Computing Borels Regulator

Computing Borel’s Regulator Zac ky Cho o, W a jid Mannan, Rub ´ en J. S´ anc hez-Garc ´ ıa Victor P . Snaith No v em ber 16 , 2018 Abstract W e present an inﬁnite series formula based on the Karoubi-H amida in tegral, for the universal Borel cl ass ev aluated on H 2 n +1 ( GL ( C )). F or a cy clotomic ﬁeld F w e d eﬁne a canonical set of elements in K 3 ( F ) and p resen t a nov el approac h ( based on a free diﬀerential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H 3 ( GL ( C )) u nder the H urewicz map. Applying our form ula to these images yields a val ue V 1 ( F ), whic h coincides with th e Borel regulator R 1 ( F ) when our set is a basis of K 3 ( F ) mo dulo t orsion. F or F = Q ( e 2 πi/ 3 ) a computation of V 1 ( F ) has b een made based on our techniques. 1 In tro duction Let F be an algebraic num b er ﬁeld and O F its ring of in tegers. F or n ≥ 1, the Borel regulator R n ( F ) is a real v alued numerical in v arian t of F . It measures the co volume of the algebraic K -theory groups K 2 n +1 ( F ) mod u lo torsion, em b edded as a lattice in R d n (where t h e in teger d n is to b e sp eciﬁed later). Although deﬁn ed in terms of th e o dd-dimensional K-theory groups, knowledge of the Borel regulator has implications for the even dimensional K-th eory groups of F and O F . In particular, the Lich tenbaum conjecture (prov en in many cases, such as ab elian extensions of the rationals [5 , 15 , 18, 23]) gives the order of K 2 n ( O F ) up to a p ow er of 2 in terms of t h e Dedekind zeta function of F , the order of T or ( K 2 n +1 ( F )) and R n ( F ). How ever, explicitly computing the Borel regulator is a very diﬃcult problem, even in th e case of cyclotomic num b er ﬁelds. The standard approach to th e Borel regula- tor is via comparison with the Beilinson regulator [4], which in turn is expressed via p olylogari thms; moreo ver, Zagier’s conjectures [26], which generalize classical results of Blo c h [2, 12], allo w one to map the higher Bloch group B n +1 ( F ) mod ulo torsion to a lattice in R d n . How ever, the identiﬁcation of B n +1 ( F ) mo dulo torsion with a full sublattice of K 2 n +1 ( F ) is d elicate [6]. W e presen t a new approach to computing R 1 ( F ) when F is a cyclotomic ﬁeld. W e ﬁrst d escribe a set of elemen ts z u ∈ K 3 ( F ) correspondin g to primitive roots of unity u ∈ F (Section 2). W e th en explain how to compute the covo lume V 1 ( F ) of the lattice generated by the images of these elements in the real vector space R d 1 . If the elemen ts z u ∈ K 3 ( F ) form a basis mo dulo torsion, th en R 1 ( F ) = V 1 ( F ) holds and we hav e computed the desired Borel regulator. Otherwise V 1 ( F ) is an integer m ultiple of R 1 ( F ). Determining for whic h cyclotomic ﬁelds the elemen ts z u form a basis modu lo torsio n is b eyond the scope of this article. Nevertheless, our formula (Theorem 5.11) allo ws one in principle to compu te V 1 ( F ) to any desired d egree of accuracy . In particular, one could numerically verify if V 1 ( F ) 6 = 0, in which case th e elemen ts z u w ould at least generate a full sub-lattice modu lo torsion. The elements z u ∈ K 3 ( F ) hav e the adv an tage that we are able to exp ress their images under th e Hurewicz map explicitly in H 3 ( GL ( F )). Th us they are amenable to This research w as funded b y the EPSRC grant EP/C549074/1. 1 application of our formula for the universal Borel class (see b elo w). Un derstanding how our elements relate to other elements in the K - theory of cyclotomic ﬁelds app ears to b e an intricate problem, and motiv ates further investig ation. Computing V 1 ( F ) breaks d o wn into tw o stages. Firstly (S ection 4) we construct the images of the z u in H 3 ( E ( C )) as explicit chains ( E ( C ) denotes the discrete group of elemen tary matrices o ver the complex num b ers). One of the diﬃculties in computing the Borel regulator of a ﬁeld F is that it is very hard to get such an explicit description of elements in t he K–theory of F o r at least t heir images in H 3 ( GL ( F )). W e overc ome this ( S ection 3) using no vel tec hniques b ased on constructing a free diﬀerential calculus (motiv ated by the one in ven ted by F ox as a tool in knot theory [10]). Then (Section 5) we show how to app ly the u nivers al Borel class to th ese chains by ex panding th e Karoubi-H amida integral [13] as an inﬁnite p ow er series (Theorem 5.11). In fact, our form ula works for any n ≥ 1 and any number ﬁeld F ; it allow s the computation of the Borel regulator class (deﬁned in § 2.1) applied to any element of H 2 n +1 ( GL ( F )) (Theorem 5.4). At p resen t the deﬁnition of V 1 ( F ) is limited to F a cyclotomic ﬁeld. Ho w ever, our introduction of a free diﬀerential and the metho ds based on it to describ e elements in K 3 ( F ) are general (cf. R emark 2.2), and could in principle b e applied to computing the Borel regulator of any ﬁ eld. F urther, the form ula d evel op ed in Section 5 w orks for ev aluating th e u nivers al Borel class on an y num ber ﬁeld (Theorem 5.4). Note that our elements lie in K 3 ( F ), so they do not allo w th e inv estigation of R n ( F ) for n > 1. In § 2.3, Remark 2.5, w e ind icate ho w one might generalize our construction to n > 1, by considering th e K–theory of spherical v arieties. Observ e that the form ula for the Karoubi-Hamida integra l w orks for any n > 1 (Theorem 5.4). The theory developed in th is p aper h as b een implemented as a computer algorithm by the 1 st author, in his PhD t hesis [7]. This gives in an actual estimate for V 1 ( F ) for the ﬁeld F = Q ( e 2 πi/ 3 ), and it is d escrib ed in App endix B. W e hav e also included some general computational aspects of our app roac h for anyone in terested in an implemen- tation (App endix A). R emark 1.1 . The 1 st and 4 th authors have results analogous to t hose in this article for the p -adic regulator [8]. 2 Deﬁning V 1 of a cyclotomic ﬁeld Let F b e th e cy clotomic num b er ﬁeld Q ( ω ), where ω is a q th root of unity ( q ≥ 3). A large part of the d iﬃcult y in computing the Borel Regulator of F comes from the inaccessibilit y of elements of in the K -theory of F . In t h is section w e will deﬁ ne a canonical set of elements in K 3 ( F ), deﬁn ed up to torsion ( § 2.2). The images of this set und er the Borel Regulator map generate a lattice and w e deﬁne V 1 ( F ) to b e its co volume ( § 2.3) . In S ection 4 we will then pro ceed to express our elements of K 3 ( F ) in the form necessary to compute V 1 ( F ). W e ﬁrst recall the deﬁ n ition of the Borel regulator of a num ber ﬁeld ( § 2.1). 2.1 The Borel regulator Let F b e a number ﬁeld, O F its ring of integers and q 1 , q 2 the num b er of rea l em- b eddings, respectively conjugate pairs of complex embed dings, F ֒ → C . The Bor el r e gulator maps are homomorphisms K 2 n +1 ( O F ) ∼ = K 2 n +1 ( F ) − → R d n ( n ≥ 1) (1) from the odd algebraic K -t heory groups of F ( or O F ) to R d n , where d n is q 1 + q 2 if n is even and q 2 if n is o dd. They can b e deﬁned in the follo wing w a y (cf. § 3.1.3 in [11]). The Hurewicz homomorphism induces the follo wing homomorphism from K-theory into the homology (with integral co eﬃcients) of the discrete group GL ( C ): h 2 n +1 : K 2 n +1 ( C ) = π 2 n +1 ( B GL ( C ) + ) / / H 2 n +1 ( B GL ( C ) + ) ∼ = H 2 n +1 ( GL ( C )) . R emark 2.1 . Suslin’s stability result [16, Corollary 2.5.3], gives th at if N ≥ 2 n + 1 then H n ( GL ( C )) ∼ = H n ( GL N ( C )) . 