A Maslov cocycle for unitary groups

A Maslo v co cycle for uni tary groups Lin us Kramer and Katri n T en t ∗ Marc h 26, 2022 Abstract W e introduce a 2-cocycle for s ymplect ic and sk ew-her m i tian hyp erb o lic groups ov er arbitrary fields and skew fields, with v alues in the Witt group of h er m it ian forms. This co c ycle has goo d functorial prop ertie s: it is natural under extension of s c alars and stable, so it ca n b e vie w ed as a universal 2-dimensional charact eristic class for these groups. Ov er R and C , it coincides with the firs t Chern class. MSC Classification: 11E70 19C09 20G 10 20 J05 In tro du c tion W e intro du ce a Maslo v index and Maslov co cy cle for symplectic and hy p erb olic unita ry gro ups o ver arbitr a ry fields and ske wfields. In the classical w or k of Lion-V e rgne [13], this is done b y asso ciating to triples ( X , Y , Z ) of Lagrangia ns in a real s ymplectic ve ctor space M a certain in tegral in v ariant, the Maslo v index. This in v arian t is used to construct a Z -v alued co cycle for the symplectic gro u p. The correspo nding group extension of the symplectic group is the top ological unive rsal co vering g roup of Sp 2 n R . In this appro ac h, it is somewhat cum b ersome that one has to deal with arbitrary t r ip les of Lagrangians. Our starting p oin t w as the idea that the whole construction should also w ork if one considers only triples of Lag rangians in ’general p osition’, that is, triples ( X, Y , Z ) in M whic h are pairwise opp osite: M = X + Y = Y + Z = Z + X . Geometrically , suc h triples are m uch easier to classify . Moreov er, these triples carry an interest- ing a lgebraic structure. T o eac h pair ( X , Y ) of o pp osite Lagrang ia ns one can asso ciate a linear map [ Y ; X ] whic h iden tifies X with the dual of Y and the dual of X with Y . In this w a y w e obtain a graph, the opp osition gra ph , whose vertice s are the Lagrangians and whose edges join opp osite Lagrangians. Concatenating the linear maps [ Y ; X ] alo n g closed paths in this graph, w e arr ive a t an in teresting group oid G M , the pro jectivit y gro upoid. A minimal closed path has length 3, and the resulting elemen t in the holo no m y group turns out to b e a complete geometric in v ariant for the tr iple consisting of the three La grangians a long t he path. This ma kes sense ∗ The authors were suppo rted by SFB 478 and SFB 701 1 and w o r ks not just for symplectic fo rms , but for arbitrary hy p erbolic sk ew-hermitian forms ov er fields or sk ewfields. In order to relate this in v a rian t to g roup cohomology , w e need a chain complex. A natural candidate is t he fla g complex of the opp osition graph, whose simplices ar e the finite complete subgraphs (cliques). If the field is infinite, this flag complex is con tractible, and the symplec- tic (or unitary) group a c ts on it, s o its equiv arian t cohomolo g y is isomorphic to the g roup cohomology . The final ingredient is the observ ation that along a closed path of length 3, the elemen t in the ho lonom y gr oup determines a nondegenerate hermitian f o rm, whic h may b e view ed as an elemen t in a Witt g r oup. In this wa y we a s so ciate to ev ery triangle in the opp osition graph an elemen t in the Witt group of hermitian forms. W e v erify that this map is indeed an in v ar ia n t co cyc le, whic h giv es us a 2- c o cycle for the unitar y group. This co cycle, which w e call the Maslov co cycle , has g o o d functorial properties. It is stable under direct sums of hermitian spaces and we ll-b eha v ed under extension of s calars. F urthermore, it coincides in the symplectic setting ov er fields of characteris tic 6 = 2 with the classical Maslo v co cycle . Our co cycle , ho wev er, ex ists ov er arbitrary fields and sk ewfields of an y characteristic . F urthermore, the co cycle can b e reduced to a subgroup of the Witt group, the k ernel of t h e signed discrimin an t. The classical Maslo v co cycle is imp ortan t, as it yields a central extension of the symplectic group. The question whic h extension is defined by our general Maslov co cy cle can by and large b e r educed to a map in algebraic K -theory . In t h e smallest case Sp 2 D = SL 2 D this is due to Nek ov a r [19] and Barge [1 ]. But ev en in the classical situation of a symplectic group Sp 2 n D o ver a field D 6 = R , our result app ears t o b e the first complete pro of for this. In general, the co cyc le is related to certain sym b ols and dep ends on algebraic prop erties of the field. W e carry this out in some detail for lo cal fields. F or R and C the Maslov co cy cle ’is’ the first Chern class c 1 and giv es t he univ ersal cov ering groups of Sp 2 n R a n d SU ( n, n ). Ov er nonarchimede an lo cal fields, w e obtain a cov ering of degree ≤ 2. A Witt g roup v alued Maslo v co cycle app ears already in [13]. Besides this, our pap er is influenced b y [19 ], [21] (but see the r emarks af te r 9.4). The idea of a ’par tially defined co cycle ’ seems to go bac k to W eil and app ears also in a top ological contex t in [16]. The opp osition graph is used (in a differen t w ay) in [20 ]. The Maslo v index itself has b een generalized in sev eral wa ys [3] [18]. Buildings [27, 10] are not men tioned in this pap er, although the motiv ation for our approac h is the opp osition relation in spherical buildings. Luring b ehind the linear alg e bra is the pro jectivit y group oid f o r spherical buildings, whic h was first studied systematically by Knarr [8] f o r spherical buildings of rank 2. W e assume that the reader is familiar with basic homological algebra, as w ell as hermitian forms and unitary groups. Apart from this, w e tried to mak e the pa p er self-contained and accessible to non-exp erts. Ac kno wledgemen t. P a rt of this w ork w as completed while the authors we re at the Sc ho ol of Mathematics, Birmingham, UK. W e thank Theo Grundh¨ ofer, Karl-Hermann Neeb, Chris P arke r, Andrew Ranic ki, and Winfried Sc harlau. 2 Con te n ts 1 Lagrangians and h yp erbolic mo dules 3 2 The opp osition graph and tr ip les of Lagrangian s 5 3 Flag complexes of graphs 7 4 The pro jectivit y gr oup oid 9 5 The Maslo v c ocycle 11 6 Naturalit y of the Maslo v co cycle 13 7 Reduction of the co cycle 15 8 Kashiw ara’s Maslo v co cycle 18 9 The Maslo v c ocycle as a central extension 22 1 Lagrangians and h yp erb ol i c mo dules In this section we intro duce some standard terminology from the theory of hermitian forms. Ev erything we need can b e found in [6, 9, 23]. W e work ov er a field or division ring D of arbitrary c haracteristic. T he mo dules w e consider are finite dimensional righ t D -modules. W e assume that J is an in volution of D , i.e. an a ntiautomorphism whose square is the iden tity (w e allo w J = id ). The in v olution extends naturally to an inv olution o f the mat r ix ring D n × n whic h w e a ls o denote by J . F or ε = ± 1 w e put D ε = { a ∈ D | a − a J ε = 0 } . 1.1 F orms A form on a right D -mo dule M is a biadditive map f : M × M with the prop ert y that f ( ua, v b ) = a J f ( u, v ) b for all u , v ∈ M and all a, b ∈ D . An ε -hermitian form h is a form with the additional prop ert y that h ( u, v ) = h ( v , u ) J ε, and ( M , h ) is called a hermitian mo dule . If f is an y form, then h f ( u, v ) = f ( u, v ) + f ( v , u ) J ε is ε -hermitian. The hermitian forms whic h arise in this w a y are called tr ac e ε -hermitian or even . If c har ( D ) 6 = 2, ev ery ε -hermitian form is automatically trace ε -hermitian; this is also true in c haracteristic 2 if J is a n in volution of the second kind, i.e. if J | Cen( D ) 6 = id, but may fail otherwise [6, 6.1.2]. Note also that h f ( u, u ) = 0 is equiv alen t to f ( u, u ) ∈ D − ε . 3 1.2 The dual M ∨ of M (whic h is a left D -mo dule ) can b e made into a righ t D -mo d ule M J b y t wisting the scalar m ultiplication with J , i.e. by setting ξ a = [ v 7− → a J ξ ( v )] (where a ∈ D , ξ ∈ M ∨ and v ∈ M ). Th us f o rms a re just linear maps M ✲ M J . A form is called non-de gener ate it the asso ciated linear map is injectiv e (and hence bijectiv e). There is a natural notion of a n isomorphism (or isometry) of forms; the automorphism group of a non-degenerate ε -hermitian form is the unitary gr oup U ( M , h ) = U ( M ) = { g ∈ GL ( V ) | h ( u , v ) = h ( g ( u ) , g ( v )) for all u , v ∈ M } . 1.3 Lagrangians F o r an y subset X ⊆ V w e ha v e the subspace X ⊥ = { u ∈ M | h ( x, u ) = 0 for all x ∈ M } , the p erp . A subspace whic h is con tained in its own p erp is called total ly isotr opic and a subspace whic h coincides with its p erp is called a L agr angian . A nondegenerate hermitian form whic h admits Lagrangians is called metab olic . 