Homological symbols and the Quillen conjecture

HOMOLOGICAL SYMBOLS AND THE QU ILLEN CONJECTURE MARIAN F. ANTON Abstract. W e formulate a ”corr ect” v ersion of the Q uillen con- jecture on linear gro up homology fo r certain arithmetic r ings a nd provide ev idence for the new conjecture. In this wa y we predict that the linear group homology has a direct summand looking like an unstable form of Milno r K-theory and we ca ll this new the- ory ”ho mo logical symbols a lgebra”. As a bypro duct we pr ov e the Quillen conjecture in homologic al degree t wo for the ra nk tw o and the pr ime 5. 1. Intr oduction Let R b e a subring with identit y of the complex num b ers C and resp. GL n , S L n the discrete gro up of n × n matrices ov er R with determinant resp. nonzero, 1. If H ( GL n ) := H ∗ ( GL n ; F ℓ ) denotes the mod ℓ group cohomology of GL n , then the canonical inc lusion R ⊂ C induces a mo dule structure of H ( GL n ) ov er the singular mo d ℓ cohomology ring of Chern classe s P n := H ∗ ( B GL n ( C ); F ℓ ) where B GL n ( C ) denotes the classifying space of the Lie group GL n ( C ) o f in v ertible n × n matrices o ver C . In [1616; 16, p. 591] Quillen conjectured that f or ce rtain primes ℓ and rings R the mo dule H ( GL n ) is fr e e ov er P n . W e call this statemen t the strong Quil l e n c onje ctur e for the r ank n and the prime ℓ . In particular, if w e fix R = Z [ 1 ℓ , ξ ℓ ] where ℓ is a r e gular prime and ξ ℓ ∈ C is a primitiv e ℓ - th ro ot of unity , then it has b een sho wn in [1212; 12, p. 51] that the strong Q uillen’s conjecture implies that the homomorphism ι np : H p ( GL × n 1 ; F ℓ ) → H p ( GL n ; F ℓ ) induc e d by the c anonic al inclusion GL × n 1 ⊂ GL n on m o d ℓ homolo gy is surjectiv e for al l p . W e call the statemen t that ι np is surjectiv e the w eak Quil len c onje ctur e in homolo gic al de gr e e p for the r ank n and the prime ℓ . This w eak conjecture w as dis pro ved in [77, 7] for n ≥ 3 2, ℓ = 2, and in [11, 1] fo r n ≥ 27, ℓ = 3, in the sense that there is an unsp ecified p dep ending on n and ℓ fo r whic h the s tatemen t fails. 1 2 MARIAN F. A NTON In this article w e form ula t e y et another conjecture for ℓ o dd and regular (Conjecture 5.1) whic h pro ve s that the w eak Quillen conjec- ture for the rank n , the prime ℓ and al l homological degrees p implies the strong Quillen conjecture f o r the same rank n and prime ℓ (see subsection § 5.1 for a full discuss io n). More s p ecifically , b y Prop osition 5.3, this new conjecture states that a certain finite set of homological classes in H ∗ ( GL 1 ; F ℓ ) v anish in H ∗ ( S L 2 ; F ℓ ) under the map induced from embedding GL 1 in S L 2 via u 7→  u − 1 0 0 u  . These classe s are called ´ etale obstruction cl a sses since t hey originate fr o m studying ´ etale mo dels [33, 3] for the classifying spaces B GL n . The bar complex cycles represen ting these class es a re given explicitly in Definition 4.3. As evidence for t he Conjecture 5.1 w e remark that this conjecture and the w eak Quillen conjecture for all p a r e true for ℓ = 3 b y direct calculations [11 , 1] and th us, t he strong Quillen conjecture holds in this case. Also the case ℓ = 2 fits into the same pattern for the ranks n = 2 [1515, 15] and n = 3 [1111, 11]. In this ar t icle we prov e a new result stating tha t Theorem 1.1. H 2 ( S L 2 ( Z [ 1 5 , ξ 5 ]; F 5 ) = 0 . As a corollary , our conjecture is true in homological degree tw o for ℓ = 5 in the sense that the ´ etale obstruction classes from H 2 ( GL 1 ; F 5 ) ob viously v anish in H 2 ( S L 2 ; F 5 ). As a b ypro duct, w e o btain t ha t the w eak Quillen conjecture in homolog ical degree tw o for the rank t w o and the prime 5 is also true and Theorem 1.2. H 2 ( GL 2 ( Z [ 1 5 , ξ 5 ]; F 5 ) ≈ F 5 ⊕ F 5 ⊕ F 5 ⊕ F 5 . The techniq ue used in pro ving Theorem 1.1 is based on prov ing a k ey result in The orem 6.2 regarding the structure o f the group S L 2 as a finitely presen ted group and using GAP [99, 9] in a clev er w a y . The main difficultie s reside in the complexit y of the combinatorial group problems asso ciated with Hopf ’s form ula a nd its generalizations [171 7, 17]. Another feature of this article is a c haracterization of a direct sum- mand (as a v ector space ) of the bigraded algebra (1.1) A := ∞ M i,j =0 H i ( GL j ; F ℓ ) where the algebra structure is induced fro m the matrix blo ck m ultipli- cation. This summ and is the bigraded subalgebra K A ⊂ A generated b y the linear subspace H ∗ ( GL 1 ; F ℓ ) ⊂ A and its structure is predicted HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 3 b y the new conjecture in t he sense that the relations in K A come from H ∗ ( GL × 2 1 ; F ℓ ) in a certain explicit w ay (see Remark 5 .2 ). W e recall [1414, 14] that the ( na iv e) Milnor K -theory of the ring R is the tensor algebra generated b y the group of units GL 1 mo dulo the Stein b erg r elat io ns u ⊗ (1 − u ) = 0 coming from GL ⊗ 2 1 for u, 1 − u ∈ GL 1 . By replacing GL 1 with H ∗ ( GL 1 ; F ℓ ), GL ⊗ 2 1 with H ∗ ( GL × 2 1 ; F ℓ ) a nd the Stein b erg relations with those relations predicted by our conjecture, w e obtain the conjectural structure of K A . F or this reason, w e call K A the algebr a of ho m olo gic al