2 Let R ( n ) = (2 π i ) n R for n ≥ 1. There e xists a universal Bor el class (see [4] for a deﬁnition) b n ∈ H 2 n +1 c ( GL ( C ); R ( n )) in the contin uous cohomology of GL ( C ) = colim r GL r ( C ). App lication of b n induces a map H 2 n +1 ( GL ( C )) → R ( n ). The universal Bor el r e gulator map r n : K 2 n +1 ( C ) − → R ( n ) is deﬁned to b e th e composition r n = b n ◦ h 2 n +1 . F or the deﬁnition on an arbitrary n umber ﬁeld F , we compose with the maps induced on K -theory by the diﬀerent emb ed dings F ֒ → C : K 2 n +1 ( F ) − → M Hom( F , C ) K 2 n +1 ( C ) − → X F ⊗ R ( n ) , where X F = Z [Hom( F, C )]. The image of this map is inv ariant under complex conju- gation acting on b oth Hom( F, C ) and R ( n ). Hence we ha ve a map K 2 n +1 ( F ) − → ( X F ⊗ R ( n )) c where ( ) c denotes the subgroup of inv aria nts under complex conjugation. I f n is odd, w e take a basis of ( X F ⊗ R ( n )) c consisting of ψ ⊗ i − ψ ⊗ i for each conjugate pair of complex embeddings ψ , ψ : F → C . If n is even we take a basis of ( X F ⊗ R ( n )) c consisting of ψ ⊗ 1 + ψ ⊗ 1 for each conjugate pair of complex em b eddings ψ , ψ : F → C , together with ψ ⊗ 1 for eac h real em bedd in g ψ : F → R . Either w ay , this y ields a natural identiﬁcatio n of ( X F ⊗ R ( n )) c with R d n . Borel pro v ed that for n ≥ 1 the Borel regulator map ( 1) is an embedding of K 2 n +1 ( F ) mod u lo torsion in R d n and its image is a full la ttice i n R d n . T he cov ol- ume of this lattice is called the Bor el r e gulator for F , written R n ( F ). 2.2 A canonical set of elemen ts in K 3 ( F ) Let R b e a ring and let E ( R ) denote the elementa ry m atrices of R . W e will make use of the follo wing result: Lemma 2.1. L et R b e a ring. Then the Hur ewicz m ap induc es a surje ctive m ap: h 3 : K 3 ( R ) = π 3 ( B E ( R ) + ) − → H 3 ( B E ( R ) + ) ∼ = H 3 ( B E ( R )) ∼ = H 3 ( E ( R )) . Pr o of. By construction, the space B E ( R ) + is simply connected, so th e surjectivity of h 3 follo ws immediately from the Hurewicz Theorem. Now let F = Q ( ω ), where ω is a q th root of unity ( q ≥ 3). Let E ( F ) d enote the group of elementary matrices o ver F . The kernel of h 3 : K 3 ( F ) → H 3 ( E ( F )) is contained in the kernel of the Borel regulator map which contains only torsion elemen ts. Thus given an element in H 3 ( E ( F )) , there exists a preimage in K 3 ( F ) (Lemma 2. 1), and it is unique up to torsion. W e may therefore sp ecify elemen ts of K 3 ( F ) up to torsion, by giving cycles in the inhomogeneous bar resolution of E ( F ) tensored with Z , whic h w e write as · · · d − → B n d − → B n − 1 d − → · · · F or later reference w e recall that t he b ou n dary map d : B 3 → B 2 is giv en b y d ([ g 1 | g 2 | g 3 ]) = [ g 2 | g 3 ] − [ g 1 g 2 | g 3 ] + [ g 1 | g 2 g 3 ] − [ g 1 | g 2 ] and the b oundary map d : B 2 → B 1 is giv en by d ([ g 1 | g 2 ]) = [ g 2 ] − [ g 1 g 2 ] + [ g 1 ] . Let A = Z [ t, t − 1 ] b e the ring of Laurent p olynomials o ver Z . F or any p rimitiv e q th root of unity u ∈ F , w e hav e a ring homomorphism α u : A → F , giv en by x 7→ x | t = u , the elemen t of F obtained b y ev aluating t at u . F or a unit λ ∈ A , let g λ ij denote t h e matrix which diﬀers from th e identit y in the ( i, i ) th and ( j, j ) th entries only , whic h are λ , λ − 1 respectively . Let a, b ∈ E ( A ) b e a = g t 12 =   t 0 0 0 t − 1 0 0 0 1   , b = g − t 13 =   − t 0 0 0 1 0 0 0 − t − 1   . 3 W e write the in h omogeneous bar resolution of E ( A ) as · · · d ′ − → B ′ n d ′ − → B ′ n − 1 d ′ − → · · · As a, b commute, we h a ve d ′ ([ a | b ] − [ b | a ]) = 0. Thus [ a | b ] − [ b | a ] is a cycle and represents an elemen t of H 2 ( E ( A )) ∼ = K 2 ( A ). In fact ([21, pp. 71– 75]) this element is trivial, so the cycle is actually the boun dary of some Z 1 ∈ B ′ 3 . That is d ′ ( Z 1 ) = [ a | b ] − [ b | a ] . Let h ′ 3 : K 3 ( A ) → H 3 ( E ( A )) den ote the su rjectiv e homomorphism of Lemma 2.1. Lemma 2.2. Given any other Z ′ 1 such that d ′ ( Z ′ 1 ) = [ a | b ] − [ b | a ] , the diﬀer enc e Z ′ 1 − Z 1 is a cycle and r epr esents a class h ′ 3 ( y ) ∈ H 3 ( E ( A )) , for some element y ∈ K 3 ( A ) . Pr o of. Clearly d ′ ( Z ′ 1 − Z 1 ) = 0, so Z ′ 1 − Z 1 represents a homology class. As h ′ 3 is surjectiv e, there exists y ∈ K 3 ( A ) such that h ′ 3 ( y ) is this homology class. Thus Z 1 is a wel l deﬁned c hain mo dulo (chains representing) the image of K 3 ( A ). An imp ortant p oint to note is th at t he construction of Z 1 made no reference to the cyclotomic ﬁeld F so w e ma y mak e the foll ow ing d eﬁnition: Deﬁnition 2.3. The universal chain (deﬁned up to chains rep resenting elements in the image of K 3 ( A )) is the chai n Z 1 ∈ B ′ 3 satisfying d ′ ( Z 1 ) = [ a | b ] − [ b | a ]. Deﬁnition 2.4. Let q ≥ 3. W e deﬁn e Z 2 ( q ) ∈ B ′ 3 as Z 2 ( q ) = q − 1 X r =0 ([ a r | a | b ] − [ a r | b | a ] + [ b | a r | a ]) . Then (writing I for the identit y matrix) we h a ve d ′ Z 2 ( q ) = q − 1 X r =0  ([ a | b ] − [ b | a ]) − ([ a r +1 | b ] − [ a r | b ]) + ([ b | a r +1 ] − [ b | a r ])  = q ([ a | b ] − [ b | a ]) − ([ a q | b ] − [ I | b ]) + ([ b | a q ] − [ b | I ]) . Note t hat Z 2 ( q ) is given explicitly ab o ve, whereas for Z 1 w e h a ve only pro ven existence and uniqueness up to the image of K 3 ( A ) under h ′ 3 . Let α u denote the induced c hain map: · · · d ′ − → B ′ n d ′ − → B ′ n − 1 d ′ − → · · · ↓ α u ↓ α u · · · d − → B n d − → B n − 1 d − → · · · Note w e use α u to denote the induced maps on matrices, chains and elements of K - theory . W e hav e dα u ( Z 2 ( q )) = α u d ′ Z 2 ( q ) = q α u ([ a | b ] − [ b | a ]) as α u ( a ) q = I . Let Z = q Z 1 − Z 2 ( q ). W e then hav e α u ( Z ) ∈ B 3 is a cy cle representing a homolog y class Z u = [ Z | t = u ] ∈ H 3 ( E ( F )) . Its preimage und er h 3 , denoted z u ∈ K 3 ( F ), is then determined up to torsion. Theorem 2.5. F or e ach primi tive q th r o ot of unity in the cyclotomic ﬁeld F = Q ( ω ) , ther e exists z u ∈ K 3 ( F ) , uni que up to torsion, satisfying h 3 ( z u ) = [( q Z 1 − Z 2 ( q )) | t = u ] , wher e Z 1 is a r epr esent ative of t he uni versal chain (Deﬁnition 2.3) and Z 2 ( q ) is as given i n Deﬁnition 2.4. Pr o of. Given a diﬀerent choice of representativ e of the universal chain, Z ′ 1 , we would hav e a homology class Z ′ u = [ α u ( q Z ′ 1 − Z 2 ( q ))]. Then Z ′ u − Z u = α u ( h ′ 3 ( y )) = h 3 ( α u ( y )) for some y ∈ K 3 ( A ). W e may take z ′ u = α u ( y ) + z u to b e a preimage of Z ′ u under h 3 . How ever, K 3 ( A ) ∼ = K 3 ( Z ) ⊕ K 2 ( Z ) ∼ = Z / 2 ⊕ Z / 48 ([22 , Theorem 5 . 3 . 30]) . Thus K 3 ( A ) is ﬁnite and α u ( y ) ∈ K 3 ( F ) is an elemen t of torsion as required. 4 Thus mo dulo torsion we hav e a w ell-deﬁned canonical set of elements z u ∈ K 3 ( F ), correspondin g to the primitiv e q th roots of unit y u ∈ F . R emark 2.2 . Even if F is a number ﬁeld other than a cyclotomic ﬁeld, we still ha ve an element z u ∈ K 3 ( F ) for each primitive ro ot of unity u ∈ F . How ev er the num b er of pairs u, u − 1 of such roots of unity no longer coincides with the number of pairs of conjugate emb eddings F ֒ → C . Thus we d o not hav e a natural candidate for a b asis for K 3 ( F ) (modulo torsion). 