1.4 The hyperb olic functor Giv en a right D -mo dule X , there is a natural form f on M = X ⊕ X J , g iv en b y f (( x, ξ ) , ( y , η )) = ξ ( y ). The a s so ciated tr a c e ε -hermitian for m h X (( x, ξ ) , ( y , η )) = ξ ( y ) + η ( x ) J ε (and ev ery isometric hermitian mo dule) is called hyp erb o l i c . Obvious ly , X is a Lagrangian, so h yp erb olic mo dules are metab olic. The conv erse is true for trace v alued hermitian fo rms , hence in particular in c har a c teristic 6 = 2 [9, I 3.7.3]. The r an k of a hyperb olic mo dule is the dimension of X (i.e. half the dimens ion of the hyperb olic mo dule). W e note t ha t t he assignmen t h yp : X 7− → ( X ⊕ X J , h X ) is a functor from D - modules to hermitian mo dules, and that h yp induces an injection GL ( X ) ✲ U ( X ⊕ X J ). 1.5 Sp ecia l cases and Lie groups Ev ery h yp erb olic form ( M , h ) can b e reduced to one o f the follo wing three t yp es. Symplectic groups: ( J, ε ) = (id , − 1). Then D is necessarily comm utativ e and U ( M ) = Sp ( M ) is the symple ctic gr oup . F or M = R 2 n , C 2 n , these Lie groups are often denoted Sp ( n, R ) and Sp ( n, C ). Hyp erbolic orthogonal groups: J = id and ε = 1 6 = − 1. Then D is comm utativ e and of c haracteristic differen t from 2. The group U ( M ) = O ( M ) is the hyperb olic ortho gonal group; for R and C , these Lie groups are often denoted O ( n, n ) and O (2 n, C ). W e will see in 2.9 b e lo w that the Maslov co cycle is unin teresting in this situation. Standard h yp erb olic unitary groups: If J 6 = id then U ( M ) is the standar d hyp erb olic unitary gr oup . Scaling the hermitian form b y a suitable constan t and c hanging the in volution, w e can assume tha t ε = − 1 (”Hilb ert 90”, see [6, p. 211]). The − 1-hermitian forms are also called skew hermitian . Examples of in v olutions are the standard conjuga tion z 7→ ¯ z on C and on the real quaternion division a lgeb ra H . Note tha t there is also t he ’nonstandard’ in v olution z α = − i ¯ z i on H . The sk ew h yp erb olic unitary gro ups corresp onding to ( C n , z 7→ ¯ z ), ( H n , z 7→ ¯ z ) and ( H n , z 7→ z α ) ar e the Lie groups denoted U ( n, n ), SO ∗ (4 n ) a nd Sp ( n, n ) in [7, X T ab. V]. 4 2 The o p p osition graph and triples of Lagrangians In this section w e construct an in v aria n t κ whic h classifi es triples of pairwise opp o s ite La- grangians in a − ε -hermitian hyperb olic mo dule up to isometry . The in v ar ia n t is a nonde- generate ε -hermitian form. In particular, w e w ill hav e to work sim ulta n eously with ε - and − ε -hermitian f o rms . W e a s sume throughout that M is a − ε -hermitian h yp erbo lic mo dule and w e let L = L ( M ) = { X ∈ M | X = X ⊥ } denote its set of Lagrangians. 2.1 Definition W e call t w o Lagrangia ns X and Y opp osite if X ∩ Y = 0 or, equiv alently , if M = X + Y . If the rank o f M is 1 , then Lagr angians are 1-dimensional, and X is opp osite Y if a nd only if X 6 = Y . 2.2 Lemma If M has rank 1 , then L has | D ε | + 1 elemen ts. Pr o of. Let x b e a nonzero v ector in the 1- dime nsional space X and let ξ ∈ X J b e its dual, i.e. ξ ( x ) = 1. Then x and ξ span X ⊕ X J ∼ = M . T he v ector v = ( xa, ξ ) spans a Lag rangian if and only if ξ ( xa ) = a ∈ D ε . There is precisely o ne additional Lag r angian, spanned by ( x, 0). ✷ Later it will b e imp ortan t that there are enough Lagrangians. W e note that D ε is infinite if D is an infinite field, unless J = id and ε = − 1 6 = 1. If D is not comm utativ e, then D ε is alw ays infinite [6, 6.1 .3]. 2.3 P r oposition I f | D ε | ≥ k , then there exists for ev ery finite collection X 1 , . . . , X k of La- grangians a La grangian Y opp osite to X 1 , . . . , X k . Pr o of. Let n denote the rank of M . W e pro ceed b y induction on k ≥ 1, mo difying the pro of in [27, 3.30]. Let X 1 , . . . , X k b e k Lag r a ngians. W e c ho ose a Lagrangian Y suc h that ℓ = dim( Y ∩ X 1 ) is as small as p ossible , a nd (b y the inductions hypothesis) such that Y is opp osite X 2 , . . . , X k . W e claim that ℓ = 0 . Otherwise, w e can c ho ose a subspace Q ⊆ X 1 of dimension n − 1, suc h that X 1 = Q + ( Y ∩ X 1 ). No w Q ⊥ can b e split as Q ⊥ H , with H h yp erb olic of rank 1. The 1-dimensional Lagrang ia ns P of H parameterize the Lagrangians of M con taining Q bijectiv ely via P 7→ Q ⊕ P . L et P 1 = Y ∩ H . F or ν = 2 , . . . , k , eac h X k determines a unique 1 -dimen sional Lag rangian P k ⊆ H with dim(( Q + P k ) ∩ X k ) 6 = 0 . By Lemma 2.2 w e ma y choo se a one-dimensional La grangian P ′ ⊆ H differen t from P 1 , . . . , P k . Then Y ′ = P ′ ⊕ Q is a Lagrang ia n o pp osite X 2 , . . . , X k with dim( Y ′ ∩ X 1 ) = ℓ − 1, a contradiction. ✷ In particular, there exists alwa ys a Lagrangian Y opp osite a giv en Lagrangian X . The map y 7→ h ( y , − ) | X is an isomorphism Y ∼ = ✲ X J and w e ha v e th us a unique isomorphism of h yp erb olic mo dules X ⊕ X J ∼ = ✲ X ⊕ Y = M extending the inclusion X ⊂ ✲ M . If ( X ′ , Y ′ ) is another suc h pair, then we can c ho ose an linear isomorphisms X ∼ = X ′ and obtain isomorphisms X ⊕ Y ∼ = ✲ X ⊕ X J ∼ = X ′ ⊕ X ′ J ✛ ∼ = X ′ ⊕ Y ′ . Hence we hav e established the follow ing result (whic h also follows from Witt’s Theorem [6 , 6.2.12]). 5 2.4 Lemma Th e unita ry group U ( M ) acts t rans itiv ely on ordered pairs of opp osite La - grangians. ✷ 2.5 W e no w study this U ( M )- action in more detail. W e fix a D -mo dule X of dime nsion n , with basis x . W e put Y = X J and w e let y denote the dual basis. Then M = X ⊕ Y is hy p erbolic of rank n , with basis x , y , and w e ma y w ork with 2 × 2 blo c k matr ices. The hermitian form h = h X on M is represen ted by the matrix h = 0 @ 0 − ε 1 0 1 A . W e find that t he U ( M )-stabilizer L of the ordered pair ( X , Y ) consis ts of matrices of the form ℓ a = 0 @ a − J 0 0 a 1 A , with a ∈ GL n D and ℓ a ℓ a ′ = ℓ aa ′ , while the U ( M )-stabilizer U of ( X, x ) consists of matrices of the form u t = 0 @ 1 t 0 1 1 A , with t − t J ε = 0, i.e. t ∈ D n × n has to b e ε -hermitian. Not e a ls o that u t u t ′ = u t + t ′ , u − 1 t = u − t , and that ℓ a u t ℓ − 1 a = u a − J ta − 1 . The U ( M )- s tabilizer P of X splits therefore as a semidirect pro duct P = LU , with L e v i f a ct or L and unip otent r adic al U ✂ P . Next, w e note that if Z is another Lagrangian opp osite X , then w e hav e a unique isomorphism X ⊕ Y ✲ X ⊕ Z fixing the basis x . This isomorphism is therefore g iv en by an elemen t of the group U , and w e hav e t he following result. 2.6 Lemma Th e group U acts regularly on the set X opp of all Lagrangia ns opp osite X . ✷ Let u t ∈ U . Then the Lagr a ngian Z = u t ( Y ) is opp osite Y if and only if M is spanned b y y , u t ( y ) . With the matrix notations we established b efore, we hav e u ( y ν ) = y ν + P µ x µ t µ,ν . A necessary and sufficien t condition for Z = u t ( Y ) b eing opp osite Y is thus that the matrix t is in v ertible. 2.7 W e let H = { t ∈ D n × n | t − t J ε = 0 } denote the set of all ε -hermitian n × n -matrices. There is a natura l left action ( a, t ) 7→ a − J ta − 1 of GL n D on H , and w e denote the orbit of t by h t i . The orbit space Herm ε ( n ) = {h t i | t ∈ H } = L \ H consists thus of the isomorphism classes of ε -hermitian forms o n D n . W e denote the subset corresp onding to the nonsingular hermitian forms by Herm ◦ ε ( n ). Then we hav e an L -equiv ar ian t bijection H ✲ X opp t 7− → u t ( Y ) . 6 F a cto r ing out the L -a ctio n, w e get bijections Herm ε ( n ) ✲ L \ X opp and Herm ◦ ε ( n ) ✲ L \ ( X opp ∩ Y opp ) While the isomorphism H ✲ U d ep ends on the c hosen basis x , these tw o ma ps are base- indep e nden t a s one can easily c hec k (this will also fo llo w from 4.3 ) . Summarizing these results, w e hav e t h e follo wing theorem. 2.8 Theorem Let L (3) ⊆ L×L×L denote the set of all triples of pairwise opp osite Lagr a ngians. Then w e ha v e a U ( M ) -in v ariant surjectiv e map L (3) κ ✲ Herm ◦ ε ( n ) whose fib ers are the U ( M ) - orbits in L (3) . The map κ is giv en b y κ ( g ( X ) , g ( Y ) , g u t ( Y ) ) = h t i , where X, Y is our fixed pair of opp o s ite Lagrangians a s in 2.5. ✷ The result will b e refined in Prop osition 4.5. 2.9 According to 1.5, w e ha ve the following cases. Symplectic groups: The triples are classified by isomorphism classes of nondegenerate sym- metric ma t r ic es. Hyp erbolic orthogonal groups: The triples are classified b y isomorphism classes of nonde- generate sk ew symmetric matrices. There is one suc h class if n is ev en, and L (3) = ∅ if n is o dd. Standard h yp erbolic unitary groups: W e may assume that ε = 1 (so the f orm is ske w hermitian), and then the triples are classified by isomorphism classes of n -dimensional nonde- generate hermitian forms. 3 Flag c omplexes of graphs W e con t in ue to assume that M is a − ε -hermitian hy p erbolic mo dule. Now w e consider the simplicial complex whose k -simplices are k + 1-sets of pairwise opp osite Lagrangians. It will b e con ve nien t to do this in the g e neral setting of graphs, flag complexes and simplicial sets. 3.1 The opp osition graph By a gr aph Γ = ( V , E ) w e understand an undirected gra ph without lo ops or multiple edges; V is it s set of vertic es , E its set of e dges , and edges ar e unordered pairs of v ertices. If { u, v } is an edge, w e call u, v adjac ent . F or the h yperb olic mo dule M , w e put V = L and O = {{ X , Y } | X, Y ∈ L and M = X + Y } . The resulting graph Γ = ( L , O ) is called the opp osition gr aph of M . 