symb ols at ℓ a sso ciated with the ring R . The pa p er is organized as follows . After reviewing some basic gr oup homology facts in § 2 and in tro ducing some algebra terminology in § 3, w e describ e the direct summand o f the algebra (1.1) and estimate from ”ab ov e” the relations of this summand in Theorem 4.6. Th e conjecture on the exact relations is form ulat ed in § 5. In § 6 w e estimate the rela- tions in S L 2 from ” b elo w” for any regular o dd prime and use them in § 7 to pro v e Theorem 1.1 (see Corollar y 7.5). Theorem 1.2 follo ws no w from Theorems 1.1 and 4.6 b y a sp ectral se quence argumen t. 2. Gr oup homology pre liminaries W e recall some standard facts a b out group homo lo gy as in [44, 4]. Let G b e a m ultiplicativ e group with neutral elemen t 1 ∈ G and k a comm utative ring with iden tit y . 2.1. The sh uffle pro duct. Let B ∗ ( G ; k ) b e the normalized bar com- plex: (2.1) B 0 ( G ; k ) ∂ ← − B 1 ( G ; k ) ... ∂ ← − B s − 1 ( G ; k ) ∂ ← − B s ( G ; k ) ∂ ← − ... where B s ( G ; k ) is the fr ee k - mo dule generated by the set of sym b ols [ x 1 | ... | x s ] with x 1 , ..., x s ∈ G \ { 1 } and ∂ is the k -homomorphism g iven b y the f o rm ula: ∂ [ x 1 | ... | x s ] = [ x 2 | ... | x s ] + s − 1 X j =1 ( − 1) j [ x 1 | ... | x j x j +1 | ... | x s ] + ( − 1) s [ x 1 | ... | x s − 1 ] with [ x 1 | ... | x j x j +1 | ... | x s ] = 0 by conv en tion if x j x j +1 = 1. By definition, the group homology H ∗ ( G ; k ) with k -co efficien ts is the homology of the c hain complex (2.1). On the other hand, the c hain complex (2.1) can b e regarded as a graded algebra B ( G ; k ) o v er k whic h is an ti-commutativ e, asso ciat ive, and unital with respect to the sh uffle pro duct (2.2) [ x 1 | ... | x i ] ∧ [ x i +1 | ... | x i + s ] = X ( − 1) σ [ x σ (1) | ... | x σ ( i + s ) ] 4 MARIAN F. A NTON where the sum is o v er all the p erm utatio ns σ of i + s letters that sh uffle { 1 , ..., i } with { i + 1 , ..., i + s } i.e. σ − 1 (1) < ... < σ − 1 ( i ) and σ − 1 ( i + 1 ) < ... < σ − 1 ( i + s ) and ( − 1) σ is the signature of σ . Nev ertheless, B ( G ; k ) is not neces sarily a differen tial algebra sinc e the Leibniz form ula (2.3) ∂ ([ x 1 | ... | x i ] ∧ [ x i +1 | ... | x i + s ]) = ( ∂ [ x 1 | ... | x i ]) ∧ [ x i +1 | ... | x i + s ] +( − 1) i [ x 1 | ... | x i ] ∧ ( ∂ [ x i +1 | ... | x i + s ]) holds if and only if x j x k = x k x j for all j ≤ i < k . As an immediate consequenc e of (2.3) w e ha v e the follo wing Lemma 2.1. If x 1 , ..., x i ar e elements of G c ommuting with one an- other, then the eleme nt of B i ( G ; k ) given by formula h x 1 , x 2 , ..., x i i = [ x 1 ] ∧ [ x 2 ] ∧ ... ∧ [ x i ] is a cycle r epr esenting a homolo gic al class in H i ( G ; k ) which is i -line ar and skew-symmetric in x 1 , ..., x i . 2.2. The B o c kstein homomorphism. If ℓ is a prime n um b er and ζ ∈ G suc h that ζ ℓ = 1, then for eac h nonnega t ive in teger s w e define an elemen t of B 2 s ( G ; k ) giv en b y the for m ula [ ζ ] ( s ) = ℓ − 1 X i 1 ,...,i s =1 [ ζ i 1 | ζ | ζ i 2 | ζ | ... | ζ i s | ζ ] where [ ζ ] (0) = [ ] is the generator of B 0 ( G ; k ). By an inductiv e argument w e can verify that (2.4) [ ζ ] ( s ) ∧ [ ζ ] ( i ) =  s + i i  [ ζ ] ( s + i ) for all nonnegativ e in tegers s, i . Again by an inductiv e ar gumen t using (2.3) and (2.4) w e can ve rify the fo rm ula (2.5) ∂ ([ ζ ] ( s ) ) = ℓ [ ζ ] ( s − 1) ∧ [ ζ ] for all p ositiv e integers s . In this con text, recall [1010; 10, p. 303] that the s hort e xact seque nce o f c hain complexes (2.6) 0 → B ∗ ( G ; Z /ℓ ) × ℓ − → B ∗ ( G ; Z /ℓ 2 ) → B ∗ ( G ; Z /ℓ ) → 0 asso ciated with the m ultiplication by ℓ map induces a ho mo lo gy long exact sequence ... → H i ( G ; Z /ℓ ) → H i ( G ; Z /ℓ 2 ) → H i ( G ; Z /ℓ ) β − → H i − 1 ( G ; Z /ℓ ) → ... HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 5 where β is the Bo c kstein homomorphism. In part icular, if F ℓ denotes the field of o r der ℓ then b y a diagram c hasing using (2.6) a nd (2.5) we obtain the follow ing Lemma 2.2. I f ζ ∈ G such that ζ ℓ = 1 and s is a p ositive inte ger, then [ ζ ] ( s ) is a cycle r epr esenting a hom olo gy class ω ∈ H 2 s ( G ; F ℓ ) such that [ ζ ] ( s − 1) ∧ [ ζ ] is a cycle r epr esenting the class β ( ω ) ∈ H 2 s − 1 ( G ; F ℓ ) . 2.3. The P ontry agin ring. If G is an abelian group then, according to (2.3), B ( G ; k ) is a differen tial graded algebra with resp ect to the sh uffle pro duct (2.2) inducing a graded algebra structure on homology H ∗ ( G ; k ). I f ℓ G denotes the ℓ -to rsion subgroup of G and Γ( ℓ G ) the algebra of divided p ow ers [44; 4, p. 1 19] o v er F ℓ generated in degree t wo by ℓ G , then the homomorphism of graded algebras (2.7) Γ( ℓ G ) → H ∗ ( G ; F ℓ ) sending eac h elemen t ζ o f ℓ G to the class o f [ ζ ] (1) in H 2 ( G ; F ℓ ) is well defined according to (2.4). Similarly , if Λ( G ⊗ Z /ℓ ) denotes the ex- terior algebra o v er F ℓ generated in degree one b y G ⊗ Z /ℓ then the homomorphism of gra ded algebras (2.8) Λ( G ⊗ Z /ℓ ) → H ∗ ( G ; F ℓ ) sending eac h elemen t g ⊗ 1 of G ⊗ Z /ℓ to the class of [ g ] in H 1 ( G ; F ℓ ) is also w ell defined according to Lemma 2.1. Prop osition 2.3 ([4 4 , 4] p. 126) . If ℓ is a prime numb er and G is an