aa r X i v : . [ m a t h . A T ] O c t ON THE EDGE OF THE STABLE RANGE
RICHARD HEPWORTH
Abstract.
We prove a general homological stability theorem for certain fami-lies of groups equipped with product maps, followed by two theorems of a newkind that give information about the last two homology groups outside the stablerange. (These last two unstable groups are the ‘edge’ in our title.) Applying ourresults to automorphism groups of free groups yields a new proof of homologicalstability with an improved stable range, a description of the last unstable groupup to a single ambiguity, and a lower bound on the rank of the penultimateunstable group. We give similar applications to the general linear groups of theintegers and of the field of order 2, this time recovering the known stablilityrange. The results can also be applied to general linear groups of arbitrary prin-cipal ideal domains, symmetric groups, and braid groups. Our methods requireus to use field coefficients throughout. Introduction
A sequence of groups and inclusions G ֒ → G ֒ → G ֒ → · · · is said to satisfy homological stability if in each degree d there is an integer n d such that the inducedmap H d ( G n − ) → H d ( G n ) is an isomorphism for n > n d . Homological stability isknown to hold for many families of groups, including symmetric groups [16], generallinear groups [17, 3, 22], mapping class groups of surfaces and 3-manifolds [8, 19, 23,12], diffeomorphism groups of highly connected manifolds [6], and automorphismgroups of free groups [11, 10]. Homological stability statements often also specifythat the last map outside the range n > n d is a surjection, so that the situationcan be pictured as follows. · · · → H d ( G n d − ) → H d ( G n d − ) → H d ( G n d − ) | {z } edge of the stable range ։ H d ( G n d ) ∼ = −→ H d ( G n d +1 ) ∼ = −→ · · · | {z } stable range The groups H d ( G n d ) , H d ( G n d +1 ) , . . . , which are all isomorphic, are said to formthe stable range . This paper studies what happens at the edge of the stable range ,by which we mean the last two unstable groups H d ( G n d − ) and H d ( G n d − ). Weprove a new and rather general homological stability result that gives exactly thepicture above with n d = 2 d + 1. Then we prove two theorems of an entirelynew kind. The first describes the kernel of the surjection H d ( G n d − ) ։ H d ( G n d ), Mathematics Subject Classification.
Key words and phrases.
Homological stability, general linear groups, automorphism groups offree groups.
1N THE EDGE OF THE STABLE RANGE 2 and the second explains how to make the map H d ( G n d − ) → H d ( G n d − ) into asurjection by adding a new summand to its domain. These general results holdfor homology with coefficients in an arbitrary field.We apply our general results to general linear groups of principal ideal domains(PIDs) and automorphism groups of free groups. In both cases we obtain newproofs of homological stability, recovering the known stable range for the generallinear groups, and improving upon the known stable range for Aut( F n ). We alsoobtain new information on the last two unstable homology groups for Aut( F n ), GL n ( Z ) and GL n ( F ), in each case identifying the last unstable group up to asingle ambiguity.Our proofs follow an overall pattern that is familiar in homological stability.We define a sequence of complexes acted on by the groups in our family, andwe assume that they satisfy a connectivity condition. Then we use an algebraicargument, based on spectral sequences obtained from the actions on the complexes,to deduce the result. The connectivity condition has to be verified separately foreach example, but it turns out that in our examples the proof is already in theliterature, or can be deduced from it. The real novelty in our paper is the algebraicargument. To the best of our knowledge it has not been used before, either in thepresent generality or in any specific instances. Even in the case of general lineargroups of PIDs, where our complexes are exactly the ones used by Charney in theoriginal proof of homological stability [3] for Dedekind domains, we are able toimprove the stable range obtained, matching the best known.1.1. General results.
Let us state our main results, after first establishing somenecessary terminology. From this point onwards homology is to be taken withcoefficients in an arbitrary field F , unless stated otherwise.A family of groups with multiplication ( G p ) p > consists of a sequence of groups G , G , G , . . . equipped with product maps G p × G q → G p + q for p, q >
0, subjectto some simple axioms. See section 2 for the precise definition. The axiomsimply in particular that L p > H ∗ ( G p ) is a graded commutative ring. Examplesinclude the symmetric groups, braid groups, the general linear groups of a PID,and automorphism groups of free groups.To each family of groups with multiplication ( G p ) p > we associate the splittingposets SP n for n >
2. If we think of G n as the group of symmetries of an ‘objectof size n ’, then an element of SP n is a splitting of that object into two orderednontrivial pieces. See section 3 for the precise definition. The stabilisation map s ∗ : H ∗ ( G n − ) → H ∗ ( G n ) is the map induced by the homomorphism G n − → G n that takes the product on the left with the neutral element of G . Our first mainresult is the following homological stability theorem. Theorem A.
Let ( G p ) p > be a family of groups with multiplication, and assumethat | SP n | is ( n − -connected for all n > . Then the stabilisation map s ∗ : H ∗ ( G n − ) −→ H ∗ ( G n ) N THE EDGE OF THE STABLE RANGE 3 is an isomorphism for ∗ n − and a surjection for ∗ n − . Here homology istaken with coefficients in an arbitrary field. Theorem A overlaps with work in progress of Søren Galatius, Alexander Ku-pers and Oscar Randal-Williams. Indeed, if we were to add the assumption that F p > G p is a braided monoidal groupoid, then it would follow from the work ofGalatius, Kupers and Randal-Williams. (The definition of family of groups withmultiplication ensures that F p > G p is a monoidal groupoid; the braiding assump-tion holds in all of our examples.) We will mention other points of overlap as theyoccur.In a given degree m , Theorem A gives us the surjection and isomorphisms inthe following sequence. · · · → H m ( G m − ) → H m ( G m − ) → H m ( G m ) | {z } edge of the stable range ։ H m ( G m +1 ) ∼ = −→ H m ( G m +2 ) ∼ = −→ · · · | {z } stable range Our next two theorems extend into the edge of the stable range.
Theorem B.
Let ( G p ) p > be a family of groups with multiplication, and assumethat | SP n | is ( n − -connected for all n > . Then the kernel of the map s ∗ : H m ( G m ) ։ H m ( G m +1 ) is the image of the product map H ( G ) ⊗ m − ⊗ ker[ H ( G ) s ∗ −→ H ( G )] −→ H m ( G m ) . Here homology is taken with coefficients in an arbitrary field.
Theorem C.
Let ( G p ) p > be a family of groups with multiplication, and assumethat | SP n | is ( n − -connected for all n > . Then the map H m ( G m − ) ⊕ H ( G ) ⊗ m ։ H m ( G m ) is surjective. Here homology is taken with coefficients in an arbitrary field. Homological stability results like Theorem A are often combined with theoremscomputing the stable homology lim n →∞ H ∗ ( G n ) to deduce the value of H ∗ ( G n ) inthe stable range. In a similar vein, Theorems B and C allow us to bound the lasttwo unstable groups H m ( G m ) and H m ( G m − ) in terms of lim n →∞ H ∗ ( G n ). Inthe following subsections we will see how this works for automorphism groups offree groups and general linear groups of PIDs. Note that our results do not ruleout the possibility of a larger stable range than the one provided by Theorem A.Nevertheless, in what follows we will refer to H m ( G m ) and H m ( G m − ) as the ‘lasttwo unstable groups’. N THE EDGE OF THE STABLE RANGE 4
Applications to automorphism groups of free groups.
The automor-phism groups of free groups form a family of groups with multiplication (Aut( F n )) n > .In this case the splitting poset SP n consists of pairs ( A, B ) of proper subgroups of F n satisfying A ∗ B = F n . By relating the splitting poset to the poset of free fac-torisations studied by Hatcher and Vogtmann in [9], we are able to show that | SP n | is ( n − Theorem D.
Let F be a field. Then the stabilisation map s ∗ : H ∗ (Aut( F n − ); F ) −→ H ∗ (Aut( F n ); F ) is an isomorphism for ∗ n − and a surjection for ∗ n − . Moreover, if char( F ) = 2 , then s ∗ is an isomorphism for ∗ n − and a surjection for ∗ n . Hatcher and Vogtmann showed in [11] that s ∗ : H ∗ (Aut( F n − )) → H ∗ (Aut( F n ))is an isomorphism for ∗ n − and a surjection for ∗ n − , where homology istaken with arbitrary coefficients. Theorem D increases this stable range one stepto the left in each degree when coefficients are taken in a field, and two steps tothe left in each degree when coefficients are taken in a field of characteristic otherthan 2. (In characteristic 0 this falls far short of the best known result [10].) Inparticular we learn for the first time that the groups H m (Aut( F m +1 ); F ) are stable.By applying Theorems B and C when F = F , we are able to learn the followingabout the last two unstable groups H m (Aut( F m ); F ) and H m (Aut( F m − ); F ). Theorem E.
Let t ∈ H (Aut( F ); F ) denote the element determined by the trans-formation x x , x x x , and let m > . Then the kernel of the stabilisationmap s ∗ : H m (Aut( F m ); F ) ։ H m (Aut( F m +1 ); F ) is the span of t m , and the map H m (Aut( F m − ); F ) ⊕ F → H m (Aut( F m ); F ) , ( x, y ) s ∗ ( x ) + y · t m is surjective. This theorem shows that the last unstable group H m (Aut( F m ); F ) is eitherisomorphic to the stable homology lim n →∞ H m (Aut( F n ); F ), or is an extensionof it by a copy of F generated by t m . It does not state which possibility holds.Galatius [5] identified the stable homology lim n →∞ H ∗ (Aut( F n )) with H ∗ (Ω ∞ S ∞ ),where Ω ∞ S ∞ denotes a path-component of Ω ∞ S ∞ = colim n →∞ Ω n S n . Thus weare able to place the following bounds on the dimensions of the last two unstablegroups for m >
1, where ǫ is either 0 or 1.dim( H m (Aut( F m ); F )) = dim( H m (Ω ∞ S ∞ ; F )) + ǫ dim( H m (Aut( F m − ); F )) > dim( H m (Ω ∞ S ∞ ; F )) N THE EDGE OF THE STABLE RANGE 5
Applications to general linear groups of PIDs.
The general linear groupsof a commutative ring R form a family of groups with multiplication ( GL n ( R )) n > .When R is a PID, the realisation | SP n | of the splitting poset is precisely the splitbuilding [ R n ] studied by Charney, who showed that it is ( n − H ∗ ( GL n − ( R )) → H ∗ ( GL n ( R )) is onto for ∗ n − andan isomorphism for ∗ n − , where homology is taken with field coefficients. Thisexactly recovers homological stability with the range due to van der Kallen [22],but only with field coefficients. Theorems B and C then allow us to learn about thelast two unstable groups H m ( GL m − ( R )) and H m ( GL m ( R )), where little seems tobe known in general. In order to illustrate this we specialise to the cases R = Z and R = F and take coefficients in F ; this is the content of our next two subsections.1.4. Applications to the general linear groups of Z . We now specialise tothe groups GL n ( Z ) and take coefficients in F . Theorems B and C give us thefollowing information about the final two unstable groups H m ( GL m ( Z ); F ) and H m ( GL m − ( Z ); F ). Theorem F.
Let t denote the element of H ( GL ( Z ); F ) determined by the matrix ( ) and let m > . Then the kernel of the stabilisation map s ∗ : H m ( GL m ( Z ); F ) ։ H m ( GL m +1 ( Z ); F ) is the span of t m , and the map H m ( GL m − ( Z ); F ) ⊕ F → H m ( GL m ( Z ); F ) , ( x, y ) s ∗ ( x ) + y · t m is surjective. This theorem shows that the last unstable group H m ( GL m ( Z ); F ) is either iso-morphic to the stable homology lim n →∞ H m ( GL m ( Z ); F ), or is an extension ofit by a copy of F generated by t m . It does not guarantee that t m = 0, and sodoes not specify which possibility occurs. The theorem also gives us the follow-ing lower bounds on the dimensions of the last two unstable groups in terms ofdim(lim n →∞ H m ( GL n ( Z ); F )), and in particular shows that they are highly non-trivial. dim( H m ( GL m ( Z ); F )) = dim (cid:16) lim n →∞ H m ( GL n ( Z ); F ) (cid:17) + ǫ dim( H m ( GL m − ( Z ); F )) > dim (cid:16) lim n →∞ H m ( GL n ( Z ); F ) (cid:17) Here ǫ is either 0 or 1.1.5. Applications to the general linear groups of F . Now let us specialiseto the groups GL n ( F ). Quillen showed that in this case the stable homologylim n →∞ H ∗ ( GL n ( F ); F ) vanishes [17, Section 11]. Combining this with Maazen’sstability result shows that H m ( GL n ( F ); F ) = 0 for n > m + 1. It is naturalto ask for a description of the final unstable homology groups H m ( GL m ( F ); F ). N THE EDGE OF THE STABLE RANGE 6
These are known to be nontrivial for m = 1 and m = 2, the latter case being dueto Milgram and Priddy (Example 2.6 and Theorem 6.5 of [15]), but to the best ofour knowledge nothing further is known. By applying Theorem B we obtain thefollowing result, which determines each of the groups H m ( GL m ( F ); F ) up to asingle ambiguity. Theorem G.
