional, spanned by Q ( σ ) .Proof. The proof of Theorem B provides a map of exact sequences H , ( A /σ ) H , ( A ) H , ( A ) 0 H , ( R F /σ ) H , ( R F ) H , ( R F ) 0 . δ σδ σ The calculations of A /σ in that proof give us part of the top row: H , ( A /σ ) = F generated by τ Q ( σ ) which the connecting homomorphism sends to Q ( σ ) .From the spectral sequence (6.1) it follows that the left vertical map is surjective.From the vanishing range of Corollary C we see that H , ( R F ) = 0. Thus thecomposite H , ( A /σ ) −→ H , ( R F /σ ) −→ H , ( R F )is a surjective map from a 1-dimensional F -vector space. By naturality its imageis generated by Q ( σ ) . (cid:3) Conditional on Szymik’s calculation that H , ( R F ) = 0, it follows that H , ( R F ) = F { Q ( σ ) } . Note that this class must destabilize (twice), as applying Q to therelation σQ ( σ ) = 0 gives the relation σ Q , ( σ ) + Q ( σ ) = 0.As announced in the introduction, the above discussion allows us to resolve theremaining case n = 3 of the Milgram–Priddy question. Lemma 6.8. The class det ∈ H ( M , ( F ); F ) GL ( F ) × GL ( F ) lies in the imageof the restriction map from H (GL ( F ); F ) .Proof. The class Q ( σ ) ∈ H (GL ( F ); F ) is represented by the matrix [ ], orequivalently by the conjugate matrix [ ]. Therefore the class Q ( σ ) ∈ H (GL ( F ); F )is carried on the subgroup G ⊂ GL ( F ) of matrices of the form ∗ ∗ ∗ . By conjugating with the permutation matrix in S ⊂ GL ( F ) which interchanges2 and 5, we see that the subgroup G is conjugate to the subgroup G consistingof matrices of the form ∗ 00 1 0 0 0 ∗ ∗ , so that H ( G ; F ) → H (GL ( F ); F ) also has image containing Q ( σ ) .The inclusion G ֒ → GL ( F ) factors over the subgroup M , ( F ). Letting x ∈ H (GL ( F ); F ) denote the cohomology class such that h x, Q ( σ ) i = 1, we con-clude that the pullback of x to H ( M , ( F ); F ) is a non-zero GL ( F ) × GL ( F )-invariant cohomology class. It is an elementary but long computation that det isthe only GL ( F ) × GL ( F )-invariant cohomology class in H ( M , ( F ); F ). (cid:3) Homology of Steinberg modules In Section 5.1 we saw that the Steinberg module St( F nq ) has the property thatSt( F nq ) ⊗ F p is a projective and irreducible F p [GL n ( F q )]-module (we stated it thereonly for q = p r with q = 2, but it also holds for q = 2 [Hum87, Sections 2 and 3]).As before, it follows that H ∗ (GL n ( F q ); St( F nq ) ⊗ F p ) = 0.What happens when we replace F p with F ℓ ? Using the methods of this paper,together with Quillen’s calculation of H ∗ (GL n ( F q ); F ℓ ) for ℓ = p , we will calculate H ∗ (GL n ( F q ); St( F nq ) ⊗ F ℓ ) for all ℓ = p . ∞ -CELLS AND GENERAL LINEAR GROUPS OF FINITE FIELDS 25 Theorem 7.1. Let q = p r , ℓ = p , and write t for the smallest number such that q t ≡ ℓ . There is an isomorphism of bigraded F ℓ -vector spaces M n ≥ H ∗ (GL n ( F q ); St( F nq ) ⊗ F ℓ )[ n ] ∼ = Λ[ sσ, sξ , sξ , . . . ] ⊗ Γ F ℓ [ sη , sη , . . . ] where | sσ | = (1 , , | sξ i | = ( t, it + 1) , and | sη i | = ( t, it ) . One may extract the following vanishing theorem from this calculation (andslight extensions are possible for specific n and q ). It improves by a factor of 2 arecent vanishing result of Ash–Putman–Sam [APS18] in the case of general lineargroups of finite fields. Corollary 7.2. H d (GL n ( F q ); St( F nq )) = 0 for d < n − . First recall that the Steinberg module is the reduced top homology of the Titsbuilding T ( F nq ) for F nq : St( F nq ) := ˜ H n − ( T ( F nq ); Z ) . On the other hand, the E -Steinberg module is the reduced top homology of thesplit Tits building: St E ( F nq ) := ˜ H n − ( S E ( F nq ); Z ) . Here T ( F nq ) is the geometric realisation of the nerve T • ( F nq ) of the poset of propersubspaces of F nq and their inclusions, whereas