ab elian gr oup, then the maps (2.7) and (2 .8) induc e an isomorphi s m of gr ade d algebr as Γ( ℓ G ) ⊗ Λ( G ⊗ Z /ℓ ) ≈ H ∗ ( G ; F ℓ ) . If G 1 , G 2 are tw o groups t hen the K ¨ unneth isomorphism [55; 5, p. 218] (2.9) κ : H ∗ ( G 1 ; F ℓ ) ⊗ H ∗ ( G 2 ; F ℓ ) ≈ − → H ∗ ( G 1 × G 2 ; F ℓ ) is indu ced b y the map sending [ x 1 | ... | x i ] ⊗ [ x i +1 | ... | x i + s ] 7→ [ x 1 × 1 | ... | x i × 1] ∧ [1 × x i +1 | ... | 1 × x i + s ] where x j is an elemen t of G 1 for j ≤ i and an elemen t of G 2 for j > i . In part icular, if b ot h G 1 and G 2 are ab elian, then κ is a graded algebra isomorphism with resp ect to the pro duct ( a 1 ⊗ b 1 )( a 2 ⊗ b 2 ) = ( − 1) | b 1 || a 2 | ( a 1 ∧ a 2 ) ⊗ ( b 1 ∧ b 2 ) defined f or homog eneous elemen ts a i , b i ∈ H ∗ ( G i ; F ℓ ) of degrees | a i | and | b i | f or i = 1 , 2. 6 MARIAN F. A NTON R emark 2.4 . If G is an ab elian group and µ : G × G → G is its group la w homomorphism, then the comp osition b etw een the induced homomorphism µ ∗ : H ∗ ( G × G ; F ℓ ) → H ∗ ( G ; F ℓ ) and the K ¨ unneth isomorphism (2.9) for G 1 = G 2 = G defines a pro duct on H ∗ ( G ; F ℓ ) that can b e easily c hec ked to b e induced by the sh uffle pro duct. In this case, H ∗ ( G ; F ℓ ) is called the Pontryagin rin g and its structure is give n b y Prop o sition 2.3. 3. Algebras of homological symbols Let k b e a fixed comm utativ e r ing with iden tit y . 3.1. Algebras of sym b o ls. If A = L ∞ i,n =0 A in is an a sso ciativ e bi- graded k -algebra, denote b y (3.1) A ∗ n := ∞ M i =0 A in ⊂ A the k - submo dule of all elemen ts with t he s econd degree n . Also, let (3.2) q : T ( A ∗ 1 ) = ∞ M n =0 A ⊗ n ∗ 1 → A b e the canonical bi-graded algebra homomorphism where T ( A ∗ 1 ) is the bi-graded tensor k -algebra generated b y the k -submo dule A ∗ 1 ⊂ A . Here ⊗ n denotes the n -fold graded tens o r pro duct o v er k . Definition 3.1. The algebr a of symb o ls a sso ciated with an asso ciative bi-graded k -a lgebra A = L ∞ i,n =0 A in is the quotien t bi-gr a ded algebra K A := T ( A ∗ 1 ) / k er q with res p ect to the k ernel of the canonical homomorphism (3.2). Definition 3.2. An a sso ciativ e bi-graded k -alg ebra A = L ∞ i,n =0 A in is quadr atic with resp ect to the second de gree if the canonical ho mo mo r - phism (3.2) is surjectiv e and its ke rnel can b e generated as a tw o-sided ideal b y a subset of A ⊗ 2 ∗ 1 . According with the ab ov e definitions the algebra of sym b ols K A as- so ciated with an asso ciat ive bi-graded k -a lg ebra A comes with a natur al bi-graded algebra monomorphism (3.3) q ′ : K A ֒ → A Some q uestions of in terest will b e to study w hen q ′ is an isomorphism and when K A is a quadratic algebra with resp ect t o the second degree. HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 7 3.2. Graded H -spaces. W e say that a to p ological space X = F ∞ n =0 X n decomp osed in to a disjoin t union of non- empt y o p en subspace s X n ⊂ X is a gr ade d H - s p ac e if there is a con tinuous map h : X × X → X with h ( X n × X m ) ⊂ X n + m for all n, m ≥ 0 suc h that X is an a sso ciativ e H -space r elat ive h in the sense of [1010; 10, p. 281] with the ho mo t o p y unit in X 0 . A contin uous map b et we en graded H -spaces f : X = ∞ G n =0 X n → Y = ∞ G n =0 Y n is a gr ade d H - map if f ( X n ) ⊂ Y n for all n ≥ 0 and f is an H - map. Definition 3.3. The k -al g e br a of homolo gic al symb ols asso ciated with a graded H - space X = F ∞ n =0 X n is the algebra of sym b o ls K H ∗ ( X ; k ) asso ciated in the sense of the Definition 3.1 with t he bi-graded k -algebra H ∗ ( X ; k ) ≈ ∞ M i,n =0 H i ( X n ; k ) where H ∗ ( ; k ) is the singular homology functor with k - co efficien ts. In the ab ov e definition, the bi-gr a ded algebra structure on H ∗ ( X ; k ) is induced from the graded H -structure on X v ia the K ¨ unneth homo- morphisms H ∗ ( X n ; k ) ⊗ H ∗ ( X m ; k ) → H ∗ ( X n × X m ; k ) and t he assignmen t X 7→ K H ∗ ( X ; k ) is obv iously natural w ith resp ect to g raded H -maps. Also we hav e a natural monomorphism (3.4) q ′ : K H ∗ ( X ; k ) ֒ → H ∗ ( X ; k ) . giv en b y (3.3) applied to A = H ∗ ( X ; k ). Notation 3.4 . F or the r est of this a rticle, if not otherwise stated, we fix ℓ := 2 r + 1 a r e gular o dd prime n umber, ξ is a primitiv e ℓ - ro ot o f unity , and R := Z [ 1 ℓ , ξ ] the ring of cyclotomic ℓ - in tegers. Also GL n , S L n will denote the groups of matrices o ver R as defined in the Introduction. 4. The main example s In this article w e are concerne d with examples of a lg ebras of homo- logical s ym b ols arising from linear groups. 