7 3.2 Flag complexes The flag c omplex F l (Γ) of a gra p h Γ is the simplicial set whose k -simplices are tuples ( x 0 , . . . , x k ) of v ertices, suc h that for all 0 ≤ µ < ν ≤ k w e hav e either x µ = x ν or { x µ , x ν } ∈ E . W e ha ve the standard Z -f re e c hain complex C ∗ ( F l (Γ)) with the usual b oundary op e rator ∂ ( x 0 , . . . , x k ) = P ν ( − 1) ν ( x 0 , . . . , ˆ x ν , . . . , x k ) and the resulting homology and cohomology g roups . W e will also use alternating chains , whic h are defined as follows [5]. Let N k denote the submo d- ule of C k ( F l (Γ)) generated by a ll elemen ts ( x 0 , . . . , x k ) with x µ = x ν for some µ < ν , and all elemen ts of the fo rm ( x 0 , . . . , x k ) − sig n ( π )( x π 0 , . . . , x π k ), fo r π ∈ S y m ( k + 1). The a lt e rnating c hain complex is defined as the quotien t c hain complex ˜ C ∗ ( F l (Γ)) = C ∗ ( F l (Γ)) / N ∗ . The natural pro jection C ∗ ( F l (Γ)) ✲ ˜ C ∗ ( F l (Γ)) is a c hain equiv alence, i.e. induces an iso- morphism in homology and cohomology , se e [5] VI.6. The cose t of ( x 0 , . . . , x k ) is denoted h x 0 , . . . , x k i , with t he relations h x 0 , . . . , x k i = 0 if x µ = x ν for some µ < ν , and h x 0 , . . . , x k i = s ig n ( π ) h x π 0 , . . . , x π k i . 3.3 Equiv arian t cohomology The unitary gro up U ( M ) acts in a natural wa y on the opp osition graph and its fla g complex. In general, when a gro up G acts (f r om the left, say) on a c hain complex C ∗ , then w e ma y consider the e quivariant homolo gy of C ∗ , whic h is defined as follows . If P ∗ ✲ Z is a pro jectiv e resolution of G ov er Z , then the equiv ariant homology H G ∗ ( C ∗ ) is defined as the total ho mo lo gy of the double complex P ∗ ⊗ G C ∗ , see [2, Ch. VI I.5]. The t w o canonical filtrations on the do uble complex yield t w o sp ectral sequence s ′ E and ′′ E con verging to H G ∗ ( C ∗ ) and the first one has on its second pag e ′ E 2 pq = H p ( G ; H q ( C ∗ )) . If C ∗ is acyclic (eg., if C ∗ = Z is concentrated in dimension 0), then ′ E collapses on the second page, and there is a na t ur a l isomorphism H G ∗ ( C ∗ ) ∼ = H ∗ ( G ). Similar remarks hold for cohomology; here, o ne lo oks at the double complex Hom G ( P ∗ , C ∗ ). Note also that if c : C ∗ ✲ A is a G - inv ariant co c hain (so G acts trivially on the co efficien t mo dule A ) and if η : P 1 ✲ Z is the augmen tation map, then c may b e view ed in a natural w ay as a co c ha in in Hom G ( P ∗ , Hom Z ( C ∗ , A )) ∼ = Hom Z ( P ∗ ⊗ G C ∗ , A ) via c ( p ⊗ z ) = η ( p ) c ( z ) . It is w ell-kno wn tha t for a complete gra ph (i.e. for E =  V 2  ) the simplicial set F L (Γ) is acyclic. The follo wing concept is a w eak ening of (infinite) complete graphs. 3.4 The star prop ert y A (no ne mpt y) graph Γ = ( V , E ) ha s the star pr op erty if for ev ery finite set x 0 , . . . , x k of ve rtices, there exis ts a v ertex y whic h is adjacen t to the x ν , for ν = 0 , . . . , k . 8 Note that we require that y 6 = x 0 , . . . , x k . A g raph with the star prop ert y is obv iously infinite. Note also t h at the opp osition graph of a h yp erb olic module has b y 2.3 the star prop ert y if D ε is infinite. 3.5 Lemma If Γ has the star prop ert y , then F l (Γ) is acyclic. Pr o of. If ( x 0 , . . . , x k ) is a k -simplex in C k ( F l (Γ)) and y if is adja c en t t o x 0 , . . . , x k put y #( x 0 , . . . , x k ) = ( y , x 0 , . . . , x k ). Supp ose that c is a k -cycle, i.e. c is a finite linear com binatio n of k -simplices and ∂ c = 0. Let y b e a verte x adjacen t to all v ertices app earing in the simplices of c . Then ∂ ( y # c ) = c − y # ∂ c = c , so c is a bo und ary . ✷ The g eometric realization | F l (Γ) | of the flag complex of a graph with the star pro perty is in fact con tra c tible. T o see this, it suffices b y Hurewicz’ Theorem to sho w that π 1 | F l (Γ) | = 0, see [26, Ch. 7.6.24 and 7.6.25]. But from the star prop ert y , any simplicial path in | F l (Γ) | is con tained in a contractible sub c omplex, and ev ery path is ho m otopic to a simplicial path [26 , 3.6]. 3.6 P r oposition I f D ε is infinite, then t he flag complex of the opposition graph of a − ε - hermitian hyperb olic mo dule is acyclic and its geometric realization is con tra ctible. Conse- quen tly , w e ha v e in equiv aria n t homology a natura l isomorphism H U ( M ) ∗ ( F l (Γ)) ∼ = ✲ H ∗ ( U ( M )) induced b y the constan t map F l (Γ) ✲ F l ( { pt } ) , and similarly for cohomology . ✷ Using this natural isomorphism, w e o f te n identify these tw o (co)homo lo gy groups. 4 The pr o ject ivit y group oid If X and Y are opp osite Lagrang ia ns in the hyperb olic mo dule M , then w e ha ve canonical isomorphisms X ⊕ X J ∼ = X ⊕ Y ∼ = Y ⊕ Y J , suc h that the first isomorphism is the iden tity on X and the second isomorphism is the iden tit y on Y . In this w ay , we a sso ciate an isomorphism X ⊕ X J ✲ Y ⊕ Y J to every oriente d edge ( X ✲ Y ) of the opp osition graph Γ. 4.1 The pro jectivit y gr oup oid Recall that a group oid is a small category where eve ry arrow is an isomorphism. The pr oje ctivity gr oup oi d G M of M is defined as fo llo ws. The ob jects of G M are 2 -graded ve ctor spaces X ∗ with X 1 = X and X − 1 = X J , where X ∈ L is a Lagrangian. T o eac h orien ted edge ( X ✲ Y ) w e asso ciate a n isomorphism [ Y ; X ] : X ∗ ✲ Y −∗ of degree − 1, the comp osite [ Y ; X ] : X ⊕ X J ∼ = ✲ X ⊕ Y ∼ = ✲ Y ⊕ Y J . These maps generate the morphisms of G M . W e not e that each ob ject X ∗ in G carries a natural structure of a h yp erbo lic mo dule with − ε - he rmitian form h X , and that the morphisms preserv e this structure. F urthermore [ X ; Y ][ Y ; X ] = id X ∗ , 9 so a morphism along a simplicial path dep ends only on the homoto p y class of the pa t h in Γ (i.e. we hav e a natural transformation from the funda mental gr oup oid π 1 Γ to G M ). F inally , w e note t ha t G M is in a natural w ay 2- graded: the pa t hs of ev en length induce maps of degree 1, a n d the pat h s of o dd length ma ps of degree − 1. 4.2 No w w e determine the morphism corresp onding to a closed pat h of length 3 . Let X, Y b e opp osite Lagrangians with bases x , y as in 2.5 and let Z = u t ( Y ). W e write [ Z ; Y ; X ] = [ Z ; Y ][ Y ; X ] and so on. Then [ Z ; X ; Y ]( y ν ) = u t ( y ν ) and [ Z ; X ; Y ]( x ν ) = u t ( x ν ) = x ν . No w h ( y λ , u t ( y ν ))) = h  y λ , y ν + P µ x µ t µ,ν  = h  y λ , P µ x µ t µ,ν  . The dual basis of y is h ( − , x ) J . With resp ect to t he graded basis ( y , h ( − , x ) J ) for Y ∗ = Y ⊕ Y J , the morphism ϕ = [ Y ; Z ; X ; Y ] is therefore giv en b y a blo c k matrix of the for m ϕ = 0 @ ∗ ∗ t ∗ 1 A . As t h is matrix has to b e unitary and of degree − 1, and because t J = tε , we obta in ϕ = 0 @ 0 − t − 1 t 0 1 A . 4.3 If h Y denotes the canonical − ε -hermitian form on Y ∗ , then h ϕ ( − , − ) = h Y ( − , ϕ ( − ))( − ε ) is the ε -hermitian form h ϕ = 0 @ t 0 0 t − J 1 A . W e note that t J t − J t = t , so b oth blo c ks r e presen t the same isomorphism t yp e h t i in H er m ◦ ε ( n ), and w e define ˜ κ ( Z , X , Y ) = h t i . Note also that this class do es not dep e nd on the ba s is y and that ˜ κ is U ( M )-in v ariant. F ur- thermore, we hav e ˜ κ ( Z , X , Y ) = κ ( X , Y , Z ) , where κ is the in v ariant from Theorem 2.8. W e will see shortly that b oth in v a rian ts agree completely . 4.4 F ro m ϕ − 1 = 0 @ 0 t − 1 − t 0 1 A w e see that ˜ κ ( X, Z , Y ) = h − t i . Next we not e that for y 1 , y 2 ∈ Y ∗ w e hav e h Y ( y 1 , ϕ ( y 2 )) = h X ([ X ; Y ] y 1 , [ X ; Y ] ϕ [ Y ; X ][ X ; Y ] y 2 ) = h X ([ X ; Y ] y 1 , [ X ; Y ][ Y , Z , X , Y ][ Y ; X ][ X ; Y ] y 2 ) = h X ([ X ; Y ] y 1 , [ X ; Y , Z , X ][ X ; Y ] y 2 ) , whence ˜ κ ( Y , Z , X ) = ˜ κ ( Z , X , Y ) , 10 i.e. ˜ κ is in v ariant under cyclic p erm utations of the argumen ts. In particular, w e ha ve ˜ κ = κ Since κ classifies b y Theorem 2.8 triples of pairwise opp osite Lagra ng ia ns , we hav e the f o llo wing sharp ening of 2.8. 4.5 P r oposition T he set-wise U ( M ) -stabilizer of a t r iple X , Y , Z of pairwis e opposite La- grangians induces (at least) the cyclic group Z / 3 on this set. It induces the full symmetric group S y m (3) if and only if a J ta = − t for some a ∈ GL n D , where κ ( X, Y , Z ) = h t i . ✷ 5 The Maslov co cycle W e wan t to turn the inv arian t κ : L (3) ✲ Herm ◦ ε ( n ) into a 2-co cycle for the flag complex F l (Γ) of the oppo s ition graph. Supp ose that A is an ab elian group and that α : Herm ◦ ε ( n ) ✲ A is a map. By the prop erties of κ derive d in 4.4 w e see that c : h X , Y , Z i 7→ α ( κ ( X , Y , Z )) is a 2-co c hain on the a lt e rnating c hain complex ˜ C 2 ( F l (Γ)), provided that w e hav e the relation α ( h− t i ) = − α ( h t i ) for all t ∈ Herm ◦ ε ( n ). Now w e inv estigate under what conditions this map is a co cycle, i.e. under what conditions c ( ∂ h X , Y , Z , Z ′ i ) = 0, i.e. when c ( h Y , Z , Z ′ i − h X , Z , Z ′ i + h X , Y , Z ′ i − h X , Y , Z i ) = 0 . 