2.3 The co v olume V 1 ( F ) Let l = ϕ ( q ) 2 (where ϕ is the Euler totient function) and let ± v 1 , · · · , ± v l b e the units of Z /q Z . Also let ξ = e 2 πi/q . W e hav e l conjugate pairs of embeddings ψ j , ψ j : F = Q ( ω ) ֒ → C , give n by ω 7→ ξ v j and ω 7→ ξ − v j for j = 1 , · · · , l . W e write ψ j to d en ote the induced map on c hains and elements of K-th eory . The rank of K 3 ( F ) is d 1 = q 2 = l . W e also hav e l conjugate pairs of primitive q th roots of unity , u k , u k = ω ± v k ∈ F for k = 1 , · · · , l . Corresp ondingly we hav e elements z u k , z u k ∈ K 3 ( F ) for k = 1 , · · · , l . F or m a unit in Z /q Z , let β m : A → C b e the ring homomorphism sending t 7→ ξ m . Again w e write β m to denote maps on c hains and elemen ts of K-theory . Note that: ψ j ◦ α u k = β v j v k Let ( L j k ) j,k b e the matrix of co ordinates of the images of the z u k under the Borel regulator map, K 3 ( F ) → R d 1 = ( X F ⊗ R ( n )) c , with respect to th e basis { ψ j ⊗ i − ψ j ⊗ i ∈ R d 1 | j = 1 , · · · , l } . Let Z = q Z 1 − Z 2 ( q ) as in § 2.2. Lemma 2.6. We have L j k = b 1 ([ Z | t = ξ v j v k ]) /i . Pr o of. The co eﬃcien t L j k is r 1 ( ψ j ( z u k )) /i (see § 2.1). W e hav e ψ j ( Z u k ) = [ ψ j α u k ( Z )] = [ β v j v k ( Z )] ∈ H 3 ( E ( C )) . Thus r 1 ( ψ j ( z u k )) = b 1 ( h 3 ( ψ j ( z u k ))) = b 1 ( ψ j ( h 3 ( z u k ))) = b 1 ( ψ j ( Z u k )) = b 1 [ β v j v k ( Z )] = b 1 ([ Z | t = ξ v j v k ]) . Hence for 1 ≤ j, k ≤ l we ha ve L j k = r 1 ( ψ j ( z u k )) /i = b 1 ([ Z | t = ξ v j v k ]) /i . Lemma 2.7. Applying the Bor el r e gul ator map to z u k , z u k ∈ K 3 ( F ) gives elements of R d 1 which di ﬀer by a sign. Thus up to torsion z u k , z u k ∈ K 3 ( F ) diﬀer by a sign. Pr o of. The inv ariance un der complex conjugation of the image of z u k in ( X F ⊗ R ( n )) c , implies that r 1 ( ψ j ( z u k )) = − r 1 ( ψ j ( z u k )) (see § 2.1). Thus r 1 ( ψ j ( z u k )) = b 1 [ β − v j v k ( Z )] = r 1 ( ψ j ( z u k )) = − r 1 ( ψ j ( z u k )) . F rom Lemma 2.7 w e hav e th at the foll ow ing are w ell deﬁned, indep endently of th e choi ce of signs on the v k : Deﬁnition 2. 8. W e deﬁne ind( F ) ∈ Z to be the index of the lattice in K 3 ( F ) mo dulo torsion, generated by the z u k , k = 1 , · · · , l . Deﬁnition 2.9. W e deﬁn e V 1 ( F ) to b e th e cov olume of the lattice in R d 1 generated by the images of the z u k , k = 1 , · · · , l , un der t he Borel R egulator map. R emark 2.3 . It is not a p riori clear that the z u k are non-zero elements of K 3 ( F ), let alone linearly indep endent. Ho weve r if one computes a non-zero v alue of V 1 ( F ) (see App endix B for a reference to a computation whic h app ears to to do this in the case q = 3) it fo llo ws that the z u k are l inearly independ en t and generate a ﬁnite ind ex subgroup of K 3 ( F ); in particular, ind ( F ) 6 = 0. Theorem 2. 10. If V 1 ( F ) 6 = 0 then ind( F ) 6 = 0 and the Bor el r e gulator satisﬁes R 1 ( F ) = V 1 ( F ) / ind( F ) . 5 In particular, whenever ind( F ) = 1 (that is, whenev er the z u k generate K 3 ( F ) mod u lo torsion), we are able to compute t h e Borel regulator R 1 ( F ). Thus if criteria w ere found to determine for which cyclotomic ﬁelds ind ( F ) = 1, then this approach w ould allo w one to compute the Borel regulator for those ﬁelds. Theorem 2.11. F or a cyclotomic ﬁeld F = Q ( ω ) , we may c ompute V 1 ( F ) in terms of the universal Bor el class b 1 : V 1 ( F ) =    det  b 1  [( q Z 1 − Z 2 ( q )) | t = ξ v j v k ]  j,k    wher e Z 1 is the universal chain (Deﬁnition 2.3) and Z 2 ( q ) as in Deﬁni ti on 2.4. Pr o of. W e know t hat V 1 ( F ) is given by the absolute val ue of th e determinant of the matrix ( L j k ) j,k of co ordinates of the images of the z u k . W e need only note that factors of i do not aﬀect the absolute val ue of the determinant. Thus to compute V 1 ( F ) w e m ust ev aluate the b 1 ([ q Z 1 − Z 2 ( q )] | t = ξ v j v k ). This com- prises t w o indep endent stages which are dealt with in Sections 3 and 5. Firstly , we need to exp licitly construct the chain q Z 1 − Z 2 ( q ). As Z 2 ( q ) is given explicitly (Deﬁnition 2.4) it remains to ﬁnd a represen tative of the unive rsal c hain Z 1 . Motiv ated by ideas in knot theory , in Section 3 w e d evise tec hniques for constructing elemen ts in the bar resolution of a group, wi th sp eciﬁed b ou n dary . Using this, in Section 4 w e obtain an expression fo r Z 1 . This expression consists of 6844 terms (chai ns of th e form ( g 1 | g 2 | g 3 ) with eac h g i ∈ M 5 ( Z [ t, t − 1 ])) and i t is a vai lable as a PDF (for visualisation) or Matlab (for compuation) ﬁle 1 . Althou gh the construction of this exp ression is an intricate pro cedure (explained in § 4), one ma y easily calculate its b oun d ary with the h elp of a computer and thus indep endently verif y that d ′ ( Z 1 ) = [ a | b ] − [ b | a ], ie. Z 1 is indeed a representativ e of the univ ersal chain (cf. Deﬁnition 2.3). Secondly , in Section 5 w e will show how the u n iv ersal Borel class b n of § 2.1 can be computed for arbitrary n ≥ 1 and any number ﬁeld by making the Karoubi-Hamida integ ral [13] exp licit. In particular we describe a form ula for computing b 1 (Theorem 5.11). R emark 2.4 . An important point to note is th at the universa l c hain Z 1 is deﬁned indep endently of t he integer q . That is, ha ving once computed it (as we do in Section 4), we may use it to compute V 1 ( F ) for an y cyclotomic ﬁeld F , merely by substitu t ing in diﬀerent ro ots of unity for the indeterminate t . R emark 2.5 . T o compute R n ( F ) for n > 1 we would n eed to construct a basis modu lo torsion of K 2 n +1 ( F ). A p otential idea for generalizing our method s in the n = 1 case is to let S = F [ x 0 , . . . , x r ] / ( x 0 x 1 . . . x r (1 − P r i =0 x i )), the co ordinate ring of an algebraic sphere ov er F . Then if we could construct elements in K 3 ( S ) we could apply the natural homomorphism K 3 ( S ) → K V 3 ( S ) to obtain elements in Karoubi- V illama y or K-theory ([24, § 3]). There is an is omorphism [9]: K V 3 ( S ) ∼ = K 3 ( F ) ⊕ K 3+ r ( F ). Pro jecting onto the second summand would yield elements of K 3+ r ( F ). On e ma y sp eculate that a pro cedure alo ng these lines could b e devised to ﬁnd a basis (modu lo torsion) for K 3+ r ( F ). This wo uld allo w our low dimensional group homology metho ds ( § 3.2) to b e used for computing R n , n > 1. 3 Boundary relations in the bar resolution As mentio ned earlier, part of the intangibleness of the Borel Regula tor of a ﬁeld F comes from the d iﬃcult y of explicitly constructing elements in the K -theory of F in the required form: their images un der the Hurewicz map represented by chains in the bar resolution of GL ( F ). In this section we describ e a nov el approach to doing p recisely that, based on ideas from knot theory ( sp eciﬁcally th e concep t of a free deriv ative). In the Section 4, we will apply these general metho ds to the case of explicitly constructing the un iv ersal chain Z 1 (Deﬁnition 2.3), a necessary step in th e calculation of V 1 ( F ) for a cyclotomic ﬁ eld, as ex plained b efore. 