8 MARIAN F. A NTON 4.1. Appro ximations to B GL n . The mo d ℓ homology o f the group GL n is naturally isomorphic to the singular mo d ℓ homology of its clas- sifying space B GL n . The classifying space B GL n can b e appro ximated b y the classifying space B GL × n 1 of the n - fold direct pro duct GL × n 1 and b y a top olog ical space B GL et n called the ´ etale mo del at ℓ , defined in [88; 8, p. 3]. These sp aces are connected b y natural con tin uous maps (4.1) B GL × n 1 ι n − → B GL n f n − → B GL ´ et n where ι n is the classifying space map induced by the canonical inclusion GL × n 1 ⊂ GL n and f n is a map defined in [88; 8, p. 3]. By taking the disjoin t union of the diagrams (4 .1) w e obtain a diagram of top ological spaces and con tinuous maps (4.2) X := ∞ G n =0 B GL × n 1 ι − → Y := ∞ G n =0 B GL n f − → Z := ∞ G n =0 B GL ´ et n suc h that eac h disjoin t union has a graded H -space structure induced b y the matrix blo c k-multiplication and the maps ι = ⊔ ι n and f = ⊔ f n are gr a ded H -maps. On mo d ℓ homology , the diagra m (4.2 ) induces a comm utative diagra m of bi-graded algebras and homomorphisms (4.3) K H ∗ ( X ; F ℓ ) K ι ∗ − − − → K H ∗ ( Y ; F ℓ ) K f ∗ − − − → K H ∗ ( Z ; F ℓ )   y q 1   y q 2   y q 3 H ∗ ( X ; F ℓ ) ι ∗ − − − → H ∗ ( Y ; F ℓ ) f ∗ − − − → H ∗ ( Z ; F ℓ ) where the a lg ebras of homolog ical sy m b o ls in t he first row are give n b y the Definition 3.3 and the monomorphisms q i are a re g iven by (3.4) for i = 1 , 2 , 3. The second row of the diagram (4.3) can written as a diagram of bi-graded algebras (4.4) T := ∞ M i,n =0 T i,n ι ∗ − → A := ∞ M i,n =0 A i,n f ∗ − → A ´ et := ∞ M i,n =0 A ´ et i,n where for eac h bi-degree ( i, n ), w e define T i,n := H i ( GL × n 1 ; F ℓ ) , A i,n := H i ( GL n ; F ℓ ) , A ´ et i,n := H i ( B GL ´ et n ; F ℓ ) . The first degree i is called the homolo gic al de gr e e and the second degree n is called the r ank . Theorem 4.1 ([88; 8, Lemma 6.2]) . The c omp ose d homom orphism f ∗ ◦ ι ∗ in the diagr am (4.4) is surje ctive. HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 9 The rank n elemen ts of T form the linear subspace T ∗ n ⊂ T (see (3.1)) suc h that: T ∗ n = H ∗ ( GL × n 1 ; F ℓ ) ≈ H ∗ ( GL 1 ; F ℓ ) ⊗ n b y the K ¨ unneth isomorphism. In particular, T = H ∗ ( X ; F ℓ ) ≈ T ( H ∗ ( GL 1 ; F ℓ )) is the tensor algebra generated b y T ∗ 1 = H ∗ ( GL 1 ; F ℓ ) and th us, q 1 in (4.3) is an isomorphism . F ro m Theorem 4.1 w e deduc e the following Corollary 4.2. The monomorphism q 3 in the diagr am (4 .3) is an iso- morphism. 4.2. ´ Etale obstruction classes. T o describe the k ernel of f ∗ ◦ ι ∗ w e observ e that according to [1818, 18] the group of units GL 1 of the ring R is the ab elian group generated by the set of cyclotomic units (4.5) {− ξ , 1 − ξ , 1 − ξ 2 , ..., 1 − ξ r } sub ject to the relation ( − ξ ) 2 ℓ = 1 . By applying Prop o sition 2.3 to GL 1 , we deduce that T ∗ 1 is a v ector space ov er F ℓ with basis the set of homology classes represen ted by cycles of the form (4.6) [ ξ ] ( s ) ∧ h v 1 , ..., v i i where s runs ov er all nonnegative in tegers a nd { v 1 , ..., v i } ov er all sub - sets of the set (4.5). In this con text, the following definition is a sligh t mo dification of [33; 3, p. 2336]: Definition 4.3. A class ǫ ∈ T ∗ 1 represen ted by a cycle of the form (4.6) is called an ´ etale obs truction class if s is a no nnegat iv e in teger and { v 1 , ..., v i } is a subset of the set (4.5) of car dina lity i suc h that i = s + 2 j for some in teger j > 0. Definition 4.4. A c lass ω ∈ T 1 ∗ represen ted b y a cycle of the f orm (4.6) is called a homo gene ous class of weigh t k ω k := s + i . R emark 4.5 . F or eac h in teger i ≥ 2 let e ( i ) denote the car dinality of the set of all integers s ≡ i mod 2 suc h that 0 ≤ s ≤ i − 2. Then the n umber e of ´ etale obstruction classes is finite and giv en b y the formula e = r +1 X i =2 e ( i )  r + 1 i  . The follo wing group homomorphisms: GL 1 t − → GL × 2 1 ρ ← − GL × 3 1 10 MARIAN F. A NTON giv en b y the form ulas (4.7) t ( u ) = u − 1 × u, ρ ( u × v × w ) = uw × v w , for u, v , w ∈ GL 1 induce homomorphisms on mo d ℓ homology: t ∗ : T ∗ 1 → T ∗ 2 , ρ ∗ : T ∗ 2 ⊗ T ∗ 1 ≈ T ∗ 3 → T ∗ 2 , where the source T ∗ 3 of ρ ∗ has b een iden t ified with T ∗ 2 ⊗ T ∗ 1 via the K ¨ unneth isomorphism. With these preparatio ns, w e hav e the follow ing imp ortant result: Theorem 4.6 ([33, 3]) . The kernel of the bi-gr ade d algebr a homomor- phism: f ∗ ◦ ι ∗ : T → A ´ et is the two-sid e d ide al of T gene r ate d by t he set of ele m ents of the form: (4.8) ρ ∗ ( t ∗ ( η ) ⊗ z ) , wher e η , z ∈ T ∗ 1 such that η runs over al l the ´ etale obstruction classes and the homo gene ous classes of o dd weight k η k , and z runs o v e r a ve ctor sp ac e b as i s fo r T ∗ 1 . The pro of o f this theorem is a direct t ranslation using Lemmas 2.1 and 2.2 of the calculations ma de in [3 3 ; 3, p. 2338]. Also, via the K ¨ unneth isomorphisms, T can b e r egarded a s the tensor a lg ebra on T ∗ 1 and T ∗ 1 can b e iden tified via f ∗ ◦ ι ∗ with A ´ et ∗ 1 (see [88; 8, Prop osition 5.2 ]). Th us, com bining the Theorems 4.1 and 4 .6 w e obtain the structure of the bi-gr a ded algebra A ´ et as a quadratic algebra w ith resp ect to the rank: Corollary 4.7. The bi-gr ade d algebr a A ´ et in (4 .4) is a quadr atic alge- br a with r esp e ct to the r ank in the sense of the Definition 3.2. R emark 4.8 . The homomorphism ρ ∗ defines a gr aded mo dule structure on T ∗ 2 o ver the P ontry agin ring T ∗ 1 (see Remark 2.4). The Theorem 4.6 say s that the kernel of f ∗ ◦ ι ∗ is generated as a t wo-sided ideal b y a submo dule of T ∗ 2 of finite rank e ov er T ∗ 1 mo dulo the classes (4.8) with k η k o dd, where e is giv en b y Remark 4.5. 