5.1 W e fix again ( X , x ) , ( Y , y ) as in 2.5. Supp ose that Z = u t ( Y ) and Z ′ = u t ′ ( Y ), and that X , Y , Z , Z ′ are pairwise opp osite. So w e ha ve κ ( Z , X , Y ) = h t i and κ ( Z ′ , X , Y ) = h t ′ i . As u − 1 t ( Z ) = Y a nd u − 1 t u t ′ = u − t + t ′ w e obtain κ ( Z ′ , X , Z ) = κ ( u − 1 t u t ′ ( Y ) , X , Y ) = h t ′ − t i . It remains to determine κ ( Z ′ , Y , Z ). Let w = 0 @ 0 1 − ε 0 1 A . Then w is unitary and in t erchanges X and Y . W e ha ve w ( Z ) = w u t ( Y ) = w u t w − 1 ( X ) and w e put v t = w u t w − 1 = 0 @ 1 0 − tε 1 1 A . Then u r v t = 0 @ 1 − r tε r − tε 1 1 A , whence u r w ( Z ) = u r v t ( X ) = Y fo r r = t − 1 ε . So fa r w e hav e ac hiev ed u r w ( Y ) = X and u r w ( Z ) = Y . 11 W e seek t ′′ suc h that u t ′′ ( Y ) = u r w ( Z ′ ) = u r w u t ′ ( Y ) , o r Y = u r − t ′′ w u t ′ ( Y ) . Now u r − t ′′ w u ′ t = 0 @ 1 r − t ′′ 0 1 1 A 0 @ 0 1 − ε − t ′ ε 1 A = 0 @ ( t ′′ − r ) ε 1 + ( t ′′ − r ) t ′ ε − ε − t ′ ε 1 A whence 1 = ( r − t ′′ ) t ′ ε , which gives t ′′ = r − t ′ − 1 ε = ( t − 1 − t ′− 1 ) ε , so κ ( Z ′ , Y , Z ) = h ( u t ′′ ( Y ) , X , Y ) = h t − J − t ′− J i . Plugging this into the b oundary form ula, w e ha v e the next result. 5.2 P r oposition Le t A b e an ab elian group. A f un ction α : Herm ◦ ε ( n ) ✲ A determines a U ( M ) -in v ariant 2 -co cycle c on the a lte rnating 2 -chains of F l (Γ) if and only if the following tw o relations hold f or all r, s, t ∈ Herm ◦ ε ( n ) : r + s = 0 implies α h r i + α h s i = 0 r + s + t = 0 implies α h r i + α h s i + α h t i + h− r − J − s − J i = 0 ✷ Recall that the Grothendiec k-Witt group K U ε 0 ( D , J ) of hermitian forms is defined as the ab elian group completion of the comm ut a tiv e monoid consisting of the isomorphism classes of nondegen- erate ε -hermitian for ms [23, p. 239]. The Witt gr oup W ε ( D , J ) is the factor group of K U ε 0 ( D , J ) b y the subgroup generated by the ε -hermitian h yp erbolic modules. W e let [ t ] denote the image of h t i in K U ε 0 ( D , J ) and W ε ( D , J ) . 5.3 Theorem Let α h t i = [ t ] ∈ W ε ( D , J ) . Then α satisfies the t wo conditions of Prop osition 5.2 and therefore m : h X , Y , Z i 7− → [ κ ( X , Y , Z )] defines a W ε ( D , J ) - v alued U ( M ) - in v ariant 2 - cocycle on the alternating chain complex ˜ C 2 ( F l (Γ)) . Pr o of. W e pr o ceed similarly a s [21, Prop 1.2] and use the fact that metab olic forms v anish in the Witt group W ε ( D , J ) , see [23, 7.3.7], and that a 2 k -dimensional nondegenerate hermitian form is metab olic if it admits a totally isotr o pic subspace of dimension k . F o r the 2 n -dimensional ε -hermitian fo rm ( r ) ⊕ ( − r ), the v ectors ( x, x ), with x ∈ D n , span an n - dimensional to t a lly isotropic subspace, so this form is metab olic and [ r ] + [ − r ] = 0. Similarly we find for r + s + t = 0 a nd the 4 n -dimensional form ( r ) ⊕ ( s ) ⊕ ( t ) ⊕ ( − r J − s J ) that the vec tors ( x, x, x, 0) and ( r − J x, s − J x, 0 , x ), with x ∈ D n , span a totally isotropic 2 n - dimensional subspace, so this form is a lso metab olic and [ r ] + [ s ] + [ t ] + [ − r J − s J ] = 0. ✷ 5.4 The Maslo v c ocycle W e call the W ε ( D , J ) - v alued co cycle m : h X , Y , Z i 7− → [ κ ( X , Y , Z )] (and the corr esp onding co cy cle for the equiv a r ian t homology of F l (Γ)) the Maslov c o cycle . 12 6 Naturalit y of the Maslov co cycle W e no w study naturality o f the Maslo v co c ycle under restriction maps. There are t w o ob vious t yp es, coming fr o m field and from v ector space inclusions. W e start with field inclusions, whic h are easier. 6.1 Extension of scalars Supp ose that D and E ar e div ision rings with inv olutions J and K , resp ectiv ely , and that ϕ : D ✲ E is a ho momo r p hism comm uting with these in volutions. If M is a h yp erbolic mo dule o ver D , then M ⊗ ϕ E is h yp erbolic o v er E . The map sending a Lagrangian X ⊆ M to X ⊗ ϕ E induces an injection L ( M ) ✲ L ( M ⊗ ϕ E ) and an injection Γ( M ) ✲ Γ( M ⊗ ϕ E ) on the respectiv e opp osition graphs. There is a natural map W D E : W ε ( D , J ) ✲ W ε ( E , K ) and ob viously , this map ta kes the Maslo v co cycle m D of M to the Maslov co cycle m E of M ⊗ ϕ E , ˜ C 2 F l (Γ( M )) ✲ ˜ C 2 F l (Γ( M ⊗ ϕ E ) W ε ( D , J ) m D ❄ W D E ✲ W ε ( E , K ) . m E ❄ This giv es the follo wing result. 6.2 Theorem Let ϕ : ( D , J ) ✲ ( E , K ) be a homomor p hism of sk ew fields with in volutions and assume tha t D ε is infinite. Consider the natura l g r o up monomorphism Φ : U ( M ) ✲ U ( M ⊗ ϕ E ) . Then ( W D E ) ∗ m D = Φ ∗ m E in the diagr am H 2 ( U ( M ); W ε ( D , J ) ) ( W D E ) ∗ ✲ H 2 ( U ( M ); W ε ( E , K )) H 2 ( U ( M ⊗ ϕ E ); W ε ( E , K )) . Φ ∗ ✻ ✷ 6.3 Supp ose now that M 1 and M 2 are hyperb olic mo dules (b oth ov er D ) with corresponding sets L 1 , L 2 of Lag rangians. Then their direct sum M = M 1 ⊕ M 2 is in a natural wa y a h yp erbolic mo dule. There is an obv ious map U ( M 1 ) ✲ U ( M ) and the question is what happ ens with the Maslo v co cycle under this map. The problem is that the opp osition gra ph Γ 1 of M 1 is not a subgraph of the o pp osition graph Γ of M . Ho wev er, 13 there is a na t u ral subgraph of Γ whic h pro jects U ( M 1 )-equiv aria n tly o nto Γ 1 and whic h yields a g oo d comparison map. The construction is a s follow s. If X 1 ⊆ M 1 and X 2 ⊆ M 2 are Lagrangians, then X 1 ⊕ X 2 is Lagrangian in M , so w e ha ve a natural injection L 1 × L 2 ✲ L . Moreo v er, X 1 ⊕ X 2 is opp osite Y 1 ⊕ Y 2 in M if and only if X ν is oppo s ite Y ν , fo r ν = 1 , 2. This leads us to the follo wing notion. 6.4 Definition Th e c ate goric al pr o duct Γ 1 × Γ 2 of t wo graphs has V 1 × V 2 as its set of v ertices and ( x 1 , x 1 ) and ( y 1 , y 2 ) are adjacen t if and only if { x 1 , y 1 } ∈ E 1 and { x 2 , y 2 } ∈ E 2 . There ar e natural maps Γ 1 ✛ Γ 1 × Γ 2 ✲ Γ 2 with the usual univ ersal prop erties. The next result is immediate. 6.5 Lemma Th e categorical pro duct of t w o graphs hav ing t he star prop ert y has again the star prop ert y . In particular, its flag complex is acyclic. ✷ Note that the categorical pro duct of the graph consisting o f one single edge with itself is not ev en connected; • • • × • • = • • • . The fact that y 6 = x 0 , . . . , x k in the star prop ert y is crucial fo r t he Lemma. 6.6 So far we ha v e for ν = 1 , 2 a diagram of U ( M 1 )-equiv aria n t maps F l (Γ 1 ) F l (Γ 1 × Γ 2 ) ✲ ✛ pr 1 F l (Γ) F l (Γ 2 ) ✛ pr 2 and if D ε is infinite, these three complexes are acyclic. Next w e note that if w e ha ve a triangle ( X 1 ⊕ X 2 , Y 1 ⊕ Y 2 , Z 1 ⊕ Z 1 ) in Γ 1 × Γ 2 and if w e ch o ose bases x 1 , x 2 , y 1 , y 2 for X 1 , X 2 , Y 1 , Y 2 , then [ κ ( X 1 ⊕ X 2 , Y 1 ⊕ Y 2 , Z 1 ⊕ Z 1 )] = [ t 1 ⊕ t 2 ] = [ t 1 ] + [ t 2 ] , with κ ( X ν , Y ν , Z ν ) = h t ν i . Thus we hav e a commutativ e diagram ˜ C 2 F l (Γ 1 ) i 1 ✲ ˜ C 2 F l (Γ 1 ) ⊕ ˜ C 2 F l (Γ 2 ) ✛ ( pr 1 , pr 2 ) ˜ C 2 F l (Γ 2 × Γ 2 ) W ε ( D , J ) . m 1 + m 2 ❄ ✛ m m 1 ✲ 14 whic h yields in cohomology H 2 U ( M 1 ) ( F l (Γ 1 )) ✛ H 2 U ( M 1 ) ( F l (Γ 1 )) ⊕ H 2 U ( M 1 ) ( F l (Γ 2 )) ✲ H 2 U ( M 1 ) ( F l (Γ 1 × Γ 2 )) [ m 1 ] ✛ [ m 1 ] + [ m 2 ] ✲ [ m ] (w e omit here the co efficien t group W ε ( D , J ) ) . Note that [ m 2 ] = 0 in H 2 U ( M 1 ) ( F l (Γ 2 )), as U ( M 1 ) acts trivially on F l (Γ 2 ). Mapping to the o ne -p oin t space { pt } , w e see that [ m 1 ] a nd [ m ] ha ve the same image in H 2 U ( M 1 ) ( { pt } ) = H 2 ( U ( M 1 )), and from H 2 U ( M 1 ) ( F l (Γ 1 )) H 2 U ( M 1 ) ( F l (Γ 1 × Γ 2 )) ✛ ∼ = H 2 U ( M 1 ) ( F l (Γ)) ✛ H 2 U ( M ) ( F l (Γ)) H 2 ( U ( M 1 )) ∼ = ✻ = = = = = = = = H 2 ( U ( M 1 )) ∼ = ✻ = = = = = = = = H 2 ( U ( M 1 )) ∼ = ✻ ✛ H 2 ( U ( M )) . ∼ = ✻ w e obtain the follo wing stabilit y result. 6.7 Theorem As sume that D ε is infinite, let M 1 , M 2 b e h yp erb olic mo dules and put M = M 1 ⊕ M 2 . Then the restriction map H 2 ( U ( M 1 ); W ε ( D , J ) ) ✛ H 2 ( U ( M ); W ε ( D , J ) ) maps the Maslov co cycle [ m ] for U ( M ) on to the Maslo v co cycle [ m 1 ] for U ( M 1 ) . ✷ 7 Reduction of the co cycle Our next aim is to show tha t the Maslo v co cycle can b e reduced to a subgroup of the Witt group. F or this, w e need a refinemen t of the La grangians and the opp osition g r a ph. W e noted in 2.9 that the Maslov co cy cle is trivial in the h yp erbo lic orthogonal situation, where J = id and ε = − 1 6 = 1, so we ma y disregard this case. By 1.5 there is no lo s s of generalit y in assuming that ε = 1 in the remaining cases, and w e will do this in this section. 