1 see ancillary ﬁles a nc/un iversalchain.tex and a nc/un iversalchain.mat 6 3.1 A free F o x ty p e deriv ativ e Let G b e a discrete group and write B n G for the degree n p art of th e inh omogeneous bar resolution [25, § 6.5]. Therefore B n G is the free left Z [ G ]-module with basis consisting of n -tuples [ g 1 | g 2 | . . . | g n ] with eac h g i ∈ G . T he b oundary map is giv en by d ([ g 1 | g 2 | . . . | g n ]) = g 1 [ g 2 | . . . | g n ] + n − 1 X i =1 ( − 1) i [ g 1 | g 2 | . . . | g i g i +1 | . . . | g n ] + ( − 1) n [ g 1 | g 2 | . . . | g n − 1 ] . Thus in particular the b oundary map B 3 G → B 2 G is d ([ g 1 | g 2 | g 3 ]) = g 1 [ g 2 | g 3 ] − [ g 1 g 2 | g 3 ] + [ g 1 | g 2 g 3 ] − [ g 1 | g 2 ] . The study of knot theory motiv ated the idea of a free d iﬀeren tial [10]: a map from a set of w ords to an ab elian ob ject, satisfying a v ariant of Le ibniz’s ru le. W e now construct t he relev ant such diﬀerential. Let F G denote t h e free group on sy mb ols s x with x ∈ G and let φ : F G → G b e the group homomorphism mapping φ ( s x ) 7→ x . Deﬁnition 3.1. There exists a free deriv ative: ∂ : F G − → Z ⊗ Z [ G ] B 2 G uniquely characteri zed by t h e p roperties: (i) ∂ ( e ) = 0 ( e is the iden tity element of F G ) and (ii) ∂ ( us x ) = ∂ ( u ) + 1 ⊗ Z [ G ] [ φ ( u ) | x ] for u ∈ F G . Lemma 3.2. F or 1 ≤ i ≤ r supp ose x i ∈ G and that ǫ i = ± 1 . Set z i = x − 1 i if ǫ i = − 1 and z i = 1 otherwise. Then for any map ∂ satisfying (i) and (ii ) as ab ove, we have: ∂ ( s ǫ 1 x 1 s ǫ 2 x 2 . . . s ǫ r x r ) = r X i =1 ǫ i ⊗ Z [ G ] [ x ǫ 1 1 x ǫ 2 2 . . . x ǫ i − 1 i − 1 z i | x i ] . (2) Thus ∂ is uniquely deﬁne d on F G , as claime d in the deﬁnition. Pr o of. Firstly , note that (ii) implies ∂ ( u ) = ∂ ( us − 1 x s x ) = ∂ ( us − 1 x ) + 1 ⊗ Z [ G ] [ φ ( u ) x − 1 | x ] whic h means that ∂ ( us − 1 x ) = ∂ ( u ) − 1 ⊗ Z [ G ] [ φ ( u ) x − 1 | x ] . Induction on r gives that ∂ ( s ǫ 1 x 1 s ǫ 2 x 2 . . . s ǫ r x r ) m ust b e given by (2). It remains to verify that (2) yields a w ell deﬁned map ∂ : F G → Z ⊗ Z [ G ] B 2 G , satisfying (i) and (ii). Tw o wo rds represen t the same element of F G precisely when they diﬀer by a series of insertions and deletions of strings s y s − 1 y and s − 1 y s y . Direct calculation sho ws that (2) is indep end en t of su ch insertions or deletions. Finally , direct calculation veriﬁes that (2) satisﬁes (i) and (ii). Deﬁnition 3.3. Given a w ord w written in the form: w = u 1 ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 u − 1 1 u 2 ( s x 2 s y 2 s − 1 x 2 y 2 ) n 2 u − 1 2 . . . u k ( s x k s y k s − 1 x k y k ) n k u − 1 k with x i , y i ∈ G , u i ∈ F G and n i ∈ Z , we deﬁ n e W ( w ) ∈ Z ⊗ Z [ G ] B 3 ( G ) by th e formula W ( w ) = k X i =1 n i ⊗ Z [ G ] [ φ ( u i ) | x i | y i ] . Note W is not a well deﬁned function on F G (unlike the free deriv ative whic h is). Lemma 3.4. The b oundary of such a chain is given by the formula (1 ⊗ Z [ G ] d )( W ( w )) = k X i =1 n i ⊗ Z [ G ] [ x i | y i ] ! − ∂ ( w ) . 7 Pr o of. F rom (2), w e may make the follo wing observ ation: ∂ ( uv ) = ∂ ( u ) + φ ( u ) · ∂ ( v ) , (3) where for g , h, k ∈ G , it is understo od that g · 1 ⊗ Z [ G ] [ h | k ] = 1 ⊗ Z [ G ] [ g h | k ]. In the case of a segmen t u i ( s x i s y i s − 1 x i y i ) n i u − 1 i , w e hav e that: φ ( u i ( s x i s y i s − 1 x i y i ) n i u − 1 i ) = φ ( u i )( x i y i ( x i y i ) − 1 ) n i φ ( u i ) − 1 = e Regarding w as the pro duct of such segments and applying (3) w e then ha ve: ∂ ( w ) = k X i =1 n i ∂ ( u i ( s x i s y i s − 1 x i y i ) u − 1 i ) . Let u i b e written out as Q l i j =1 s m j z j , where m j is either 1 or − 1. Then by (2) ∂ ( u i ( s x i s y i s − 1 x i y i ) u − 1 i ) = X j | m j =1 1 ⊗ Z [ G ] [ j − 1 Y p =1 z m p p | z j ] − X j | m j = − 1 1 ⊗ Z [ G ] [ j Y p =1 z m p p | z j ] + 1 ⊗ Z [ G ] [ φ ( u i ) | x i ] + 1 ⊗ Z [ G ] [ φ ( u i ) x i | y i ] − 1 ⊗ Z [ G ] [ φ ( u i ) | x i y i ] − X j | m j =1 1 ⊗ Z [ G ] [ j − 1 Y p =1 z m p p | z j ] + X j | m j = − 1 1 ⊗ Z [ G ] [ j Y p =1 z m p p | z j ] = 1 ⊗ Z [ G ] [ φ ( u i ) | x i ] + 1 ⊗ Z [ G ] ( φ ( u i ) x i | y i ] − 1 ⊗ Z [ G ] [ φ ( u i ) | x i y i ] . Thus we ha ve ∂ ( w ) = k X i =1 n i (1 ⊗ Z [ G ] [ φ ( u i ) | x i ] + 1 ⊗ Z [ G ] [( φ ( u i ) x i | y i ] − 1 ⊗ Z [ G ] [ φ ( u i ) | x i y i ]) and (1 ⊗ Z [ G ] d )( W ( w )) = k X i =1 n i (1 ⊗ Z [ G ] [ x i | y i ] − 1 ⊗ Z [ G ] [( φ ( u i ) x i | y i ] + 1 ⊗ Z [ G ] [ φ ( u i ) | x i y i ] − 1 ⊗ Z [ G ] [ φ ( u i ) | x i ]) = k X i =1 n i (1 ⊗ Z [ G ] [ x i | y i ] ! − ∂ ( w ) as required. 3.2 Constructing b oundary relations W e now d escribe a meth od, based on Lemma 3.4 for constructing b oundary relations: identities of the form (1 ⊗ Z [ G ] d ) α = β , for α ∈ Z ⊗ Z [ G ] B 3 ( G ) and β ∈ Z ⊗ Z [ G ] B 2 ( G ). Lemma 3.5. Given u ∈ F G we may write u = ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 ( s x 2 s y 2 s − 1 x 2 y 2 ) n 2 . . . ( s x k s y k s − 1 x k y k ) n k s φ ( u ) wher e x i , y i ∈ G . Pr o of. F or contradiction let l be th e smallest integer such that there is some u ∈ F G of length l contradicting th e lemma. As e = ( s e s e s − 1 e ) − 1 s e , we know l > 0. Then either u = u ′ s a or u = u ′ s − 1 a with u ′ of length l − 1 and a ∈ G . Thus either u = P s ( φ ( u ′ ) s a = P ( s ( φ ( u ′ ) s a s − 1 φ ( u ) ) s φ ( u ) or u = P s ( φ ( u ′ ) s − 1 a = P ( s ( φ ( u ′ ) s − 1 a s − 1 φ ( u ) ) s φ ( u ) = P ( s φ ( u ) s a s − 1 ( φ ( u ′ ) ) − 1 s φ ( u ) , where P is a p roduct of ( s x i s y i s − 1 x i y i ) n i . Deﬁnition 3.6. A r elator is an element of Ker( φ : F G − → G ). 8 Lemma 3.7. Given a r elator R ∈ F G we m ay write R = ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 ( s x 2 s y 2 s − 1 x 2 y 2 ) n 2 . . . ( s x k s y k s − 1 x k y k ) n k . Pr o of. W e hav e R = ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 ( s x 2 s y 2 s − 1 x 2 y 2 ) n 2 . . . ( s x k s y k s − 1 x k y k ) n k s e = ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 ( s x 2 s y 2 s − 1 x 2 y 2 ) n 2 . . . ( s x k s y k s − 1 x k y k ) n k ( s e s e s − 1 e ) . Now let C 1 , · · · , C k +1 , C ′ 1 , · · · C ′ l +1 b e pro ducts of commutators of the form [ R, u ] ± 1 = ( RuR − 1 u − 1 ) ± 1 , where R is a relator and u ∈ F G . Given an identit y in F G of the form C 1 v 1 ( s x 1 s y 1 s − 1 x 1 y 1 ) n 1 v − 1 1 · · · C k v k ( s x k s y k s − 1 x k y k ) n k v − 1 k C k +1 = C ′ 1 v ′ 1 ( s x ′ 1 s y ′ 1 s − 1 x ′ 1 y ′ 1 ) n ′ 1 v ′− 1 1 · · · C ′ l v ′ l ( s x ′ l s y ′ l s − 1 x ′ l y ′ l ) n ′ l v ′− 1 l C ′ l +1 (4) w e ma y use Lemma 3.7 to express each relator R as a produ ct of ( s a s b s − 1 ab ) m and each uR − 1 u − 1 as a pro duct of u ( s a s b s − 1 ab ) m u − 1 . Thus we may express th e left and right hand sides of (4) as w ords w 1 , w 2 respectively , to whic h w e ma y apply W . W e get (1 ⊗ Z [ G ] d )( W ( w 1 )) = k X i =1 n i ⊗ Z [ G ] [ x i | y i ] ! − ∂ ( w 1 ) and (1 ⊗ Z [ G ] d )( W ( w 2 )) = l X i =1 n ′ i ⊗ Z [ G ] [ x ′ i | y ′ i ] ! − ∂ ( w 2 ) as the remaining terms coming from eac h relator R are canceled b y the corresponding terms from eac h uR − 1 u − 1 . F rom (4) w e ha ve that w 1 = w 2 as elemen ts of F G , so ∂ ( w 1 ) − ∂ ( w 2 ) = 0. Thus: Theorem 3.8. We have a b oundary r elation: (1 ⊗ Z [ G ] d )( W ( w 1 ) − W ( w 2 )) = k X i =1 n i ⊗ Z [ G ] [ x i | y i ] ! − l X i =1 n ′ i ⊗ Z [ G ] [ x ′ i | y ′ i ] ! . 