5. The main conjecture 5.1. The statemen t. The ma ps in the diagra m (4.3) ha ve the follo w- ing known prop erties: (1) K ι ∗ and K f ∗ are surjectiv e. This is immediate fro m t he fact tha t K ι ∗ and K f ∗ are bijectiv e in rank 1 and their targets are g enerated as algebras b y rank 1 eleme nts. HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 11 (2) f ∗ is surjectiv e but not an isomorphism. The first part follows from t he T heorem 4.1 while the last part w as pro v en in [22, 2]. (3) ι ∗ is s urjectiv e if the Quillen conjecture [16 1 6; 16, p. 591] holds true for the ring R and all the ranks n . This fact w as pro v en in [1212; 12, p. 51 ]. (4) q 1 and q 3 are isomorphisms. The se facts follow fro m the Corollary 4.2 and its prec eding pro of. (5) q 2 is an isomorphism if ι ∗ is surjectiv e. This follo ws from (4) by c hasing the dia gram (4.3). (6) f ∗ is an isomorphism if K f ∗ is bijectiv e and ι ∗ is surjectiv e. This follo ws fro m (4) and (5) b y ch asing the diagram (4.3). In this article w e conjecture that: Conjecture 5.1. The map K f ∗ : K A → K A ´ et ≈ A ´ et in the diagr am (4.3) is an isom orphism. By (2), (3 ) and (6 ), our Conjecture 5.1 implies tha t the Quillen con- jecture [161 6 ; 1 6, p. 591] for the ring R defined in Notation 3.4 cannot b e true in a ll the ra nks n . In this sense, our conjecture can b e regarded as a ”correction” of t he Quillen conjecture. Also o ur conjecture implies that ι ∗ is not surjectiv e and q 2 is not an isomorphism. R emark 5.2 . The Conjecture 5.1 and the Theorems 4.1 and 4.6 (se e also Remark 4.8) compute the direct summand K A of the m ysterious algebra (1.1). This summand is an alg ebra o f homological sym b ols whic h is q uadratic with resp ect to the rank b y the C orollary 4.7 . 5.2. A useful reduction. Recalling t , ρ defined in (4.7), we hav e a comm utative diagra m GL 1 × GL 1 t × I d − − − → GL × 2 1 × GL 1 τ × I d   y   y e ρ S L 2 × GL 1 µ − − − → GL 2 where I d is the iden tity map, e ρ is ρ comp osed with the canonical in- clusion GL × 2 1 ⊂ GL 2 , (5.1) τ ( u ) =  u − 1 0 0 u  and µ ( A × u ) = A  u 0 0 u  (matrix product) for all u ∈ GL 1 and A ∈ S L 2 . By pa ssing to mo d ℓ homology w e hav e the follo wing Prop osition 5.3. The Conje ctur e 5.1 i s true if and only if τ ∗ ( ǫ ) = 0 in H ∗ ( S L 2 ; F ℓ ) for al l ´ etale obs truction classes ǫ ∈ H ∗ ( GL 1 ; F ℓ ) . 12 MARIAN F. A NTON Pr o of. The cyc le [ ξ − 1 ] (1) is homologous to − [ ξ ] (1) as w e deduc e from ∂ [ ξ i | ξ − 1 | ξ ] = [ ξ − 1 | ξ ] − [ ξ i − 1 | ξ ] − [ ξ i | ξ − 1 ] b y taking the sum o v er i = 1 , ..., ℓ . If σ ∗ : T ∗ 1 → T ∗ 1 is the homo- morphism induced b y σ : GL 1 → GL 1 , u 7→ u − 1 , and η is represen ted b y (4.6) then w e can pro v e inductiv ely that σ ∗ ( η ) = ( − 1) k η k η where k η k is giv en b y Definition 4.4. Because σ extends to an inner auto- morphism of S L 2 via τ , we conclude that τ ∗ ◦ σ ∗ is the identit y map on H ∗ ( S L 2 ; F ℓ ). Hence, the classes τ ∗ ( η ) with η ∈ T ∗ 1 and k η k o dd v anish in H ∗ ( S L 2 ; F ℓ ). The nec essit y follows now from the equation e ρ ∗ ( t ∗ ( η ) ⊗ z ) = µ ∗ ( τ ∗ ( η ) ⊗ z ) b y c hasing the diagram (4.3) and using the Theorem 4.6. The suffi- ciency follo ws b y a spectral sequence argumen t as in [33; 3, Lemma 4.8].  6. A gro up theoretical appro ach The aim of this sec tion is to pro vide a gro up theoretical metho d pro ducing ev idence for the Conjecture 5.1. This metho d is ba sed on a finitely presen ted group defined next. 6.1. A finitely presen ted group. L et S E 2 b e the gr o up generated b y t he sym b ols D ( u ) and E ( x ) sub ject to the following relations [66, 6]: T yp e I. E ( x ) E (0) E ( y ) = D ( − 1) E ( x + y ) (6.1) T yp e I I. E ( x ) = D ( u ) E ( xu 2 ) D ( u ) T yp e I I I. E ( u − 1 ) E ( u ) E ( u − 1 ) = D ( − u ) T yp e IV. D ( u ) D ( v ) = D ( uv ) where u , v ∈ GL 1 and x, y ∈ R run ov er all elemen ts. W e in tro duce the follo wing la b els: (6.2) z := D ( ξ ) , u i := D ( ǫ i ) , a := E (0) , b := E (1) where ǫ i := 1 − ξ i for i = 1 , 2 , ..., r are give n b y (4.5), and w e define: (6.3) b t := z r t bz r t a, w := z c u 1 u 2 ...u r where t = 0 , 1 , 2 , ..., 2 r and c ≥ 0 is the smallest in teger suc h that 2 c ≡ r 2 + r ( r + 1) 2 mo d ℓ. W e will o ccasionally use b t with t an arbitr ary in teger where b t = b s if t ≡ s mod ℓ and the follo wing notation [ x, y ] = xy x − 1 y − 1 . HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 13 Definition 6.1. F or eac h non-empt y subset I ⊂ { 1 , 2 , ..., r } define c ( I ) := ( 2 r Y t =0 b c t ( I ) t ) a − 1 Y i ∈ I u i where c t ( I ) ∈ Z s uc h that in R w e ha v e the follo wing iden tit y: ǫ I := Y i ∈ I ǫ i = Y i ∈ I (1 − ξ i ) = 2 r X t =0 c t ( I ) ξ t . F or instance, if I = { i } is a singleton, then c ( I ) = b 0 b − 1 i a − 1 u i and if I = { i, j } has t w o elemen ts then c ( I ) = b 0 b − 1 i b − 1 j b i + j a − 1 u i u j . Theorem 6.2. The gr oup S E 2 define d ab ove is gener ate d by z , u 1 , u 2 , ..., u r , a, b subje ct to the fol lowin g r elations: z ℓ = [ z , u i ] = [ u i , u j ] = 1 (6.4) a 4 = [ a 2 , z ] = [ a 2 , u i ] = 1 (6.5) a = z az = u i au i (6.6) [ b s , b t ] = 1 (6.7) b 3 = a 2 = b 0 b 1 ...b 2 r (6.8) b ℓ t = w − 1 b ( − 1) r t w (6.9) c ( I ) 3 = 1 (6.10) ba 2 = u i bz − r i b − 1 b − 1 0 z r i bz − i u i (6.11) wher e i, j ∈ { 1 , 2 , ..., r } , s, t ∈ { 0 , 1 , 2 , ..., 2 r } , and I ⊂ { 1 , 2 , ..., r } runs over al l no nempty subsets. The theorem implies that S E 2 has a finite presen tation with r + 3 generators and 6 + 6 . 