Homology of general linear groups over infinite fields
aa r X i v : . [ m a t h . K T ] N ov HOMOLOGY OF GENERAL LINEAR GROUPS OVERINFINITE FIELDS
BEHROOZ MIRZAII
Abstract.
For an infinite field F , we study the cokernel of the mapof homology groups H n +1 (GL n − ( F ) , k ) → H n +1 (GL n ( F ) , k ), where k is a field such that ( n − ∈ k × , and the kernel of the natural map H n (cid:0) GL n − ( F ) , Z (cid:2) n − (cid:3)(cid:1) → H n (cid:0) GL n ( F ) , Z (cid:2) n − (cid:3)(cid:1) . We give conjec-tural estimates of these cokernel and kernel and prove our conjecturesfor n ≤ Introduction
Let F be a field. For any positive integer r , GL r ( F ) embeds naturallyin GL r +1 ( F ). The sequence of group embeddings GL ( F ) ⊆ GL ( F ) ⊆ GL ( F ) ⊆ · · · induces the sequence of homomorphisms of homology groups H n (GL ( F ) , Z ) → H n (GL ( F ) , Z ) → H n (GL ( F ) , Z ) → · · · . By an unpublished work of Quillen [20, p. 10], if F has more than twoelements, then the map(0.1) H n (GL r ( F ) , Z ) → H n (GL r +1 ( F ) , Z )is surjective for r ≥ n and is bijective for r ≥ n + 1 (see also [23]). Witha different method, Suslin showed that if the field is infinite, then the map(0.1) is an isomorphism for r ≥ n . Moreover he showed that the cokernel of H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z )is isomorphic to the n -th Milnor K -group K Mn ( F ) [24] (see Theorem 1.1below). Furthermore Suslin and Nesterenko [19] and independently Totaro[26] demonstrated that K Mn ( F ) is isomorphic to the higher Chow groupCH n (Spec( F ) , n ). (For the theory of higher Chow groups see [3].) Thus H n (GL n ( F ) , Z ) /H n (GL n − ( F ) , Z ) ≃ K Mn ( F ) ≃ CH n (Spec( F ) , n ) . Not much is known about the map (0.1) bellow the stable range. Thefollowing conjecture is attributed to Suslin.
Injectivity Conjecture.
For any infinite field F and any positive integers n > r the natural map H n (GL r ( F ) , Q ) → H n (GL r +1 ( F ) , Q ) is injective. MSC(2010): Primary: 19D55, 19D45; Secondary: 20J06.Keywords: Homology of groups, general linear groups, Milnor K -groups . The conjecture is trivial for n = 1 ,
2. It was known for global fields bythe work of Borel and Yang [4]. It was proved for ( n = 3 , r = 2) by Sah [21]and Elbaz-Vincent [5] and for ( n = 4 , r = 3) by the author in [13]. Recentlythe conjecture has been proved for r = n − n and r ( r ≤ n ) one can construct (see [2], [7], [19]) a naturalmap H n (GL r ( F ) , GL r − ( F ); Z ) → CH r (Spec( F ) , n ) . By the work of Suslin and Nesterenko this map is an isomorphism for r = n .In [8], Gerdes has proved that it has torsion cokernel for r = n − Surjectivity Conjecture.
For any infinite field F and any n > r the nat-ural map H n (GL r ( F ) , GL r − ( F ); Q ) → CH r (Spec( F ) , n ) ⊗ Z Q is surjective. In this article we study the cokernel of the map H n +1 (GL n − ( F ) , k ) → H n +1 (GL n ( F ) , k ) , k a field such that ( n − ∈ k × and the kernel of the map H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) . We will show that both of them are related to the second homology of acomplex which we introduce now. The chain of maps(0.2) F ×⊗ n ⊗ K M ( F ) δ ( n ) n −→ F ×⊗ ( n − ⊗ K M ( F ) δ ( n ) n − −→· · · δ ( n )3 −→ F ×⊗ ⊗ K Mn − ( F ) δ ( n )2 −→ F × ⊗ K Mn − ( F ) δ ( n )1 −→ K Mn ( F ) → δ ( n ) i ( a ⊗· · ·⊗ a i ⊗{ b , . . . , b n − i } ) = i X j =1 a ⊗· · ·⊗ b a j ⊗· · ·⊗ a i ⊗{ a j , b , . . . , b n − i } , is a complex. If n ≥
3, it is easy to see that ker( δ ( n )1 ) = im( δ ( n )2 ) (see Section4 below). For any n ≥
1, let B n ( F ) := n = 1 K ind3 ( F ) if n = 2.ker( δ ( n )2 ) / im( δ ( n )3 ) if n ≥ H r GL ( F, n ) := H n (GL r ( F ) , GL r − ( F ); Z ) . Our first main result concerns the group H n GL ( F, n + 1) ⊗ Z k = H n +1 (GL n ( F ) , k ) /H n +1 (GL n − ( F ) , k ) , OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 3 where k is a field such that( n − ∈ k × . Inductively, the study of thegroup H n GL ( F, n + 1) k can be reduced to the study of the quotient group H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k ).For an infinite field F and any positive integer n we construct a naturalmap χ n : B n ( F ) ⊗ Z k → H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k ) , where k is a field such that ( n − ∈ k × . For this construction we use theresult of Galatius, Kupers and Randal-Williamas on the injectivity conjec-ture (see Theorem 1.2 below). We conjecture that χ n always is surjectiveand proof the conjecture for n ≤ κ n : B n ( F ) → ker (cid:0) H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) (cid:1) , where its image is ( n − κ n is surjective andproof the conjecture for n ≤ κ n holds, then the natural mapinc ∗ : H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) )is injective. In particular we demonstrate that the natural homomorphismsinc ∗ : H (GL ( F ) , Z (cid:2) (cid:3) ) → H (GL ( F ) , Z (cid:2) (cid:3) )and inc ∗ : H (GL ( F ) , Z (cid:2) (cid:3) ) → H (GL ( F ) , Z (cid:2) (cid:3) )are injective. The first injectivity (the case n = 3) was already known by[12] or [16]. The second injectivity is new.Furthermore we improve the construction of χ n for n ≤ χ : B ( F ) → H (GL ( F ) , Z ) /H ( F × × GL ( F ) , Z ) ,χ : B ( F ) (cid:2) (cid:3) → H (GL ( F ) , Z (cid:2) (cid:3) ) /H ( F × × GL ( F ) , Z (cid:2) (cid:3) )and show that both are surjective.Finally we show that the above results can be improved over algebraicallyclosed fields. We also discuss the case of real, local and global fields. Notation. If A → B is a homomorphism of abelian groups, by B/A wemean coker( A → B ). We denote an element of B/A represented by b ∈ B again by b . Any inclusion of groups H ⊆ G is denoted by inc : H → G . If k is a commutative ring and A an abelian group, by A k we mean A ⊗ Z k .Moreover by A (cid:2) n (cid:3) we mean A ⊗ Z Z (cid:2) n (cid:3) = A Z [1 /n ] . We always assume that F is an infinite field. We denote the i -th summand of F × k = F × × · · · × F × | {z } k -times by F × i . BEHROOZ MIRZAII The homology of general linear groups and Milnor K -groups For an arbitrary group G , let B • ( G ) ε → Z denote the right bar resolutionof G . For any left G -module N , the homology group H n ( G, N ) coincideswith the homology of the complex B • ( G ) ⊗ G N . In particular H n ( G, Z ) = H n ( B • ( G ) ⊗ G Z ) = H n ( B • ( G ) G ) . For any n -tuple ( g , . . . g n ) of pairwise commuting elements of G let c ( g , g , . . . , g n ) := X σ ∈ Σ n sign( σ )[ g σ (1) | g σ (2) | . . . | g σ ( n ) ] ∈ H n ( G, Z ) , where Σ n is the symmetric group of degree n .Let F be an infinite field. For any a ∈ F × and any 1 ≤ i ≤ n , let D i,n ( a )be the diagonal matrix of size n with a in the i -th position of the diagonaland 1 everywhere else. It is not difficult to show that the map c n : F × ⊗ · · · ⊗ F × → H n (GL n ( F ) , Z ) /H n (GL n − ( F ) , Z ) ,a ⊗ · · · ⊗ a n c ( D ,n ( a ) , . . . , D n,n ( a n )) (mod H n (GL n − ( F ) , Z )) . factors through the n -th Milnor K -group K Mn ( F ):¯ c n : K Mn ( F ) → H n (GL n ( F ) , Z ) /H n (GL n − ( F ) , Z ) . Theorem 1.1 (Suslin [24], Nesterenko-Suslin [19]) . (i) The natural map inc ∗ : H n (GL r ( F ) , Z ) → H n (GL r +1 ( F ) , Z ) is an isomorphism for any r ≥ n . (ii) There exists a natural map s n : H n (GL n ( F ) , Z ) → K Mn ( F ) such thatthe complex H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) s n −→ K Mn ( F ) → is exact. Moreover ¯ s n is the inverse of ¯ c n . (iii) The composite K Mn ( F ) → K n ( F ) h n −→ H n (GL( F ) , Z ) inc − ∗ −→ H n (GL n ( F ) , Z ) s n −→ K Mn ( F ) coincides with multiplication by ( − n − ( n − , where h n is the Hurewiczmap. In [13, Proposition 4] the author showed that if n ≥ k is afield such that ( n − ∈ k × , then the injectivity of H n (GL n − ( F ) , k ) → H n (GL n ( F ) , k ) follows, by an induction process, from the the exactness ofthe complex(1.1) H n ( F × × GL n − ( F ) , k ) α ∗ − α ∗ −−−−−→ H n ( F × × GL n − ( F ) , k ) inc ∗ −→ H n (GL n ( F ) , k ) → , where α (diag( a, b, A )) = diag( b, a, A ) and α = inc.The exactness of the above complex was known for n = 3 [12, Corol-lary 3.5] and n = 4 [13, Theorem 2]. Recently Galatius, Kupers and Randal-Williams have proved, by completely new and interesting methods, that thiscomplex is exact and proved Suslin’s injectivity conjecture for r = n − OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 5
Theorem 1.2 (Galatius, Kupers, Randal-Williams [6, Section 9.6]) . Let k be a field such that ( n − ∈ k × . (i) For any n ≥ , the complex (1 . is exact. (ii) For any n , inc ∗ : H n (GL n − ( F ) , k ) → H n (GL n ( F ) , k ) is injective. For any a ∈ F × , let A i,n ∈ GL n ( F ), 1 ≤ i ≤ n , be defined as follow A i,n := A i,n ( a ) = ( diag( a, I n − ) i = 1 , diag( a, . . . , a, a − ( i − , I n − i ) 2 ≤ i ≤ n. For b ∈ F × , we denote the corresponding matrices with B i,n , etc. Observethat for any 1 ≤ i ≤ n − A i,n = diag( A i,n − , . The following proposition plays a crucial role in the proof of (i) ⇒ (ii) in theprevious theorem. Proposition 1.3.