3.3 Examples Let G be a group and let x, y ∈ G commute. Then w = [ s x , s y ] is a relator and we h a ve w = ( s x s y s − 1 xy )( s y s x s − 1 y x ) − 1 . Then W ( w ) = 1 ⊗ Z [ G ] [ e | x | y ] − 1 ⊗ Z [ G ] [ e | y | x ] and (1 ⊗ Z [ G ] d ) W ( w ) = 1 ⊗ Z [ G ] [ x | y ] − 1 ⊗ Z [ G ] [ y | x ] − ∂ ( w ) . W e use the n otation { x, y } to den ote 1 ⊗ Z [ G ] [ x | y ] − 1 ⊗ Z [ G ] [ y | x ]. Example 1. As our ﬁ rst example we consider th e id entity ( s e s y s − 1 y )( s y s e s − 1 y ) − 1 = [ s e , s y ] . Letting w 1 , w 2 denote the left and righ t sides of this iden tity as b efore, w e get W ( w 1 ) − W ( w 2 ) = 1 ⊗ Z [ G ] [ e | e | y ] − 1 ⊗ Z [ G ] [ e | y | e ] − 1 ⊗ Z [ G ] [ e | e | e ] + 1 ⊗ Z [ G ] [ y | e | e ] . Thus by Theorem 3.8 (1 ⊗ Z [ G ] d )( W ( w 1 ) − W ( w 2 )) = 1 ⊗ Z [ G ] [ e | y ] − 1 ⊗ Z [ G ] [ y | e ] = { e, y } . In fact (1 ⊗ Z [ G ] d )  1 ⊗ Z [ G ] [ e | e | y | e ] + 1 ⊗ Z [ G ] [ e | e | e | e ]  = 1 ⊗ Z [ G ] [ e | y | e ] + 1 ⊗ Z [ G ] [ e | e | e ]. So adding th is b oundary of a 4-chain to W ( w 1 ) − W ( w 2 ) lea v es the b ound ary relation (1 ⊗ Z [ G ] d )  1 ⊗ Z [ G ] [ e | e | y ] + 1 ⊗ Z [ G ] [ y | e | e ]  = { e, y } . Example 2. Let x, y , c ∈ G satisfy [ x, c ] = [ y , c ] = e . C onsider the identit y s x [ s y , s c ] s − 1 x [ s x , s c ] = [ s x s y , s c ] = ( s x s y s − 1 xy )[ s xy , s c ] s c ( s x s y s − 1 xy ) − 1 s − 1 c . Let w 1 , w 2 denote the left and righ t hand sides of this iden tity . Then W ( w 1 ) = 1 ⊗ Z [ G ] [ x | y | c ] − 1 ⊗ Z [ G ] [ x | c | y )] + 1 ⊗ Z [ G ] [ e | x | c ] − 1 ⊗ Z [ G ] [ e | c | x ] 9 and W ( w 2 ) = 1 ⊗ Z [ G ] [ e | xy | c ] − 1 ⊗ Z [ G ] [ e | c | xy ] + 1 ⊗ Z [ G ] [ e | x | y ] − 1 ⊗ Z [ G ] [ c | x | y ] . Thus by Theorem 3.8 w e hav e the b oundary relation (1 ⊗ Z [ G ] d )( W ( w 1 ) − W ( w 2 )) = { x, c } + { y , c } − { xy , c } . Adding the bou n dary of th e 4-c hain 1 ⊗ Z [ G ] [ e | x | y | c ] + 1 ⊗ Z [ G ] [ e | c | x | y ] − 1 ⊗ Z [ G ] [ e | x | c | y ] to W ( w 1 ) − W ( w 2 ) w e get a 3-c hain with the same boun dary: (1 ⊗ Z [ G ] d )  1 ⊗ Z [ G ] [ x | y | c ] − 1 ⊗ Z [ G ] [ x | c | y )] + 1 ⊗ Z [ G ] [ c | x | y ]  = { x, c } + { y , c } − { xy , c } . Example 3. Let x, y , c ∈ G satisfy cxc − 1 = y , cy c − 1 = x, xy = y x . Then we ha ve s c ( s x s y s − 1 xy )( s y s x s − 1 y x ) − 1 s − 1 c = [( s c s x s − 1 cx )( s y s c s − 1 y c ) − 1 , s y s c s y s − 1 c s − 1 y ]( s y s x s − 1 y x )( s x s y s − 1 xy ) − 1 [ s x s y s − 1 x , ( s c s y s − 1 cy )( s x s c s − 1 xc ) − 1 ] . Again let w 1 , w 2 denote the left and righ t hand sides of this identity . Theorem 3.8 giv es (1 ⊗ Z [ G ] d )( W ( w 1 ) − W ( w 2 )) = 2 { x, y } . Applying W to w 1 , w 2 giv es W ( w 1 ) = 1 ⊗ Z [ G ] [ c | x | y ] − 1 ⊗ Z [ G ] [ c | y | x )] W ( w 2 ) = 1 ⊗ Z [ G ] [ e | c | x ] − 1 ⊗ Z [ G ] [ e | y | c ] − 1 ⊗ Z [ G ] [ x | c | x ] + 1 ⊗ Z [ G ] [ x | y | c ] + 1 ⊗ Z [ G ] [ e | y | x ] − 1 ⊗ Z [ G ] [ e | x | y ] + 1 ⊗ Z [ G ] [ y | c | y ] − 1 ⊗ Z [ G ] [ y | x || c ] − 1 ⊗ Z [ G ] [ e | c | y ] + 1 ⊗ Z [ G ] [ e | x | c ] . W e ha ve a 4-chai n T = 1 ⊗ Z [ G ] [ e | y | x | c ] + 1 ⊗ Z [ G ] [ e | x | c | x ] + 1 ⊗ Z [ G ] [ e | c | x | y ] − 1 ⊗ Z [ G ] [ e | x | y | c ] − 1 ⊗ Z [ G ] [ e | y | c | y ] − 1 ⊗ Z [ G ] [ e | c | y | x ] . Let P = W ( w 1 ) − W ( w 2 ) + (1 ⊗ Z [ G ] d ) T = 1 ⊗ Z [ G ] [ c | x | y ] − 1 ⊗ Z [ G ] [ x | y | c ] − 1 ⊗ Z [ G ] [ c | y | x ] + 1 ⊗ Z [ G ] [ y | x | c ] + 1 ⊗ Z [ G ] [ x | c | x ] − 1 ⊗ Z [ G ] [ y | c | y ] . Thus we ha ve a b oundary relation (1 ⊗ Z [ G ] d )( P ) = 2 { x, y } . 4 Constructing the univ ersal c hain As in § 2.2 let A = Z [ t, t − 1 ] and a = g t 12 , b = g − t 13 . O u r aim in § 4.1 is to d escribe the idea b ehind th e construction of the univers al chain Z 1 , which satisﬁes d ′ ( Z 1 ) = [ a | b ] − [ b | a ]. Then in § 4.2 w e describ e the actual process of constructing it. Recall (Remark 2.4 on page 6) that once constructed, we may use it to compute V 1 ( F ) for any cyclotomic ﬁeld F , by su b stituting in the relev ant ro ots of unity for th e indeterminate t . 10 4.1 The strategy for the univ ersal cha in In order to construct this b oundary relation we will employ the method given in § 3.2 and used in Examp les 1, 2, 3 of § 3.3. In this case our group is E ( A ) and we seek an identit y in the letters of the free group generated by elements of E ( A ): C 1 ( s a s b s − 1 ab )( s b s a s − 1 ab ) − 1 C 2 = C 3 (5) where, as b efore C 1 , C 2 , C 3 are pro ducts of comm utators [ R, u ] ± 1 , with R a relator. In this subsection we will describe the idea b ehind the construction of (5). Then in § 4.2 w e will go though the stages in its construction. F or i 6 = j an d µ ∈ A , let E µ ij ∈ E ( A ) diﬀer from the identit y in E ( A ) in the ( i, j ) th entry only , whic h is µ . Let F E ⊂ F E ( A ) b e the subgroup generated by the s E µ ij . The St ein b erg group S t ( A ) is deﬁned t o b e the gro up generated by letters X µ ij , sub ject to the Steinberg relations S µ,ν ij = T µ,ν ij k = U µ,ν ij kl = e , where S µ,ν ij = ( X µ + ν ij ) − 1 X µ ij X ν ij , i 6 = j T µ,ν ij k = X µ ij X ν j k ( X µ ij ) − 1 ( X ν j k ) − 1 ( X µν ik ) − 1 , i, j, k distinct U µ,v ij kl = X µ ij X ν kl ( X µ ij ) − 1 ( X ν kl ) − 1 , i 6 = l, j 6 = k , i 6 = j, k 6 = l. The h omomorphism ψ : S t ( A ) → E ( A ) mapping X µ ij 7→ E µ ij is clearly surjective and its kernel K 2 ( A ) is cen tral in S t ( A ) ([21, p.40, Theorem 5.1] ). T hus giv en x , y ∈ E ( A ) and p reimages x ′ , y ′ ∈ S t ( A ), the commutator [ x ′ , y ′ ] is indep endent of the choice of preimages, and ma y b e denoted { x, y } (this diﬀers from our earlier notation). The ﬁrst step in our construction is to ex press { a, b } exp licitly as a pro duct of the X µ ij . This is d one by writing a and b as pro ducts of the E µ ij , then replacing eac h E µ ij with X µ ij , to get a ′ , b ′ ∈ S t ( A ). Then { a, b } is represented by t he w ord [ a ′ , b ′ ]. In [21, pp. 71 –75] it is shown that { a, b } = e in S t ( A ). This means that w e ha ve an identit y in the free group generated by the letters X µ ij : [ a ′ , b ′ ] = ( w 1 R ± 1 1 w − 1 1 )( w 2 R ± 1 2 w − 1 2 ) · · · ( w m R ± 1 m w − 1 m ) (6) where the R i are of the form S µ,ν ij , T µ,ν ij k or U µ,ν ij kl . Let R E ⊂ F E b e the subgroup k er( φ | F E ), and let ˆ θ : F E → S t ( A ) b e giv en b y s E µ ij 7→ X µ ij . Then ˆ θ is surjective and by construction the follo wing diagram commutes: F E φ | F E " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ˆ θ / / S t ( A ) ψ   E ( A ) . Hence ˆ θ ( R E ) ⊂ k er( ψ ) whic h is, as we hav e said, central. Hence ˆ θ ( [ F E , R E ]) = { e } so ˆ θ induces a w ell deﬁned map θ : F E / [ F E , R E ] → S t ( A ) mapping s E µ ij 7→ X µ ij . Note that [ F E , R E ] ⊂ R E so φ F E induces a well deﬁned map φ : F E / [ F E , R E ] → E ( A ) and th e follow ing diagram also comm utes: F E / [ F E , R E ] φ & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ θ / / S t ( A ) ψ   E ( A ) where φ here is u nderstoo d to denote the map induced by th e restriction. The kernel of θ is contained in ker( ψ θ ) = R E / [ F E , R E ], so is central in F E / [ F E , R E ]. Thus θ is a central exten sion. F rom [21, pp.48–51] applied to θ we get: Lemma 4.1. i) The element [ s E 1 ik , s E µ kj ] ∈ F E / [ F E , R E ] is indep endent of k 6 = i, j . ii) The map θ has a se ction c , given by c ( X µ ij ) = [ s E 1 ik , s E µ kj ] , k 6 = i, j . In particular c resp ects the Steinberg relations. Thus each c ( R i ) = e ∈ F E / [ F E , R E ] and ma y b e expressed as a p roduct of commutators [ u, v ] ± 1 , with v ∈ R E . W e apply c to (6 ) and t ake conjugation by c ( w i ) inside the comm utators to get [ c ( a ′ ) , c ( b ′ )] = C 3 11 where C 3 has the required form of a pro duct of comm utators, eac h in volving a relator. W e hav e φc ( a ′ ) = ψ θ c ( a ′ ) = ψ ( a ′ ) = a and φc ( b ′ ) = ψ θ c ( b ′ ) = ψ ( b ′ ) = b , so K a = c ( a ′ ) s − 1 a and K b = c ( b ′ ) s − 1 b are relators. W e hav e [ c ( a ′ ) , c ( b ′ )] = [ K a s a , K b s b ] = [ K a , s a s b s − 1 a ]( s a s b s − 1 ab )( s b s a s − 1 ab ) − 1 [ s b K a s a s − 1 b , K b ] . Let C 1 = [ K a , s a s b s − 1 a ] and C 2 = [ s b K a s a s − 1 b , K b ]. Then we hav e constructed (5) as C 1 ( s a s b s − 1 ab )( s b s a s − 1 ab ) − 1 C 2 = [ c ( a ′ ) , c ( b ′ )] = C 3 . (7) 4.2 Computing the free group iden tit y W e shall no w describe ho w to implement th e strategy of § 4.1. The ﬁrst s tep is to ﬁnd a ′ , b ′ ∈ S t ( A ). F or λ a un it in A , let Y λ ij = X λ ij X − ( λ − 1 ) j i X λ ij . F actorizing g λ ij into matrices of t he form E µ ij and rep lacing eac h E µ ij with X µ ij , we get Y λ ij Y − 1 ij , which we will denote h λ ij . Thus ψ ( h λ ij ) = g λ ij and in particular a ′ = h t 12 , b ′ = h − t 13 . In [21, pp . 