7.1 Based Lagrangians Let Γ = ( V , E ) b e a g raph and f : X ✲ V a map. The induc e d gr aph f ∗ Γ o n X is the graph whose v ertices are the elemen ts of X , and { x, x ′ } is a n edge if and only if { f ( x ) , f ( x ′ ) } is an edge of Γ. If f is surjectiv e and if Γ has the star prop e rt y , then f ∗ Γ a ls o has the star prop e rt y . In what follo ws, w e consider the set b L of b ase d L agr angians , i.e. pairs ( X , x ) where X ⊆ M is a La grangian and x is a basis for X . There is a forgetful surjection F : b L ✲ L a nd w e let b Γ = F ∗ Γ 15 denote the induced graph o n this vertex set. W e call b Γ the b ase d opp osition gr aph . Because the U ( M )-stabilizer P induces the f ull group GL ( X ) on X , we see that U ( M ) acts transitiv ely on b L . With the notation of 2.5, the stabilizer of ( X , x ) is the group U . The map b Γ ✲ Γ is equiv ariant, and F l ( b Γ) is a cyclic if D 1 is infinite. In particular, w e may use F l ( b Γ) to compute the group cohomology of U ( M ). W e also ha ve a ba s ed v ersion b G M of the pro jectivity group oid. The ob jects are again the 2-graded spaces X ⊕ X J , but no w with a preferred gra ded basis consisting of x and the dual basis of x . The morphisms in b G M are th us giv en by unita r y matrices. 7.2 W e re-calculate the Maslo v co cy cle in terms of the based spaces. In 4.2 w e saw that w e ha ve in terms of our standard basis x , y the matrices ( Y ∗ , y )  0 − 1 1 0  ✲ ( X ∗ , x ) ✛  0 − 1 1 0  ( Z ∗ , u t ( y ) )  0 − t − 1 t 0  ✲ ( Y ∗ , y ) . If we add base c hang e s through matrices a, b, c ∈ GL n D for X , Y and Z and rev erse the middle arro w, w e arriv e at the dia gram ( Y ∗ , b y )  0 − ab J a − J b − 1 0  ✲ ( X ∗ , a x )  0 ca J − c − J a − 1 0  ✲ ( Z ∗ , cu t ( y ) )  0 − bt − 1 c J b − J tc − 1 0  ✲ ( Y ∗ , b y ) (and cu t ( y ) = u c − J tc − 1 ( c y )). With resp ec t to the basis b y , we ha ve [ Y ; Z ; X ; Y ] =  0 − bt − 1 b J b − J tb − 1 0  . Using in v arian ts of these matrices, w e no w construct a refined co cycle. 7.3 In v arian ts of hermitian forms The dimension induces a natural homomorphism dim : K U 1 0 ( D , J ) ✲ Z . Since the dimension of an y h yp erb olic mo dule is ev en, there is an induced map W 1 ( D , J ) ✲ {± 1 } mapping the class [ t ] to ( − 1) dim( t ) . W e denote its kernel by I ( D , J ); its elemen ts are represen ted b y ev en dimensional hermitian forms. In the quadratic case ( J = id and ε = 1 6 = − 1), I D = I ( D , id) is called the fundamen t al ide al in the Witt ring WD = W 1 ( D , id) [1 2, Ch. I I.1]. Recall tha t the determinant is a homomorphism from GL n D to K 1 ( D ), the ab elianiza- tion of D ∗ = GL 1 D . The in v olution J induces an automorphism J on K 1 ( D ). W e let N denote the subgroup of K 1 ( D ) consisting of elemen ts of the form x J x and put S = K 1 ( D ) / N . Since det( g J tg ) = det( g J g ) det( t ), w e hav e a w ell-defined homomorphism [ t ] 7− → det( t ) N fro m K U 1 0 ( D , J ) t o S . Ho wev er, this map cannot b e factored through W 1 ( D , J ) . Similarly as in [12 , Ch. II.2] w e in tro duce therefore the ab elian group b S = S × {± 1 } , endo w ed with the comm utativ e g r o up la w ( x, ( − 1) m ) + ( y , ( − 1) n ) = ( xy ( − 1) mn , ( − 1) m + n ) , 16 and w e define the signe d discriminant as disc ( t ) = ( d et( t ) N ( − 1) n ( n − 1) / 2 , ( − 1) n ) , where n = dim( t ). This map v anishes o n h yp erbolic forms and induces therefore a homomor - phism d i sc : W 1 ( D , J ) ✲ b S . W e let I I ( D , J ) ⊆ W 1 ( D , J ) denote the subgroup generated by all elemen ts [ t ], where dim( t ) ∈ 4 Z and det( t ) = 1. Ob viously , I I ( D , J ) ⊆ k er ( disc ). 7.4 Lemma Th e sequence 0 ✲ I I ( D , J ) ✲ W 1 ( D , J ) ✲ b S is exact. Pr o of. Let [ t ] b e a form in the ke rnel of disc . Then dim( t ) is ev en and w e distinguish t wo cases. If dim( t ) = 4, then det( t ) = x J x ∈ N . Cho ose g ∈ GL n D with det( g ) = x − 1 , then det( g J tg ) = 1 and [ t ] = [ g J tg ] ∈ I I ( D , J ). F or dim( t ) = 4 ℓ + 2 w e ha v e det( t ) = − x J x and w e consider the 4 ℓ + 4-dimensional form t ⊕ h , fo r h =  0 − 1 1 0  . Then det( t ⊕ h ) = x J x . By the previous remark, [ t ⊕ h ] = [ t ] ∈ I I ( D , J ). ✷ In the quadratic case, I I ( D , J ) is the square I 2 D of the fundamen tal ideal [12, Ch. I I,2.1] 7.5 W e define an b S -v alued equiv ariant 1 - coch ain f on ˜ C 1 F l ( b Γ) b y f h ( X , a x ) , ( Y , b y ) i = (det( − ab J )( − 1) n ( n − 1) / 2 N , ( − 1) n ) ∈ b S , where the nota tion is as in 7.2. Note that t h is is indeed an alternating co c ha in: (det( g ) N , ( − 1) n ) + (det( − g J ) N , ( − 1) n ) = (det( − g g J )( − 1) n 2 N , ( − 1) 2 n ) = ( N , 1) . Then d f = f ∂ is an b S - v alued 2-cob oundary , and d f h ( Z , cu t ( y ) ) , ( X , a x )) , ( Y , b y ) i = f h ( X , a x ) , ( Y , b y )) i − f h ( Z, cu t y ) , ( Y , b y )) i + f h Z, cu t ( y ) )( X, a x )) i = (det( − ab J )( − 1) n ( n − 1) / 2 N , ( − 1) n ) + (det( − bt − 1 c J )( − 1) n ( n − 1) / 2 N , ( − 1) n ) + (det( ca J )( − 1) n ( n − 1) / 2 N , ( − 1) n ) = (det( aa J bb J cc J t )( − 1) n ( n − 1) / 2 N , ( − 1) n ) = (det( t )( − 1) n ( n − 1) / 2 N , ( − 1) n ) whence disc ∗ m + d f = 0 where d isc ∗ denotes t he co efficien t ho mo mor p hism induced b y disc : W 1 ( D , J ) ✲ b S . Conse- quen tly , the image of m v anishes in H 2 ( U ( M ); b S ). 17 7.6 Recall that E U ( M ) ⊆ U ( M ) is the in v ariant subgroup g e nerated b y the Eic hler tra nsfor - mations. T his g r oup is p erfect if D 1 is infinite [6, 6 .3 .15] and consequen tly H 1 ( EU ( M ); A ) = Hom( EU ( M ) , A ) = 0 for any co efficien t g r oup A with trivial EU ( M )-action. W e put b S 0 = disc ( W 1 ( D , J ) ) ⊆ b S . The long exact cohomology sequences for the coefficien t maps 0 ✲ b S 0 ✲ b S ✲ b S / b S 0 ✲ 1 0 ✲ I I ( D , J ) ✲ W 1 ( D , J ) ✲ b S 0 ✲ 0 yield therefore mono m orphisms 0 ✲ H 2 ( EU ( M ); b S 0 ) ✲ H 2 ( EU ( M ); b S ) 0 ✲ H 2 ( EU ( M ); I I ( D , J )) ✲ H 2 ( EU ( M ); W 1 ( D , J ) ) . This gives us the next Theorem. T o k eep no tation simple, we denote the restriction of m to the subgroup EU ( M ) also b y m . 7.7 Theorem As sume that ε = 1 and that D 1 is infinite. There exis ts a unique coho- mology class [ e m ] ∈ H 2 ( EU ( M ); I I ( D , J )) wh ic h maps under the coefficien t homomorphism I I ( D , J ) ✲ W 1 ( D , J ) on to [ m ] . W e call t his class the reduced Maslo v co cyc le . Pr o of. As we prov ed in 7.5, disc ∗ [ m ] + [ d f ] = 0 in H 2 (EU( M ); b S ), whence disc ∗ [ m ] = 0 in H 2 ( EU ( M ); b S 0 ). Th erefore [ m ] has a preimage [ e m ] in H 2 ( EU ( M ); I I ( D , J )). The map H 2 ( EU ( M ); I I ( D , J )) ✲ H 2 ( EU ( M ); W 1 ( D , J ) ) is injectiv e, so the preimage is unique. ✷ 7.8 In the symplectic situation ( J, ε ) = (id , 1) it is p ossible to give an explicit form ula fo r the reduced co cycle e m . Then b S = b S 0 and EU ( M ) = U ( M ) = Sp 2 n D and w e can directly define a W 1 ( D , J ) - v alued 1- c o c hain on F l ( b Γ) b y ˜ f h ( X , a x ) , ( Y , b y ) i = h det( − ab ) , 1 , . . . , 1 i , where the right-hand side denotes as usual t h e n -dimensional symmetric bilinear form with the giv en en tries on the diagonal. Under the map p : F l ( e Γ) ✲ F l (Γ) this is a lift of f and w e ha ve disc ∗ d ˜ f = p ∗ d f . Th us e m = p ∗ m + d ˜ f is the reduced Maslo v co cycle on F L ( b Γ) in the symplectic case. Explicitly , it reads as e m h ( X , a x ) , ( Y , b y ) , ( Z , cu t y ) i = h det( − ab ) , 1 , . . . , 1 i + h det( ca ) , 1 , . . . , 1 i + h det( − btc ) , 1 , . . . , 1 i − h t i . 8 Kashiw ara’s Maslo v co cycle In the symplectic situation ov er a field D of c haracteristic 6 = 2, the Maslo v index is classically defined through a differen t quadratic form [13]. (A v ariant is used in [21], while a top ological generalization for b ounded symmetric domains of tub e t yp e is g iv en in [18]. See [3] for a surv ey of top ological Maslo v indices.) 18 8.1 Kashiw ara’s Maslo v index Let D b e a field of c haracteristic 6 = 2. W e assume that w e are in the symplectic situation ε = 1, J = id. G iven three La grangian X , Y , Z (not necessarily pairwise opp osite) w e consider the follo wing 3 n -dimensional quadratic f rom q X,Y ,Z on the direct sum X ⊕ Y ⊕ Z : q X,Y ,Z ( x, y , z ) = h ( x, y ) + h ( y , z ) + h ( z , x ) . If the Lagrangians are not pairwise opp osite, the quadratic form is going to ha v e a radical. The Kashiwar a-Maslov index of ( X , Y , Z ) is the class in the Witt group W D whic h is represen ted b y the nondegenerate part q + X,Y ,Z of q X,Y ,Z . F o r D = R , the Witt group W R is isomorphic to Z via the signature and the Maslov index can directly b e defined as the signature of q X,Y ,Z (ev en if the form is degenerate). This is essen tially Kashiw ara’s definition of the symplectic Maslo v index as dev elop ed in [13, 1.5.1]. If X , Y , Z are pa ir w ise o pposite, w e find tha t with resp ec t to our standard basis x , y , u t y for X ⊕ Y ⊕ Z the quadratic f orm is represen ted b y the matrix q X,Y ,Z =   0 − 1 0 0 0 t 1 0 0   . W e note that X ⊕ Y ⊕ 0 is a hyperb olic submo dule in X ⊕ Y ⊕ Z whose ortho g onal complemen t is spanned b y vec tors of the form ( tz , z , z ) ∈ D 3 n . The restriction of q X,Y ,Z to this subspace is giv en by z 7− → ( z T tz ), so q X,Y ,Z = q + X,Y ,Z is represe n ted b y [ t ] in WD . T his is our first result. 