Let t denote the element of H ( GL ( F ); F ) determined by the ma-trix ( ) . Then H m ( GL m ( F ); F ) is either trivial, or is a copy of F generatedby the class t m . We hope that by extending the techniques of the present paper we will be ablein future to prove the following conjecture. We anticipate that the known non-vanishing of t and t will be an essential ingredient in its proof. Conjecture.
For every m > the group H m ( GL m ( F ); F ) is a single copy of F generated by the class t m , where t ∈ H ( GL ( F ); F ) is the element determined bythe matrix ( ) . A proof of this conjecture would, via the homomorphisms Aut( F n ) → GL n ( Z ) → GL n ( F ), also resolve the ambiguities in Theorems E and F, showing that the finalunstable homology groups H m (Aut( F m ) , F ) and H m ( GL m ( Z ); F ) are extensionsby F of lim n →∞ H m (Aut( F n ); F ) and lim n →∞ H m ( GL n ( Z ); F ) respectively. Inparticular this would confirm that the known homological stability ranges aresharp.Theorem G is relevant to questions about the groups H m ( GL m ( F ); F ) raisedby Milgram and Priddy in [15, p.301], and posed explicitly by Priddy in [2, section5]. Let M mm denote the subgroup of GL m ( F ) consisting of matrices of the form (cid:18) I m ∗ I m (cid:19) . Milgram and Priddy describe an element det m ∈ H m ( M mm ; F ) that is invariantunder the action of N GL m ( F ) ( M mm ) /M mm = GL m ( F ) × GL m ( F ), and so poten-tially lifts to an element of H m ( GL m ( F ); F ). Priddy asks whether det m lifts to H m ( GL m ( F ); F ), and if so, whether it spans H m ( GL m ( F ); F ). As explainedto us by David Sprehn, t m is the image of a class in H m ( M mm ; F ), and det m spans the invariants H m ( M mm ; F ) GL m ( F ) × GL m ( F ) . Theorem G therefore showsthat H m ( GL m ( F ); F ) is either trivial, or is a single copy of F generated by alift of det m .1.6. Decomposability beyond the stable range.
Let ( G p ) p > be a family ofgroups with multiplication, and consider the bigraded commutative ring A = L p > H ∗ ( G p ). Homological stability tells us that any element of H ∗ ( G p ) thatlies in the stable range decomposes as a product of elements in the augmentationideal of A . (In fact it tells us that such an element decomposes as a product withthe generator of H ( G ).) We believe that connectivity bounds on the splitting N THE EDGE OF THE STABLE RANGE 7 complex can yield decomposability results far beyond the stable range. The follow-ing conjecture was formulated after studying explicit computations for symmetricgroups and braid groups [4], in which cases it holds.
Conjecture.
Let ( G p ) p > be a family of groups with multiplication. Suppose that | SP n | is ( n − -connected for all n > . Then the map µ : M p + q = np,q > H ∗ ( G p ) ⊗ H ∗ ( G q ) −→ H ∗ ( G n ) is surjective in degrees ∗ ( n − , and its kernel is the image of α : M p + q + r = np,q,r > H ∗ ( G p ) ⊗ H ∗ ( G q ) ⊗ H ∗ ( G r ) −→ M p + q = np,q > H ∗ ( G p ) ⊗ H ∗ ( G q ) in degrees ∗ ( n − . Here µ and α are defined by µ ( x ⊗ y ) = x · y and α ( x ⊗ y ⊗ z ) = ( x · y ) ⊗ z − x ⊗ ( y · z ) . We are able to prove the surjectivity statement in degrees ∗ n and the in-jectivity statement in degrees ∗ n − , both of which are half a degree betterthan the stable range (Lemmas 11.3 and 11.4), and Theorems B and C are the‘practical’ versions of these facts. We hope that in future work we will be able toobtain information further beyond the stable range.1.7. Organisation of the paper.
In the first half of the paper we introduce theconcepts required to understand the statements of Theorems A, B and C and then,assuming these theorems for the time being, we give the proofs of the applicationsstated earlier in this introduction. Section 2 introduces families of groups withmultiplication, and introduces four main examples: the symmetric groups, generallinear groups of PIDs, automorphism groups of free groups, and braid groups.Section 3 introduces the splitting posets SP n associated to a family of groupswith multiplication, and identifies them in the four examples. In section 4 weshow that for these four examples, the realisation | SP n | of the splitting poset is( n − B n obtained from a family of groups with multiplication.In section 8 we show that, under the hypotheses of Theorems A, B and C thereis a spectral sequence with E -term B n and converging to 0 in total degrees ( n − B n . The filtration allowsus to understand the homology of B n inductively within a range of degrees. Thensections 10, 11 and 12 give the proofs of the three theorems. N THE EDGE OF THE STABLE RANGE 8
Acknowledgements.
My thanks to Rachael Boyd, Anssi Lahtinen, Mar-tin Palmer, Oscar Randal-Williams, David Sprehn and Nathalie Wahl for usefuldiscussions. 2.
Families of groups with multiplication
In this section we define the families of groups with multiplication to which ourmethods will apply, and we provide a series of examples.
Definition 2.1. A family of groups with multiplication ( G p ) p > is a sequence ofdiscrete groups G , G , G , . . . equipped with a multiplication map G p × G q −→ G p + q , ( g, h ) g ⊕ h for each p, q >
0. We assume that the following axioms hold:(1)
Unit:
The group G is the trivial group, and its unique element e acts asa unit for left and right multiplication. In other words e ⊕ g = g = g ⊕ e for all p > g ∈ G p .(2) Associativity:
The associative law( g ⊕ h ) ⊕ k = g ⊕ ( h ⊕ k ) . holds for all p, q, r > g ∈ G p , h ∈ G q and k ∈ G r . Consequently,for any sequence p , . . . , p r > iterated multiplicationmap G p × · · · × G p r −→ G p + ··· + p r . (3) Commutativity:
The product maps are commutative up to conjugation, inthe sense that there exists an element τ pq ∈ G p + q such that the squares G p × G q / / ∼ = (cid:15) (cid:15) G p + qc τpq (cid:15) (cid:15) G q × G p / / G p + q commute, where c τ pq denotes conjugation by τ pq . (We do not impose anyfurther conditions upon the τ pq .)(4) Injectivity:
The multiplication maps are all injective. It follows that theiterated multiplication maps are also injective. Using this, we henceforthregard G p × · · · × G p r as a subgroup of G p + ··· + p r for each p , . . . , p r > Intersection:
We have( G p + q × G r ) ∩ ( G p × G q + r ) = G p × G q × G r , for all p, q, r >
0, where G p + q × G r , G p × G q + r and G p × G q × G r are allregarded as subgroups of G p + q + r .We denote the neutral element of G p by e p . N THE EDGE OF THE STABLE RANGE 9
Remark 2.2.
We could delete the intersection axiom from Definition 2.1, at theexpense of working with the splitting complex of section 6 instead of the splittingposet. See Remark 6.5 for further discussion.
Example 2.3 (Symmetric groups) . For p > p denote the symmetricgroup on n letters. Then we may form the family of groups with multiplication(Σ p ) p > , equipped with the product mapsΣ p × Σ q → Σ p + q , ( f, g ) f ⊔ g where f ⊔ g is the automorphism of { , . . . , p + q } ∼ = { , . . . , p }⊔{ , . . . , q } given by f on the first summand and by g on the second. Then the axioms of a multiplicativefamily are all immediately verified. In the case of commutativity, the element τ pq is the permutation that interchanges the first p and last q letters while preservingtheir ordering. Example 2.4 (General linear groups of PIDs) . Let R be a PID. For n >
0, let GL n ( R ) denote the general linear group of n × n invertible matrices over R . Thenwe may form the family of groups with multiplication ( GL p ( R )) p > , equipped withthe product maps GL p ( R ) × GL q ( R ) → GL p + q ( R ) , ( A, B ) (cid:18) A B (cid:19) given by the block sum of matrices. The unit, associativity, commutativity, injec-tivity and intersection axioms all hold by inspection. In the case of commutativity,the element τ pq is the permutation matrix = (cid:16) I q I p (cid:17) . (It would have been enoughto assume that R is a commutative ring here. However, as we will see later, wewill only be able to apply our results when R is a PID.) Example 2.5 (Automorphism groups of free groups) . For p > F p denotethe free group on p letters, and we let Aut( F p ) denote the group of automorphismsof F p . Then we may form the family of groups with multiplication (Aut( F p )) p > ,equipped with the product mapsAut( F p ) × Aut( F q ) → Aut( F p + q ) , ( f, g ) f ∗ g. Here f ∗ g is the automorphism of F p + q ∼ = F p ∗ F q given by f on the first freefactor and by g on the second. Then the unit, associativity and connectivityaxioms all hold by inspection. In the case of commutativity, the element τ pq is theautomorphism that interchanges the first p generators with the last q generators.The injectivity axiom is also clear. We prove the intersection axiom as follows.Suppose that f p ∗ f q + r = f p + q ∗ f r where each f α lies in Aut( F α ). We would liketo show that f q + r = f q ∗ f r for some f q ∈ Aut( F q ). Let x i be one of the middle q generators. Then f q + r sends x i to a reduced word in the first p + q generators andto a reduced word in the last q + r generators. Since an element of a free grouphas a unique reduced expression, it follows that x i is sent to a word in the middle N THE EDGE OF THE STABLE RANGE 10 q generators. Thus f q + r = f q ∗ f r for some f q : F q → F q . By inverting the originalequation we see that in fact f q ∈ Aut( F q ). Example 2.6 (Braid groups) . Given p >
0, let B p denote the braid group on p strands. This is defined to be the group of diffeomorphisms of the disk D thatpreserve the boundary pointwise and that preserve (not necessarily pointwise) aset X p ⊂ D of p points in the interior of D , arranged from left to right, all takenmodulo isotopies relative to ∂D and X p . D X The product maps are B p × B q → B p + q , ( β, γ ) β ⊔ γ where β ⊔ γ denotes the braid obtained by juxtaposing β and γ . More precisely, wechoose an embedding D ⊔ D ֒ → D that embeds two copies of D ‘side by side’in D , in such a way that X p ⊔ X q is sent into X p + q preserving the left-to-rightorder.Then β ⊔ γ is defined to be the map given by β and γ on the respective embeddedpunctured discs, and by the identity elsewhere. Then the unit, associativity andinjectivity axioms are immediate. The commutativity axiom holds when we take τ pq to be the class of a diffeomorphism that interchanges the two embedded discs,passing the left one above the right. The intersection axiom follows from the factthat we may identify the subgroup B p × B q + r ⊆ B p + q + r with the set of isotopyclasses of diffeomorphisms that fix an arc that cuts the disc in two, separating thefirst p punctures from the last q + r punctures, and similarly for B p + q × B r and B p × B q × B r . 3. The splitting poset
In this section we define the splitting posets associated to a family of groupswith multiplication, and identify them in the case of symmetric groups, braidgroups, general linear groups of PIDs, and automorphism groups of free groups.
N THE EDGE OF THE STABLE RANGE 11
Conditions on the connectivity of these posets are the key assumptions in all ofour main theorems.
Definition 3.1 (The splitting poset) . Let ( G p ) p > be a family of groups withmultiplication. Then for n >
2, the n -th splitting poset SP n of ( G p ) p > is definedto be the set SP n = G n G × G n − ⊔ G n G × G n − ⊔ · · · ⊔ G n G n − × G ⊔ G n G n − × G equipped with the partial ordering with respect to which g ( G p × G n − p ) h ( G q × G n − q )if and only if p q and there is k ∈ G n such that g ( G p × G n − p ) = k ( G p × G n − p ) and h ( G q × G n − q ) = k ( G q × G n − q ) . Lemma 3.2 below verifies that the relation is transitive. Lemma 3.2.