71–75], it is shown t hat the commutator [ h t 12 , h − t 13 ] may b e reduced via t he Steinberg relations to e . This p roof dep ends on the identities h − t 13 Y λ 12 ( h − t 13 ) − 1 = Y − tλ 12 for λ = t or − 1, and Y t 12 Y − 1 12 Y − t 12 = Y − t 2 12 , which themselves dep end on several identities prov ed in [21, Lemma 9.2]. The proof of each of these is given by t ak ing w ords in the X µ ij and simplifying them using the Steinberg iden tities. Thus the entire pro of may b e written out as a sequence of equalities in the Steinberg group, where at each step a simpliﬁcation is made using one of th e Steinb erg identities. Of course suc h a pro of wo uld b e ex tremely long, as each step in p ro ving an identity needs to b e rep eated every time the identit y is used t o p rove a consequential id en tity . If the conseq uentia l identit y is u sed several times to prove an identit y higher up the chai n of consequences then one can appreciate ho w th e length of such a pro of gro ws exp onentia lly with the length of the proof given in [21]. Thus we hav e a long c hain of equalities in th e Steinberg group [ h t 12 , h − t 13 ] = · · · = e , where at eac h step w e essentially factor oﬀ a conjugate of one of the relators S µ,ν ij , T µ,ν ij k , U µ,ν ij kl or their inver ses. Next we must write t he entire pro of out again, this time not suppressing these factors, so w e hav e a seq u ence of equalities in the free group on th e letters X µ ij . This long nested sequence of op erations, t ogether with the v ast amount of data needed to store all these factors, made it natural to emp lo y a computer to construct the resulting identit y : [ a ′ , b ′ ] = [ h t 12 , h − t 13 ] = · · · = m Y i =1 ( w i R ± 1 i w − 1 i ) (8) where the R i are words of the form S µ,ν ij , T µ,ν ij k or U µ,ν ij kl and m = 392. The sequen ce of letters in the pro duct w ould ﬁll ab ou t 40 pages. W e note that non e of the relations used required us to include extra indices, so only 1, 2, and 3 were used. W e n ext apply the homomorphism c . Explicitly , this means replacing eac h X µ ij in (8) by [ s E 1 i 4 , s E µ 4 j ]. This is now a free group identit y in F E ⊂ F E ( A ) : [ c ( a ′ ) , c ( b ′ )] = m Y i =1 ( c ( w i ) c ( R i ) ± 1 c ( w i ) − 1 ) . (9) F rom [21, pp. 48-51] we kn o w that c : S t ( A ) → F E / [ F E , R E ] is a well deﬁned homo- morphism. Thus as the words S µ,ν ij , T µ,ν ij k and U µ,ν ij kl represent e ∈ S t ( A ), we ha ve th at the w ords c ( S µ,ν ij ) , c ( T µ,ν ij k ) , c ( U µ,ν ij kl ) ∈ F E represent e ∈ F E / [ F E , R E ]. The pro ofs of these th ree identiti es [21, p p.49-51] can b e written as a sequence of equalities in F E / [ F E , R E ], w here at each step we factor oﬀ a word in [ F E , R E ]. As b efore, by not suppressing these factors we obtain iden tities in t he free g roup F E , equating the c ( S µ,ν ij ) , c ( T µ,ν ij k ) , c ( U µ,ν ij kl ) ∈ F E with elemen ts of [ F E , R E ]. F or example: c ( U µ,ν ij kl ) = h [ s E 1 i 4 , s E µ 4 j ] , [ s E 1 k 4 , s E ν 4 l ] i = [ L ′ , s E µ ij [ s E 1 k 4 , s E ν 4 l ]( s E µ ij ) − 1 ] [ L ′′ , s E 1 k 4 [ s E µ ij , s E ν 4 l ] s E ν 4 l ( s E 1 k 4 ) − 1 ] [ L ′′′ , s E 1 k 4 s E ν 4 l ( s E 1 k 4 ) − 1 ( s E ν 4 l ) − 1 ( s E 1 k 4 ) − 1 ] where L ′ is the relator [ s E 1 i 4 , s E µ 4 j ]( s E µ ij ) − 1 , L ′′ is the relator [ s E µ ij , s E 1 k 4 ] and L ′′′ is the relator s E 1 k 4 [ s E µ ij , s E ν 4 l ]( s E 1 k 4 ) − 1 . Thus c ( U µ,ν ij kl ) ma y b e exp ressed as the prod uct 12 of 3 comm utators, eac h inv olving a relator. S imilarly , we may derive ex p ressions for c ( S µ,ν ij kl ), c ( T µ,ν ij kl ) expressing them as the pro duct of 4 and 18 comm utators resp ectiv ely , all in vol ving a relator. During the expansion of T µ,ν ij kl it w as n ecessary to introduce the index 5, so from no w on w e w ork with 5 × 5 matrices. Eac h c ( R i ) in (9 ) may n o w be expressed as a p ro duct: c ( R i ) = Q m i j =1 [ L ij , y ij ] ± 1 , so ( c ( w i ) c ( R i ) ± 1 c ( w i ) − 1 ) = c ( w i ) m i Y j =1 [ L ij , y ij ] ± 1 ! ± 1 c ( w i ) − 1 = m i Y j =1  c ( w i ) L ij c ( w i ) − 1 , c ( w i ) y ij c ( w i ) − 1  ± 1 ! ± 1 and from (9) [ c ( a ′ ) , c ( b ′ )] = m Y i =1 m i Y j =1  c ( w i ) L ij c ( w i ) − 1 , c ( w i ) y ij c ( w i ) − 1  ± 1 ! ± 1 = C 3 . Here C 3 is the p ro duct of 2392 comm utators, each inv olving a lengthy relator and word. Con versel y C 1 = [ K a , s a s b s − 1 a ] and C 2 = [ s b K a s a s − 1 b , K b ] are m uch shorter. In order to apply the operation W to b oth sides of (7) w e m ust rewrite the relator in each comm utator in C 1 , C 2 , C 3 as a prod u ct of words of the form ( s x s y s − 1 xy ) ± 1 . W e ma y use the ind u ctive pro cess from the proof of Lemma 3.5 to do this. The number of terms of the form ( s x s y s − 1 xy ) ± 1 needed to express eac h relator will b e app ro ximately the length of the relator, and each such term will add 2 terms to the 3-cy cle produced by the application of W (one coming from the relator and t he oth er from its in verse). With the help of a compu ter we w ork through th e left and right h and sides of (7) (whic h we will denote u 1 , u 2 respectively) a pplying W . Clearly this w ould prod uce millions of t erms. Whenever a n ew t erm was p roduced , we stored it in memory , and thereafter merely kept a running total of the coeﬃcient on it. Thus we obtained W ( u 2 ) as a 3-chain with 11123 t erms, which b etw een th em invo lve 3691 distinct 5 × 5 matrices. Subtracting from the m uch smaller W ( u 1 ) w e apply Theorem 3.8 to get: d ′ ( W ( v 1 ) − W ( v 2 )) = { a, b } . W e next ran an algorithm on W ( u 1 ) − W ( u 2 ) searching for b ound aries of 4-cycles whic h could b e added or subtracted to shorten it. Let Z 1 denote the result after adding and sub t racting those b oundaries. This h as merely 6844 distinct terms, in volving be- tw een th em 3265 matrices. Th us w e ha ve attained th e desired b oundary relation: d ′ Z 1 = [ a | b ] − [ b | a ] . (10) R emark 4.1 . The operator d ′ w as app lied to Z 1 by computer, to indep endently verify this identit y . R emark 4 .2 . Clearly it is not p ossible to include in th is article all t he steps req uired to prod uce our ex pression for Z 1 . How eve r, this ex pression is av ailable from the authors and one can chec k t h at it satisﬁes Equation ( 10) (see t h e end of § 2.3), that is, it is indeed a represen tative of the un iversal c hain (cf. Deﬁnition 2.3). 5 Computing the univ ersal Borel class As an indep endent result we show how t o compu te th e u niver sal Borel class b n ev aluated in the homology group H 2 n +1 ( GL ( C )) by expand ing the Karoubi-Hamida integ ral of [13]. This approac h allows t he calculation of the Borel regulator map for any number ﬁeld after eval uation of the Hu rewicz h omomorphism. In particular for n = 1 and F a cyclotomic ﬁeld, we can compute V 1 ( F ) (Theorem 2.11). This section is organised as follo ws. W e b egin by recalling the deﬁnition o f th e Karoubi-Hamida integra l ( § 5.1). W e expand this integral in an arbitrary o dd dimension as an inﬁnite series ( § 5.2). The form ula requ ires a certain matrix A to hav e norm less than 1; in § 5 .3 w e explain ho w to guarantee this condition. W e then simplify the form ula in dimension 3 ( § 5.4) in a w ay that can be implemen ted straigh taw ay in a computer algorithm. 