The composite K Mn ( F ) → K n ( F ) h n −→ H n (GL n ( F ) , Z ) ,denoted by ι n , is given by { a , . . . , a n } 7→ [ a , . . . , a n ] := c ( A ,n , . . . , A n,n ) ,where A i,n = A i,n ( a i ) . Moreover the composite K Mn ( F ) ι n −→ H n (GL n ( F ) , Z ) s n −→ K Mn ( F ) coincides with multiplication by ( − n − ( n − .Proof. See [13, Proposition 3] and [18, Corollary 2.2]. (cid:3)
The following corollary which follows from Theorems 1.1, 1.2 and Propo-sition 1.3 will play very important role in this article.
Corollary 1.4.
Let k be a field such that ( n − ∈ k × . Then we have thedecomposition H n (GL n ( F ) , k ) ≃ H n (GL n − ( F ) , k ) ⊕ K Mn ( F ) k , where the splitting map K Mn ( F ) k → H n (GL n − ( F ) , k ) is given by { a , . . . , a n } 7→ ( − n − ( n − [ a , . . . , a n ] . The cokernel of H n +1 (GL n − ( F ) , Z ) → H n +1 (GL n ( F ) , Z )For any n and r , let H r GL ( F, n ) := H n (GL r ( F ) , GL r − ( F ) , Z ). By The-orem 1.1, H r GL ( F, n ) = 0 for r > n and H n GL ( F, n ) ≃ K Mn ( F ). By Theo-rem 1.2, H n GL ( F, n + 1) k ≃ H n +1 (GL n ( F ) , k ) /H n +1 (GL n − ( F ) , k ) , where k is a field such that ( n − ∈ k × .Let k be a field. From the inclusions F × × GL n − ( F ) ⊆ F × × GL n − ( F ) ⊆ GL n ( F ) we obtain the exact sequence H n +1 ( F × × GL n − ( F ) , k ) H n +1 (GL n − ( F ) , k ) Φ → H n +1 (GL n ( F ) , k ) H n +1 (GL n − ( F ) , k ) → H n +1 (GL n ( F ) , k ) H n +1 ( F × × GL n − ( F ) , k ) → . BEHROOZ MIRZAII
Let H n +1 ( F × × GL n − ( F ) , k ) = L i =1 S i , where S = H n +1 (GL n − ( F ) , k ) ,S = F × ⊗ H n (GL n − ( F ) , k ) ,S = H ( F × , k ) ⊗ H n − (GL n − ( F ) , k )= S ′ ⊕ S ′′ ,S ′ = H ( F × , k ) ⊗ H n − (GL n − ( F ) , k ) ,S ′′ = H ( F × , k ) ⊗ K Mn − ( F ) k ,S = L ni =3 H i ( F × , k ) ⊗ H n − i +1 (GL n − ( F ) , k ) , where the decomposition S = S ′ ⊕ S ′′ follows from Corollary 1.4. It iseasy to see that Φ( S ⊕ S ′ ) = 0. By homological stability we have theisomorphism H n − i +1 (GL n − ( F ) , k ) ≃ H n − i +1 (GL n − ( F ) , k ) for i ≥ S ) = 0. Moreover Φ (cid:0) im( F × ⊗ H n (GL n − ( F ) , k ) id F × ⊗ inc ∗ −−−−−−→ S ) (cid:1) = 0.Thus the above exact sequence finds the form F × ⊗ H n − ( F, n ) k ⊕ V Z F × ⊗ K Mn − ( F ) k Φ −→ H n GL ( F, n + 1) k → H n +1 (GL n ( F ) , k ) H n +1 ( F × × GL n − ( F ) , k ) → , where K Mn − ( F ) ≃ H n − (GL n − ( F ) , Z ) /H n − (GL n − ( F ) , Z ) by Theorem 1.1.In the rest of this section, we will study the group H n − ( F, n ) for n = 2and n = 3. The group H ( F,
2) has a simple structure: H ( F,
2) = H (GL ( F ) , Z ) /H (GL ( F ) , Z ) ≃ V Z F × . The group H ( F, ≃ H (GL ( F ) , Z ) /H (GL ( F ) , Z ) has much richerstructure. By studying the Lyndon/Hochschild-Serre spectral sequence as-sociated to the extension 1 → SL ( F ) → GL ( F ) det −→ F × → H (SL ( F ) , Z ) F × → H ( F, → H ( F × , H (SL ( F ) , Z )) → . The inclusion SL −→ SL induces the short exact sequence0 → H ( F × , H (SL ( F ) , Z )) → H ( F × , H (SL ( F ) , Z )) → K M ( F ) / → F × acts trivially on H (SL ( F ) , Z ), we have H ( F × , H (SL ( F ) , Z )) ≃ F × ⊗ K M ( F ) . Moreover the map F × ⊗ K M ( F ) ≃ H ( F × , H (SL ( F ) , Z )) → K M ( F ) / F × ⊗ K M ( F ) → K M ( F ). Thus we have theexact sequence H (SL ( F ) , Z ) F × → H ( F, ψ → F × ⊗ K M ( F ) → K M ( F ) / → . By [12, Theorem 6.1], the map H (SL ( F ) , Z [ ]) F × → H ( F, (cid:2) (cid:3) is injec-tive. Moreover H (SL ( F ) , Z (cid:2) (cid:3) ) F × ≃ K ind3 ( F ) (cid:2) (cid:3) by [12, Proposition 6.4]or [16, Theorem 3.7]. Thus we have the exact sequence0 → K ind3 ( F )[ ] → H ( F, (cid:2) (cid:3) → F × ⊗ K M ( F )[ ] → . OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 7
Proposition 2.1.
For any infinite field F we have the decomposition H ( F, (cid:2) (cid:3) ≃ K ind3 ( F )[ ] ⊕ F × ⊗ K M ( F )[ ] . Proof.