Given an arbitrary chain g ( G p × G n − p ) g ( G p × G n − p ) · · · g r ( G p r × G n − p r ) (1) in SP n we may assume, after possibly choosing new coset representatives, that g = · · · = g r . It follows that g i ( G p i × G n − p i ) g j ( G p j × G n − p j ) for any i j .Proof. We prove by induction on s = 1 , , . . . , r that given an arbitrary chain (1)we may assume, after choosing new coset representatives, that g = · · · = g s = g for some g ∈ G n , the case s = r being our desired result.When s = 1, the claim is immediate from the definition of .For the induction step, suppose that the claim holds for s . Take an arbitrarychain (1) and use the induction hypothesis to choose new coset representatives sothat g = · · · = g s = g . Since g ( G p s × G n − p s ) g s +1 ( G p s +1 × G n − p s +1 ) we mayassume, after replacing g s +1 if necessary, that g ( G p s × G n − p s ) = g s +1 ( G p s × G n − p s ).Then there are γ ∈ G p s and δ ∈ G n − p s such that g − g s +1 = γ ⊕ δ . Since e p s ⊕ δ liesin G p t × G n − p t for t s , we may replace g with g ( e p s ⊕ δ ). And since γ ⊕ e n − p s liesin G p s +1 × G n − p s +1 , we may replace g s +1 with g s +1 ( γ − p s ⊕ e n − p s ). But then g s +1 = g .So g = · · · = g s +1 as required. (cid:3) Now we will identify the splitting posets associated to the symmetric groups,general linear groups of PIDs, automorphism groups of free groups, and braidgroups.
Proposition 3.3 (Splitting posets for symmetric groups) . For the family of groupswith multiplication (Σ p ) p > , the n -th splitting poset SP n is isomorphic to the posetof proper subsets of { , . . . , n } under inclusion.Proof. We define a bijection φ from SP n to the poset of proper subsets of { , . . . , n } by the rule φ ( g (Σ p × Σ n − p )) = { g (1) , . . . , g ( p ) } . N THE EDGE OF THE STABLE RANGE 12
This φ is a well-defined bijection, and we must show that g (Σ p × Σ n − p ) h (Σ q × Σ n − q ) ⇐⇒ { g (1) , . . . , g ( p ) } ⊆ { h (1) , . . . , h ( q ) } . If the first condition holds then p q and we may assume that g = h , so that thesecond condition follows immediately. If the second condition holds then p q and, replacing h by h ◦ ( k × Id) and g by g ◦ (Id × l ) for an appropriate k ∈ Σ q and l ∈ Σ n − p , we may assume that g = h , so that the first condition holds. (cid:3) Let R be a PID. To identify the splitting posets associated to the family ( GL p ( R )) p > ,recall that Charney in [3] defined S R ( R n ) to be the poset of ordered pairs ( P, Q ) ofproper submodules of R n satisfying P ⊕ Q = R n , equipped with the partial order defined by ( P, Q ) ( P ′ , Q ′ ) ⇐⇒ P ⊆ P ′ and Q ⊇ Q ′ . Charney then defined the split building of R n , denoted by [ R n ], to be the realisation | S R ( R n ) | . (Note that Charney worked with arbitrary Dedekind domains.) Proposition 3.4 (Splitting posets for general linear groups of PIDs) . Let R be aPID. For the family of groups with multiplication ( GL n ( R )) n > , the splitting posetSP n is isomorphic to S R ( R n ) , so that | SP n | is isomorphic to the split building [ R n ] .Proof. Define s , . . . , s n − ∈ SP n and t , . . . , t n − ∈ S R ( R n ) by s p = e n ( GL p ( R ) × GL n − p ( R )) , t p = (span( x , . . . , x p ) , span( x p +1 , . . . , x n )) , where e n ∈ GL n ( R ) denotes the identity element and x , . . . , x n is the standardbasis of R n . Then the following three properties hold for the elements s i ∈ SP n ,and their analogues hold for the t i ∈ S R ( R n ).(1) s , . . . , s n − are a complete set of orbit representatives for the GL n ( R ) ac-tion on SP n .(2) The stabiliser of s p is GL p ( R ) × GL n − p ( R ).(3) x y if and only if there is g ∈ GL n ( R ) such that x = g · s p and y = g · s q where p q .It follows immediately that there is a unique isomorphism of posets SP n → S R ( R n )satisfying s i t i for all i .The three properties hold for s i ∈ SP n by definition. We prove them for t i ∈ S R ( R n ) as follows. For (1), the fact that R is a PID guarantees that if ( P, Q ) ∈ S R ( R n ) then P and Q are free, of ranks p and q say, such that p + q = n . If wechoose bases of P and Q and concatenate them to form an element A ∈ GL n ( R ),then A · t p = ( P, Q ) as required. Property (2) is immediate. For (3), suppose that(
P, Q ) ( P ′ , Q ′ ) and let p = rank( P ) and p ′ = rank( P ′ ), so that p p ′ . Then R n = P ⊕ ( P ′ ∩ Q ) ⊕ Q ′ , P ⊕ ( P ′ ∩ Q ) = P ′ , ( P ′ ∩ Q ) ⊕ Q ′ = Q. Let g denote the element of GL n ( R ) whose columns are given by a basis of P ,followed by a basis of ( P ′ ∩ Q ), followed by a basis of Q ′ . Again this is possible N THE EDGE OF THE STABLE RANGE 13 since R is a PID. Then ( P, Q ) = g · t p and ( P ′ , Q ′ ) = g · t p ′ where p p ′ , asrequired. (cid:3) Let us now identify the splitting posets for automorphism groups of free groups.The situation is closely analogous to that for general linear groups. Define S ( F n ),for each n >
2, to be the poset of ordered pairs (
P, Q ) of proper subgroups of F n satisfying P ∗ Q = F n . It is equipped with the partial order under which( P, Q ) ( P ′ , Q ′ ) if and only if ( P, Q ) = ( J , J ∗ J ) and ( P ′ , Q ′ ) = ( J ∗ J , J ) forsome proper subgroups J , J , J of F n satisfying J ∗ J ∗ J = F n . (Note that thecondition in the definition of is stronger than assuming that P ⊆ P ′ and Q ′ ⊇ Q ).The proof of the following proposition is similar to that of Proposition 3.4, and weleave the details to the reader. Proposition 3.5 (Splitting posets for automorphism groups of free groups) . Forthe family of groups with multiplication (Aut( F n )) n > , the splitting poset SP n isisomorphic to S ( F n ) . Let us now identify the splitting posets associated to the family ( B p ) p > of braidgroups. See Example 2.6 for the relevant notation. Given n >
2, let us define aposet A n as follows. The elements of A n are the arcs embedded in D \ X n , startingat the ‘north pole’ of the disc and ending at the ‘south pole’, such that X n meetsboth components of their complement, all taken modulo isotopies in D \ X n thatpreserve the endpoints. α Given α, β ∈ A n , we say that α β if α and β have representatives a and b that meet only at their endpoints, and such that a lies ‘to the left’ of b . (Moreprecisely, a and b must meet the north pole in anticlockwise order and the southpole in clockwise order.) α β Again, the proof of the following is similar to that of Proposition 3.4, and we leaveit to the reader to provide details if they wish.
N THE EDGE OF THE STABLE RANGE 14
Proposition 3.6 (Splitting posets for braid groups) . For the family of groups withmultiplication ( B p ) p > , we have SP n ∼ = A n . Examples of connectivity of | SP n | Our Theorems A, B and C apply to a family of groups with multiplicationonly when the associated splitting posets satisfy the connectivity condition thateach | SP n | is ( n − poset of free factorisations of F n in [9]; and for braid groups,where the claim is a variant of known results on arc complexes.Let us fix our definitions and notation for realisations of posets. If P is a poset,then its order complex (or flag complex or derived complex ) ∆( P ) is the abstractsimplicial complex whose vertices are the elements of P , and in which vertices p , . . . , p r span an r -simplex if they form a chain p < · · · < p r after possiblyreordering. The realisation | P | of P is then defined to be the realisation | ∆( P ) | of ∆( P ). We will usually not distinguish between a simplicial complex and itsrealisation. So if P is a poset, then the simplicial complex | P | and topologicalspace | ( | P | ) | will both be denoted by | P | . When we discuss topological propertiesof a poset or of a simplicial complex, we are referring to the topological propertiesof its realisation as a topological space.4.1. Symmetric groups.
The result for symmetric groups is elementary.
Proposition 4.1 (Connectivity of | SP n | for symmetric groups) . For the family ofgroups with multiplication (Σ p ) p > we have | SP n | ∼ = S n − , and in particular | SP n | is ( n − -connected.Proof. Let ∂ ∆ n − denote the simplicial complex given by the boundary of thesimplex with vertices 1 , . . . , n . Then the face poset F ( ∂ ∆ n − ) of ∂ ∆ n − is exactlythe poset of proper subsets of { , . . . , n } ordered by inclusion. But we saw inProposition 3.3 that the latter is isomorphic to SP n . Thus | SP n | ∼ = | F ( ∂ ∆ n − ) | ∼ = | ∂ ∆ n − | ∼ = S n − as required. (cid:3) General linear groups of PIDs.
Let R be a PID. In Proposition 3.4 wesaw that for the family of groups with multiplication ( GL p ( R )) p > there is anisomorphism SP n ∼ = S R ( R n ), where S R ( R n ) is the poset whose realisation is thesplit building [ R n ]. Since R is in particular a Dedekind domain, Theorem 1.1 of [3]shows that [ R n ] has the homotopy type of a wedge of ( n − Proposition 4.2 (Connectivity of | SP n | for general linear groups of PIDs) . Let R be a PID. For the family of groups with multiplication ( GL p ( R )) p > , and for any N THE EDGE OF THE STABLE RANGE 15 n > , | SP n | has the homotopy type of a wedge of ( n − -spheres, and in particularis ( n − -connected. Automorphism groups of free groups.
Now we give the proof of theconnectivity condition on the splitting posets for automorphism groups of freegroups. This is the most involved of our connectivity proofs.
Definition 4.3.
Let F be a free group of finite rank. Define P ( F ) to be the posetof ordered tuples H = ( H , . . . , H r ) of proper subgroups of F such that r > H ∗ · · · ∗ H r = F . It is equipped with the partial order in which H > K if K canbe obtained by repeatedly amalgamating adjacent entries of H . Theorem 4.4. If F has rank n , then | P ( F ) | has the homotopy type of a wedge of S n − -spheres. Corollary 4.5 (Connectivity of | SP n | for automorphism groups of free groups) . For the family of groups with multiplication (Aut( F p )) p > , the splitting poset | SP n | has the homotopy type of a wedge of ( n − -spheres, and in particular is ( n − -connected. This result has been obtained independently, and with the same proof, as part ofwork in progress by Kupers, Galatius and Randal-Williams. (See also the remarksafter Theorem A.)
Proof of Corollary 4.5. If P is a poset then we denote by P ′ the derived poset ofchains p < · · · < p r in P ordered by inclusion. Its realisation satisfies | P ′ | ∼ = | P | .Recall from Proposition 3.5 that SP n is isomorphic to the poset S ( F n ) definedthere. So it will suffice to show that P ( F n ) is isomorphic to S ( F n ) ′ , for then | SP n | ∼ = | S ( F n ) | ∼ = | S ( F n ) ′ | ∼ = | P ( F n ) | and the result follows from Theorem 4.4.Consider the maps λ : P ( F n ) → S ( F n ) ′ , µ : S ( F n ) ′ → P ( F n )defined by λ (cid:16) H , . . . , H r +1 (cid:17) = h ( H , H ∗ · · · ∗ H r +1 ) < · · · < ( H ∗ · · · ∗ H r , H r +1 ) i and µ h ( A , B ) < · · · < ( A r , B r ) i = (cid:16) A , A ∩ B , A ∩ B , . . . , A r ∩ B r − , B r (cid:17) . Then one can verify that λ and µ are mutually inverse maps of posets. Theverification requires one to use the fact that if ( X , Y ) < ( X , Y ) < ( X , Y ), then X ∗ ( X ∩ Y ) = X , Y ∗ ( Y ∩ X ) = Y and ( X ∩ Y ) ∗ ( X ∩ Y ) = X ∩ Y ,which follow from the definition of the partial ordering on S ( F n ). (cid:3) We now move towards the proof of Theorem 4.4. In order to do so we requireanother definition.
N THE EDGE OF THE STABLE RANGE 16
Definition 4.6.