5 r + 2 . 5 r 2 + 2 r relators. Its pro of will b e giv en as a sequence of lemmas. F or con v enience, w e will refer to the relatio ns (6.1) only b y t yp e. Also we will tacitly use (6.2), ( 6 .3), the relations in GL 1 giv en at the b eginning o f § 4.2, and when appropria tely , T yp e IV. Lemma 6.3. z , u 1 , u 2 , ..., u r , a, b gener ate S E 2 . Pr o of. By T yp e I I with u = − 1, it follow s that D ( − 1) is cen tral and b y T yp e I with x = y = 0, w e ha v e (6.12) a 2 = D ( − 1) . Since each v ∈ GL 1 can be written as v = ( − ξ ) j ǫ a 1 1 ...ǫ a r r for some in tegers j , a 1 ,..., a r , w e ha v e (6.13) D ( v ) = a 2 j z j u a 1 1 ...u a r r . 14 MARIAN F. A NTON By T ype I I with u = ξ r t and x = ξ − 2 rt = ξ t , (6.14) b t = E ( ξ t ) E (0 ) . By T ype I with y = − x and (6.12), E ( x ) E (0) E ( − x ) E (0) = 1 and hence, (6.15) b − 1 t = E ( − ξ t ) E (0 ) . If x ′ = P 2 r t =0 m t ξ t in R with m t in tegers, then, b y T yp e I, E ( x ′ ) = [ 2 r Y t =0 ( E ( ξ t ) E (0 ) ) m + t ( E ( − ξ t ) E (0 ) ) m − t ] E (0) − 1 D ( − 1) m − 1 where m = P 2 r t =0 m t , and m t = m + t − m − t with m + t , m − t nonnegativ e in tegers. Combinin g (6.12), (6 .1 4), (6.15) with the equation a b ov e, w e deduce that (6.16) E ( x ′ ) = ( 2 r Y t =0 b m t t ) a 2 m − 3 . W e remark that a p erm uta tion of the ξ t -terms in x ′ corresp onds to a p erm uta t io n of the b t -factors in E ( x ′ ). An y ring elemen t x ∈ R can b e written in the form x = x ′ v − 2 for some x ′ ∈ Z [ ξ ] and v ∈ GL 1 . By T yp e I I, w e ha v e (6.17) E ( x ) = D ( v ) E ( x ′ ) D ( v ) with D ( v ) giv en b y (6.13) and E ( x ′ ) b y (6.16), concluding the pro of.  Lemma 6.4. The r e lations (6.4) - (6.8) ar e necessary . Pr o of. W e ha v e the follo wing list of short argumen ts: (6.4) follo ws from Ty p e IV. (6.5) follo ws from (6.1 2). (6.6) follo ws from Ty p e I I with x = 0 . (6.7) follo ws from (6.1 6) w ith x ′ = ξ t + ξ s = ξ s + ξ t in R . (6.8) the first p art fo llows from T yp e I I I with u = 1 and (6.12). (6.8) the se c o n d p art f o llo ws b y (6.16 ) with x ′ = P 2 r j =0 ξ j = 0 in R .  Lemma 6.5 ([1818, 18]) . In R we have ℓ = ( − 1) r λ 2 wher e λ := ξ c ǫ 1 ǫ 2 ...ǫ r . Lemma 6.6. (6.9) is neces sary . HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 15 Pr o of. By Lemma 6.5 w e can apply (6 .17) to x = ℓξ t , x ′ = ( − 1) r ξ t , v = λ − 1 and get E ( ℓξ t ) = D ( λ ) − 1 E (( − 1) r ξ t ) D ( λ ) − 1 . By (6.13) a nd (6.16), the eq uation ab ov e can b e rewritten as b ℓ t a 4 r − 1 = w − 1 b ( − 1) r t a − 1 w − 1 . No w w e can use (6.5) and (6.6) pro ven in Lemma 6.4.  Lemma 6.7. (6.10) is necess ary . Pr o of. F or I ⊂ { 1 , 2 , ..., r } recall that ǫ I := Q i ∈ I ǫ i . Then, the Defini- tion 6 .1 g iv es b y (6.1 6) w ith x ′ = ǫ I the follo wing form ula c ( I ) = D ( − 1) E ( ǫ I ) D ( ǫ I ) . By (6.17) with x = ǫ − 1 I and x ′ = v = ǫ I w e ha ve E ( ǫ − 1 I ) = D ( ǫ I ) E ( ǫ I ) D ( ǫ I ) . By T ype I I I with u = ǫ I , w e ha v e D ( − 1) E ( ǫ I ) E ( ǫ − 1 I ) E ( ǫ I ) D ( ǫ I ) = 1 . The conclusion follow s b y com bining the t hree equations ab ov e.  Lemma 6.8. (6.11) is necess ary . Pr o of. W e start with ǫ 2 i = ξ i ( ξ − i − 2 + ξ i ) in R and b y T yp e I I with u = ǫ − 1 i , x = ǫ 2 i and (6 .1 7) w ith x ′ = ξ − i − 2 + ξ i , v = ξ r i , x = ξ i x ′ w e get u − 1 i bu − 1 i = z r i b − i b − 1 0 b − 1 0 b i a − 3 z r i The desired relation no w follo ws b y (6.3) and (6 .5).  Lemma 6.9. The r elations ( 6.4) - (6.11 ) ar e sufficien t to verify that 1) the r ela tion (6.13) is wel l define d for v ∈ GL 1 , 2) T yp e IV holds true, and 3) a 2 = D ( − 1) is c entr al. The pro of is immediate by (6.4), (6 .5), and ( 6.8) the first p art . In what follows we will use this lemma tacitly . Lemma 6.10. The r elations (6.4) - (6.11) ar e sufficien t to verify that (6.16) is wel l define d for x ′ ∈ Z [ ξ ] . 16 MARIAN F. A NTON Pr o of. Let x ′ = P 2 r t =0 m t ξ t = P 2 r t =0 n t ξ t in R with m t , n t ∈ Z . Then m t − n t = j is independen t of t . F rom (6.7) and ( 6 .8) the se c ond p art w e deduce that the right hand side of (6.16) remains unc hang ed under the transforma t io n m t = n t + j or a p erm utatio n of the b t -factors.  Lemma 6.11. The r elations (6.4) - (6 .11) ar e sufficien t for T yp e I with x, y ∈ Z [ ξ ] . Pr o of. Let x = P 2 r t =0 m t ξ t and y = P 2 r t =0 n t ξ t with m t , n t in tegers. By Lemma 6.1 0 we can choose x + y = P 2 r t =0 ( m t + n t ) ξ t and T yp e I fo llows from ( 6.16) and (6.7).  