It is enough to show that the above short exact sequence splits. Let φ be the composite F × ⊗ K M ( F ) id F × ⊗ ι −−−−−→ F × ⊗ H (GL ( F ) , Z ) ∪ → H ( F × × GL ( F ) , Z ) µ ∗ −→ H (GL ( F ) , Z ) → H ( F, , where µ : F × × GL ( F ) → GL ( F ) is given by ( a, A ) aA . By [17,Lemma 3.2 (ii)], the composite F × ⊗ K M ( F ) φ → H ( F, ψ → F × ⊗ K M ( F )coincides with multiplication by 2. Now it is easy to construct a splittingmap. (cid:3) In the next section we will describe H ( F, (cid:2) (cid:3) in a different way(Proposition 3.2), but still will be very close to the above description (seeRemark 3.3). Remark 2.2. (i) It is an open question, asked by Suslin, whether the map H (SL ( F ) , Z ) F × → K ind3 ( F )is an isomorphism [21, Question 4.4]. Hutchinson and Tao have proved thatit is surjective [9, Lemma 5.1]. See [17] for more on this question.(ii) For another way of looking at H ( F, The cokernel of H n +1 ( F × × GL n − ( F ) , Z ) → H n +1 (GL n ( F ) , Z )To study the quotient group H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k ),we look at a certain spectral sequence introduced and studied in [5], [12] and[13].Let k be a commutative ring. For any l ≥
0, let D l ( F n ) be the free k -module with a basis consisting of ( l + 1)-tuples ( h w i , . . . , h w l i ), where0 = w i ∈ F n , h w i i = F w i and h w i i 6 = h w j i when i = j . Set D − ( F n ) := k .The group GL n ( F ) acts naturally on D l ( F n ). We consider D − ( F n ) = k asa trivial GL n ( F )-module.Let define ∂ : D ( F n ) → D − ( F n ) = k by P i n i ( h w i i ) P i n i . For l ≥
1, we define the l -th differential operator ∂ l : D l ( F n ) → D l − ( F n ), asan alternating sum of face operators which throws away the i -th componentof generators.For any integer l ≥
0, set M l = D l − ( F n ). It is easy to see that thecomplex of GL n ( F )-modules M • : 0 ← M ← M ← · · · ← M l ← · · · is exact (see the proof of [24, Lemma 2.2]). BEHROOZ MIRZAII
Take a projective resolution P • → Z of Z over GL n ( F ). From the dou-ble complex M • ⊗ GL n ( F ) P • we obtain the first quadrant spectral sequenceconverging to zero with E • , • -terms E p,q ( n, k ) = ( H q ( F × p × GL n − p ( F ) , k ) if 0 ≤ p ≤ H q (GL n ( F ) , M p ) if p ≥ , (see [12, Section 3], [13, Section 5]). It is easy to see that d ,q ( n, k ) = inc ∗ .In particular E ,q ( n, k ) = H q (GL n ( F ) , k ) H q ( F × × GL n − ( F ) , k ) · Moreover d ,q ( n, k ) = α ∗ − α ∗ which is discusssed in the exact sequence(1.1). Now we would like to describe E ,q ( n, k ). The orbits of the action ofGL n ( F ) on M = D ( F n ) are represented by w = ( h e i , h e i , h e i ) and w = ( h e i , h e i , h e + e i ) . Thus E ,q ( n, k ) ≃ H q (Stab GL n ( F ) ( w ) , k ) ⊕ H q (Stab GL n ( F ) ( w ) , k ) ≃ H q ( F × × GL n − ( F ) , k ) ⊕ H q ( F × I × GL n − ( F ) , k ) , (see [19, Theorem 1.11]). Moreover d ,q ( n, k ) | H q ( F × × GL n − ( F ) , k ) = σ ∗ − σ ∗ + σ ∗ ,d ,q ( n, k ) | H q ( F × I × GL n − ( F ) , k ) = inc ∗ , where diag( a, b, c, A ) σ diag( b, c, a, A ), diag( a, b, c, A ) σ diag( a, c, b, A ) and σ = inc : diag( a, b, c, A ) diag( a, b, c, A ).When k is a field, with a similar method as in [10, Lemma 4.2], one canshow that E p,q ( n, k ) = 0 for p = 1 , q ≤ n −
1. Moreover by Theorem 1.2,this is also true for q = n if ( n − ∈ k × . Conjecture 3.1.
Let n ≥ . Then for any ≤ i ≤ n +2 , E i,n − i +2 ( n, Z ) = 0 .In particular E ,n ( n, Z ) = 0 and d ,n ( n, Z ) : E ,n ( n, Z ) → E ,n +1 ( n, Z ) issurjective. This conjecture has been proved for n = 3 in [12] and for n = 4 in [13]. Aswe will see in the next section, for the construction of the map χ n , discussedin the introduction, we will use the differential d ,n ( n, k ) : E ,n ( n, k ) → E ,n +1 ( n, k ).Now we study the group H (GL ( F ) , Z (cid:2) (cid:3) ) /H ( F × × GL ( F ) , Z (cid:2) (cid:3) ). Infact this group already has been studied in [25] by studying the spectralsequence E p,q (2 , Z ) (see also [14] for similar results over rings with manyunits).In [25, Theorem 2.1] Suslin showed that there is a natural homomorphism H (GL ( F ) , Z ) → B ( F ) such that the sequence H (GM ( F ) , Z ) → H (GL ( F ) , Z ) → B ( F ) → OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 9 is exact. Here B ( F ) is the Bloch group of F and GM ( F ) is the subgroupof monomial matrices in GL ( F ). Note that GM ( F ) ≃ ( F × × F × ) ⋊ Σ ,where Σ = (cid:26)(cid:18) (cid:19) , (cid:18) (cid:19)(cid:27) . From the previous exact sequence andan analysis of the Lyndon/Hochschild-Serre spectral sequence associated tothe extension 1 → F × × F × → GM → Σ → Z / ⊕ Z / → H (GL ( F ) , Z ) /H ( F × × GL ( F ) , Z ) → B ( F ) → . Therefore H (GL ( F ) , Z (cid:2) (cid:3) ) /H ( F × × GL ( F ) , Z (cid:2) (cid:3) ) ≃ B ( F ) (cid:2) (cid:3) . Sincethe images of F × ⊗ H (GL ( F ) , Z ) and V Z F × ⊗ H (GL ( F ) , Z ) coincide in H ( F, Proposition 3.2.
For any infinite field F , we have the exact sequence F × ⊗ H ( F, (cid:2) (cid:3) → H ( F, (cid:2) (cid:3) → B ( F ) (cid:2) (cid:3) → . Remark 3.3.
Putting the exact sequences of Propositions 2.1 and 3.2 inone diagram we get the commutative diagram F × ⊗ H ( F, (cid:2) (cid:3) −→ K ind3 ( F ) (cid:2) (cid:3) H ( F, (cid:2) (cid:3) F × ⊗ K M ( F ) (cid:2) (cid:3) −→ ,B ( F ) (cid:2) (cid:3) θ where θ ( c ⊗ ( a ∧ b )) = − a ⊗ { b, c } + b ⊗ { a, c } [17, Lemma 3.2]. Note thatby Suslin’s generalisation of the Bloch-Wigner exact sequence we have theexact sequence0 → Tor Z ( µ ( F ) , µ ( F )) ∼ → K ind3 ( F ) → B ( F ) → , where the group Tor Z ( µ ( F ) , µ ( F )) ∼ is the unique nontrivial extension ofTor Z ( µ ( F ) , µ ( F )) by Z / The map χ n : B n ( F ) k → H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k )Consider the complex (0.2) introduced in the introduction. For any n ≥ δ ( n )1 ) = im( δ ( n )2 ). In fact the map K Mn ( F ) → F × ⊗ K Mn − ( F ) / im( δ ( n )2 ) , { a , a , . . . , a n } 7→ a ⊗ { a , . . . , a n } (mod δ ( n )2 ) is well-defined and is the inverse of δ ( n )1 : F × ⊗ K Mn − ( F ) / im( δ ( n )2 ) → K Mn ( F ).The second homology of this complex, B n ( F ), seems to be an interestinginvariant of the field. Proposition 4.1.
For any positive integers n and any field k such that ( n − ∈ k × , there is a natural map χ n : B n ( F ) k → H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k ) . If Conjecture . holds, then χ n is surjective and it implies a surjective map F × ⊗ H n − ( F, n ) k ⊕ V Z F × ⊗ K Mn − ( F ) k ⊕ B n ( F ) k ։ H n GL ( F, n + 1) k . Proof.