Let F be a free group of finite rank. Define Q ( F ) to be the posetof unordered tuples H = ( H , . . . , H r ) of proper subgroups of F such that r > H ∗ · · · ∗ H r = F . Give it the partial order in which H > K if K can beobtained by repeatedly amalgamating entries of H , adjacent or otherwise. Let f : P ( F ) → Q ( F ) denote the map that sends an ordered tuple to the same tuple,now unordered.The poset Q ( F n ) is exactly the opposite of the poset of free factorisations of F n . This poset was introduced and studied by Hatcher and Vogtmann in section 6of [9], where it was shown that its realisation has the homotopy type of a wedgeof ( n − F is a free group of rank m then | Q ( F ) | hasthe homotopy type of a wedge of ( m − | P ( F ) | fromthe known connectivity of | Q ( F ) | . In order to do this we will use a poset fibretheorem due to Bj¨orner, Wachs and Welker [1]. Let us recall some necessarynotation. Given a poset P and an element p ∈ P , we define P
p and P > p similarly. The length ℓ ( P ) of a poset P is defined to be the maximum ℓ such that there is a chain p < p < · · · < p ℓ in P ; the length of the empty poset is defined to be −
1. Theorem 1.1 of [1] statesthat if f : P → Q is a map of posets such that for all q ∈ Q the fibre | f − Q q | is ℓ ( f − Q q | where ∗ denotes the join . See the introduction to [1] for further details. Proof of Theorem 4.4.
The proof is by induction on the rank of F . When rank( F ) =2 we need only observe that P ( F ) is an infinite set with trivial partial order, sothat | P ( F ) | is an infinite discrete set, and in particular is a wedge of 0-spheres.Suppose now that rank( F ) > F . Since rank( F ) > | Q ( F ) | is connected. Suppose that H = ( H , . . . , H r H ) ∈ Q ( F ). Then Lemmas 4.7, 4.8, 4.9 and 4.10 below tell us thefollowing. • ℓ ( f − ( Q ( F )
Since S r H − is ( r H − | P ( F ) | ≃ | Q ( F ) | ∨ _ H ∈ Q ( F ) (cid:0) | f − ( Q ( F ) H ) | ∗ | Q ( F ) >H | (cid:1) . ≃ _ S n − ∨ _ H ∈ Q ( F ) (cid:16)(cid:16)_ S n − r H − (cid:17) ∗ S r H − (cid:17) ≃ _ S n − ∨ _ H ∈ Q ( F ) _ (cid:0) S n − r H − ∗ S r H − (cid:1) ≃ _ S n − ∨ _ H ∈ Q ( F ) _ S n − ≃ _ S n − as required. (cid:3) Lemma 4.7.
Let F be a free group of finite rank and let H = ( H , . . . , H r ) ∈ Q ( F ) .Then ℓ ( f − ( Q ( F ) 1) adjacententries before obtaining a 2-tuple. This shows that ℓ ( f − ( Q ( F ) H )) = r − H itself (with some ordering) it followsthat ℓ ( f − ( Q ( F ) Let F be a free group of finite rank and let H = ( H , . . . , H r ) ∈ Q ( F ) .Then | f − ( Q ( F ) H ) | ∼ = S r − .Proof. The poset f − ( Q ( F )) H is the subposet of P ( F ) consisting of tuples K =( K , . . . , K s ) where each K j is an amalgamation of some of the H i . It is isomorphicto the poset X r of sequences F = ( F ⊂ F ⊂ · · · ⊂ F s − ) of proper subsets of { , . . . , r } , where F ′ F if F ′ can be obtained from F by forgetting terms of thesequence. The isomorphism X r ∼ = −−→ f − ( Q ( F )) H sends F = ( F ⊂ · · · ⊂ F s − ) to K = ( K , . . . , K s ) where for i s − K j is thesubgroup generated by the H i for i ∈ F j \ F j − , and where K s is the subgroupgenerated by the H j for j F s − . Now X n is isomorphic to the poset of faces ofthe barycentric subdivision of ∂ ∆ r , as we see by identifying F ⊂ · · · ⊂ F s − withthe face whose vertices are the barycentres of the simplices spanned by the F i . So | X n | ∼ = ∂ ∆ r ∼ = S r − as claimed. (cid:3) Before stating the next lemma we introduce some notation. Given a poset P ,let CP denote the poset obtained by adding a new minimal element − . N THE EDGE OF THE STABLE RANGE 18 Lemma 4.9. Let F be a free group of finite rank and let H = ( H , . . . , H r ) ∈ Q ( F ) .Then | Q ( F ) >H | ∼ = | Q ( H ) | ∗ · · · ∗ | Q ( H r ) | . Proof. There is an isomorphism Q ( F ) > H ∼ = CQ ( H ) × · · · × CQ ( H r ) . It simply takes a tuple K = ( K , . . . , K s ) and sends it to the element of CQ ( H ) ×· · · × CQ ( H r ) whose CQ ( H i )-component is the tuple consisting of those K j whichare contained in H i if there are more than one such, and which is − otherwise, inwhich case H i itself appears as one of the K j . This isomorphism identifies H itselfwith the tuple ( − , . . . , − ), so that we obtain a restricted isomorphism Q ( F ) >H ∼ = CQ ( H ) × · · · × CQ ( H r ) \ ( − , . . . , − ) . Now the realisation of the right hand side is exactly | Q ( H ) | ∗ · · · ∗ | Q ( H r ) | , so theresult follows. (cid:3) Lemma 4.10. Let F be a free group of finite rank and let H = ( H , . . . , H r ) ∈ Q ( F ) . Then | Q ( H ) | ∗ · · ·∗ | Q ( H r ) | has the homotopy type of a wedge of ( n − r − -spheres.Proof. Write s i for the rank of H i , so that | Q ( H i ) | has the homotopy type of awedge of ( s i − | Q ( H ) | ∗ · · · ∗ | Q ( H r ) | has the homotopy type of awedge of copies of S s − ∗ · · · ∗ S s r − . But then S s − ∗ · · · ∗ S s r − ∼ = S ( s − ··· +( s r − r = S ( s + ··· + s r ) − r +1)+ r = S n − r − as required. (cid:3) Braid groups. Now we investige the connectivity of the realisations of thesplitting posets for braid groups. In this case we will appeal to well-known con-nectivity results for complexes of arcs. Proposition 4.11 (Connectivity of | SP n | for braid groups) . For the family ofgroups with multiplication ( B p ) p > , and for any n > , | SP n | has the homotopytype of a wedge of ( n − -spheres, and in particular is ( n − -connected.Proof. Recall from Proposition 3.5 that we identified SP n with the poset of arcs A n defined there. Thus | A n | is (the realisation of) the simplicial complex with verticesthe elements of A n , in which vertices α , . . . , α r span a simplex if and only if, afterpossibly reordering, α < · · · < α r . Now α < · · · < α r holds if and only if the α i have representatives a i that are disjoint except at their endpoints, and such that a , . . . , a r meet the north pole in anticlockwise order. Thus | A n | is the realisationof the simplicial complex whose vertices are isotopy classes of nontrivial (they donot separate a disc from the remainder of the surface) arcs in D \ X n from thenorth pole to the south, where a collection of vertices form a simplex if they have N THE EDGE OF THE STABLE RANGE 19 representing arcs that can be embedded disjointly except at their endpoints. Inthe notation of section 4 of [23], this is exactly the complex B ( S, ∆ , ∆ ) where S = D \ X n , ∆ ⊂ ∂D is the set containing just the north pole, and ∆ ⊂ ∂D is the set containing just the south pole. Now, replacing S with the complementof n open discs in D does not change the isomorphism type of the complex. Butin that case, Lemma 4.7 of [23] applies to show that | A n | has connectivity ( n − | A | , which is ( − (cid:3) Proofs of the applications In this section we will assume that Theorems A, B and C hold, and we will provethe remaining theorems stated in the introduction. We begin with three closelyanalogous lemmas about the groups GL n ( Z ), GL n ( F ) and Aut( F n ). Lemma 5.1. Define elements of GL n ( Z ) , n = 1 , , as follows s = (cid:0) − (cid:1) , s = (cid:18) − (cid:19) , s = − , t = (cid:18) (cid:19) . Use the same symbols to denote the corresponding elements of H ( GL n ( Z ); Z ) = GL n ( Z ) ab . Then the H ( GL n ( Z ); Z ) for n = 1 , , are elementary abelian -groups with generators s ∈ H ( GL ( Z ); Z ) , s , t ∈ H ( GL ( Z ); Z ) and s ∈ H ( GL ( Z ); Z ) , and the stabilisation maps have the following effect. H ( GL ( Z ); Z ) s ∗ / / H ( GL ( Z ); Z ) s ∗ / / H ( GL ( Z ); Z ) s ✤ / / s ✤ / / s t ✤ / / Proof. There are split extensions SL n ( Z ) −→ GL n ( Z ) det −→ {± } with section determined by − s n , so that we have isomorphisms H ( GL n ( Z ); Z ) ∼ = H ( SL n ( Z ); Z ) {± } ⊕ Z / Z , where Z / Z is generated by the class of s n . This isomorphism respects the stabil-isation maps. Now H ( SL ( Z ); Z ) obviously vanishes, and H ( SL ( Z ); Z ) vanishessince SL n ( Z ) is perfect for n > 3. So it suffices to show that H ( SL ( Z ); Z ) {± } isa group of order 2 generated by t .Let us write u = (cid:18) − 11 1 (cid:19) , v = (cid:18) − (cid:19) . Then H ( SL ( Z ); Z ) ∼ = Z / Z , where v ↔ u ↔ s vs − = v − and s us − = v − u − v , so that {± } acts on H ( SL ( Z )) N THE EDGE OF THE STABLE RANGE 20 by negation. Consequently H ( SL ( Z ); Z ) {± } = ( Z / Z ) {± } has order 2 withgenerator t = vu as required. (cid:3) Lemma 5.2. H ( GL n ( F )) = GL n ( F ) ab is trivial for n = 1 , , and is generatedby the element t determined by the matrix ( ) for n = 2 .Proof. For n = 1 this is trivial, and for n = 3 it follows from the fact that GL ( F ) = SL ( F ) is perfect. For n = 2, we simply observe that GL ( F ) isa dihedral group of order 6 generated by the involutions (cid:18) (cid:19) and (cid:18) (cid:19) , so that the abelianization is a group of order 2 generated by either of the involu-tions. (cid:3) Lemma 5.3. Define elements of Aut( F n ) , n = 1 , , as follows. For n = 1 , , let s i denote the transformation that inverts the first letter and fixes the others. And let t ∈ Aut( F ) denote the transformation x x , x x x . Use the same symbolsto denote the corresponding elements of H (Aut( F n ); Z ) = Aut ( F n ) ab . Then the H (Aut( F n ); Z ) for n = 1 , , are elementary abelian -groups with generators s ∈ H (Aut( F ); Z ) , s , t ∈ H (Aut( F ); Z ) and s ∈ H (Aut( F ); Z ) , and thestabilisation maps have the following effect. H (Aut( F ); Z ) s ∗ / / H (Aut( F ); Z ) s ∗ / / H (Aut( F ); Z ) s ✤ / / s ✤ / / s t ✤ / / Proof. The linearisation map Aut( F n ) → GL n ( Z ) is an isomorphism on abelianisa-tions for all n . In the case n = 1 this is because the map itself is an isomorphism.In the case n = 2 this is because the map Out( F ) → GL ( Z ) is an isomorphism,so there is an extension F → Aut( F ) → GL ( Z ) in which the action of GL ( Z )on ( F ) ab = Z is the tautological one, so that the coinvariants (( F ) ab ) GL ( Z ) van-ish, and the claim follows. And for n > SL n ( Z ) is perfect, asis the subgroup SA n of Aut( F n ) consisting of automorphisms with determinantone. (For the last claim we refer to the presentation of SA n given in Theorem 2.8of [7].) The linearisation map sends the generators s , s , s , t listed here to thecorresponding generators from Lemma 5.1, so the claim follows. (cid:3) Proof of Theorem F. Let F be a field of characteristic 2. We will use the K¨unnethisomorphism H ( − ; F ) ∼ = H ( − ; Z ) ⊗ F without further mention. Theorem B statesthat the kernel of the map s ∗ : H m ( GL m ( Z ); F ) ։ H m ( GL m +1 ( Z ); F ) (2)is the image of the product map H ( GL ( Z ); F ) ⊗ m − ⊗ ker[ H ( GL ( Z ); F ) s ∗ −→ H ( GL ( Z ); F )] −→ H m ( GL m ( Z ); F ) . N THE EDGE OF THE STABLE RANGE 21 By Lemma 5.1, H ( GL ( Z ); F ) is spanned by the classes s and t , and ker[ H ( GL ( Z ); F ) s ∗ −→ H ( GL ( Z ); F )] is spanned by t . Any product involving both s and t vanishes,since s · t = s ∗ ( s ) · t = s · s ∗ ( t ) = 0. So it follows that the image of the givenproduct map is precisely the span of t m , which gives us the claimed description ofof kernel of (2). Next, Theorem C states that the map H m ( GL m − ( Z ); F ) ⊕ H ( GL ( Z ); F ) ⊗ m ։ H m ( GL m ( Z ); F )is surjective. The second summand of the domain is spanned by the words in s and t , but the image of any word involving s = s ∗ ( s ) lies in the image of H m ( GL m − ( Z ); F ). Thus the image of the given map is in fact spanned by theimage of H m ( GL m − ( Z ); F ) and of t m , as required. (cid:3) Proof of Theorem G. Since H m ( GL m +1 ( F ); F ) vanishes, Theorem B shows that H m ( GL m ( F ); F ) is spanned by the image of H ( GL ( F ); F ) ⊗ ( m − ⊗ ker[ s ∗ : H ( GL ( F ); F ) → H ( GL ( F ); F )] . But by Lemma 5.2, this image is precisely the span of t m . (cid:3) Proof of Theorem D. The first claim is immediate from Theorem A. For the secondclaim, when char( F ) = 2 we have H (Aut( F ); F ) = 0 by Lemma 5.3, so thatTheorem B shows that s ∗ : H ∗ ( G n − ) → H ∗ ( G n ) is injective for ∗ = n − , andTheorem C shows that s ∗ : H ∗ ( G n − ) → H ∗ ( G n ) is surjective for ∗ = n . (cid:3) Proof of Theorem E. This is entirely analogous to the proof of Theorem F, thistime making use of Lemma 5.3. (cid:3) The splitting complex In this section we identify the realisation of the splitting poset SP n with therealisation of a semisimplicial set that we call the ‘splitting complex’. It is thesplitting complex, rather than the splitting poset, that will feature in our argu-ments from this section onwards. In this section we will make use of semisimplicialsets; see section 2 of [18] for a general discussion of semisimplicial sets (and spaces)and their realisations.We have borrowed the name ‘splitting complex’ from work in progress of Galatius,Kupers and Randal-Williams. See also the remarks after Theorem A. Definition 6.1 (The splitting complex) . Let n > 2. The n -th splitting complex of a family of groups with multiplication ( G p ) p > is the semisimplicial set SC n defined as follows. Its set of r -simplices is( SC n ) r = G q + ··· + q r +1 = nq ,...,q r +1 > G n G q × · · · × G q r +1 if r n − 2, and is empty otherwise. And the i -th face map d i : ( SC n ) r −→ ( SC n ) r − , N THE EDGE OF THE STABLE RANGE 22 G G × G G G × G × G d z z ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ d o o G G × G G G × G × G ❍❍❍❍❍❍❍❍ d d d ❍❍❍❍❍❍❍❍ ✈✈✈✈✈✈✈✈ d z z ✈✈✈✈✈✈✈✈ G G × G × G × G d f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ d o o d x x qqqqqqqqqqqqqqqqqqqq G G × G G G × G × G d d d ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ d o o Figure 1. The splitting complex SC is defined by d i ( g ( G q × · · · × G q r +1 )) = g ( G q × · · · × G q i + q i +1 × · · · × G q r +1 )for g ∈ G n . Example 6.2. Figure 1 illustrates the splitting complex SC . Taking the disjointunion of the terms in each column gives the 0-, 1- and 2-simplices. And thearrows leaving each term represent the face maps on that term, ordered from topto bottom. Remark 6.3. In the expression G q × · · · × G q r +1 appearing in Definition 6.1, wecan imagine the symbols × as being labelled from 0 , . . . , r , so that the i -th facemap d i simply ‘erases the i -th × ’.Let P be a poset. The semisimplicial nerve N P of P is defined to be thesemisimplicial set whose r -simplices are the chains p < · · · < p r of length ( r + 1)in P , and whose face maps are defined by d i ( p < · · · < p r ) = p < · · · b p i · · · < p r .The realisation k N P k of the semisimplicial nerve is naturally homeomorphic tothe realisation | P | of the poset. Proposition 6.4. Let ( G p ) p > be a family of groups with multiplication and let n > . Then SC n ∼ = N ( SP n ) . In particular | SP n | ∼ = k SC n k .Proof. Let φ : SC n → N ( SP n ) denote the map that sends an r -simplex g ( G q ×· · · × G q r +1 ) of SC n to the r -simplex g ( G q × G q + ··· + q r +1 ) < g ( G q + q × G q + ··· + q r +1 ) < · · · < g ( G q + ··· + q r × G q r +1 ) N THE EDGE OF THE STABLE RANGE 23 of N ( SP n ). One can verify that φ is indeed a semi-simplicial map. Surjectivityfollows from Lemma 3.2. Injectivity follows from the fact that r \ i =0 G q + ··· + q i × G q i +1 + ··· + q r +1 = G q × · · · × G q r +1 , which follows by induction from the intersection axiom. (cid:3) Remark 6.5 (Splitting posets or splitting complexes?) . The results of this sectionshow that if we wish we could replace | SP n | with k SC n k in the statements ofTheorems A, B and C. In doing so, we could jettison the intersection axiom fromDefinition 2.1, possibly admitting more examples in the process. However, it isarguably simpler to work with the splitting poset, and that was certainly the casein sections 3 and 4 where we studied specific examples. Moreover, the examplesof interest to us here all satisfy the intersection axiom. We therefore decided towrite our paper with splitting posets at the forefront.7. A bar construction In this section we introduce a variant of the bar construction which takes asits input an algebra like L p > H ∗ ( G p ) and produces a graded chain complex (thatis, a chain complex of graded vector spaces) called B n . We will see in the nextsection that B n is the E -term of the spectral sequence around which all of ourproofs revolve. We fix a field F throughout.For the purposes of this section we fix a field F and a commutative graded F -algebra A equipped with an additional grading that we call the charge . Thus A = M p > A p where A p is the part of A with charge p . We will call the natural grading of A the topological grading, and we will suppress it from the notation whereverpossible. We require that the multiplication on A respects the charge grading,and that each charge-graded piece A p is concentrated in non-negative degrees. Wefurther require that A is a copy of F concentrated in topological degree 0 and(necessarily) generated by the unit element 1. In particular, A is augmented.Finally we assume that ( A ) , the part of A of charge 1 and topological degree 0,is a copy of F generated by an element σ . Example 7.1. Our only examples of such algebras will be A = M p > H ∗ ( G p )where ( G p ) p > is a family of groups with multiplication. Here the topologicalgrading is the grading of homology, and the charge grading is obtained from themultiplicative family. The element σ ∈ ( A ) = H ( G ) is defined to be thestandard generator. N THE EDGE OF THE STABLE RANGE 24 Definition 7.2 (The chain complex B n ) . Let A be an F -algebra as described atthe start of the section. For n > B n to be the chain complex of gradedabelian groups whose b -th term is( B n ) b = M q + ··· + q b = nq ,...,q b > A q ⊗ · · · ⊗ A q b and whose differential is defined by d b ( x ⊗ · · · ⊗ x b ) = b − X i =0 ( − i x ⊗ · · · ⊗ x i · x i +1 ⊗ · · · ⊗ x b . For n = 0 we define B by letting all groups vanish except for ( B ) , which consistsof a single copy of F .Note that B n is bigraded. Its homological grading is the grading that is explicitin the definition, and which is reduced by the differential d b . Its topological gradingis the grading obtained from the topological grading of A , and is preserved bythe differential d b . We say that the part of B n with homological grading b andtopological grading d lies in bidegree ( b, d ). We reserve the notation ( B n ) b for thepart of B n that lies in homological degree b . Remark 7.3 ( B n and the bar complex) . Regarding F as a left and right A -module via the projection A → ( A ) = F , we may form the two-sided normalised barcomplex B ( F , ¯ A, F ) F ⊗ F ←− F ⊗ ¯ A ⊗ F ←− F ⊗ ¯ A ⊗ ¯ A ⊗ F ←− F ⊗ ¯ A ⊗ ¯ A ⊗ ¯ A ⊗ F · · · or, more simply, F ←− ¯ A ←− ¯ A ⊗ ¯ A ←− ¯ A ⊗ ¯ A ⊗ ¯ A ←− · · · where all tensor products are over F . This is naturally trigraded : there is thehomological grading explicit in the the expressions above, together with chargeand topological gradings inherited from A . Writing [ B ( F , ¯ A, F )] charge= n for thehomogeneous piece with charge grading n inherited from A , then we have thefollowing: ( B n ) b = [ B ( F , ¯ A, F ) b +1 ] charge= n . See Remark 8.2 for further discussion. N THE EDGE OF THE STABLE RANGE 25 Example 7.4. Here is a diagram of B . A ⊗ A { { ✇✇✇✇✇✇✇✇✇✇✇✇✇✇ A ⊗ A ⊗ A x x qqqqqqqqqqqqqqqqq o o A A ⊗ A o o A ⊗ A ⊗ A ▼▼▼▼▼▼▼▼ f f ▼▼▼▼▼▼▼▼ qqqqqqqq x x qqqqqqqq A ⊗ A ⊗ A ⊗ A h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ o o v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ A ⊗ A c c ●●●●●●●●●●●●●● A ⊗ A ⊗ A f f ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ o o The first column of the diagram represents ( B ) , the direct sum of the terms inthe next column represent ( B ) , and so on. The effect of the differential d b onan element of one of the summands is the alternating sum (taken from top tobottom) of its images under the arrows exiting that summand. The arrows are allconstructed using the product of A in the evident way.8. The spectral sequence The complex B n is our main tool in proving the theorems stated in the intro-duction. The aim of the present section is to prove the following result, whichdemonstrates the connection between B n and the splitting poset. Throughoutthis section we fix a family of groups with multiplication ( G p ) p > and the algebra A = L H ∗ ( G p ), which is of the kind described at the start of section 7. Through-out this section homology is to be taken with coefficients in an arbitrary field F . Theorem 8.1. Let ( G p ) p > be a family of groups with multiplication that satisfiesthe connectivity axiom, and let A = L p > H ∗ ( G p ) . Then there is a first quadrantspectral sequence with E -term ( E , d ) = ( B n , d b ) for which E ∞ vanishes in bidegrees ( b, d ) satisfying b + d ( n − . Remark 8.2 (The spectral sequence and Tor) . In Remark 7.3, we identified B n in terms of a two-sided bar complex. It follows that we may therefore identify the E -term of the above spectral sequence in terms of a Tor group: E i,j = Tor Ai +1 ( F , F ) charge= n topological= j This observation potentially allows us to use the machinery of derived functors tounderstand the E -term of our spectral sequence. We do not do this in the presentversion of this paper. Instead, our arguments are all done explicitly on the levelof B n itself. We hope that in a future version of this paper we will rephrase ourarguments in terms of Tor wherever possible. N THE EDGE OF THE STABLE RANGE 26 The rest of the section is devoted to the proof of Theorem 8.1. To begin, weintroduce a topological analogue of B n . Observe that the multiplication map G a × G b → G a + b induces a map of classifying spaces BG a × BG b → BG a + b . Wecall it the product map on classifying spaces and denote it by ( x, y ) x · y . We willuse the product maps on classifying spaces to create an augmented semisimplicialspace from which we can recover B n . See section 2 of [18] for conventions aboutsemisimplicial spaces, augmented semisimplicial spaces, and their realisations. Definition 8.3 (The augmented semisimplicial space t B n ) . Given a family ofgroups with multiplication ( G p ) p > , and given n > 2, we let t B n denote the aug-mented semisimplicial set whose set of r -simplices is given by( t B n ) r = G q + ··· + q r +1 = nq ,...,q r +1 > BG q × · · · × BG q r +1 for r = − , . . . , ( n − d i : ( t B n ) r → ( t B n ) r − is defined by d i ( x , . . . , x r +1 ) = ( x , . . . , x i · x i +1 , . . . , x r +1 ) , where · denotes the product map on classifying spaces. Example 8.4. Here is a diagram of t B . BG × BG z z ✉✉✉✉✉✉✉✉✉✉✉✉✉✉ BG × BG × BG x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ o o BG BG × BG o o BG × BG × BG ◆◆◆◆◆◆◆◆◆ g g ◆◆◆◆◆◆◆◆ ♣♣♣♣♣♣♣♣ x x ♣♣♣♣♣♣♣♣♣ BG × BG × BG × BG h h ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ o o v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ BG × BG d d ■■■■■■■■■■■■■■ BG × BG × BG g g ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ o o The four columns correspond to the r -simplices of t B for r = − , , , t B n is a topological analogue of B n . Proposition 8.5 (From t B n to B n ) . There is a spectral sequence with E -term ( E , d ) = ( B n , d b ) and converging to H ∗ ( k t B n k ) . N THE EDGE OF THE STABLE RANGE 27 Proof. As in section 2.3 of [18], but with a shift of grading, the augmented semisim-plicial set t B n gives rise to a spectral sequence, converging to H ∗ ( k t B n k ), andwhose E -term is given by E s,t = H t (( t B n ) s − ) , with d given by the alternating sum of the maps induced by the face maps of t B n . Writing each ( t B n ) s − as a product of spaces and applying the K¨unnethisomorphism (which applies because homology is taken with coefficients in thefield F ) we see that this is isomorphic to B n equipped with the differential d b . (cid:3) Proposition 8.6. Suppose that the realisation of the n -th splitting poset SP n is ( n − -connected. Then the realisation k t B n k is ( n − -connected.Proof. In order to give this proof, we must be precise about our construction ofclassifying spaces. Given a group G , we define EG to be the realisation of thecategory obtained from the action of G on itself by right multiplication. (So it is BG in the notation of [21].) Then we define BG = EG/G . The map EG → BG isa locally trivial principal G -fibration, and EG is itself contractible. The assignment G → EG is functorial, and respects products in the sense that if G and H aregroups then the map E ( G × H ) → EG × EH obtained from the projections is anisomorphism. We can therefore construct a homotopy equivalence as follows. BG q × · · · × BG q r +1 = EG q G q × · · · × EG q r +1 G q r +1 = EG q × · · · × EG q r +1 G q × · · · × G q r +1 ∼ = −→ E ( G q × · · · × G q r +1 ) G q × · · · × G q r +1 ≃ −→ EG n G q × · · · × G q r +1 Here the first arrow comes from the compatibility with products. The second mapcomes from the iterated product map G q × · · · × G q r +1 → G n , and it is a homotopyequivalence because it lifts to a map of principal ( G q × · · · × G q r +1 )-bundles whosetotal spaces are both contractible. There is an isomorphism EG n G q × · · · × G q r +1 ∼ = −−→ EG n × G n (cid:18) G n G q × · · · × G q r +1 (cid:19) sending the orbit of an element x to the orbit of ( x, e n ( G q × · · · × G q r +1 )). Com-bining the two maps just constructed gives us a homotopy equivalence: BG q × · · · × BG q r +1 ≃ −−→ EG n × G n (cid:18) G n G q × · · · × G q r +1 (cid:19) (3) N THE EDGE OF THE STABLE RANGE 28 Now let SC + n denote the augmented semisimplicial set obtained from SC n byadding a single point as a − t B n ) r ≃ −−→ EG n × G n ( SC + n ) r . These equivalences in turn assemble to a levelwise homotopy equivalence t B n ≃ −−→ EG n × G n SC + n and consequently induce a homotopy equivalence k t B n k ≃ −−→ k EG n × G n SC + n k . By assumption, | SP n | is ( n − k SC n k (to which it is isomor-phic by Proposition 6.4) is also ( n − k SC + n k , whichis just the suspension of k SC n k , is ( n − ∗ ֒ → k SC + n k is an ( n − EG n × G n ∗ → EG n × G n k SC + n k is also an ( n − EG n × G n k SC + n k EG n × G n ∗ is ( n − k t B n k ∼ = k EG n × G n SC + n k ∼ = EG n × G n k SC + n k EG n × G n ∗ is ( n − (cid:3) Relating B n to the stabilisation maps Let A be an F -algebra of the kind described at the start of section 7. Thus A has a natural topological grading with respect to which it is commutative, it hasan additional charge grading A = L p > A p , A consists of a single copy of F intopological degree 0, ( A ) is a copy of F generated by an element σ , and eachpiece A p is concentrated in non-negative topological degrees. Definition 9.1 (The stabilisation map.) . The stabilisation map s : A n − → A n isdefined by s ( a ) = σ · a . Example 9.2. In the case A = L p > H ∗ ( G p ) where ( G p ) p > is a family of groupswith multiplication, we take σ to be the standard generator of ( A ) = H ( G ), andthen s : A n − → A n is nothing other than the stabilisation map s ∗ : H ∗ ( G n − ) → H ∗ ( G n ) defined in the introduction.The aim of this section is to relate the complex B n to the stabilisation maps. Inorder to do so, we introduce complexes S n whose homology quantifies the injectivityand surjectivity of the stabilisation maps. N THE EDGE OF THE STABLE RANGE 29 Definition 9.3 (The complex S n ) . For n > 1, let S n denote the graded chaincomplex defined as follows. If n > 2, then S n is the complex.( S n ) ( S n ) d o o A n A n − s o o concentrated in homological degrees 0 and 1. And for n = 1, S is the complexconcentrated in homological degree 0, where it is given by the part of A lying inpositive degrees, which we denote by A ,> .In the case where A = L p > H ∗ ( G p ) comes from a family of groups with multi-plication ( G n ) n > , the complex S n for n > H ∗ ( G n ) s ∗ ←−−−−−− H ∗ ( G n − ) , so that injectivity and surjectivity of the stabilisation map s ∗ in certain ranges ofdegrees can be expressed as the vanishing of the homology of S n in certain rangesof bidegrees. All of our results on the stabilisation map are proved from this pointof view.Our aim now is to relate the stabilisation maps, via the complexes S n , to thecomplex B n . We do this using the following filtration. Definition 9.4. Given n > 2, define a filtration F ⊆ F ⊆ · · · ⊆ F n − = B n of B n by defining F n − = B n , and by defining F r for r ( n − 2) to be thesubcomplex of B n spanned by summands of the form A n − s ⊗− and A , ⊗ A n − s − ⊗− for s r . As usual A , denotes the part of A lying in bidegree (1 , A . Example 9.5. Let us illustrate the above definition in the case n = 3, i.e. for thefiltration F ⊆ F ⊆ F = B . A F A , ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄ A ⊗ A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A F A , ⊗ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A , ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄ A ⊗ A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A F A ⊗ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄ N THE EDGE OF THE STABLE RANGE 30 Example 9.6. In the case n = 4, we can depict B as follows. A ⊗ A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A ⊗ A ⊗ A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ o o A A ⊗ A o o A ⊗ A ⊗ A ❄❄❄❄❄❄❄❄❄ _ _ ❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧ A ⊗ A ⊗ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ o o (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ A ⊗ A ⊗ A _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ o o Then we can depict the filtration F ⊆ F ⊆ F ⊆ F = B symbolically in the form · ·• · · ·◦ _ _ ❅❅❅❅❅ · ⊆ • (cid:127) (cid:127) ⑦⑦⑦⑦⑦ ·• · ◦ ❅❅ _ _ ❅❅ ⑦⑦ (cid:127) (cid:127) ⑦⑦ ·◦ _ _ ❅❅❅❅❅ · ⊆ • (cid:127) (cid:127) ⑦⑦⑦⑦⑦ • (cid:127) (cid:127) ⑦⑦⑦⑦⑦ o o • • o o ◦ ❅❅ _ _ ❅❅ ⑦⑦ (cid:127) (cid:127) ⑦⑦ ◦ _ _ ❅❅❅❅❅ o o (cid:127) (cid:127) ⑦⑦⑦⑦⑦ ◦ _ _ ❅❅❅❅❅ ◦ _ _ ❅❅❅❅❅ o o ⊆ • (cid:127) (cid:127) ⑦⑦⑦⑦⑦ • (cid:127) (cid:127) ⑦⑦⑦⑦⑦ o o • • o o • ❅❅ _ _ ❅❅ ⑦⑦ (cid:127) (cid:127) ⑦⑦ • _ _ ❅❅❅❅❅ o o (cid:127) (cid:127) ⑦⑦⑦⑦⑦ • _ _ ❅❅❅❅❅ • _ _ ❅❅❅❅❅ o o where a bullet • indicates that the relevant summand of B is included in thatterm of the filtration, a circle ◦ indicates a summand A ⊗ − of B that has beenreplaced by A , ⊗ − , and a dot · indicates an omitted summand.The next proposition will describe the filtration quotients of the filtration wehave just defined. In order to state it we need the following definition. Definition 9.7. Let C be a chain complex of graded F -vector spaces (such as B n or S n ). The homological suspension of C , denoted Σ b C , is defined to be the chaincomplex of graded F -vector spaces obtained by increasing the homological gradingof each term by 1. In other words(Σ b C ) b,d = C b − ,d for b, d > Proposition 9.8. For r > there is an isomorphism F r /F r − ∼ = Σ b [ S n − r ⊗ B r ] , while F ∼ = S n . N THE EDGE OF THE STABLE RANGE 31 Example 9.9. Let us illustrate the result of of Proposition 9.8 in the case n = 4and r = 2. Following on from Example 9.6, we see that F /F can be depicted likethis: · [ A ] ⊗ [ A ⊗ A ] − (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · [ A ] ⊗ [ A ] [ A , ⊗ A ] ⊗ [ A ⊗ A ]+ _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ + (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · [ A , ⊗ A ] ⊗ [ A ]+ _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ The signs on the arrows indicate whether the arrow is the one obtained from theobvious multiplication map, or is the negative of that map. Observing now that S = ( A s ←− A ) ∼ = ( A ←− A , ⊗ A )and that B = ( A ←− A ⊗ A ) , where the unmarked arrows are obtained from multiplication maps, we see that F /F is isomorphic to the complex depicted as follows. · ( S ) ⊗ ( B ) − (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · ( S ) ⊗ ( B ) ( S ) ⊗ ( B ) + _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ + (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · ( S ) ⊗ ( B ) + _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ The signs on the arrows now indicate whether the arrow is equal to the tensorproduct of a differential from S or B with an identity map, or to the negativeof such. On the other hand, Σ b [ S ⊗ B ] is exactly the same, but where now the N THE EDGE OF THE STABLE RANGE 32 signs are governed by the Koszul sign convention. · ( S ) ⊗ ( B ) + (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · ( S ) ⊗ ( B ) ( S ) ⊗ ( B ) + _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ − (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ · ( S ) ⊗ ( B ) + _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ The last two complexes are isomorphic via the identity map on the summands( S ) ⊗ ( B ) and ( S ) ⊗ ( B ) , and via the negative of the identity map on thesummands ( S ) ⊗ ( B ) and ( S ) ⊗ ( B ) , as claimed in Proposition 9.8. Proof of Proposition 9.8. For the purposes of the proof, for m > F -modules ¯ S m as follows. For m > 2, ¯ S m is¯ S ¯ S d o o A m A , ⊗ A m − s o o concentrated in homological degrees 0 and 1. For m = 1, we define ¯ S to be thegraded submodule A , > of A consisting of the terms in positive degree. Observethat ¯ S m is isomorphic to S m via the identity map A m → A m in homological degree0, and via the isomorphism A , ⊗ A m − ∼ = −→ A m − , σ ⊗ x x in homological degree 1. We will prove the result with ¯ S m in place of S m .We begin with the case r n − 2. By definition, ( F r /F r − ) b is the direct sumof the terms A q ⊗ · · · ⊗ A q b where q + · · · + q b = n , q , . . . , q b > q = n − r , together with the terms