13 5.1 Karoubi-Hamida’s integral By a result of Hamida [13], the universal Borel class b m has th e follo wing description as an in tegral of d iﬀeren tial forms. Let n = 2 m + 1 and let X 0 , . . . , X n ∈ GL N ( C ), for some N ≥ 2 n + 1 (cf. Remark 2.1). Let ∆ n b e th e standard n -simplex ∆ n = n ( x 0 , . . . , x n ) ∈ R n +1   x i ≥ 0 , X i x i = 1 o . (11) Deﬁne for ev ery point x = ( x 0 , . . . , x n ) ∈ ∆ n ν ( x ) = x 0 X ∗ 0 X 0 + . . . + x n X ∗ n X n , (12) where X ∗ denotes the conjugate transp ose of X . Thus ν is a matrix of 0-forms (complex functions) on th e n -manifold ∆ n . F or an y x and an y non-zero vector ~ u ∈ C N , we ha ve ~ u ∗ ν ( x ) ~ u a p ositive real n umber. That is ν ( x ) i s positive deﬁ n ite hermitian and in particular inv ertible. C onsider the matrix of d iﬀeren tial n -forms ( ν − 1 dν ) n , where ν − 1 denotes matrix inv ersion, d is the exterior d eriv ative applied to each entry of ν and we multiply individu al diﬀeren tial forms using the w edge product. Deﬁne ϕ ( X 0 , X 1 , . . . , X n ) = T r Z ∆ n ( ν − 1 dν ) n , (13) the trace of a matrix of integrals of diﬀerential n -forms. Theorem 5.1 (H amida [13]) . L et m ≥ 1 and n = 2 m + 1 . The map deﬁne d on tuples in the homo gene ous b ar r esolution, sending ( X 0 , . . . , X n ) 7→ ( − 1) m +1 2 3 m +1 ( π i ) m ϕ ( X ∗ 0 , X ∗ 1 , . . . , X ∗ n ) (14) is a c o cycle r epr esenting the universal Bor el class b m : H 2 m +1 ( GL ( C )) → R ( m ) . R emark 5.1 . Note that our conv en tion X ∗ i X i is diﬀerent from [13]: the d eﬁnition of ν ( x ) there is P i x i X i X ∗ i . R emark 5.2 . In fact ϕ is an alternating map. R emark 5.3 . The cocy cle (14) is homogeneous and unitarily normalized [13], that is, ϕ ( X 0 g , . . . , X n g ) = ϕ ( X 0 , . . . , X n ) for all g ∈ GL N ( C ) , (15) ϕ ( u 0 X 0 , . . . , u n X n ) = ϕ ( X 0 , . . . , X n ) for all u i ∈ U N ( C ) . (16) In particular, w e can assume X n = 1 by (15), and all the X i to b e p ositiv e deﬁ n ite hermitian matrices by (16), via th e p olar decomp osition: every inv ertible matrix X can b e written as X = U P where U is un itary and P is p ositive deﬁnite hermitian. In d eed, X ∗ X = P ∗ P = P 2 ; compare with (12). 5.2 The inﬁnite series form ula Our goal is to make the compu tation of the Karoub i- Hamida integra l (13) exp licit. Namely , w e will transform t h e integral into an inﬁnite series whose v alue w e can arbi- trarily appro ximate. Step 1 Expr ess the i nte gr and in terms of n c o or dinates r ather than n + 1 W e ha ve a homeomorphism from the n -simplex in R n ∆ n = { ( y 1 , y 2 , . . . , y n ) | y i ≥ 0 for all i , n X j =1 y j ≤ 1 } to ∆ n ⊂ R n +1 giv en by the map ( y 1 , . . . , y n ) 7→ (1 − n X j =1 y j , y 1 , y 2 , . . . , y n ) . Therefore, in terms of the y -coordinates, we h a ve ν = X ∗ 0 X 0 + y 1 ( X ∗ 1 X 1 − X ∗ 0 X 0 ) + . . . + y n ( X ∗ n X n − X ∗ 0 X 0 ) . (17) 14 Step 2 Change of variables Next, w e p erform a c hange of v ariables by means of th e map T : [0 , 1] × ∆ n − 1 − → ∆ n giv en by T ( t, s 1 , s 2 , . . . , s n − 1 ) = ( s 1 t, s 2 t, . . . , s n − 1 t, 1 − t ) . F or each ﬁxed non- zero v alue of t the corresp onding horizontal ( n − 1)-simplex is mapp ed diﬀeomorphically onto its image, while { 0 } × ∆ n − 1 is collapsed to the vertex (0 , 0 , . . . , 0 , 1). The Jacobian of T is equal to ( − 1) n t n − 1 . Therefore when t 6 = 0 we hav e J ( T ) > 0 if n is ev en and J ( T ) < 0 if n is o dd. F or an y compact n -manifold with b oundary M ⊂ [0 , 1] × ∆ n − 1 having image T ( M ) in the n -simplex and con tinuous map f : T ( M ) → R we hav e t h e sub stitution rule [3, p. 28] Z M ( f ◦ T )( t, s 1 , . . . , s n − 1 ) t n − 1 dt ds 1 . . . d s n − 1 = Z T ( M ) f ( y 1 , . . . , y n ) dy 1 . . . dy n , since the absolute v alue of the Jacobian is just t n − 1 . W e shall b e interested in the integral T r R ∆ n (( ν ) − 1 dν ) n whic h may b e written as the limit of integrals ov er man ifolds of t h e form T ( M ) as they tend to w ards the n -simplex. (F or example, when M α = [ α, 1] × ∆ n − 1 and α → 0 + .) Therefore w e can compute th is integ ral as a limit of correspond ing integrals ov er M ⊂ [0 , 1] × ∆ n − 1 , provided t hat this limit exists. Ho w ever the integral ov er M , invo lving the Jacobian of T , is merely the integral ov er M where ν ◦ T is ν written in terms of t , s 1 , . . . , s n − 1 and dν is also computed in these coordinates. Th us from (17) and the d eﬁnition of T w e have ν ◦ T = X ∗ 0 X 0 + n − 1 X j =1 ts j ( X ∗ j X j − X ∗ 0 X 0 ) + (1 − t )( X ∗ n X n − X ∗ 0 X 0 ) = X ∗ n X n + tA ( s 1 , . . . , s n − 1 ) , where A ( s 1 , s 2 , . . . , s n − 1 ) = ( X ∗ 0 X 0 − X ∗ n X n ) + n − 1 X j =1 s j ( X ∗ j X j − X ∗ 0 X 0 ) . (18) Therefore w e shall compute T r Z M  ( ν ′ ) − 1 dν ′  n ( − 1) n t n − 1 where ν ′ = ν ◦ T , that is, ν ′ ( t, s 1 , . . . , s n ) = X ∗ n X n + tA ( s 1 , . . . , s n − 1 ) for all ( t, s 1 , . . . , s n − 1 ) ∈ M where M ⊂ [0 , 1] × ∆ n − 1 is an arbitrary compact n -manifold with b oundary . Step 3 Assume X n = 1 T aking g = X − 1 n in (15) we may from now on assume X n = 1 (we write 1 for t he identit y matrix when ev er there is no possibility of confusion), and hence ν ( t, s 1 , . . . , s n − 1 ) = 1 + tA ( s 1 , . . . , s n − 1 ) , A ( s 1 , . . . , s n − 1 ) = ( X ∗ 0 X 0 − 1) + n − 1 X j =1 s j ( X ∗ j X j − X ∗ 0 X 0 ) . (19) R emark 5.4 . Alternatively , Prop osition 5.7 allow s u s to alwa ys assume X n = 1. Therefore dν = dt A + t dA , with dA = P n − 1 j =1 ds j ( X ∗ j X j − X ∗ 0 X 0 ) and as M v aries, w e must examine the integral T r Z M t n − 1  (1 + tA ) − 1 ( dt A + t dA )  n . (20) 15 Step 4 Com muting factors and cyclic p ermutations W e ha ve T r Z M t n − 1  (1 + tA ) − 1 ( dt A + t dA )  n = T r Z M t n − 1  (1 + tA ) − 1 dt A + (1 + tA ) − 1 t dA  n . W rite Y = (1 + tA ) − 1 dt A and Z = (1 + tA ) − 1 t dA . The 1- form dt commutes with the 0-forms (1 + tA ) − 1 and t , and an ticomm utes with the 1-form dA . In the expansion of ( Y + Z ) n , any monomial inv olving more than one Y will v anish, as it contains dt d t = 0. In addition, Z n = 0 as it is an n -form on n − 1 v ariables s 1 , . . . , s n − 1 . Consequently , T r Z M ( Y + Z ) n = n − 1 X j =0 T r Z M Z j Y Z n − 1 − j . (21) Next w e observe th at if W 1 , . . . , W n are m × m matrix-v alued fun ctions then T r ( W 1 dx 1 W 2 dx 2 . . . W n dx n ) = ( − 1) n − 1 T r ( W 2 dx 2 . . . W n dx n W 1 dx 1 ) (22) b ecause the trace of a p roduct of n matrices is inv ariant und er cyclic permutations b ut the n -form dx 1 . . . dx n changes by the sign of the n -cycle, which is ( − 1) n − 1 . Accordingly eac h integral in t he sum (21) equals ( − 1) n − 1 the next one. This implies that if n is even the sum is zero, that is, ϕ ( X 0 , X 1 , . . . , X 2 n ) = T r Z ∆ 2 n ( ν − 1 dν ) 2 n = 0 . F or th e rest of this section we shall assume that n ≥ 3 is o dd (for the case n = 1 see Remark 5.7). In this case all the summands in (21) are equal and consequently T r Z M t n − 1  (1 + tA ) − 1 ( dt A + t dA )  n = n T r Z M t n − 1 (1 + tA ) − 1 dt A  (1 + tA ) − 1 t dA  n − 1 = n T r Z M t 2 n − 2 (1 + tA ) − 1 dt A  (1 + tA ) − 1 dA  n − 1 . Now A comm utes with 1 + tA and hence with its inv erse. Th us the previous integral equals n T r Z M t 2 n − 2 dt A (1 + t A ) − 1  (1 + tA ) − 1 dA  n − 1 = n T r Z M t 2 n − 2 dt A (1 + t A ) − 2  dA (1 + t A ) − 1  n − 2 dA. (23) Step 5 I nvert 1 + tA Recall th e geometric series form u la for a matrix A [14, 5.6.16]: if k · k is a matrix norm and k A k < 1 then 1 − A is inv ertible and P ∞ k =0 A k = (1 − A ) − 1 with resp ect to k · k . (A m atrix norm on M N ( C ) is a vector norm which satisﬁes k X Y k ≤ k X kk Y k .) In order to inv ert 1 + tA , we assume from now on that k A k < 1 through th e domain of integ ration. There is a justiﬁcation, explained in § 5.3, which allows us to do so. R emark 5.5 . By k A k < 1 we formally mean k A ( s 1 , . . . , s n − 1 ) k < 1 for all ( s 1 , . . . , s n − 1 ) ∈ ∆ n − 1 . Eq u iv alently , we may d eﬁne k A k as the maximum of this fun ction ov er th e com- pact set ∆ n − 1 . The geometric series for − tA giv es (1 + tA ) − 1 = ∞ X k =0 ( − tA ) k = ∞ X k =0 ( − 1) k t k A k , and it follo ws that (1 + tA ) − 2 = ∞ X k =0 ( − 1) k ( k + 1) t k A k . 