It is sufficient to construct the map χ n . The other claims followeasily. The map χ is the trivial map . Moreover one can show that K ind3 ( F ) α ≃ H (GL ( F ) , Z ) / (cid:0) H (GL ( F ) , Z ) + F × ∪ H (GL ( F ) , Z ) (cid:1) := ˜ H (SL ( F ) , Z )(see [18, Remark 5.2]). We define χ := η ◦ α , where η : ˜ H (SL ( F ) , Z ) → H (GL ( F ) , Z ) /H ( F × × GL ( F ) , Z )is the usual quotient map. Observe that χ is surjective.So let n ≥
3. We construct a surjective map χ ′ n : B n ( F ) k ։ E ,n ( n, k ).Then χ n is defined as the composite of χ ′ n with the differential d ,n ( n, k ): χ n := d ,n ( n, k ) ◦ χ ′ n . The group E ,n ( n, k ) is homology of the complex H n ( F × I × GL n − ( F ) , k ) ⊕ H n ( F × × GL n − ( F ) , k ) d ,n ( n, k ) −−−−→ H n ( F × × GL n − ( F ) , k ) d ,n ( n, k ) −−−−→ H n ( F × × GL n − ( F ) , k ) . By the K¨unneth formula and Corollary 1.4, we have H n ( F × × GL n − ( F ) , k ) = T ⊕ T ⊕ T ,H n ( F × × GL n − ( F ) , k ) = U ⊕ U ⊕ U ⊕ U ⊕ U , where T = H n (GL n − ( F ) , k ) ,T = F × ⊗ H n − (GL n − ( F ) , k ) = T ′ ⊕ T ′′ ,T ′ = F × ⊗ H n − (GL n − ( F ) , k ) ,T ′′ = F × ⊗ K Mn − ( F ) k ,T = L ni =2 H i ( F × , k ) ⊗ H n − i (GL n − ( F ) , k ) , OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 11 and U = H n (GL n − ( F ) , k ) ,U = L ni =1 H i ( F × , k ) ⊗ H n − i (GL n − ( F ) , k ) ,U = L ni =1 H i ( F × , k ) ⊗ H n − i (GL n − ( F ) , k ) ,U = F × ⊗ F × ⊗ H n − (GL n − ( F ) , k ) = U ′ ⊕ U ′′ ,U ′ = F × ⊗ F × ⊗ H n − (GL n − ( F ) , k ) ,U ′′ = F × ⊗ F × ⊗ K Mn − ( F ) k ,U = L i + j ≥ i,j> H i ( F × , k ) ⊗ H j ( F × , k ) ⊗ H n − i − j (GL n − ( F ) , k ) . Let u = ( u , u , u , u , u ) be in the kernel of d ,n ( n, k ) = α ∗ − α ∗ , and let u = ( u ′ , u ′′ ). Let H n ( F × I × GL n − ( F ) , k ) = V ⊕ V , where V = H n (GL n − ( F ) , k ) ,V = L ni =1 H i ( F × I , k ) ⊗ H n − i (GL n − ( F ) , k ) . Since d ,n ( n, k ) | H n ( F × I × GL n − ( F ) , k ) = inc ∗ , we may assume u = u = 0.Let W be the following summand of H n ( F × × GL n − ( F ) , k ): W = M i + j ≥ i,j> W i,j = M i + j ≥ i,j> H i ( F × , k ) ⊗ H j ( F × , k ) ⊗ H n − i − j (GL n − ( F ) , k ) . Note that d ,n ( n, k ) | H n ( F × × GL n − ( F ) , k ) = σ ∗ − σ ∗ + σ ∗ . We have d ,n ( n, k ) : W i,j → U ⊕ U ,x ⊗ y ⊗ z (cid:0) ( − σ ∗ + σ ∗ )( x ⊗ y ⊗ z ) , x ⊗ y ⊗ inc ∗ ( z ) (cid:1) . By homology stability H n − i − j (GL n − ( F ) , k ) → H n − i − j (GL n − ( F ) , k ) is anisomorphism for i + j ≥
3. Thus we may assume u = 0.The restriction of the differential d ,n ( n, k ) on the summand W ′ = F × ⊗ F × ⊗ H n − (GL n − ( F ) , k ) of H n ( F × × GL n − ( F ) , k ) is given by d ,n ( n, k ) : W ′ → U ⊕ U ′ ,a ⊗ b ⊗ z (cid:0) ( − σ ∗ + σ ∗ )( a ⊗ b ⊗ z ) , a ⊗ b ⊗ z (cid:1) . Thus we may assume that u ′ = 0. Therefore u = ( u , u ′′ ) ∈ U ⊕ U ′′ . Let U = L ni =1 U ,i and set u = ( u ,i ) ≤ i ≤ n . If T = L ni =1 T ,i , then d ,n ( n, k ) : U , → T ⊕ T ′ , s ⊗ z ( − s ∪ z, s ⊗ z ) ,d ,n ( n, k ) : U ,i → T ⊕ T ,i , r ⊗ v ( − r ∪ v, r ⊗ inc ∗ ( v )) , ≤ i ≤ n. Moreover d ,n ( n, k ) : U ′′ → T = T ′ ⊕ T ′′ , a ⊗ b ⊗ { c , . . . , c n − } 7→ t = ( t ′ , t ′′ ) , where t = − ( − n − ( n − (cid:0) b ⊗ c (diag( a, I n − ) , diag(1 , C ,n − ) , . . . , diag(1 , C n − ,n − ))+ a ⊗ c (diag( b, I n − ) , diag(1 , C ,n − ) , . . . , diag(1 , C n − ,n − )) (cid:1) = n − (cid:0) b ⊗ [ c , . . . , c n − , a ] + a ⊗ [ c , . . . , c n − , b ] − b ⊗ c (diag( C ,n − , , . . . , diag( C n − ,n − , , diag( aI n − , − a ⊗ c (diag( C ,n − , , . . . , diag( C n − ,n − , , diag( bI n − , (cid:1) t ′ = − ( − n − ( n − (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) ,t ′′ = b ⊗ { a, c , . . . , c n − } + a ⊗ { b, c , . . . , c n − } . Since u ∈ ker( d ,n ( n, k )), we have u ,i = 0 for all 2 ≤ i ≤ n (for i ≥ H n − i (GL n − ( F ) , k ) ≃ H n − i (GL n − ( F ) , k )).Therefore u = u , and hence we may assume u = ( u , , u ′′ ) ∈ U , ⊕ U ′′ . If u , = P s ⊗ z and u ′′ = P a ⊗ b ⊗ { c , . . . , c n − } , then d ,n ( n, k )( u ) = ( t , t ′ , t ′′ ) = 0 ∈ T ⊕ T ′ ⊕ T ′′ , where t = − P s ∪ z = 0 ,t ′ = u , = P s ⊗ z = ( − n − ( n − P (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) ,t ′′ = P (cid:0) b ⊗ { a, c , . . . , c n − } + a ⊗ { b, c , . . . , c n − } (cid:1) = 0 . These calculations show that the map χ ′ n : ker( δ ( n )2 ) k −→ E ,n ( n, k ), givenby P a ⊗ b ⊗ { c , . . . , c n − } 7→ ( u , , P a ⊗ b ⊗ { c , . . . , c n − } )is surjective, where u , = ( − n − ( n − P (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) . To finish the proof, we should check that this map factors through B n ( F ) k and for this we should show that χ ′ n (cid:0) b ⊗ c ⊗{ a, d , . . . , d n − } + a ⊗ c ⊗{ b, d , . . . , d n − } + a ⊗ b ⊗{ c, d , . . . , d n − } (cid:1) is trivial. Consider the following summands of H n ( F × × GL n − ( F ) , k ): W ′′ = F × ⊗ F × ⊗ F × ⊗ H n − (GL n − ( F ) , k )= W ′′ ⊕ W ′′ ,W ′′ = F × ⊗ F × ⊗ F × ⊗ H n − (GL n − ( F ) , k ) ,W ′′ = F × ⊗ F × ⊗ F × ⊗ K Mn − ( F ) k . Then d ,n ( n, k ) : W ′′ → U = U ′ ⊕ U ′′ , a ⊗ b ⊗ c ⊗ { d , . . . , d n − } 7→ ( u ′ , u ′′ ) , OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 13 where u ′ = ( − n − ( n − (cid:0) b ⊗ c ⊗ c ( aI n − , D ,n − , . . . , D n − ,n − )+ a ⊗ c ⊗ c ( bI n − , D ,n − , . . . , D n − ,n − )+ a ⊗ b ⊗ c (cid:0) cI n − , D ,n − , . . . , D n − ,n − ) (cid:1) ,u ′′ = − b ⊗ c ⊗ { a, d , . . . , d n − } − a ⊗ c ⊗ { b, d , . . . , d n − }− a ⊗ b ⊗ { c, d , . . . , d n − } . On the other hand, d ,n ( n, k ) : W ′ ⊕ W ′′ → U , ⊕ U ′ ⊕ U ′′ , ( − u ′ , a ⊗ b ⊗ c ⊗ { d , . . . , d n − } ) ( − ( − n − ( n − z, , u ′′ ) , where z = c ⊗ c (diag( I n − , b ) , diag( aI n − , , D ,n − , . . . , D n − ,n − )+ b ⊗ c (diag( I n − , c ) , diag( aI n − , , D ,n − , . . . , D n − ,n − )+ c ⊗ c (diag( I n − , a ) , diag( bI n − , , D ,n − , . . . , D n − ,n − )+ a ⊗ c (diag( I n − , c ) , diag( bI n − , , D ,n − , . . . , D n − ,n − )+ b ⊗ c (diag( I n − , a ) , diag( cI n − , , D ,n − , . . . , D n − ,n − )+ a ⊗ c (diag( I n − , b ) , diag( cI n − , , D ,n − , . . . , D n − ,n − ) . (Note that D i,n − = diag( D i,n − ,
1) = diag( D i,n − ( d i ) , W ′′ → U ′ ⊕ U ′′ → T ′ ⊕ T ′′ we have a ⊗ b ⊗ c ⊗ { d , . . . , d n − } 7→ ( u ′ , u ′′ ) ( − ( − n − ( n − z + u ′′′′ , t ′′′ ) = 0 , where u ′′′ = ( − n − ( n − (cid:0) c ⊗ c ( bI n − , A ,n − , D ,n − , . . . , D n − ,n − )+ b ⊗ c ( cI n − , A ,n − , D ,n − , . . . , D n − ,n − )+ c ⊗ c ( aI n − , B ,n − , D ,n − , . . . , D n − ,n − )+ a ⊗ c ( cI n − , B ,n − , D ,n − , . . . , D n − ,n − )+ b ⊗ c ( aI n − , C ,n − , D ,n − , . . . , D n − ,n − )+ a ⊗ c ( bI n − , C ,n − , D ,n − , . . . , D n − ,n − ) (cid:1) . Thus u ′′′ = ( − n − ( n − z and we have χ ′ n ( u ′′ ) = ( − u ′′′ , u ′′ ) = ( ( − n − ( n − z, u ′′ )which is trivial in E ,n ( n, k ). This induces a well defined surjective map B n ( F ) k → E ,n ( n, k ), which we denote it again by χ ′ n . This completes theproof of the proposition. (cid:3) Theorem 4.2.