Lemma 6.12. The r elations (6.4) - (6.11) ar e sufficien t to verify that (6.17) is wel l define d for x = x ′ v − 2 with x ′ ∈ Z [ ξ ] and v ∈ GL 1 . Pr o of. It suffi ces to prov e that the follo wing statemen t P ( x ′ , v ) : If y ′ := x ′ v − 2 ∈ Z [ ξ ] then D ( v ) E ( x ′ ) D ( v ) = E ( y ′ ) is a c onse quenc e of the r elations (6.4) - (6.11) . is true for a ll x ′ ∈ Z [ ξ ] and v ∈ GL 1 where E ( x ′ ), E ( y ′ ), and D ( v ) ar e giv en b y ( 6 .16) and (6.13). By Lemmas 6.10 and 6 .9, t hese fo r mulas are indep endent of the w a y x ′ , y ′ , and v a re presen ted. Also, w e recall that b t = b s if t ≡ s mo d ℓ . P ( ± ξ t , − ξ ) is true . If x ′ = ξ t and y ′ = ξ t − 2 , w e c hec k that z b t a − 1 z = b t − 2 a − 1 holds true b y definitions. The case x ′ = − ξ t is s imilar. P ( ± ξ t , ǫ − 1 i ) is true . If x ′ = ξ t and y ′ = ξ t − 2 ξ t + i + ξ t +2 i , w e use (6.6) to r educe the equation u − 1 i b t a − 1 u − 1 i = b s b − 2 t + i b t +2 i a − 3 to ( 6.11) as in the pro of of Lemma 6.8. The case x ′ = − ξ t is s imilar. P ( ± ℓξ t , λ ) is true . Here λ is defined in Lemma 6.5 suc h that y ′ = x ′ ℓλ − 2 = ± ( − 1) r ξ t The statemen t no w follo ws from (6.6) and (6.9). By (6.6) and Lemma 6.11, P ( x ′ 1 , v ) and P ( x ′ 2 , v ) imply P ( x ′ 1 + x ′ 2 , v ). So, P ( x ′ , − ξ ), P ( x ′ , ǫ − 1 i ), and P ( ℓx ′ , λ ) a re true for all x ′ ∈ Z [ ξ ] and i = 1 , 2 , ..., r . If v − 1 1 , v − 1 2 ∈ Z [ ξ ] suc h that P ( x ′ , v 1 ) and P ( x ′ , v 2 ) are true for all x ′ ∈ Z [ ξ ], then P ( x ′ , v 1 v 2 ) is also true for all x ′ ∈ Z [ ξ ]. Since (4.5) is a generating set for GL 1 it follows that P ( x ′ , v ) is true for all x ′ ∈ Z [ ξ ] and all v ∈ GL 1 suc h that v − 1 ∈ Z [ ξ ]. The pro of can no w b e concluded b y the observ ation that ev ery elemen t of G L 1 is of the form v λ s with v − 1 ∈ Z [ ξ ] and s a nonnegativ e in teger.  HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 17 Lemma 6.13. The r elations (6.4) - (6.11) ar e sufficien t fo r T yp e I, T yp e II, and T yp e I II. Pr o of. T yp e I: Give n t wo ring ele men ts x, y ∈ R there exists v ∈ GL 1 suc h t ha t x = x ′ v − 2 and y = y ′ v − 2 with x ′ , y ′ ∈ Z [ ξ ]. By (6.6) and Lemma 6.12 w e get E ( x ) E (0) E ( y ) = D ( v ) E ( x ′ ) E (0 ) E ( y ′ ) D ( v ) . So Type I is reduced to Lemma 6.11. T yp e I I follo ws from Lemma 6.12. T yp e I I I: By Lemmas 6.11 and 6.12 w e can r ev erse the pro of of Lemma 6.7 to conclude that Ty p e II I with u = ǫ I = Q i ∈ I ǫ i follo ws from (6.10) for I ⊂ { 1 , 2 , ..., r } non-empt y and from (6 .8) t he first p art if I is empt y i.e. u = 1. Com bining this with T yp e I I, w e deduce that T yp e I I I holds with u = ǫ I v 2 for any v ∈ GL 1 and any subse t I . Moreo ve r, the Type I implies E ( − u ) = E (0) − 1 E ( u ) − 1 E (0) − 1 and henc e, if T yp e I I I holds for u ∈ GL 1 then it ho lds for − u as we ll. Since ± ǫ I ’s form a s et of coset represen tatives for GL 1 mo dulo the squares, T ype I I I holds in general.  7. Hopf’s form ula calcula t ions There is a group homo mo r phism π : S E 2 → S L 2 giv en b y D ( u ) 7→  u − 1 0 0 u  , E ( x ) 7→  x 1 − 1 0  for all u ∈ GL 1 and x ∈ R . Regarding D : G L 1 → S E 2 as a group homomorphism, w e ha v e the follo wing comm utativ e diagram (7.1) H p ( GL 1 ; F ℓ ) D ∗ − − − → H p ( S E 2 ; F ℓ )   y τ ∗    H p ( S L 2 ; F ℓ ) π ∗ ← − − − H p ( S E 2 ; F ℓ ) where p is a p ositiv e inte ger and τ ∗ is induced by (5.1). Chasing this diagram, b y Propo sition 5.3 and Definition 4.3 w e deduce that Prop osition 7.1. The Conje ctur e 5.1 is true if fo r e ach subset { e 1 , ..., e i } of { z , u 1 , ..., u r } with 2 ≤ i ≤ r + 1 elements and for e ach p air ( s, j ) of nonne gative inte gers with i = s + 2 j and j > 0 , the standar d c ycle (7.2) [ z ] ( s ) ∧ h e 1 , ..., e i i 18 MARIAN F. A NTON r epr esents the zer o class in H p ( S E 2 ; F ℓ ) wher e z , u 1 ,..., u r ar e ele ments of S E 2 define d by (6.2) and p = 3 s + 2 j . According to this prop osition, for eac h prime ℓ = 2 r + 1 , Conjecture 5.1 fo llo ws from a verific ation that a certain finite set of explicitly giv en cycles (7 .2) r epresen t the zero class in H ∗ ( S E 2 ; F ℓ ). In particular, this set of cycles in H 2 ( S E 2 ; F ℓ ) is giv en by h e 1 , e 2 i for e 1 , e 2 in { z , u 1 , ..., u r } . Theorem 6.2 giv es a short exact sequence 1 → K → F → S E 2 → 1 where F is the free group generated b y z , u i , a , b , b t , and w for 1 ≤ i ≤ r and 0 ≤ t ≤ 2 r , and K ⊂ F is the normal subgroup give n b y the relators associated with the relatio ns (6.3) and (6.4) - (6.11). Asso ciated with this fr ee presen tation, Hopf ’s form ula [44; 4, p. 42] iden tifies (7.3) H 2 ( S E 2 ; Z ) ≈ K ∩ [ F , F ] [ F , K ] suc h that the standard cycle h e 1 , e 2 i with integer co efficien ts corre- sp onds to the comm utator [ e 1 , e 2 ] mo d [ F , K ]. Here [ X , Y ] denotes the group generated by the comm utator s [ x, y ] with x ∈ X a nd y ∈ Y . Lemma 7.2. S E 2 is a p erfe ct gr oup. Pr o of. F or x, y ∈ S E 2 , let x ≡ y mean that xy − 1 is a pro duct of comm utators in S E 2 . By (6.4 ) and (6.6) w e deduce that z ℓ ≡ z 2 ≡ 1 and hence, z ≡ 1 since ℓ is o dd. No w (6.3) implies b t ≡ ba for all t . Com bining this with (6 .6) and (6.11), we get ba 3 ≡ u 2 i ≡ 1. Since a 4 ≡ 1 b y (6.5), w e conclude that b ≡ a , and since b 3 ≡ a 2 b y (6.8) w e conclude that b ≡ a ≡ 1. Finally , from (6.10) with I = { i } singleton (see Definition 6.1) w e get u 3 i ≡ 1 and since u 2 i ≡ 1 w e deduc e that u i ≡ 1 for a ll i = 1 , 2 , ..., r . Th us, all generators of S E 2 are ≡ 1.  