8.2 P r oposition I f X , Y , Z are pairwise opp osite L a grangians, then the Kashiw a ra-Maslo v index of ( X , Y , Z ) agrees with the image [ t ] of h t i = κ ( X, Y , Z ) in the Witt group W D . ✷ Next we get to Kashiw ara’s Maslo v co cycle, whic h is defined as follow s. W e fix a Lagrangian X 0 ∈ L and define τ : Sp 2 n D × Sp 2 n D ✲ W D via τ ( g , h ) = h q + X 0 ,g ( X 0 ) ,g h ( X 0 ) i . W e w a nt to r e late this group co cycle to o ur Maslo v co cycle defined in terms of the flag complex of the opp osition graph. 8.3 Recall the bar nota tion [2, I.5] for the standard free resolution of a g r oup G ov er Z . Its c hain complex is giv en as F n = Z G n +1 and the generator (1 , g 1 , g 1 g 2 , g 1 g 2 g 3 , . . . , g 1 · · · g n ) ⊗ 1 ∈ F n ⊗ G Z is denoted [ g 1 | . . . | g n ]. Then τ can b e view ed as the W D -v alued 2-ch o c hain [ g | h ] 7→ τ ( g , h ) for G = Sp 2 n D and one v erifies the cocycle iden tity [13, 1.5.8] . In general, supp ose that X is a set on whic h a group G acts, and that c : X × X × X ✲ A is a G - in v aria n t map ta king v alues in an a belian group A , suc h that c satisfies the co cycle iden tity c ( x, y , z ) − c ( w , y , z ) + c ( w , x, z ) − c ( w , x, y ) = 0. If w e choose a base p oin t o ∈ X , it is not difficult to see that the cocycle ( g 1 , g 2 , g 3 ) 7− → c ( g 1 ( o ) , g 2 ( o ) , g 3 ( o )) defined on the standard free resolution F ∗ of G ov er Z and the co cyc le g ⊗ ( x, y , z ) 7− → c ( x, y , z ) defined on 19 F 0 ⊗ G C 2 ⊆ F ∗ ⊗ G C ∗ are homologous ( C ∗ is the standard complex of k + 1-tuples of elemen ts of X ). How ev er, w e cannot use t his directly t o compare our Maslov co cycle with its classical coun terpart, s ince our co cycle is defined only on sp ec ial triples of Lagrangians. W e need to refine this idea, using some elemen tary homological algebra. W e do this in general, as w e need it also in the next section. 8.4 Let Γ = ( V , E ) b e a g raph with the star prop ert y . Supp ose that G is a g roup acting transitiv ely on the v ertices of Γ. Let o ∈ V b e a base p oin t and consider the induced graph Γ G on G under the map G ✲ V , g 7− → g ( o ) and its flag complex F ′ ∗ = C ∗ F l (Γ G ) ⊆ F ∗ . Ob viously , this c hain complex is a free resolution of G ov er Z and a sub c omplex of the standard free resolution F ∗ of G . Both c hain complexes F ∗ and F ′ ∗ can b e used to determine the group (co)homology of G . Supp ose no w that c : C 2 F l (Γ) ✲ A is a G -inv ariant co cycle. Then w e can construct t w o 2-co cycles for G , o ne via ˆ c : ( g 0 , g 1 , g 2 ) 7− → c ( g 0 ( o ) , g 1 ( o ) , g 2 ( o )) on F ′ 2 ⊗ G Z , and the other via c : g ⊗ ( x, y , z ) 7− → c ( x, y , z ) on F ′ 0 ⊗ G C 2 F l (Γ) ⊆ F ′ ∗ ⊗ G C 2 F l (Γ). Our first aim is to prov e t h at b oth co cy cles a re homolog ous . W e put C ∗ = C ∗ F l (Γ) a nd w e call a generator ( g 0 , . . . , g m ) ⊗ ( x 0 , . . . , x n ) ∈ F ′ m ⊗ G C n admissible if { g 0 ( o ) , . . . , g m ( o ) , x 0 , . . . , x n } consists of pairwise adjacen t elemen ts in Γ. This is a well- defined notion, i.e. in v aria n t under the left diagonal action of G . Let D ∗∗ ⊆ F ′ ∗ ⊗ G C ∗ denote the submo dule generated b y the admissible elemen ts. W e note that this submo dule is Z -free and closed under the v ertical and horizon tal differen tials, so it is a double complex. 8.5 Lemma Th e inclusion D ∗∗ ⊂ ✲ F ′ ∗ ⊗ G C ∗ induces an isomorphism in homology and co- homology (for coefficien t groups with trivial G - action). Pr o of. W e sho w that the relativ e homology gro ups of t h e pair ( F ′ ∗ ⊗ G C ∗ , D ∗∗ ) v anish. Let z ∈ L m + n = k F ′ m ⊗ G C n b e a r elat ive k -cycle and let ˜ z ∈ L m + n = k F ′ m ⊗ Z C n b e an elemen t whic h maps o nto z . W e c ho ose a group elemen t j suc h tha t for all terms ( g 0 , . . . , g m ) ⊗ ( x 0 , . . . , x n ) app earing in ˜ z , the v ertex j ( o ) is adjacen t to g 0 ( o ) , . . . , g m ( o ) , x 0 , . . . , x m (this is a w ell-defined condition as w e w ork with ˜ z ∈ L m + n = k F ′ m ⊗ Z C n where t he G -action is not factored out). Consider the k +1 -c hain j # ˜ z , whose ( m + 1 , n )-terms are o f the f orm ( j, g 0 , . . . , g m ) ⊗ ( x 0 , . . . , x n ). The total differential is ∂ ( j # ˜ z ) = ˜ z − j #( ∂ ˜ z ) . Pro jecting this equation back to F ′ m ⊗ G C n , w e see that the image of j # ∂ ˜ z is in D ∗ +1 , ∗ . Th us z is a relative b oundary and H ∗ ( F ′ ∗ ⊗ G C ∗ , D ∗∗ ) = 0. F rom the long exact homolog y sequence w e get a n isomorphism H ∗ ( D ∗∗ ) ∼ = ✲ H ∗ ( F ′ ∗ ⊗ G C ∗ ). Since b oth F ′ ∗ ⊗ G C ∗ and D ∗∗ are Z - free, the univers al co efficien t theorems and the 5-Lemma yield isomorphisms for homology and cohomology with arbitrary co efficien t g roups A (with trivial G -actio n), see [26, 5.3.15,5 .5 .3]. ✷ 20 The remaining part of the comparison is routine. W e denote elemen ts o f G by g , h, i and v ertices of Γ by u, v , w . W e define t w o 1- coc hains f 1 , f 2 on D ∗∗ b y f 1 (( g ) ⊗ ( u, v )) = c ( g ( o ) , u, v ) and f 2 (( g , h ) ⊗ ( u )) = c ( g ( o ) , h ( o ) , u ) , where c is the giv en G -inv arian t 2-co cycle on F l (Γ). Then d f ν = f ν ∂ a nd using the co cycle iden tity for c , w e obtain d f 1 (( g ) ⊗ ( u, v , w )) = ( c ( g ( o ) , v , w ) − c ( g ( o ) , u, w ) + c ( g ( o ) , u, v ) = c ( u , v , w ) d f 1 (( g , h ) ⊗ ( u , v )) = c ( h ( o ) , u , v ) − c ( g ( o ) , u, v ) d f 2 (( g , h ) ⊗ ( u , v )) = − c ( g ( o ) , h ( o ) , v ) + c ( g ( o ) , h ( o ) , u ) = d f 1 (( h, i ) ⊗ ( x, y )) d f 2 (( g , h, i ) ⊗ ( u )) = c ( h ( o ) , i ( o ) , u ) − c ( g ( o ) , i ( o ) , u ) + c ( g ( o ) , h ( o ) , u ) = c ( g ( o ) , h ( o ) , i ( o )) whic h sho ws that d f 1 − d f 2 = c − ˆ c. 8.6 Theorem Let G b e a gr o up acting verte x-transitiv ely on a gra ph Γ ha ving the star pro p - ert y , let c : F l 2 (Γ) ✲ A b e a G -in v ariant A -v alued 2 - cocycle (where G acts trivially on A ). Fix a v ertex o of Γ and let F ′ ∗ ⊆ F ∗ and C ∗ b e as in 8.4 . Then the co cycles ˆ c : F ′ 2 ⊗ G Z ✲ A, ( g 0 , g 1 , g 2 ) ⊗ 1 7− → c ( g 0 ( o ) , g 1 ( o ) , g 2 ( o )) and c : F 0 ⊗ G C 2 ✲ A, g ⊗ ( x, y , z ) 7→ c ( x, y , z ) are homologous under the isomorphism H 2 ( G ; A ) ∼ = ✲ H 2 G ( C ∗ ; A ) . Moreo ve r, there exists a co cycle ˆ ˆ c : F 2 ⊗ G Z ✲ A extending ˆ c , i.e. ˆ c = ˆ ˆ c | F ′ ∗ . Pr o of. Only the la st claim remains to b e pro ve d. Since the inclusion F ′ ∗ ⊆ F ∗ induces a n isomorphism in cohomo lo gy , w e find a co cycle ˜ ˆ c on F ∗ ⊗ G Z suc h that ˆ c − ˜ ˆ c | F ′ ∗ ⊗ G Z = da is a cob oundary . Now F ′ ∗ ⊗ G Z is a direct summand in the Z -free mo dule F ∗ ⊗ G Z , so w e can extend a to a 1-co c hain ˜ a on F ∗ ⊗ G Z . Then ( ˜ ˆ c + d ˜ a ) | F ′ ∗ ⊗ G Z = ˆ c . ✷ 8.7 Corollary F or a field D of c haracteristic 6 = 2 , Kashiw ar a ’s Maslov co cycle and our Maslo v co cyc le yield the same cohomology class in H 2 ( Sp 2 n D ; W D ) . ✷ W e obtain also t he following general result for unitary gr o ups o ver arbitrary sk ew fields. 8.8 Corollary If D ε is infinite and o ∈ L is a fixed Lagr a ngian, then there exists a group co cyc le τ : U ( M ) × U ( M ) ✲ W 1 ( D , J ) suc h that τ ( g , h ) = h κ ( o , g ( o ) , g h ( o )) i holds for all pairs g , h with o , g ( o ) , g h ( o ) pairwise opp osite. ✷ 21 9 The Maslov co cycle as a cen tral e xtension The reduced Maslov co cycle defines a cen tral extension [2, IV.3] [6, 1.4C] 1 ✲ I I ( D , J ) ✲ \ EU ( M ) ✲ EU ( M ) ✲ 1 of EU ( M ) by I I ( D , J ). This extension is uniquely determined by the homomorphism [ e m ] ∈ H 2 ( EU ( M ); I I ( D , J )) ∼ = Hom( H 2 ( EU ( M )) , I I ( D , J )); our aim is to determine this homomorphism H 2 ( EU ( M )) ✲ I I ( D , J ) algebraically . In view of the naturality we prov ed in Section 6, we b egin with the smallest case Sp 2 D = SL 2 D , where D is an infinite field. W e do a llow fields of c haracteristic 2, as w e rely on results in [15] and [17] whic h are v alid o ver ar b itrary (infinite) fields. Note, how ever, that in our set-up the Witt group W 1 ( D , id) is alwa ys the Witt group of symmetric bilinear forms (a nd not of quadratic forms). 