Let k be a field. (i) The map χ is surjective and there is a surjective map F × ⊗ H ( F, k ⊕ V Z F × ⊗ K M ( F ) k ⊕ B ( F ) k ։ H ( F, k . (ii) If char( k ) = 2 , then χ is surjective and there is a surjective map F × ⊗ H ( F, k ⊕ V Z F × ⊗ K M ( F ) k ⊕ B ( F ) k ։ H ( F, k . Proof.
These follow from Proposition 4.1 and the fact that Conjecture 3.1holds for n = 3 [12] and n = 4 [13]. (cid:3) Based on these results we make the following conjecture.
Conjecture 4.3.
For any n and any field k such that ( n − ∈ k × , thenatural map χ n : B n ( F ) k → H n +1 (GL n ( F ) , k ) /H n +1 ( F × × GL n − ( F ) , k ) issurjective. The proof of Proposition 4.1 shows that the above conjecture follows fromConjecture 3.1.5.
On the kernel of H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z )As we will see in this section the group B n ( F ) also is related to the kernelof inc ∗ : H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ). This connection first wasrealised by the author in [16, Remark 3.5] for n = 3, where the complex(0.2) was introduced (for n = 3). Proposition 5.1. (i)
For any positive integer n , there is a natural map ϕ n : B n ( F ) → ker (cid:0) H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) (cid:1) , which factors through the map B n ( F ) → ( n − B n ( F ) . Moreover im( ϕ n ) is ( n − -torsion. (ii) For any n there is a natural map κ n : B n ( F ) ⊗ Z (cid:2) n − (cid:3) → ker (cid:0) H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) )) were its image is ( n − -torsion.Proof. (i) We define ϕ and ϕ as trivial maps. So let n ≥
3. Let x = P a ⊗ b ⊗ { c , . . . , c n − } represent an element of B n ( F ). Let y be the imageof x by the mapid F × ⊗ id F × ⊗ ι n − : F × ⊗ F × ⊗ K Mn − ( F ) → F × ⊗ F × ⊗ H n − (GL n − ( F ) , Z ) . Thus y = P a ⊗ b ⊗ [ c , . . . , c n − ] = P a ⊗ b ⊗ c ( C ,n − , . . . , C n − ,n − ). If werestrict d ,n ( n, Z ) to F × ⊗ F × ⊗ H n − (GL n − ( F ) , Z ) ⊆ E ,n ( n, Z ) we have d ,n ( y ) = − P(cid:0) b ⊗ c (diag( a, I n − ) , diag(1 , C ,n − )) , . . . , diag(1 , C n − ,n − ))+ a ⊗ c (diag( b, I n − ) , diag(1 , C ,n − ) , . . . , diag(1 , C n − ,n − )) (cid:1) . On the other hand, the composite F × ⊗ F × ⊗ K Mn − ( F ) δ ( n )2 −−→ F × ⊗ K Mn − ( F ) id F × ⊗ ι n − −−−−−−→ F × ⊗ H n − (GL n − ( F ) , Z )takes x to 0 = P (cid:0) b ⊗ [ a, c , . . . , c n − ] + a ⊗ [ b, c , . . . , c n − ] (cid:1) . We have d ,n (( n − y )= ( − n − P(cid:0) b ⊗ c ( C ,n − , . . . , C n − ,n − , diag( I n − , a − ( n − ))+ a ⊗ c ( C ,n − , . . . , C n − ,n − , diag( I n − , b − ( n − )) (cid:1) = ( − n − P(cid:0) b ⊗ [ c , . . . , c n − , a ] + a ⊗ [ c , . . . , c n − , b ] − b ⊗ c ( C ,n − , . . . , C n − ,n − , diag( aI n − , − a ⊗ c ( C ,n − , . . . , C n − ,n − , diag( bI n − , (cid:1) = − P (cid:0) b ⊗ c (diag( aI n − , , C ,n − , . . . , C n − ,n − )+ a ⊗ c (diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 15
Let T , T and U , U be the following summands of E ,n ( n, Z ) = H n ( F × × GL n − ( F ) , Z ) and E ,n ( n, Z ) = H n ( F × × GL n − ( F ) , Z ), respectively: T = H n (GL n − ( F ) , Z ) , U = F × ⊗ H n − (GL n − ( F ) , Z ) ,T = F × ⊗ H n − (GL n − ( F ) , Z ) , U = F × ⊗ F × ⊗ H n − (GL n − ( F ) , Z ) . Consider the following element of U : z = P (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) . The differential d ,n : U ⊕ U → T ⊕ T maps ( z, ( n − y ) to ( w, w = − P (cid:0) c (diag( I n − , b ) , diag( aI n − , , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , a ) , diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . Clearly w is in the kernel of H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ). Nowdefine ϕ n : B n ( F ) → ker (cid:0) H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) (cid:1) , ϕ ( x ) = w. It is not difficult to see that this map is well defined (see the proof of Propo-sition 4.1).To prove that w is ( n − F × ⊗ K Mn − ( F ) id F × ⊗ ι n − −−−−−→ F × ⊗ H n − (GL n − ( F ) , Z ) ∪ → H n ( F × × GL n − ( F ) , Z ) µ ∗ −→ H n (GL n − ( F ) , Z ) , where µ : F × × GL n − ( F ) → GL n − ( F ) is the multiplication ( a, A ) aA = ( aI n − ) A . Under this composite 0 = P (cid:0) b ⊗ [ a, c , . . . , c n − ] + a ⊗ [ b, c , . . . , c n − ] (cid:1) maps to0 = ( − n − P(cid:0) c ( bI n − , C ,n − , . . . , C n − ,n − , A n − ,n − )+ c ( aI n − , C ,n − , . . . , C n − ,n − , B n − ,n − ) (cid:1) = P (cid:0) c (diag( bI n − , b ) , diag( aI n − , a − ( n − ) , C ,n − , . . . , C n − ,n − )+ c (diag( aI n − , a ) , diag( bI n − , b − ( n − ) , C ,n − , . . . , C n − ,n − ) (cid:1) = P (cid:0) c (diag( bI n − , , diag( I n − , a − ( n − ) , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , b ) , diag( aI n − , , C ,n − , . . . , C n − ,n − )+ c (diag( aI n − , , diag( I n − , b − ( n − ) , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , a ) , diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) = − ( n − w. (Observe that C i,n − = diag( C i,n − ,
1) for 1 ≤ i ≤ n − w = ϕ ( x ) is ( n − κ n is similar to that of ϕ n . By putting κ = 0 and κ = 0, we may assume n ≥
3. Let x = P a ⊗ b ⊗ { c , . . . , c n − } representan element of B n ( F ) ⊗ Z (cid:2) n − (cid:3) . Since the composite K Mn − ( F ) ι n − −→ H n − (GL n − ( F ) , Z ) s n − −→ K Mn − ( F ) coincide with multiplication by ( − n − ( n − K Mn − ( F ) ⊗ Z (cid:2) n − (cid:3) → H n − (GL n − ( F ) , Z (cid:2) n − (cid:3) ) , { c , . . . , c n − } 7→ ( − n − ( n − [ c , . . . , c n − ]is injective. Thus y = ( − n − ( n − P a ⊗ b ⊗ [ c , . . . , c n − ] is the image of x through the above map. As in above one sees that d ,n ( y ) = − ( − n − ( n − P (cid:0) b ⊗ c (diag( aI n − , , C ,n − , . . . , C n − ,n − )+ a ⊗ c (diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . Let T ′ , T ′ and U ′ , U ′ be defined as T , T and U , U in above with Z replaced by Z (cid:2) n − (cid:3) . If y ′ = ( − n − ( n − P (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) ∈ U ′ , then d ,n : U ′ ⊕ U ′ → T ′ ⊕ T ′ , maps ( y ′ , y ) to ( ( − n − ( n − w, , κ n ( x ) = ( − n − ( n − w . Since w is ( n − φ n ( x ) also is ( n − (cid:3) Note that ϕ n = κ n for n ≤
3. The above proof shows that:
Corollary 5.2.
For any n ≥ , κ n is well defined and is given by P a ⊗ b ⊗ { c , . . . , c n − }− ( − n − ( n − P (cid:0) c (diag( I n − , b ) , diag( aI n − , , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , a ) , diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . Proposition 5.3.