S E ( F nq ) is the geometric realisation ofthe nerve S E • ( F nq ) of the poset S E ( F nq ) of ordered pairs ( V , V ) of complementarysubspaces of F nq , where the V ’s are ordered by inclusion and the V ’s by reverseinclusion. Forgetting the V ’s defines a map of posets, and so a map φ : S E ( F nq ) −→ T ( F nq ) . This induces in particular a map φ ∗ : St E ( F nq ) → St( F nq ). This is far from being anisomorphism, but we do have the following comparison. Lemma 7.3. For ℓ = p the induced map φ ∗ : H ∗ (GL n ( F q ); St E ( F nq ) ⊗ F ℓ ) −→ H ∗ (GL( F nq ); St( F nq ) ⊗ F ℓ ) is an isomorphism.Proof. As both the Steinberg and E -Steinberg modules are the reduced top ho-mology of the spaces T ( F nq ) and S E ( F nq ) respectively, which are each wedges ofspheres, it suffices to prove that the induced map φ// GL n ( F q ) : S E ( F nq ) // GL n ( F q ) −→ T ( F nq ) // GL n ( F q )on homotopy orbits induces an isomorphism on homology with F ℓ -coefficients. Thegroup GL( F nq ) acts simplicially on the semi-simplicial sets S E • ( F nq ) and T • ( F nq ), andtaking levelwise homotopy orbits gives a map of semi-simplicial spaces.The p -simplices of S E • ( F nq ) are given by the set of splittings V ⊕ V ⊕· · ·⊕ V p = F nq of F nq into p + 1 non-zero subspaces, whereas the p -simplices of T • ( F nq ) is the set ofproper flags 0 < W < W < · · · < W p − < F nq . The map S E p ( F nq ) → T p ( F nq ) is a bijection on GL n ( F q )-orbits. The stabiliser of adecomposition V ⊕ V ⊕ · · · ⊕ V p is the subgroup GL( V ) × · · · × GL( V p ), whichwe may think of as block-diagonal matrices BD ≤ GL n ( F q ) with certain blocksizes, whereas the stabiliser of the associated flag with W i := V ⊕ · · · ⊕ V i is theassociated subgroup of block-upper triangular matrices BU T ≤ GL n ( F q ). Theinclusion i : BD → BU T is split by the homomorphism ρ : BU T → BD whichrecords the diagonal blocks. The kernel of ρ is a p -group, so has vanishing F ℓ -homology: thus ρ is a F ℓ -homology equivalence, and so i is too. It follows that the semi-simplicial map S E • ( F nq ) // GL n ( F q ) → T • ( F nq ) // GL n ( F q )is a levelwise F ℓ -homology equivalence, so the map on geometric realisations istoo. (cid:3) Using the identification in this lemma, (3.2) says that H n,d ( Q E L ( R F ℓ )) = H E n,d ( R F ℓ ) ∼ = H d − ( n − (GL n ( F q ); St( F nq ) ⊗ F ℓ ) . On the other hand, Theorem E k .13.1.7 gives a pointed weak equivalence S ∧ Q E L ( R F ℓ ) ≃ B E ( R F ℓ ) /S , . We may compute the homology of B E ( R F ℓ ) using the bar spectral sequence ofTheorem E k .14.1.1, which takes the form E ∗ ,p,q = Tor p,qH ∗ , ∗ ( R F ℓ ) ( F ℓ [ ] , F ℓ [ ]) = ⇒ H ∗ ,p + q ( B E ( R F ℓ )) . At this point we invoke Quillen’s calculation of the F ℓ -homology of the groupsGL( F nq ). Quillen shows [Qui72, Theorem 3] that there is a ring isomorphism H ∗ , ∗ ( R F ℓ ) ∼ = F ℓ [ σ, ξ , ξ , . . . ] ⊗ Λ[ η , η , . . . ]for classes of bidegrees | σ | = (1 , | ξ i | = ( t, it ), and | η i | = ( t, it − E ∗ , ∗ , ∗ = Λ[ sσ, sξ , sξ , . . . ] ⊗ Γ F ℓ [ sη , sη , . . . ]for classes of tridegrees | sσ | = (1 , , | sξ i | = ( t, , it ), and | sη i | = ( t, , it − R F ℓ is an E -algebra (in fact an E ∞ -algebra), by Lemma E k .14.1.2 the bar spectralsequence is a spectral sequence of F ℓ -algebras. 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Yagita, The mod p cohomology ring of GL ( F p ), J. Algebra (1983), no. 2, 295–303.[vdK75] W. van der Kallen, The Schur multipliers of SL(3 , Z ) and SL(4 , Z ), Math. Ann. (1974/75), 47–49. E-mail address : [email protected] Department of Mathematics, University of Copenhagen, Denmark E-mail address : [email protected] Department of Mathematics, One Oxford Street, Cambridge MA, 02138, USA E-mail address : [email protected]@dpmms.cam.ac.uk