By Lem ma 7.2 a nd the unive rsal co efficien ts, fr om (7 .3) w e ha ve (7.4) H 2 ( S E 2 ; F ℓ ) ≈ ( K ∩ [ F , F ]) K ℓ [ F , K ] K ℓ where K ℓ is the normal subgroup w hose relators are the ℓ - th p ow ers of the relato rs of K . With these preparations, the follow ing result is evidence for the Conjecture 5.1: Prop osition 7.3. I f ℓ ∈ { 3 , 5 } , then [ e 1 , e 2 ] ∈ [ F , K ] K ℓ for a l l e 1 , e 2 in { z , u 1 , ..., u r } . HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 19 The pro o f of t his prop osition is giv en next ba sed on G AP [99, 9]. W e remark that the case ℓ = 3 is kno wn [11 , 1] but the pro of giv en here and the case ℓ = 5 a re new . The Case ℓ = 3 . The free group F is giv en b y F:=FreeGrou p(8) z:=F.1; u1:=F.2; a:=F.3; b:=F.4; b0:=F.5; b1:=F.6; b2:=F.7; w:=F.8; The relato rs of K are giv en in Theorem 6.2 for ℓ = 3 b y the list k:=[b0^-1*b *a, b1^-1*z*b*z *a, b2^-1*z^2*b *z^2*a, w^-1*z*u1, z^3, z*u1*z^-1* u1^-1, a^4, a^2*z*a^-2 *z^-1, a^2*u1*a^-2 *u1^-1, z*a*z*a^-1, u1*a*u1*a^- 1, b0*b1*b0^-1 *b1^-1, b0*b2*b0^-1 *b2^-1, b1*b2*b1^-1 *b2^-1, b^3*a^-2, b0*b1*b2*a ^-2, b0^-3*w^-1* b0^-1*w, b1^-3*w^-1* b1^-1*w, b2^-3*w^-1* b2^-1*w, (b0*b1^-1*a ^-1*u1)^3, a^2*b^-1*u1 *b*z^2*b^- 1*b0^-1*z*b*z^2*u1]; The relato rs for K 3 are giv en b y the list k3:=List(k, x->x^3); The relato rs for [ F , K ] are given by the follo wing algorithm c:=function (i,j) return Comm(i,j);end;; f:=Generato rsOfGroup( F); fk:=ListX(f ,k,c); The only comm utato r o f the form [ e 1 , e 2 ] in Prop osition 7.3 for ℓ = 3 is the w ord ”k[6]”, i.e. the sixth on the list ”k”. T o c hec k that ”k[6]” b elongs to [ F , K ] K 3 w e use t he follo wing algorithm: H:=F/Concat enation(fk ,k3); RequirePack age("kbmag "); RH:=KBMAGRe writingSys tem(H); OR:=Options RecordOfKB MAGRewritingSystem(RH); 20 MARIAN F. A NTON OR.maxeqns: =500000; OR.tidyint: =1000; OR.confnum: =100; MakeConflue nt(RH); ReducedWord (RH,k[6]); The Case ℓ = 5 . The free group F is giv en b y F:=FreeGrou p(11); z:=F.1; u1:=F.2; u2:=F.3; a:=F.4; b:=F.5; b0:=F.6; b1:=F.7; b2:=F.8; b3:=F.9; b4:=F.10; w:=F.11; The relato rs of K are giv en in Theorem 6.2 for ℓ = 5 b y the list k:=[b0^-1*b *a, b1^-1*z^2*b *z^2*a, b2^-1*z^4*b *z^4*a, b3^-1*z*b*z *a, b4^-1*z^3*b *z^3*a, w^-1*z*u1*u 2, z^5, z*u1*z^-1* u1^-1, z*u2*z^-1*u 2^-1, u1*u2*u1^-1 *u2^-1, a^4, a^2*z*a^-2 *z^-1, a^2*u1*a^-2 *u1^-1, a^2*u2*a^-2 *u2^-1, z*a*z*a^-1, u1*a*u1*a^- 1, u2*a*u2*a^- 1, b0*b1*b0^-1 *b1^-1, b0*b2*b0^-1 *b2^-1, b0*b3*b0^-1 *b3^-1, b0*b4*b0^-1 *b4^-1, b1*b2*b1^-1 *b2^-1, b1*b3*b1^-1 *b3^-1, b1*b4*b1^-1 *b4^-1, b2*b3*b2^-1 *b3^-1, b2*b4*b2^-1 *b4^-1, b3*b4*b3^-1 *b4^-1, b^3*a^-2, b0*b1*b2*b 3*b4*a^-2, b0^-5*w^-1* b0*w, b1^-5*w^-1* b1*w, b2^-5*w^-1* b2*w, HOMOLOGICAL SYMBOLS AND THE QUILLEN CONJECTURE 21 b3^-5*w^-1* b3*w, b4^-5*w^-1* b4*w, (b0*b1^-1*a ^-1*u1)^3, (b0*b2^-1*a ^-1*u2)^3, (b0*b1^-1*b 2^-1*b3*a^ -1*u1*u2)^3, a^2*b^-1*u1 *b*z^3*b^- 1*b0^-1*z^2*b*z^4*u1, a^2*b^-1*u2 *b*z*b^-1* b0^-1*z^4*b*z^3*u2]; The relato rs of K 5 are giv en b y the list k5:=List(k, x->x^5); The relators of [ F , K ] are giv en b y a list ”f k” via the same algorithm as for the case ℓ = 3 but applied to the new ”f” and ”k”. The only com- m utator s of the form [ e 1 , e 2 ] in Prop o sition 7.3 are the words ”k[8]”, ”k[9]”, and ”k[10]” but the algorithm used in the case ℓ = 3 is inconclu- siv e in the case ℓ = 5 due to it s increased complexit y . F or this reason, w e show that thes e w ords b elong to [ F , K ] K 5 b y proving the following Lemma 7.4. [ F, F ] ∩ K ⊂ [ F , K ] K 5 . Pr o of. By trial and error w e find a sublis t ” e ⊂ k” of 11 eleme nts e:=k{[5,6,1 5,16,17,30 ,31,32,33,34,37]}; suc h that the complemen tary sublist ”n ⊂ k” n:=k{[1,2,3 ,4,7,8,9,1 0,11,12,13,14,18,19,20, 21,22,23,24 ,25,26,27, 28,29,35,36,39,38]}; consists of elemen ts v anishing ” mo d e” i.e. represen t zero in the g roup t:=F/Concat enation(fk ,k5,e); according to the follo wing algorithm: RequirePack age("kbmag "); Rt:=KBMAGRe writingSys tem(t);; OR:=Options RecordOfKB MAGRewritingSystem(Rt); OR.maxeqns: =500000; OR.tidyint: =1000; OR.confnum: =100; MakeConflue nt(Rt); nt:=List([1 ..Length(n )],i->ReducedWord(Rt,n[i])); This means that the gro up K / [ F , K ] K 5 is generated by the elemen ts in ”e”. The c omm utator group [ F , F ] is giv en b y the list of relators ff:=ListX(f ,f,\<,c); 22 MARIAN F. ANTON The ”r educed” group F / [ F , F ] K 5 is giv en b y h:=F/Concat enation(ff ,k5); Observ e that ” h” is a v ector space of dime nsion 11 ov er F 5 b y using typeh:=Abel ianInvaria nts(h); Moreo ve r the elemen t s in the list ”e” generate ”h” since s = 1 w here s:=Size(F/C oncatenati on(ff,k5,e)); Putting these facts together and us ing form ula (7.4) w e conclude that there is a s hort exact sequence 0 → H 2 ( S E 2 ; F 5 ) → K [ F , K ] K 5 → F [ F , F ] K 5 → 0 where the last term is a ve cto r space of dimens ion 11 while the middle term is a v ector space of dimension at most 11 b eing generated b y the elemen ts in t he list ”e”. So that H 2 ( S E 2 ; F 5 ) = 0.  By [66; 6, p. 7], the canonical homomorphism π : S E 2 → S L 2 is a group isomorphism if the ring R is Euclidean and b y [1313, 13] the ring R is indeed Euclidean f o r ℓ = 5 . Hence, w e deduce the follo wing Corollary 7.5. H 2 ( S L 2 ; F 5 ) = 0 . Reference s [1] ]cite.MR17 23188 1Maria n F. 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