A , ⊗ A q ⊗ · · · ⊗ A q b where 1 + q + · · · + q b = n , q , . . . , q b > 1, and q = n − r − 1. In other words,( F r /F r − ) b is the direct sum of the terms A n − r ⊗ [ A q ⊗ · · · ⊗ A q b − ]where q + · · · + q b − = r , q , . . . , q b − > 1, which is exactly (¯ S n − r ) ⊗ ( B r ) b − ,together with the direct sum of the terms[ A , ⊗ A n − r − ] ⊗ [ A q ⊗ · · · ⊗ A q b − ] N THE EDGE OF THE STABLE RANGE 33 where q + · · · + q b − = r , q , . . . , q b − > 1, which is exactly (¯ S n − r ) ⊗ ( B r ) b − .But that is exactly (¯ S n − r ⊗ B r ) b − = (Σ[¯ S n − r ⊗ B r ]) b . Thus we may construct adegree-wise isomorphism between F r /F r − and Σ[¯ S n − r ⊗ B r ] by simply identifyingcorresponding direct summands. However, the map constructed this way respectsthe differential only up to sign. To correct this, we map from Σ b [¯ S n − r ⊗ B r ] to F r /F r − by taking ( − b times the identity map on the summands coming from(¯ S n − r ) b − ⊗ ( B r ) b . One can now check that this gives the required isomorphismof chain complexes.The proof in the case r = n − (cid:3) Proof of Theorem A For the purposes of this section, we let ( G p ) p > be a family of groups withmultiplication satisfying the hypotheses of Theorems A, B and C, and we define A = L n > H ∗ ( G n ). In this section we will prove the following. Theorem 10.1. The complexes S n for n > , and B n for n > , are acyclic inthe range b n − d − . Here and in what follows, the phrase “in the range” should be understood tomean “in the range of bidegrees ( b, d ) for which”. So for example, the theoremstates that for n > S n and B n are acyclic in all bidegrees ( b, d )for which b n − d − S n vanishes in bidegrees (0 , d ) for d n − , and in bidegrees (1 , d ) for d n − . Unwinding the definition of S n and A , we see that this states that s ∗ : H ∗ ( G n − ) → H ∗ ( G n ) is surjective in degrees ∗ n − , and injective in degrees ∗ n − . In other words, it exactly recovers thestatement of Theorem A.Our proof of Theorem 10.1 will be by strong induction on n . The case n = 1simply states that the homology of S is concentrated in positive degrees, whichholds by definition. The case n = 2 is immediately verified since it states that themaps s ∗ : H ∗ ( G ) → H ∗ ( G ) and H ∗ ( G ) ⊗ H ∗ ( G ) → H ∗ ( G ) are isomorphisms indegree ∗ = 0. For the rest of the section we will assume that Theorem 10.1 holdsfor all integers smaller than n , and will will prove that it holds for n . Lemma 10.2. Assume that Theorem 10.1 holds for all integers smaller than n .Then the composite F ֒ → F ֒ → · · · ֒ → F n − ֒ → F n − = B n is a surjection on homology in the range b n − d and an isomorphism in therange b n − d − .Proof. For r in the range n − > r > 2, the inductive hypothesis tells us that S n − r and B r are acyclic in the ranges b ( n − r ) − d − b r − d − N THE EDGE OF THE STABLE RANGE 34 respectively. Consequently S n − r ⊗ B r is acyclic in the range b n − d − 1, sothat F r /F r − ∼ = Σ b ( S n − r ⊗ B r ) is acyclic in the range b n − d . It follows that F r − → F r is a surjection on homology in the range b n − d and an isomorphismin the range b n − d − S n − r ⊗ B r is seen as follows. The K¨unnethTheorem tells us that the homology of S n − r ⊗ B r is the tensor product of thehomologies of S n − r and B r . Nonzero elements x and y of these respective ho-mologies must lie in bidegrees ( b , d ) and ( b , d ) satisfying b > ( n − r ) − d and b > r − d , so that x ⊗ y lies in bidegree ( b + b , d + d ) satisfying( b + b ) > n − d + d ), so that S n − r ⊗ B r is acyclic in the range b n − d − (cid:3) Lemma 10.3. The inclusion F ֒ → F is an isomorphism in homology in the range b n − d − .Proof. Consider the chain complex corresponding to the square A n − ⊗ A , (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A n A , ⊗ A n − ⊗ A , _ _ ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A , ⊗ A n − _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ in which the arrows are induced by the multiplication maps of A . This is a sub-complex S q n of B n , and indeed of F . Restricting the filtration F ⊂ F of F to S q n gives a filtration ¯ F ⊂ ¯ F of S q n for which ¯ F = F . There results a commutativediagram with short exact rows and left column an isomorphism.0 / / ¯ F / / ∼ = (cid:15) (cid:15) S q n / / (cid:15) (cid:15) ¯ F / ¯ F / / (cid:15) (cid:15) / / F / / F / / F /F / / S n − ⊗ H ∗ > ( G )] . Since S n − is acyclic in the range b ( n − − d − 1, this cokernel is acyclicin the range b [( n − − d − − 1] + 1 = n − d + 1, so that the right-hand map in the diagram is a surjection in homology in the same range. Theconnecting homomorphism for the top row is zero, since S q n is isomorphic to the N THE EDGE OF THE STABLE RANGE 35 chain complex obtained from the square A n − (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A n A n − _ _ ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A n − _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄ in which each map is multiplication by σ ∈ A , , where triviality of the connect-ing homomorphism is evident. The connecting homomorphism for the bottomsequence is therefore zero in the range (of bidegrees for its domain) b n − d + 1.It follows that in the range b n − d we have short exact sequences0 → H ∗ ( F ) → H ∗ ( F ) → H ∗ ( F /F ) → . In the smaller range b n − d − H ∗ ( F ) → H ∗ ( F ) is an isomorphism as claimed. (cid:3) We can now complete the proof of Theorem 10.1. It follows from the last twolemmas that in the range b n − d − S n = F ֒ → B n is anisomorphism in homology. The homology of S n is concentrated in the range b B n vanishes in the range 2 b n − d − 1. It remainsto prove that H ∗ ( S n ) = H ∗ ( B n ) = 0 in the range where b n − d − b H ∗ ( B n ) = E ∗ , ∗ . No nonzero differentials d r , r > 2, of the spectral sequence affectterms in the range b n − d − b 1. This is because any differential withsource in this range has target outside the first quadrant. And any differential d r with target in this range has source E rb + r,d − r +1 , where b + r n − d − r n − d − r + 1) − , so that E rb + r,d − r +1 = 0. Thus H ∗ ( S n ) = H ∗ ( B n ) = E ∞∗ , ∗ in the range b n − d − b 1. Recall that E ∞∗ , ∗ = 0 in the range d n − − b . Now for n > b = 0 , d n − b − 12 = ⇒ d n − − b. (The case d > d = 0, which is vacuous.)Thus H ∗ ( S n ) = H ∗ ( B n ) = E ∞∗ , ∗ = 0 as required. N THE EDGE OF THE STABLE RANGE 36 Proof of Theorem C For the purposes of this section, we let ( G p ) p > be a family of groups withmultiplication satisfying the hypotheses of Theorems A, B and C, and we define A = L n > H ∗ ( G n ). In this section we will prove Theorem C, essentially by ex-tracting a little extra data from the proof of Theorem A, and then exploiting acheap trick. Throughout the section we will write A i,j for the part of A with charge i and topological degree j . In other words, A i,j = ( A i ) j . Lemma 11.1. For m > , the graded chain complex B m +1 is acyclic in the range b (2 m + 1) − d .Proof. Lemma 10.2 shows that the inclusion F ֒ → B n is a surjection on homologyin the range b n − d . However, F is concentrated in homological degrees b = 0 , , 2, and so is acyclic in the range b > 3. Combining the two facts gives theresult. (cid:3) Lemma 11.2. In the spectral sequence of Theorem 8.1, for n = 2 m + 1 , there areno differentials affecting the term in bidegree (1 , m ) from the E page onwards.Proof. Certainly there are no such differentials with source in this bidegree, sincethe spectral sequence is concentrated in the first quadrant. Since E = B m +1 ,Lemma 11.1 shows that E vanishes in the range3 b (2 m + 1) − d. If r > 2, then any differential d r with target in bidegree (1 , m ) has source inbidegree ( b, d ) = (1 + r, m − r + 1), so that b = (2 m + 1) − d − [ r − (2 m + 1) − d, and consequently the source term vanishes. (cid:3) Lemma 11.3. Let m > . Then the complex B m +1 is acyclic in bidegree (1 , m ) .Proof. In the spectral sequence of Theorem 8.1 for n = 2 m + 1, we know that E ,m = E ∞ ,m by Lemma 11.2, and that E ∞ ,m = 0 since m > m (2 m + 1) − 2. So E ,m = 0, but this is simply the homology of B m +1 inbidegree (1 , m ). (cid:3) Lemma 11.4. Let m > . Then B m is acyclic in bidegree (0 , m ) .Proof. Consider the following composite.Σ b A m θ −→ B m +1 φ −−→ Σ b B m ⊗ B ψ −−→ Σ b B m Here θ is the map that sends x ∈ A m to the element x ⊗ σ − σ ⊗ x ∈ ( B m +1 ) .To check that θ is a chain map, we need only check that the differential vanisheson its image, which holds because d ( x ⊗ σ − σ ⊗ x ) = x · σ − σ · x = 0 . N THE EDGE OF THE STABLE RANGE 37 Next, Σ b ( B m ⊗ B ) can be identified with the submodule of B m +1 consisting ofsummands of the form − ⊗ A , and φ is the projection onto these summands. Itis a chain map. Finally, ψ is the map that projects B = A onto its degree 0 part A , ∼ = F . In homology in bidegree (1 , m ) this map is zero since it factors throughthe homology of B m +1 , which vanishes in that bidegree. On the other hand, inthis bidegree the composite is simply the suspension of the map A m → B m ,which is a surjection in homological degree b = 0. It follows that the target of thismap, which is the homology of B m in bidegree (0 , m ), is zero. (cid:3) Proof of Theorem C. We have seen that B m is acyclic in bidegree (0 , m ). Thismeans that the map M p + q =2 mp,q > M p ′ + q ′ = mp ′ ,q ′ > A p,p ′ ⊗ A q,q ′ −→ A m,m is surjective. Now, suppose that p, q, p ′ , q ′ are as in the summation above, with p ′ p − . Then we have the commutative diagram A p,p ′ ⊗ A q,q ′ / / A m,m A p − ,p ′ ⊗ A q,q ′ s ⊗ id O O / / A m − ,ms O O in which the left-hand map is surjective by Theorem A, so that the image of A p,p ′ ⊗ A q,q ′ is contained in the image of s . Similarly, if q ′ q − , then the imageof A p,p ′ ⊗ A q,q ′ is contained in the image of s . The only summands to which theseobservations do not apply are those indexed by p, q, p ′ , q ′ as in the summation,satisfying also that p ′ > p − , q ′ > q − . Adding these inequalities shows that we have m = p ′ + q ′ > m − . Thus the only possibility it that p ′ is greater than p − by exactly 1 / 2, and similarlyfor q ′ . In other words, we must have p = 2 p ′ and q = 2 q ′ . So we have shown thatthe map A m − ,m ⊕ M p ′ + q ′ = mp ′ ,q ′ > A p ′ ,p ′ ⊗ A q ′ ,q ′ −→ A m,m is surjective. In the case m = 2 this proves the claim, and for m > (cid:3) N THE EDGE OF THE STABLE RANGE 38 Proof of Theorem B For the purposes of this section, we let ( G p ) p > be a family of groups withmultiplication satisfying the hypotheses of Theorems A, B and C, and we define A = L n > H ∗ ( G n ). The aim of this section is to prove Theorem B, which is animmediate consequence of Theorem C and the following.The section will deal with complexes like B n which have a homological andtopological grading. Given such a complex C , we will write H i,j ( C ) for the part of H i ( C ) that lies in topological grading j , in other words H i,j ( C ) = H i ( C ) j . Theorem 12.1. Let m > . Then the images of the maps ker h s ∗ : H m − ( G m − ) → H m − ( G m − ) i ⊗ H ( G ) −→ ker h s ∗ : H m ( G m ) → H m ( G m +1 ) i (4) H m − ( G m − ) ⊗ ker h s ∗ : H ( G ) → H ( G ) i −→ ker h s ∗ : H m ( G m ) → H m ( G m +1 ) i (5) together span ker [ s ∗ : H m ( G m ) → H m ( G m +1 )] . The main ingredient in the proof of Theorem 12.1 is Lemma 11.3, which statesthat H ,m ( B m +1 ) = 0 for m > 1, and of which it is an entirely algebraic con-sequence. However our argument is significantly more unpleasant than we wouldlike. Here is the general outline: Theorem 12.1 is a statement about H ,m ( S m +1 ),which is by definition the kernel ker [ s ∗ : H m ( G m ) → H m ( G m +1 )]. We will usethe filtration S m +1 = F ⊆ F ⊆ · · · ⊆ F m = B m +1 from Definition 9.4 to get from what we know about H ,m ( B m +1 ) to what we needto know about H ,m ( S m +1 ). We will do this by using the spectral sequence arisingfrom the filtration in topological degree m . E i,j = H i + j,m ( F i /F i − ) = ⇒ H i + j,m ( B m +1 )The point is to identify the differentials affecting the term E , = H ,m ( S m +1 )with the maps (4) and (5).Let us begin the proof in detail. We are interested in the values of H r,m ( F i /F i − )in the cases r = 0 , , 2. Recall from Proposition 9.8 that for i > F i /F i − ∼ = Σ b [ S m +1 − i ⊗ B i ]so that H r,m ( F i /F i − ) ∼ = H r − ,m [ S m +1 − i ⊗ B i ] ∼ = M r + r = r − m + m = m H r ,m ( S m +1 − i ) ⊗ H r ,m ( B i ) . We have the following. N THE EDGE OF THE STABLE RANGE 39 Lemma 12.2. For r = 0 , , and i = 0 , . . . , m , the only nonzero groups H r,m ( F i /F i − ) are as follows. H ,m ( F ) ∼ = H ,m ( S m +1 ) H ,m ( F ) ∼ = H ,m ( S m +1 ) H ,m ( F /F ) ∼ = H ,m ( S m ) ⊗ H , ( B ) H ,m ( F /F ) ∼ = H ,m ( S m ) ⊗ H , ( B ) H ,m ( F /F ) ∼ = H ,m − ( S m − ) ⊗ H , ( B ) H ,m ( F /F ) ∼ = H ,m − ( S m − ) ⊗ H , ( B ) Proof. Case i = 0 . In this case we have H r,m ( F ) = H r,m ( S m +1 ), and by Theo-rem 10.1 this is nonzero only for r > Case i = 1 . In this case we have H r,m ( F /F ) ∼ = H r,m (Σ b [ S m ⊗ B ]) ∼ = H r − ,m ( S m ⊗ B ) ∼ = M m + m = m H r − ,m ( S m ) ⊗ H ,m ( B )since B is concentrated in homological degree b = 0. Now by Theorem 10.1 theterm H r − ,m ( S m ) vanishes for m m − r/ 2. So for r = 0 we require m > m ,which is impossible, and for r = 1 , m = m , m = 0. Sothe possible terms are H ,m ( F /F ) ∼ = H ,m ( S m ) ⊗ H , ( B )and H ,m ( F /F ) ∼ = H ,m ( S m ) ⊗ H , ( B ) . Case i m . In this case we have H r,m ( F i /F i − ) ∼ = H r,m (Σ b [ S m +1 − i ⊗ B i ]) ∼ = H r − ,m ( S m +1 − i ⊗ B i ) ∼ = M r + r = r − m + m = m H r ,m ( S m +1 − i ) ⊗ H r ,m ( B i ) . Now from Theorem 10.1 we know that H r ,m ( B i ) = 0 for r i − m − H r ,m ( S m +1 − i ) = 0 for r m + 1 − i − m − r = 2 m − i − m + δ and r = i − m − ǫ for δ, ǫ > 0. Then the constraints r + r = r − m + m = m give us r = δ + ǫ .Thus, to find a nonzero group when i > r = 0 , , 2, the only possibility isthat r = 2 and δ = ǫ = 1. But then ( r , r ) = (1 , 0) or ( r , r ) = (0 , N THE EDGE OF THE STABLE RANGE 40 namely H ,m ( F i /F i − ) = (cid:26) H ,m − ( i − / ( S m +1 − i ) ⊗ H , ( i − / ( B i ) for i odd, H ,m − i/ ( S m +1 − i ) ⊗ H ,i/ ( B i ) for i even.However, Lemmas 11.3 and 11.4 guarantee that the second factors vanish for i > H ,m ( F /F ) = H ,m − ( S m − ) ⊗ H , ( B )and H ,m ( F /F ) = H ,m − ( S m ) ⊗ H , ( B ) . This completes the proof. (cid:3) Thus the spectral sequence is as follows. H ,m ( S m +1 ) • • • • H ,m ( S m +1 ) H ,m ( S m ) ⊗ H , ( B ) • • H ,m ( S m ) ⊗ H , ( B ) H ,m − ( S m − ) ⊗ H , ( B ) • • H ,m − ( S m − ) ⊗ H , ( B ) • H ,m ( S m +1 ). We will need thefollowing preliminary result. Lemma 12.3. An arbitrary element of H , ( B ) has a representative of the form ( x ⊗ σ − σ ⊗ x ) + q ⊗ σ where x ∈ A , and q ∈ ker( s : A , → A , ) , and σ ∈ A , is the stabilising element.Proof. A and A are concentrated in non-negative degrees, and in degree 0 theyare spanned by σ and σ respectively. Thus an arbitrary cycle of B in bidegree(1 , 1) has form j ⊗ σ + k ⊗ σ + σ ⊗ l + σ ⊗ m for j, l ∈ A , and k, m ∈ A , . Byadding d ( k ⊗ σ ⊗ σ − σ ⊗ σ ⊗ m ), we may assume that k = m = 0, so that ourcycle has the form j ⊗ σ + σ ⊗ l . This can be rewritten in the required form with x = − l and q = j + l . (cid:3) Lemma 12.4. The span of the images of the differentials with target H ,m ( S m +1 ) ∼ =ker( s : H m ( G m ) → H m ( G m +1 ) is precisely the span of the maps (4) and (5) . N THE EDGE OF THE STABLE RANGE 41 Proof. There are just three positions in the spectral sequence supporting differen-tials with the given target. We will compute the differentials case by case. Case 1: Differentials with domain H ,m ( S m ) ⊗ H , ( B ) . An element l of the domain can be represented by a cycle l = x ⊗ σ in S m ⊗ B , where x ∈ ker( s : A m,m → A m +1 ,m ) and σ ∈ A , is the stabilising element. Thenunder the isomorphism of Proposition 9.8, l corresponds to the element l = σ ⊗ x ⊗ σ of F /F . We lift this to the element l = σ ⊗ x ⊗ σ of F . Then d ( l ) = σ · x ⊗ σ − σ ⊗ x · σ = 0. Thus all differentials d r vanish on l . (In fact thereis only one possibility, d .) Case 2: Differentials with domain H ,m − ( S m − ) ⊗ H , ( B ). An element l of the domain can be represented by a linear combination of cycles of the form x ⊗ y in S m − ⊗ B , where x ∈ ker( s : A m − ,m − → A m − ,m − ) and y ∈ A , .Let us assume without loss that l is in fact represented by l = x ⊗ y . Thenunder the isomorphism of Proposition 9.8, l corresponds to the element l = σ ⊗ x ⊗ y of F /F , which we lift to the element l = σ ⊗ x ⊗ y of F . Now d ( l ) = σ · x ⊗ y − σ ⊗ x · y = − σ ⊗ x · y , which lies in F . Thus d ( l ) = 0,while d ( l ) is the class represented by − σ ⊗ x · y , which under the isomorphism ofProposition 9.8 corresponds to the element − x · y of A m,m = H m ( G m ). This isprecisely the image of − x · y under the map (4) above. Thus the image of d isprecisely the image of (4). Case 3: Differentials with domain H ,m − ( S m − ) ⊗ H , ( B ) . By Lemma 12.3,an element l of the domain has a representative of the form l = X α x α ⊗ ( y α ⊗ σ − σ ⊗ y α ) + X β p β ⊗ ( q β ⊗ σ )where x α , p β ∈ A m − ,m − , y α ∈ A , and q β ∈ ker( s : A , → A , ). Under theisomorphism of Proposition 9.8, l corresponds to the element l = X α ( x α ⊗ y α ⊗ σ − x α ⊗ σ ⊗ y α ) + X β p β ⊗ q β ⊗ σ of F /F . We lift this to the element l = X α ( x α ⊗ y α ⊗ σ − x α ⊗ σ ⊗ y α + σ ⊗ x α ⊗ y α ) + X β p β ⊗ q β ⊗ σ of F . (The apparently new terms lie in F .) Then d ( l ) = X α ( x α · y α ⊗ σ − σ ⊗ x α · y α ) + X β p β · q β ⊗ σ. This lies in F , so that d ( l ) = 0, and its image in F /F is X α x α · y α ⊗ σ + X β p β · q β ⊗ σ N THE EDGE OF THE STABLE RANGE 42 so that applying the isomorphism of Proposition 9.8 shows that d ( l ) = "X α x α · y α + X β p β · q β ⊗ [ σ ] ∈ H ,m ( S m ) ⊗ H , ( B ) . Thus l lies in the kernel of d if and only if "X α x α · y α + X β p β · q β is zero in H ,m ( S m ), or in other words if and only if there is w ∈ A m − ,m suchthat P α x α · y α + P β p β · q β = σ · w . In this case, we may again represent l by l , which again corresponds to the element l of F /F , but which we now lift tothe element l − σ ⊗ w ⊗ σ of F . (The additional term lies in F .) But then d ( l − σ ⊗ w ⊗ σ ) is precisely the element X β σ ⊗ p β · q β of F . Applying the isomorphism of Proposition 9.8, we find that d ( l ) = "X β p β · q β ∈ H ,m ( S m +1 ) . But then it follows that the image of d is precisely the span of the map (5). (cid:3) We may now complete the proof. Since H ,m ( B m +1 ) = 0, it follows that theinfinity-page of the spectral sequence must vanish in total degree 1. So then inparticular we must have E ∞ , = 0, or in other words, the differentials with target H ,m ( S m +1 ) must span. But we have identified the (nonzero) differentials withthe maps (4) and (5). Thus it follows that together, the images of these two mapsmust span. This completes the proof of Theorem 12.1. References [1] Anders Bj¨orner, Michelle L. Wachs, and Volkmar Welker. Poset fiber theorems. Trans. Amer.Math. Soc. , 357(5):1877–1899, 2005.[2] Carles Broto, Nguyen H. V. Hu’ng, Nicholas J. Kuhn, John H. Palmieri, Stewart Priddy,and Nobuaki Yagita. The problem session. In Proceedings of the School and Conferencein Algebraic Topology , volume 11 of Geom. Topol. Monogr. , pages 435–441. Geom. Topol.Publ., Coventry, 2007.[3] Ruth M. Charney. Homology stability for GL n of a Dedekind domain. Invent. Math. , 56(1):1–17, 1980.[4] Frederick R. Cohen, Thomas J. Lada, and J. Peter May. The homology of iterated loopspaces . Lecture Notes in Mathematics, Vol. 533. Springer-Verlag, Berlin-New York, 1976.[5] Søren Galatius. Stable homology of automorphism groups of free groups. Ann. of Math. (2) ,173(2):705–768, 2011. N THE EDGE OF THE STABLE RANGE 43 [6] Søren Galatius and Oscar Randal-Williams. Homological stability for moduli spaces of highdimensional manifolds. I. arXiv:1403.2334v1 , 2014.[7] S. M. Gersten. A presentation for the special automorphism group of a free group. J. PureAppl. Algebra , 33(3):269–279, 1984.[8] John L. Harer. Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2) , 121(2):215–249, 1985.[9] Allen Hatcher and Karen Vogtmann. Cerf theory for graphs. J. London Math. Soc. (2) ,58(3):633–655, 1998.[10] Allen Hatcher and Karen Vogtmann. Rational homology of Aut( F n ). Math. Res. Lett. ,5(6):759–780, 1998.[11] Allen Hatcher and Karen Vogtmann. Homology stability for outer automorphism groups offree groups. Algebr. Geom. Topol. , 4:1253–1272, 2004.[12] Allen Hatcher and Nathalie Wahl. Stabilization for mapping class groups of 3-manifolds. Duke Math. J. , 155(2):205–269, 2010.[13] Kevin P. Knudson. Homology of linear groups , volume 193 of Progress in Mathematics .Birkh¨auser Verlag, Basel, 2001.[14] Hendrik Maazen. Homology stability for the general linear group . 1979. Thesis (Ph.D.)–Utrecht University.[15] R. James Milgram and Stewart B. Priddy. Invariant theory and H ∗ (GL n ( F p ); F p ). In Pro-ceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985) ,volume 44, pages 291–302, 1987.[16] Minoru Nakaoka. Decomposition theorem for homology groups of symmetric groups. Ann.of Math. (2) , 71:16–42, 1960.[17] Daniel Quillen. Finite generation of the groups K i of rings of algebraic integers. In Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) ,pages 179–198. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973.[18] Oscar Randal-Williams. Homological stability for unordered configuration spaces. Q. J.Math. , 64(1):303–326, 2013.[19] Oscar Randal-Williams. Resolutions of moduli spaces and homological stability. J. Eur.Math. Soc. (JEMS) , 18(1):1–81, 2016.[20] Oscar Randal-Williams and Nathalie Wahl. Homological stability for automorphism groups. arXiv:1409.3541 , 2014.[21] Graeme Segal. Classifying spaces and spectral sequences. Inst. Hautes ´Etudes Sci. Publ.Math. , (34):105–112, 1968.[22] Wilberd van der Kallen. Homology stability for linear groups. Invent. Math. , 60(3):269–295,1980.[23] Nathalie Wahl. Homological stability for mapping class groups of surfaces. In Handbook ofmoduli. Vol. III , volume 26 of Adv. Lect. Math. (ALM) , pages 547–583. Int. Press, Somerville,MA, 2013. 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