Let the sequence H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) α ∗ − α ∗ −−−−→ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) inc ∗ −→ H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) → be exact for any ≤ n ≤ s . Then for any ≤ n ≤ s , (i) The map κ n is surjective and its image is ( n − -torsion. (ii) The map H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) is in-jective. (iii) We have the decomposition H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) ≃ H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ⊕ K Mn ( F ) ⊗ Z (cid:2) n − (cid:3) , where the splitting map K Mn ( F ) ⊗ Z (cid:2) n − (cid:3) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) isgiven by { a , . . . , a n } 7→ ( − n − ( n − [ a , . . . , a n ] . OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 17
Proof.
The claims are trivial for n = 1 ,
2. So let 3 ≤ n ≤ s . The proof isby induction on all parts simultaneously. By induction we assume that theclaims hold for any m < n . Let H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) ≃ L i =1 T i ,H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) ≃ L i =1 U i , where T = H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,T = F × ⊗ H n − (GL n − ( F ) , Z (cid:2) n − (cid:3) ) = T ′ ⊕ T ′′ ,T ′ = F × ⊗ H n − (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,T ′′ = F × ⊗ K Mn − ( F ) ⊗ Z (cid:2) n − (cid:3) ,T = L ni =2 H i ( F × , Z ) ⊗ H n − i (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,T = L n − i =1 Tor Z ( H i ( F × , Z ) , H n − i − (GL n − ( F ) , Z )) ⊗ Z (cid:2) n − (cid:3) , and U = H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,U = L ni =1 H i ( F × , Z ) ⊗ H n − i (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,U = L ni =1 H i ( F × , Z ) ⊗ H n − i (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,U = F × ⊗ F × ⊗ H n − (GL n − ( F ) , Z (cid:2) n − (cid:3) ) = U ′ ⊕ U ′′ ,U ′ = F × ⊗ F × ⊗ H n − (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,U ′′ = F × ⊗ F × ⊗ K Mn − ( F ) ⊗ Z (cid:2) n − (cid:3) ,U = L i + j ≥ i,j> H i ( F × , Z ) ⊗ H j ( F × , Z ) ⊗ H n − i − j (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,U = L n − i =1 Tor Z ( H i ( F × , Z ) , H n − i − (GL n − ( F ) , Z )) ⊗ Z (cid:2) n − (cid:3) ,U = L n − i =1 Tor Z ( H i ( F × , Z ) , H n − i − (GL n − ( F ) , Z )) ⊗ Z (cid:2) n − (cid:3) ,U = L n − i =1 Tor Z ( H i ( F × , Z ) , H n − i − ( F × , Z )) ⊗ Z (cid:2) n − (cid:3) , where the the decompositions of T and U follow by induction.Let t ∈ ker (cid:0) inc ∗ : H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) (cid:1) .Then t = ( t , , ,
0) is in the kernel ofinc ∗ : H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) → H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) . Thus there exists u = ( u , . . . , u ) ∈ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) suchthat ( α ∗ − α ∗ )( u ) = t .Let α = α ∗ − α ∗ and consider the complex H n ( F × I × GL n − ( F ) , Z (cid:2) n − (cid:3) ) ⊕ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) β −→ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) α −→ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) , where β = (inc ∗ , σ ∗ − σ ∗ + σ ∗ ). The restriction of β on the summand W ′′′ = L n − i =1 Tor Z ( H i ( F × , Z ) , H n − i − ( F × , Z )) ⊗ Z (cid:2) n − (cid:3) of H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) is as follow: β : W ′′′ → U ⊕ U , x ( − σ ∗ + σ ∗ ( x ) , x ) . So we may assume u = 0. Let H n ( F × I × GL n − ( F ) , Z (cid:2) n − (cid:3) ) ≃ V ⊕ V ⊕ V , where V = H n (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,V = L ni =1 H i ( F × I , Z ) ⊗ H n − i (GL n − ( F ) , Z (cid:2) n − (cid:3) ) ,V = L n − i =1 Tor Z ( H i ( F × I , Z ) , H n − i − (GL n − ( F ) , Z )) ⊗ Z (cid:2) n − (cid:3) . Since β | H n ( F × I × GL ( F ) , Z (cid:2) n − (cid:3) ) = inc ∗ , we may assume that u = u = u = 0. Moreover, with an argument similar to the proof of Proposition 4.1,we may assume that u ′ = u = 0 and u ,i = 0 for 2 ≤ i ≤ n , where U = L ni =1 U ,i . Thus u = ( u , , u ′′ , u ), where u , = P s ⊗ z , u ′′ = P a ⊗ b ⊗ { c , . . . , c n − } . We have α ( u , , u ′′ , u ) = ( t ′ , ( t ′ , t ′′ ) , t ′ , t ′ ) = ( t , , , t ′ = t = − P s ∪ z − α ∗ ( u ) ,t ′ = 0 = P s ⊗ z − ( − n − ( n − P (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) ,t ′′ = 0 = P (cid:0) b ⊗ { a, c , . . . , c n − } + a ⊗ { b, c , . . . , c n − } (cid:1) ,t ′ = 0 ,t ′ = 0 = α ∗ ( u ) . Since α ∗ : U → T is induced by the homology stability isomorphisms H n − i − (GL n − ( F ) , Z ) ≃ H n − i − (GL n − ( F ) , Z ) for 1 ≤ i ≤ n −
2, we have u = 0. Therefore t = − ( − n − ( n − P(cid:0) c (diag( I n − , b ) , diag( aI n − , , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , a ) , diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . This shows that κ n is surjective. But by the previous proposition im( κ n )is ( n − (cid:3) Based on the above proposition we make the following conjecture.
Conjecture 5.4.
For any n ≥ , the sequence H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) α ∗ − α ∗ −−−→ H n ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) inc ∗ −→ H n (GL n ( F ) , Z (cid:2) n − (cid:3) ) → is exact. Remark 5.5. (i) In the proof of Proposition 5.3 we do not use the spectralsequence E • , • ( n, Z (cid:2) n − (cid:3) ) introduced in Section 3.(ii) Conjecture 5.4 follows from Conjecture 3.1. In fact by Conjecture 3.1, E ,n ( n, Z ) = E ,n ( n, Z ) = 0, which from it the exactness of H n ( F × × GL n − ( F ) , Z ) α −→ H n ( F × × GL n − ( F ) , Z ) inc ∗ −→ H n (GL n ( F ) , Z ) → OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 19 follows, where α = α ∗ − α ∗ .(iii) Galatius, Kupers and Randal-Williams showed that the above con-jectures holds if we replace Z (cid:2) n − (cid:3) with a field k such that ( n − ∈ k × (see Theorem 1.2). Theorem 5.6.
For any n ≤ , the natural map κ n is surjective. In par-ticular the natural maps inc ∗ : H (GL ( F ) , Z (cid:2) (cid:3) ) → H (GL ( F ) , Z (cid:2) (cid:3) ) and inc ∗ : H (GL ( F ) , Z (cid:2) (cid:3) ) → H (GL ( F ) , Z (cid:2) (cid:3) ) are injective.Proof. Clearly κ and κ are surjective. Conjecture 5.4 holds for n = 3 by[12, Corollary 3.5 (ii)] and for n = 4 by [13, Theorem 2]. Now the claimfollows from Proposition 5.3. (cid:3) Remark 5.7. (i) The case n = 3 of the above theorem already was known.In fact in [16, Remark 3.5], κ is defined and is shown to be surjective.Moreover by [12, Theorem 5.4 (ii)] or [16, Corollary 3.3] the map inc ∗ : H (GL ( F ) , Z (cid:2) (cid:3) ) → H (GL ( F ) , Z (cid:2) (cid:3) ) is injective.(ii) It is an open problem whether κ is trivial, or equivalently, if the mapinc ∗ : H (GL ( F ) , Z ) → H (GL ( F ) , Z )is injective. This is closely related to the map H (SL ( F ) , Z ) F × → K ind3 ( F )discussed in Remark 2.2 (see [17, Theorem 4.4]).Let n ≥ χ n : B n ( F ) ⊗ Z (cid:2) n − (cid:3) −→ H n +1 (GL n − ( F ) , Z (cid:2) n − (cid:3) ) H n +1 ( F × × GL n − ( F ) , Z (cid:2) n − (cid:3) ) . If Conjecture 3.1 holds, then the differential d ,n ( n, Z ) : E ,n ( n, Z ) → E ,n +1 ( n, Z )is surjective and it follows that the above map is surjective. Theorem 5.8.
The maps χ : B ( F ) → H (GL ( F ) , Z ) /H ( F × × GL ( F ) , Z ) ,χ : B ( F ) (cid:2) (cid:3) → H (GL ( F ) , Z (cid:2) (cid:3) ) /H ( F × × GL ( F ) , Z (cid:2) (cid:3) ) are surjective.Proof. We know that Conjecture 3.1 holds for n = 3 by [12, Section 3] andfor n = 4 by [13, Section 6]. Therefore we have the desired results. (cid:3) Homology of GL n ( F ) over algebraically closed fields In this section we will show that many of the above results can be im-proved over algebraically closed fields. The main property that make thispossible is the fact that K Mn ( F ) is uniquely divisible for any n ≥ K M ( F ) = F × is divisible. A fact that we use frequently,without mentioning, is that if A and B are m -divisible groups, then A ⊗ B is uniquely m -divisible (see the proof of [1, Proposition 1.2]). In particular, B n ( F ) is uniquely divisible for any n ≥ Proposition 6.1.