9.1 The Sch ur mu ltiplier of SL 2 D and the Stein b erg co cycle W e put u t =  1 t 0 1  a r =  r 0 0 r − 1  b r =  0 r − r − 1 0  Since SL 2 D is a tw o-transitiv e group, ev ery elemen t is either of the form a r u t or of the fo rm u s b r u t . W e define K S p 2 D a s the ab elian group g enerated b y sym b ols { x, y } , for x, y ∈ D ∗ , (the symple ctic Steinb er g symb ols ), sub ject to the relations { st, r } + { s, t } = { s, tr } + { t, r } { s, 1 } = { 1 , s } = 0 { s, t } = { t − 1 , s } { s, t } = { s, − st } { s, t } = { s, (1 − s ) t } if s 6 = 1 . According to [1 7 , p. 199] [15, 5.11] the Sc h ur m ultiplier o f SL 2 D is H 2 ( SL 2 D ) ∼ = K S p 2 D . Moreo ve r, the Stein b erg normal f orm of the univers al group co cycle stbg : SL 2 D × SL 2 D ✲ H 2 ( SL 2 D ) is giv en for ’generic’ group elemen ts b y stbg ( g ( s 1 , r 1 , t 1 ) , g ( s 2 , r 2 , t 2 )) = { t r 1 r 2 , − r 1 r 2 } − {− r 1 , − r 2 } where t = t 1 + s 2 6 = 0 and g ( s, r, t ) = u s b r u t , cp. [17, p. 198 (1)], [15, 5.12] and [11] in a more sp e cial situation. W e note that the formula giv en in [17, p. 1 98 (1)] is incorrect. The form ula ab o v e is due to Sch w arze [2 4 , 5 .9] and agrees with Matsumoto’s calculations. 9.2 Giv en x, y ∈ D , w e denote b y ( x, y ) D the 4-dimensional symmetric bilinear fo rm ( x, y ) D = h 1 , − x, − y , xy i . 22 If char( D ) 6 = 2, t h is is the norm form o f the quaternion algebra  x,y D  [23, 2. § 11]. Ob viously , ( x, y ) D ∈ I I ( D , id), and ( x, y ) D = ( y , x ) D = ( xz 2 , y ) D . Us ing the fact that the metabolic form h x, − x i v anishes in W 1 ( D , id), it is routine to verify that these elemen ts satisfy the first four defining relatio ns of K S p 2 D ; for example ( s, − st ) D = h 1 , − s, st, − s 2 t i ∼ = h 1 , − s, st, − t i ∼ = ( s, t ) D . F or the last relation, it suffices to chec k that h − t, st i ∼ = h− (1 − s ) t, (1 − s ) st i for s 6 = 1. This follo ws from  1 1 s 1   − t 0 0 st   1 s 1 1  =  − (1 − s ) t 0 0 (1 − s ) st  Th us w e hav e a homomorphism R : K S p 2 ( D ) ✲ I I ( D , id) ⊆ W 1 ( D , id) whic h maps the symplectic Stein b erg sym b ol { u, v } t o t he 4- dimensional symmetric bilinear form R ( { u, v } ) = ( u , v ) D . Applying R t o the Stein b erg co cyc le, w e o b tain (with t he same notation as b efore) for ’generic’ group eleme n ts R ◦ stbg ( g ( s 1 , r 1 , t 1 ) , g ( s 2 , r 2 , t 2 )) = ( t r 1 r 2 , − r 1 r 2 ) D − ( − r 1 , − r 2 ) D = ( r 1 r 2 t, r 1 r 2 ) D − ( − r 1 , − r 2 ) D = ( − r 1 r 2 , t ) D − ( − r 1 , − r 2 ) D = h 1 , r 1 r 2 , − t, − r 1 r 2 t i − h 1 , r 1 , r 2 , r 1 r 2 i = h 1 , r 1 r 2 , − t, − r 1 r 2 , t, − 1 , − r 1 , − r 2 , − r 1 r 2 i = h− t, − r 1 r 2 t, − r 1 , − r 2 i = −h t, r 1 r 2 t, r 1 , r 2 i . 9.3 W e compare this expression with the reduced Maslo v co cy cle. In M = D 2 w e put x =  1 0  , X = x D , a nd o = ( X , x ). Using the notat io n o f 8.4, w e ha v e for F ′ 2 the form ula τ ( g 1 , g 2 ) = τ ([ g 1 | g 2 ]) = e m ( o, g 1 ( o ) , g 1 g 2 ( o )) where three v ertices o, g 1 ( o ) , g 1 g 2 ( o ) ha ve to b e pairwise opp osite. F or the first pair o f v ertices, this condition gives g 1 = u s 1 b r 1 u t 1 , and for the second pair g 2 = u s 2 b r 2 u t 2 . Then τ ([ g 1 | g 2 ]) = e m ( o, g 1 ( o ) , g 1 g 2 ( o )) = e m ( g − 1 1 ( o ) , o, g 2 ( o )) = − e m ( o, g − 1 1 ( o ) , g 2 ( o )) = − e m ( o, g ( − t 1 , − r 2 , − s 1 )( o ) , g ( s 2 , r 2 , t 2 )( o )) = − e m ( o, u − t 1 b − r 1 ( o ) , u s 2 b r 2 ( o )) = − e m ( o, b − r 1 ( o ) , u t 1 + s 2 b r 2 ( o )) whic h yields the additiona l conditio n t = t 1 + s 2 6 = 0 t ha t ensures that the first and third v ertex are opp osite. Note that b y 8.4 t he class o f a n y 2-co cycle is completely determined b y its v alues on F ′ 2 , so it suffices indeed to w ork with ’generic’ elemen ts. The explicit form ula in 7.8 for the reduced Maslo v co cycle yields no w a = 1, b = r − 1 1 and c = − r − 1 2 , whence τ ([ g 1 | g 2 ]) = −h t, r 1 , r 2 , r 1 r 2 t i = R ∗ stbg ([ g 1 | g 2 ]) . 23 F o r SL 2 D ov er fields of c haracteristic 6 = 2, the follow ing result w as pro v ed in [19, Sec. 5] and [1]. 9.4 Theorem Let D b e an infinite field. The cen tral extension of Sp 2 n D determined by the reduced Maslo v co cycle is giv en by the homomorphism R : K S p 2 D ✲ I I ( D , id) . Pr o of. F or n = 1 w e show ed this in 9.3 ab ov e. In general, the standard inclusion Sp 2 n D ⊂ ✲ Sp 2 n +2 D induces for all n ≥ 1 an isomorphism in 2- dim ensional homology , suc h that the univ ersal Steinberg co cyc le fo r Sp 2 n +2 D r e stricts to the univ ersal Stein b erg co cycle for Sp 2 n D [15, 5.11]. The result follo ws no w b y induction on n . ✷ F o r fields o f characteristic 6 = 2, this is stated in [21, 3.1]. Ho w ev er, the pro of has a gap: the authors ev aluate t he reduced Maslov co cyc le on the t o rus (the diagonal ma t r ic es) and compare it there with the univ ersal Stein b erg co cycle. But they f a il to sho w that the reduced Maslov co cyc le is a Stein b erg co cycle, so they cannot use the comparison theorem [15, 5.10]. In any case, this result settles the situation for symplectic gro ups ov er infinite fields of arbitrary c haracteristic. Note that for fields of c haracteristic 6 = 2, the map R is surjectiv e [23, 4.5.5], so \ Sp 2 n D is an epimorphic image of the univ ersal cen tra l extension. 9.5 Lo cal fields By a lo cal field w e mean a lo cally compact (nondiscrete) field; the connected lo cal fields are R , C and the totally disconnected ones are the finite extensions of the p -adic fields Q p and, in p o s itiv e c haracteristic, the fields F q (( X )) of formal Lauren t series ov er finite fields [29, 1.3]. Being a closed subgroup of the g e neral linear group, a symplectic or unitary gro up o v er a lo cal field is in a natural w a y a lo cally compact group. 9.6 The Maslo v c ocycle ov er R F o r D = R , the Witt group W R = W 1 ( R , id) is isomorphic to Z via the signature sig : W R ✲ Z [23, 2.4.8 ]; the fundamen tal ideal I R has index 2 , and I I R = I 2 R has index 4. W e note that sig (( x, y ) D ) = ( 4 if x, y < 0 0 else By [17, 10.4] [1 5 , p. 51], this 4 Z -v alued co cycle yields precisely t he univ ersal cov ering gro up ^ Sp 2 n R of Sp 2 n R . W e compare the relev an t classifying spaces. Let B Sp 2 n R δ denote the classifying space for Sp 2 n R , view ed as a discrete top ological group, and B Sp 2 n R the classifying space for the Lie group Sp 2 n R ; the lat t er is homotopy equiv alen t to B U ( n ), as U ( n ) ⊆ Sp 2 n R is b y [7, X T ab. V] and Iw asaw a’s Theorem [7, VI § 2] a homotopy equiv alence. T he classifying space B Sp 2 n R δ is an Eilen b erg-MacLane space of t yp e K ( Sp 2 n R , 1) whose cohomology is naturally isomorphic to the abstract group cohomolo gy of Sp 2 n R [2, I I.4]. The iden tity map from the discrete group to t h e Lie gr o up induces a contin uous map betw een the classifying spaces F : B Sp 2 n R δ ✲ B Sp 2 n R . On the right, the univ ersal cov ering is classified b y the first Chern class c 1 . This shows that under the forg e tful map F ∗ the fir st univ ersal Chern class c 1 ∈ H 2 ( B SU ( n )) ∼ = H 2 ( B Sp 2 n R ) 24 pulls bac k to the Maslo v co cyc le, F ∗ ( c 1 ) = [ e m ] (if the sign for the Chern classes is c hosen appro priately ). The real Maslo v co cycle ma y b e view ed therefore as a com binatoria l description of the first Chern class; this w as observ ed in [28]. 9.7 P r oposition U nder the forgetful functor from top ological groups to abstract g roups , the first Chern class for Sp 2 n R maps to the reduced Maslov co cycle. ✷ As I I ( C , id) = 0 the Maslo v co cycle fo r Sp 2 n C v anishes. Now w e turn to nonarchimede an lo cal fields, cp. [13 , p. 104-115]. 9.8 W e assume that D is a nonarchime dean and nondyadic lo cal field (i.e., the c haracteristic of the residue field o f D is 6 = 2). The Witt group WD has 16 elemen ts, the group b S of extended square classes 8, and thus I I ( D , id) = I 2 D is cyclic of order 2 [12, VI.2.2]. Its nontrivial elemen t is represen ted by the norm form of the unique quaternion division algebra ov er D . Let S denote the group of square classes of D , and ( − , − ) H : S × S ✲ {± 1 } the Hilb ert sym b ol [12, p. 159]: ( x, y ) H = − 1 if ( x, y ) D is anisotropic, i.e. the norm form of a quaternion division a lgebra. Put e : I 2 D ∼ = ✲ {± 1 } , then we clearly ha v e e ◦ R ◦ stbg ( x, y ) = ( x, y ) H for SL 2 D . Thus the r educed Maslov co cycle fo r SL 2 D is the reduction of the univ ersal co cy cle stbg to {± 1 } via the Hilb ert sym b ol. As in the pro of of 9.4, this carr ies ov er to Sp 2 n D . The follo wing result is partially con tained in [13, p. 104-115]. 