Let F be algebraically closed and let the sequence H n ( F × × GL n − ( F ) , Z ) α ∗ − α ∗ −−−−→ H n ( F × × GL n − ( F ) , Z ) inc ∗ −→ H n (GL n ( F ) , Z ) → be exact for any ≤ n ≤ s . Then for any ≤ n ≤ s , (i) H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) is injective, (ii) H n (GL n ( F ) , Z ) ≃ H n (GL n − ( F ) , Z ) ⊕ K Mn ( F ) , where the splittingmap K Mn ( F ) → H n (GL n ( F ) , Z ) is given by ( − n − ( n − { a , . . . , a n } 7→ [ a , . . . , a n ] . or euivalently { a , . . . , a n } 7→ [ a ( − n − n − , . . . , a n ] = [ a , a − , a − , . . . , a − n − n ] . (iii) There is a natural map χ n : B n ( F ) → H n +1 (GL n ( F ) , Z ) /H n +1 ( F × × GL n − ( F ) , Z ) . Proof.
The proof is similar to the proof of Proposition 5.3, replacing thecoefficients Z (cid:2) n − (cid:3) with Z , but we need some adjustments in few places.The claims are trivial for n = 1 ,
2. So let 3 ≤ n ≤ s . The proof is byinduction. By induction we assume that the claims hold for any m < n .Let t ∈ ker( H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z )). As in the proof ofProposition 5.3 we can show that there is u = ( u , , u ′′ ) ∈ U , ⊕ U ′′ , suchthat d ,n ( u ) = ( t , , , ∈ L i =1 T i = H n ( F × × GL n − ( F ) , Z ) . Let u , = P s ⊗ z and u ′′ = ( − n − ( n − P a ⊗ b ⊗ { c , . . . , c n − } . Thenby similar calculations as in the proof of Proposition 5.3, we get u , = − ( n − (cid:0) b ⊗ c ( aI n − , C ,n − , . . . , C n − ,n − )+ a ⊗ c ( bI n − , C ,n − , . . . , C n − ,n − ) (cid:1) , − n − ( n − P (cid:0) b ⊗ { a, c , . . . , c n − } + a ⊗ { b, c , . . . , c n − } (cid:1) . Now it is easy to see that t = ( n − P(cid:0) c (diag( I n − , b ) , diag( aI n − , , C ,n − , . . . , C n − ,n − )+ c (diag( I n − , a ) , diag( bI n − , , C ,n − , . . . , C n − ,n − ) (cid:1) . OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 21
But by the proof of Proposition 5.1 this element is trivial. This implies theinjectivity of the map in (i).(ii) This follows from (i) and the fact that K Mn ( F ) is uniquely divisiblefor n ≥ n ≥
3. The proof is very similar to the proof ofProposition 4.1, with some help from the proof of Proposition 5.3. In fact χ n = d ,n ( n, Z ) ◦ χ ′ n , where χ ′ n : B n ( F ) ։ E ,n ( n, Z ) is surjective. We leavethe details to the reader. (cid:3) Corollary 6.2.
Let F be algebraically closed. If Conjecture . holds, thenfor any n ≥ , (i) H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) is injective, (ii) H n (GL n ( F ) , Z ) ≃ H n (GL n − ( F ) , Z ) ⊕ K Mn ( F ) , (iii) χ n , defined in the previous proposition, is surjective, (iv) H n GL ( F, n + 1) is divisible.Proof.
By Conjecture 3 .
1, for any n ≥
3, the sequence H n ( F × × GL n − ( F ) , Z ) α ∗ − α ∗ −−−−→ H n ( F × × GL n − ( F ) , Z ) inc ∗ −→ H n (GL n ( F ) , Z ) → d ,n ( n, Z ) is surjective. Now (i) and (ii) followfrom the previous proposition and (iii) follows from the fact that χ n iscomposition of the surjective maps d ,n ( n, Z ) and χ ′ n : B n ( F ) ։ E ,n ( n, Z ).To prove (iv) first note that B n ( F ) is uniquely divisible and thus the group H n +1 (GL n ( F ) , Z ) /H n +1 ( F × × GL n − ( F ) , Z ) is divisible. Now the claimfollows, by induction, from the exact sequence F × ⊗ H n − ( F, n ) ⊕ V Z F × ⊗ K Mn − ( F ) → H n GL ( F, n + 1) → H n +1 (GL n ( F ) , Z ) H n +1 ( F × × GL n − ( F ) , Z ) → . (cid:3) Conjecture 3.1 holds for n = 3 by [12, Section 3] and n = 4 by [13,Section 6]. Thus we have the following corollary. Corollary 6.3.
Let F be algebraically closed. Then for any n ≤ , (i) H n (GL n − ( F ) , Z ) → H n (GL n ( F ) , Z ) is injective, (ii) H n (GL n ( F ) , Z ) ≃ H n (GL n − ( F ) , Z ) ⊕ K Mn ( F ) , (iii) χ n , defined in the previous proposition, is surjective, (iv) H n GL ( F, n + 1) is divisible.
Remark 6.4.
For any n ≥ K Mn ( R ) is direct sum of a cyclic group of order2 generated by {− , . . . , − } and a uniquely divisible subgroup generatedby all { a , . . . , a n } , a , . . . , a n >
0. Hence K Mn ( R ) (cid:2) (cid:3) is uniquely divisible.Now the results of Proposition 6.1 and Corollaries 6.2, 6.3 still are valid ifwe replace F with R , Z with Z (cid:2) (cid:3) , K Mn ( F ) with K Mn ( R ) (cid:2) (cid:3) , B n ( F ) with B n ( R ) (cid:2) (cid:3) , etc. Remark 6.5. (i) By a theorem of Galatius, Kuper and Randal-Williams(Theorem E in [6]), if F is algebraically closed, then H d (GL n ( F ) , GL n − ( F ) : Z /p ) = 0 , in degree d < n/
3, for all prime p . This in particular implies that H n +1 (GL n ( F ) , GL n − ( F ) : Z /p ) = 0for any n ≥
2. Therefore H n GL ( F, n + 1) is divisible for any n ≥ Homology of GL n ( F ) over local and global fields The homology of general linear groups over global fields is well studied.For example in [4, Corollary 7.6] Borel and Yang has shown that for anumber field F , the natural map H d (GL n − ( F ) , Q ) → H d (GL n ( F ) , Q ) isinjective for any d and is surjective if d ≤ n −
3. Furthermore recently,Galatius, Kupers, Randal-Williams have shown [6, Theorem E] that if F isa field with torsion second K -group (global fields have this property), then H d (GL n − ( F ) , Q ) → H d (GL n ( F ) , Q ) is surjective if d < (4 n − / d < (4 n − / F is a global field, by a theorem of Bass and Tate [1, Theorem 2.1], K Mm ( F ) ≃ ( Z / r for any m ≥
3, where r is the number of embeddings of F in R . Thus for any n ≥ B n ( F ) (cid:2) (cid:3) = 0. Proposition 7.1.
Let F be a global field. If the sequence H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) α ∗ − α ∗ −−−−→ H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) inc ∗ −→ H n (GL n ( F ) , Z (cid:2) (cid:3) ) → is exact for any ≤ n ≤ s , then H n (GL n − ( F ) , Z (cid:2) (cid:3) ) → H n (GL n ( F ) , Z (cid:2) (cid:3) ) is bijective.Proof. The claim for n = 3 , n ≥
5. Let the claim be true for all 4 ≤ m < n . Observe thatthe surjectivity of the map follows from Theorem 1.1 and the fact that K Mn ( F ) (cid:2) (cid:3) = 0 for n ≥
3. Let H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) ≃ L i =1 T i ,H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) ≃ L i =1 U i , where T i and U i are as in the proof of Proposition 5.3, Z (cid:2) n − (cid:3) replaced by Z (cid:2) (cid:3) . Note that T = T ′ and U = U ′ . Let t ∈ ker( H n (GL n − ( F ) , Z (cid:2) (cid:3) ) → H n (GL n ( F ) , Z (cid:2) (cid:3) )). Then as in the proof of Proposition 5.3, one can showthat there exists u , ∈ U , , such that ( α ∗ − α ∗ )( u , ) = ( t , , , u , = P s ⊗ z , then( α ∗ − α ∗ )( u , ) = ( − P s ∪ z, P s ⊗ z, ,
0) = ( t , , , . Thus u , = 0, which implies that t = − P s ∪ z = 0. (cid:3) OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 23 If F is a local field, then K Mm ( F ) is uniquely divisible for any m ≥ n ≥ B n ( F ) is uniquely divisible. Now as in the caseof algebraically closed fields we have the following result. Proposition 7.2.