9.9 P r oposition Le t D b e a nonarc himedean nondyadic lo cal field. The reduced Maslo v co cyc le defines a tw ofold non trivial cov ering of Sp 2 n D whic h is determined b y the Hilb ert sym b ol K S p 2 D ✲ {± 1 } . The corresp onding co vering group \ Sp 2 n D is a lo cally compact group; it is the unique non trivial tw ofold cov ering of Sp 2 n D in the category o f lo cally compact groups. Pr o of. Only the top ological result remains to b e pro ve d. It is sho wn in [17, 10.4] that in the category of lo cally compact groups, Sp 2 n D admits a univ ersal cen tral extension ^ Sp 2 n D ; the extending group is the g roup µ ( D ) o f a ll ro ots of unit y in D . (See [22] for a mo dern account and a m uch more general result.) This group µ ( D ) is a finite cyclic gro up [17, Ch. I I] and of ev en order 2 n , as it contains the in volution − 1. The cor r esp onding St einberg co cycle is given b y the norm residue sym b ol K S p 2 D ✲ µ ( D ) [17, Ch. I I]. But the n th p o w er of the norm residue sym b ol is the Hilb ert sym b ol. This sho ws that \ Sp 2 n D is a con tinuous quotien t of ^ Sp 2 n D . As the cyclic gro up µ ( D ) has a unique subgroup of index 2, the extension is the unique nonsplit t wofold top ological extension. ✷ 25 9.10 Finally , w e consider unitary groups ov er fields. W e assume that E is a field with an automorphism J 6 = id of order 2; the fixed field is D ⊆ E . W e denote t he hy p erbolic unitary group by U 2 n E ; then EU 2 n E = SU 2 n E = U 2 n E ∩ SL 2 n E [6, 6 .4.25,6.4.27]. As w e noted in Section 6, there is a natural injection Φ : Sp 2 n D ⊂ ✲ SU 2 n E and w e hav e a comm utativ e diagram H 2 ( Sp 2 n D ) Φ ∗ ✲ H 2 ( SU 2 n E ) I I ( D , id) [ e m D ] ❄ W D E ✲ I I ( E , J ) . [ e m E ] ❄ Unfortunately , the Sc hur multiplier H 2 ( SU 2 n E ) seems to b e less understo o d than its symplectic coun terpart. Ho we v er it is prov ed in [4, 2.1,2.5] (and in a weak er form in [6, 6.5.12 ]) tha t the map Φ ∗ is surjectiv e, so H 2 ( SU 2 n E ) is a quotient of K S p 2 D . ( SU 2 n E is the g roup of D -p oin ts of a quasisplit absolutely simple and simply connected a lg e braic group ov er D , so the results from [4 ] apply .) The following facts concerning W D E w ere kindly p oin ted out b y W. Sc harlau. Firstly , the map W D E : W 1 ( D , id) ✲ W 1 ( E , J ) is an epimorphism, because ev ery hermitian f orm can b e diago na liz ed (ev en in c hara c teristic 2 [9, I.6.2 .4]) and th us is the image of a diagonal symmetric bilinear form ov er D . Assume no w t ha t c har( D ) 6 = 2 a nd E = D ( √ δ ). Pass ing from a hermitian form h o ve r E to its trace form b h o ver D [23 , p. 3 48], we hav e an monomorphism trf : W 1 ( E , J ) ✲ W 1 ( D , id) = W D ; explicitly , trf h a 1 , . . . , a n i = h 1 , − δ i ⊗ h a 1 , . . . , a n i . In particular, trf ◦ W D E (( x, y ) D ) = h 1 , − δ i ⊗ h 1 , − x i ⊗ h 1 , − y i , and W D E ( I 2 D ) is isomorphic to a subgroup of I 3 D . It fo llows that the Maslo v co cycle fo r the unitary g roup o ver a nonarc himedean nondyadic lo cal field E v anishes, b ecaus e I 3 D = 0 [12, VI.2.15(3)]. The case of the complex n umbers is more in teresting. 9.11 Complex unitary groups F o r E /D = C / R the map W R C : W R ✲ W 1 ( C , ¯ ) and its restriction I 2 R ✲ I I ( C , ¯ ) is an isomorphism. W e use the standard Lie group notatio n SU 2 n C = SU ( n, n ) [7] (note that m ultiplication by i tr ans forms ske w hermitian in to hermitian matrices). The maximal compact subgroup is S ( U ( n ) × U ( n )). As in 9.6 w e compare the classifying space f o r t he discre te gro up (whose homology is the abstract group homology) with the classifying space B SU ( n, n ) for the Lie group. F or n = 1 w e hav e an isomorphism Sp 2 R = SU 2 C , whence a big comm utativ e diagram 26 H 2 ( Sp 2 R ) = = = = = = = = = = = = = = = H 2 ( SU (1 , 1)) F ∗ ✲ H 2 ( B SU (1 , 1)) I 2 R ϕ ∗ ∼ = [ e m R ] ✲ ✲ I I ( C , ¯ ) 1 4 sig ∼ = [ e m C ] ✲ ✲ Z c 1 ∼ = ✲ H 2 ( Sp 2 n R ) ❄ ✲ [ e m R ] ✲ H 2 ( SU ( n, n )) ❄ F ∗ ✲ [ e m C ] ✲ H 2 ( B SU ( n, n )) . ∼ = ❄ ∼ = c 1 ✲ 9.12 P r oposition I f w e iden tify the first Chern class c 1 with the generator of H 2 ( B SU ( n, n )) , it pulls under the forg e tful map F bac k to the reduced Maslo v co cycle f or SU ( n, n ) . Th us \ SU ( n, n ) is the univ ersal co ve ring g roup of t he Lie group SU ( n, n ) . ✷ 9.13 Sharp e [25] [6, 5.6D*] has constructed an exact seque nce K 2 ( D ) ✲ K U − 1 2 ( D , J ) ✲ L 1 0 ( D , J ) ✲ 0 The L -group L 1 0 ( D , J ) maps on to I I ( D , J ) and we conjecture that the compo s ite K U − 1 2 ( D , J ) ✲ I I ( D , J ) ’is’ (in most cases) the reduced Maslo v co cycle e m : H 2 ( EU ( M )) ✲ I I ( D , J ). In the sym- plectic situation ov er fields of c haracteristic 6 = 2, this is indeed t he case b y 9.4 and [6, 5.6.8]. Ho we v er, a pro of w ould certainly require a differen t description of the relev an t maps than the one in [25]. References [1] J. Barge, Co cycle d’Euler et K 2 , K -Theory 7 (1993 ) , no. 1, 9–16. MR1220423 (94d:19003) [2] K. S. Brow n, Cohomolo gy o f gr oups , Corrected reprin t of the 1982 original, Springer, New Y ork, 1994. MR1324339 (96a:20072) [3] S. E. Capp ell, R. Lee and E. Y. Miller, On the Maslo v index, Comm. Pure Appl. Math. 47 (1994 ), no . 2 , 121 –186. MR1 263126 (95f:570 4 5) [4] Vinay V. D eodhar, On cen tr a l extensions o f rational p oin ts of algebraic groups, Amer. J. Math. 100 (1978), no. 2, 303 – 386. MR48 9962 ( 80c:20058) [5] S. Eilen b erg a n d N. Steenro d, F oundations of algebr aic top olo gy , Princeton Univ. Press, Princeton, New Jersey , 1952. MR0050886 (14,398b) [6] A. J. Hahn and O. T. O’Meara, The clas s i c al gr oups and K -the ory , Springer, Berlin, 1989. MR10 07302 (90i:20002 ) 27 [7] S. Helgason, Differ ential ge om et ry, Lie gr oups, and symm e t ric sp ac es , Corrected reprin t o f the 1978 orig inal, Amer. Math. So c., Pro vidence, RI, 2001 . MR183 4454 (2002b:5308 1) [8] N. Knarr, Pro jectivities of generalized p olygons, Ars Comb in. 25 (1988), B, 265–275 . MR0942482 (89e:20008) [9] M.-A. Kn us, Quadr atic and Hermitian forms o ver rings , Springer, Berlin, 1991. MR1096299 (92i:11039) [10] L . Kramer, Buildings and classical groups, in Tits buildings and the mo del the ory o f gr oups (W ¨ urzbur g, 2000) , 5 9–101, Cambridge Univ. Press, Cambridge. MR2018 3 82 (2005b:2005 8) [11] T. Kub ota, T op ological co ve ring of SL(2) ov er a lo cal field, J. Math. So c. Japan 19 (1967), 1 14–121. MR0204422 (34 #4 2 64) [12] T. Y. Lam, Intr o duction to quadr atic forms over fields , Amer. Math. So c., Prov idence, RI, 20 05. MR2104929 (2005h:11075 ) [13] G . Lion and M. V ergne, The W eil r epr esentation, Maslov index and t heta series , Progr. Math., 6, Birkh¨ auser, Boston, Mass., 198 0. MR0573448 (81j:58075) [14] S. Mac Lane, Homolo gy , Reprin t of the 1975 edition, Springer, Berlin, 1 995. MR1344215 (96d:18001) [15] H. Matsumoto, Sur les sous-group es arit h m ´ etiques des group es semi-simples d ´ eplo y´ es, Ann. Sci. ´ Ecole Norm. Sup. (4) 2 ( 1 969), 1–62. MR0240214 (39 #15 66) [16] A. Mazzoleni, Partially defined co cyc les and the Maslo v index f or a lo cal ring, Ann. Inst. F ourier (Grenoble) 54 (2004), no. 4, 875–885. MR2 1 11015 (200 5h:20119) [17] C. C. Mo ore, Group extensions of p -adic and adelic linear groups, Inst. Hautes ´ Etudes Sci. Publ. Math. No. 35 ( 1968), 157–222. MR0244258 (39 #55 75) [18] K .- H. Neeb and B. Ø r s t e d, A top ological Maslov index for 3- graded Lie groups, J. F unct. Anal. 233 (2 006), no . 2, 426–477. MR2214583 (2007h:53127 ) [19] Y a. Nek ov a r, Maslov index and Clifford algebras, F unktsional. Anal. i Prilozhen. 24 (1990), no. 3, 36–4 4 , 96; tra n slation in F unct. Anal. Appl. 24 (1990), no. 3, 196–2 0 4 (1991). MR1 082029 (92b:1102 4 ) [20] M. V. Nori, The unive rsal prop ert y of the Maslo v index, J. Raman uja n Math. So c. 13 (19 98), no. 2, 1 11–124. MR1666437 (2000e:11039 [21] R . P a r im ala, R. Preeti and R. Sridharan, Maslov index and a cen tral extension of the symplectic group, K -Theory 19 (2000), no. 1, 29–45 . MR17408 81 (20 01c:11053a) R. P arima la , R. Preeti and R. Sridharan, Errata : “Maslov index and a cen tr a l extension of the s ymplectic group”, K -Theory 19 (20 0 0), no. 4, 403. MR1763935 (2001c:11053 b) 28 [22] G . Pra sad, Deligne’s top ological cen tral extension is univ ersal, Adv. Math. 181 (2004), no. 1 , 160 –164. MR2 0 20658 (2004k:2009 7) [23] W. Sc harlau, Qu adr atic and Hermitian forms , Springer, Berlin, 1 985. MR077 0063 (86k:11022) [24] R . Sch w arze, Gr oup extens i ons and the Maslov ind e x , Diploma Thesis, Bielefeld, 2008. [25] R . W. Sharp e, On the structure of the unita r y Stein b erg group, Ann. of Math. (2) 96 (1972), 4 44–479. MR0320076 (47 #8 6 17) [26] E. H. Spanier, Algebr aic top olo gy , Corrected reprin t , Springer, New Y ork, 198 1 . MR0666554 (83i:55001) [27] J. Tits, Buildings of spheric al typ e a nd fi nite BN-p airs , Lecture Notes in Math., 3 8 6, Springer, Berlin, 19 7 4. MR0470099 (57 #986 6) [28] V. G . T uraev, A co cycle of the symplectic first Chern class a nd Maslov indices, F unk- tsional. Anal. i Prilozhen. 18 (1984), no. 1, 43–48. MR0739088 (85m:58191) [29] A. W eil, B asic numb er the ory , Reprint of the second (1973) edition, Springer, Berlin, 1995. MR13 44916 (96c:11002) Lin us Kramer Katrin T en t Mathematisc hes Institut, Univ ersit¨ at M ¨ unster, Einsteinstr. 62, 48149 M ¨ unster, German y e-mail: linus.kramer@u ni-muenster .de e-mail: tent@uni-muens ter.de 29