Let F be a local field and let the sequence H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) α ∗ − α ∗ −−−−→ H n ( F × × GL n − ( F ) , Z (cid:2) (cid:3) ) inc ∗ −→ H n (GL n ( F ) , Z (cid:2) (cid:3) ) → be exact for any ≤ n ≤ s . Then for any ≤ n ≤ s , (i) H n (GL n − ( F ) , (cid:2) (cid:3) ) → H n (GL n ( F ) , (cid:2) (cid:3) ) is injective, (ii) H n (GL n ( F ) , (cid:2) (cid:3) ) ≃ H n (GL n − ( F ) , (cid:2) (cid:3) ) ⊕ K Mn ( F ) , where the splittingmap K Mn ( F ) → H n (GL n ( F ) , (cid:2) (cid:3) ) is given by ( − n − ( n − { a , . . . , a n } 7→ [ a , . . . , a n ] , (iii) there is a natural map χ n : B n ( F ) (cid:2) (cid:3) → H n +1 (GL n ( F ) , (cid:2) (cid:3) ) /H n +1 ( F × × GL n − ( F ) , (cid:2) (cid:3) ) , which is surjective if Conjecture . holds.Proof. The proof is by induction and is similar to the proof of Proposi-tion 6.1. To start the induction we need the cases n = 3 and n = 4. Thesecases follow from Theorems 5.6 and 5.8. (cid:3) Some remarks
We know of another group which behaves similar to B n ( F ). Let C h ( F n )be the free Z -module with basis consisting of ( h + 1)-tuples of vectors( v , . . . , v h ), where the vectors v , . . . , v h are in general positions, that isevery min { h + 1 , n } of them are linearly independent. We define the dif-ferentials ∂ i similar to those defined in Section 3. It is easy to see that thecomplex0 ← Z ← C ( F n ) ∂ ←− · · · ∂ n − ←− C n − ( F n ) ∂ n ←− C n ( F n ) ← · · · is exact. Let K n := ker( ∂ n − ). Let L ′• and L • be the exact complexes0 ← ← ← Z ← C ( F n − ) ← · · · ← C n − ( F n − ) ← K n − ← ← Z ← C ( F n ) ← C ( F n ) ← · · · · · · · · · · · · ← C n − ( F n ) ← K n ← L ′• θ ∗ −→ L • by( v , . . . , v j ) θ j ( e , e , v , . . . , v j ) − ( e , e + e , v , . . . , v j )+( e , e + e , v , . . . , v j ) . This induces a morphism of double complexes L ′• ⊗ GL n − ( F ) P • → L • ⊗ GL n ( F ) P • , where P • → Z is a free GL n ( F )-resolution of Z (which is also a freeGL n − ( F )-resolution of Z ). The double complex( L • ⊗ GL n ( F ) P • ) / ( L ′• ⊗ GL n − ( F ) P • ) induces the first quadrant spectral sequence converging to zero with E p,q ( n ) = H q (GL n − p ( F ) , Z ) if p = 0 , H q (GL n ( F ) , GL n − ( F ); K n , K n − ) if p = n + 1 . d p,q ( n ) = ( inc ∗ if p = 10 otherwise . Now by an easy analysis of this spectralsequence, for any natural numbers n, k , we obtain the exact sequence0 → H n + k (GL n ( F ) , Z ) H n + k (GL n − ( F ) , Z ) → H k (GL n ( F ) , GL n − ( F ); K n , K n − ) → ker( H n + k − (GL n − ( F ) , Z ) inc ∗ −→ H n + k − (GL n ( F ) , Z )) → . By Theorem 1.1, H (GL n ( F ) , GL n − ( F ); K n , K n − ) ≃ K Mn ( F ) . Moreover by Theorem 1.2, H (GL n ( F ) , GL n − ( F ); K n , K n − , k ) ≃ H n GL ( F, n + 1) k , where k is a field such that ( n − ∈ k × . If n = 2, then the map H l (GL ( F ) , Z ) → H l (GL ( F ) , Z ) always is injective and thus for any naturalnumber k , H k (GL ( F ) , K ) ≃ H ( F, k + 2) . There should be a natural map from H (GL n ( F ) , GL n − ( F ); K n , K n − ) to B n ( F ), as there is a natural morphism L • −→ M • . For an exact sequenceinvolving H (GL ( F ) , K ) ≃ H (GL ( F ) , Z ) /H (GL ( F ) , Z ) see [15, Theo-rem 2.5]. Remark 8.1.
When n ≥
3, we expect other homologies of Complex (0.2)be related to the quotient groups H n + k (GL n ( F ) , Z ) /H n + k ( F × × GL n − ( F ) , Z )and maybe to the kernel of the mapsinc ∗ : H n + k − (GL n − ( F ) , Z ) → H n + k − (GL n ( F ) , Z )for 0 ≤ k ≤ n −
1. For example if k = 0, then all the groups ker( δ ( n )1 ) / im( δ ( n )2 ), H n (GL n ( F ) , Z ) /H n ( F × × GL n − ( F ) , Z ) and ker(inc ∗ ) are trivial. We do notknow how to make these connections for k ≥ B n ( F ) really is needed for the study of thecokernel of the map H n +1 ( F × × GL n − ( F ) , Q ) → H n +1 (GL n ( F ) , Q ). Question 8.2.
Let F be an infinite field and let n ≥
3. Is the natural map H n +1 ( F × × GL n − ( F ) , Q ) → H n +1 (GL n ( F ) , Q ) surjective? OMOLOGY OF GENERAL LINEAR GROUPS OVER INFINITE FIELDS 25
References [1] Bass, H., Tate, J. The Milnor ring of a global field. Lecture Notes in Math., Vol. ,349–446 (1973) 20, 22[2] Beilinson, A., MacPherson, R., Schechtman, V. Notes on motivic cohomology. DukeMath. J. (1987), no. 2, 679–710 2[3] Bloch, S. Algebraic Cycles and Higher K -theory. Advances in Math. (1986), 267–304 1[4] Borel, A., Yang, J. The rank conjecture for number fields. Math. Res. Lett. (1994),no. 6, 689–699 2, 22[5] Elbaz-Vincent, P. The indecomposable K of rings and homology of SL . J. PureAppl. Algebra (1998), no. 1, 27–71 2, 7[6] Galatius, S. Kupers, A., Randal-Williams, O. E ∞ -cells and general linear groups ofinfinite fields. https://arxiv.org/abs/2005.05620 2, 5, 22[7] Gerdes, W. Affine Grassmannian homology and the homology of general linear groups.Duke Math. J. (1991), no. 1,85–103 2[8] Gerdes, W. The linearization of higher Chow cycles of dimension one. Duke Math. J. (1991), no. 1, 105–129 2[9] Hutchinson, K., Tao, L. The third homology of the special linear group of a field. J.Pure Appl. Algebra (2009), no. 9, 1665–1680 6, 7[10] Mirzaii, B. Homology stability for unitary groups II. K -Theory (2005) , 305–326 8[11] Mirzaii, B. Homology of GL n over algebraically closed fields. J. London Math. Soc. (2007), 605-621 22[12] Mirzaii, B. Third homology of general linear groups. J. Algebra (2008), no. 5,1851–1877 3, 4, 6, 7, 8, 13, 19, 21, 22[13] Mirzaii, B. Homology of GL n : injectivity conjecture for GL . Math. Ann. (2008),no.1, 159-184 2, 4, 5, 7, 8, 13, 19, 21, 22[14] Mirzaii, B. Bloch-Wigner theorem over rings with many units. Math. Z. (2011),329–346. Erratum to: Bloch-Wigner theorem over rings with many units. Math. Z. (2013), 653–655. 8[15] Mirzaii, B. A note on third Homology of GL . Comm. Algebra, (2011), no. 5,1595–1604 7, 24[16] Mirzaii, B. Third homology of general linear groups over rings with many units.Journal of Algebra (2012), no. 1, 374–385. 3, 6, 14, 19[17] Mirzaii, B. Third homology of SL and the indecomposable K . J. Homotopy Relat.Struct. (2015), 673–683 7, 9, 19[18] Mirzaii, B. Mokari, F. Y. A Bloch-Wigner theorem over rings with many units II.Journal of Pure and Applied Algebra (2015), 5078–5096. Math. Ann. (2008),no.1, 159–184 5, 9, 10[19] Nesterenko Yu. P., Suslin A. A. Homology of the general linear group over a localring, and Milnor’s K -theory. Math. USSR-Izv. (1989), no. 3, 269–312 2, 7[22] Sivitskii, I. Ya. On torsion in Milnor’s K -groups for a local field, Mat. Sb. (N.S.),1985,
126 (168) , no. 4, 576–583 23[23] Sprehn, D., Wahl, N. Homological stability for classical groups. Trans. Amer. Math.Soc. (2020), 4807–4861 1[24] Suslin, A. A. Homology of GL n , characteristic classes and Milnor K -theory. Proc.Steklov Math. (1985), 207–225 1, 4, 7[25] Suslin A. A. K of a field, and the Bloch group. Proc. Steklov Inst. Math. 1991, no. 4, 217–239 8, 9 [26] Totaro, B. Milnor K -Theory is the simplest part of algebraic K -theory. K -Theory1992,6