16 Hence und er the assumpt ion k A k < 1 we ma y express the in tegral (20) as a converge nt inﬁnite series in the foll ow ing manner. T r Z M t n − 1  (1 + tA ) − 1 ( dtA + tdA )  n ( 23 ) = n T r Z M t 2 n − 2 dt A (1 + tA ) − 2 dA (1 + tA ) − 1 dA . . . (1 + tA ) − 1 dA = n T r Z M t 2 n − 2 dt A X m i ≥ 0 ( − 1) m 1 ( m 1 + 1)( tA ) m 1 dA . . . ( − 1) m n − 1 ( tA ) m n − 1 dA = n T r Z M X m i ≥ 0 ( − 1) m ( m 1 + 1) t m +2 n − 2 dt A m 1 +1 dA A m 2 dA . . . A m n − 1 dA = n T r Z M X m i ≥ 0 ( − 1) m − 1 m 1 t m +2 n − 3 dt A m 1 dA A m 2 dA . . . A m n − 1 dA , where m is our short notation for m ( m 1 , . . . , m n − 1 ) = m 1 + . . . + m n − 1 . F or 0 < α < 1 write M α = [ α, 1] × ∆ n − 1 . The conditional converg ence and F ubini theorems guaran tee that the original integral ov er ∆ n equals lim α → 0 + Z M α X m i ≥ 0 ( − 1) m − 1 m 1 t m +2 n − 3 dt A m 1 dA A m 2 dA . . . A m n − 1 dA = X m i ≥ 0 ( − 1) m − 1 m 1 m + 2 n − 2 Z ∆ n − 1 A m 1 dA A m 2 dA . . . A m n − 1 dA , since lim α → 0 + Z 1 α t m +2 n − 3 dt = 1 m + 2 n − 2 . Step 6 Exp and the p owers of A Let us write A = U 0 + n − 1 X j =1 U j s j where U 0 = X ∗ 0 X 0 − 1, U j = X ∗ j X j − X ∗ 0 X 0 for 1 ≤ j ≤ n − 1 as in Eq. (19). R emark 5 .6 . I f X n 6 = 1 then we h ave U 0 = X n ( X ∗ 0 X 0 − 1) X − 1 n and, for 1 ≤ j ≤ n − 1, U j = X n  X ∗ j X j − X ∗ 0 X 0  X − 1 n . Then dA = n − 1 X j =1 U j ds j and w e can write A m 1 dA . . . A m n − 1 dA = n − 1 Y i =1 U 0 + n − 1 X j =1 U j s j ! m i n − 1 X j =1 U j ds j ! = X | l |≤| m | U ( l 1 , l 2 , . . . , l n − 1 ) s l 1 1 s l 2 2 . . . s l n − 1 n − 1 ds 1 ds 2 . . . ds n − 1 . Here the sum is ov er all nonnegative integ er vectors l = ( l 1 , . . . , l n − 1 ) such that | l | = P i l i ≤ P i m i = | m | , and U ( l 1 , . . . , l n − 1 ) is deﬁned as th e matrix co eﬃcien t of s l 1 1 . . . s l n − 1 n − 1 in the expansion of the previous line. Since this matrix d ep end s on b oth l and m = ( m 1 , . . . , m n − 1 ), we will sometimes write U ( m , l ). It can b e describ ed more explicitly: Lemma 5.2. The matrix U ( m , l ) e quals the sum of al l the matric es of the form s · V 1 1 . . . V 1 m 1 W 1 V 2 1 . . . V 2 m 2 W 2 . . . V n − 1 1 . . . V n − 1 m n − 1 W n − 1 wher e 1. V i j ∈ { U k : 0 ≤ k ≤ n − 1 } f or al l i, j ; 2. ( W 1 , . . . , W n − 1 ) is a p ermutation of ( U 1 , . . . , U n − 1 ) of signatur e s ∈ {± 1 } ; 3. l k is the c ar dinality of the set { ( i, j ) | V i j = U k } , for e ach k . 17 Step 7 Re move the r emaining inte gr al Since the matrices U ( m , l ) are constant with resp ect to the v ariables s i , w e hav e Z ∆ n − 1 A m 1 dA A m 2 dA . . . A m n − 1 dA = X | l |≤| m | U ( m , l ) Z ∆ n − 1 s l 1 1 s l 2 2 . . . s l n − 1 n − 1 ds 1 ds 2 . . . ds n − 1 . Lemma 5.3. Z ∆ n − 1 s l 1 1 . . . s l n − 1 n − 1 ds 1 . . . ds n − 1 = l 1 ! l 2 ! . . . l n − 1 ! ( l 1 + l 2 + . . . + l n − 1 + n − 1)! . T o ease n otation we write the num b er ab o ve as fact( l 1 , . . . , l n − 1 ) or fact( l ). W e leav e the proof of t he lemma as a multiv aria ble calculus exercise. On the whole w e ha ve prov en the fol lo wing. Theorem 5.4. L et n ≥ 3 o dd and X 0 , . . . , X n ∈ GL N ( C ) . L et A = U 0 + n − 1 X j =1 U j s j wher e U 0 = X ∗ n ( X ∗ 0 X 0 − 1) X − 1 n and U j = X n  X ∗ j X j − X ∗ 0 X 0  X − 1 n for 1 ≤ j ≤ n − 1 . Supp ose that k A k < 1 . Then Hamida’s function ϕ ( X 0 , . . . , X n ) e quals the lim it of the c onver gent series n ∞ X | m | =0 ( − 1) | m |− 1 | m | + 2 n − 2 m 1 X | l |≤| m | fact( l ) T r ( U ( m , l )) wher e 1. the outer sum is over al l nonne gative inte ger ve ctors m = ( m 1 , . . . , m n − 1 ) , that is, the limi t when k → ∞ of the ﬁnite sums over | m | ≤ k ; 2. the ( ﬁnite) inner sum is over al l non ne gative inte ger ve ctors l = ( l 1 , . . . , l n − 1 ) such that | l | ≤ | m | ; 3. fact( l ) is as deﬁne d in L emma 5.3; 4. U ( m , l ) is as deﬁne d i n L emma 5.2. R emark 5.7 (Case n = 1) . In this case we only need Steps 1 to 3 and the geometric series formula to inv ert 1 + tA . The map T can b e tak en t o b e the identity map, M = [0 , 1] and note that the matrix A is constant. F rom (20) and assuming k A k < 1, T r Z ∆ 1 ( ν − 1 dν ) = T r Z 1 0 (1 + tA ) − 1 A dt = T r Z 1 0 X m ≥ 0 ( − 1) m t m A m A dt = T r X m ≥ 0 ( − 1) m A m +1 Z 1 0 t m dt = T r (log (1 + A ) ) . In Section 5.4 w e w ill consider th e case n = 3 in more detail (the relev an t case for the computation of V 1 ( F )). Before that, w e explain ho w to ensure the condition k A k < 1. 5.3 Con trolling the norm In order to use the geometric series t o in vert 1 + tA (Step 5 in § 5.2) we need k A k < 1 for an y matrix norm k · k . W e ensure t h is condition for th e sp ectral norm by using a homological tric k. Fi rst we brieﬂy discuss matrix norms. Let k · k a matrix norm , that is, a vector norm in M N ( C ) which satisﬁes k X Y k ≤ k X kk Y k . Our main example will b e the sp e ctr al norm [14, 5.6.6] k X k 2 = max n √ λ : λ is an eigenv alue of X ∗ X o . (Note that X ∗ X is positive semideﬁnite (hermitian) and hence all eigenv alues are real and nonnegative .) T his norm satisﬁes 18 (i) k X ∗ k = k X k for all X ∈ M N ( C ); (ii) k X ∗ X k = k X k 2 for all X ∈ M N ( C ); (iii) if X is hermitian then k X k = max {| λ | : λ is an eigenv alue of X } ; (iv) if X is p ositiv e semideﬁnite then k X k = λ max ( X ) the maximum eigenv alue. If X is hermitian, all its eigen v alues are real and w e will write them in increasing order as λ min ( X ) = λ 1 ( X ) ≤ λ 2 ( X ) ≤ · · · ≤ λ N ( X ) = λ max ( X ) . Lemma 5.5. L et X , Y b e hermitian matric es. T hen λ max ( X + Y ) ≤ λ max ( X )+ λ max ( Y ) and λ min ( X + Y ) ≥ λ min ( X ) + λ min ( Y ) . This Lemma follo ws from Theorem 4.3.1 in [14]. If X is a matrix of functions ov er a set ∆ ⊂ R n such that X ( s ) is hermitian for eac h s ∈ ∆, w e deﬁne λ max ( X ) = sup s ∈ ∆ λ max ( X ( s )) , λ min ( X ) = inf s ∈ ∆ λ min ( X ( s )) , k X k = sup s ∈ ∆ k X ( s ) k . This deﬁn ition is consis tent with our previous notation (Remark 5.5) . Note that if X ( s ) is p ositiv e deﬁnite for all s then k X k = λ max ( X ). Let X 0 , . . . , X n ∈ GL N ( C ) and consider for eac h s = ( s 1 , . . . , s n ) ∈ ∆ n A ( s ) = X ∗ 0 X 0 − I + n X j =1 s j  X ∗ j X j − X ∗ 0 X 0  . (This is the matrix A asso ciated to the tu ple ( X 0 , X 1 , . . . , X n , I ) as in § 5.2.) W e may write A ( s ) = H ( s ) − I where H ( s ) = X ∗ 0 X 0  1 − n X j =1 s j  + n X j =1 s j X ∗ j X j is a p ositive deﬁnite hermitian matrix (a p ositive linear combination of p ositiv e d eﬁnite hermitian matrices). D eﬁ ne λ max = max 0 ≤ j ≤ n λ max ( X ∗ j X j ) = max i,j λ i ( X ∗ j X j ) and λ min = min 0 ≤ j ≤ n λ min ( X ∗ j X j ) = min i,j λ i ( X ∗ j X j ) . Lemma 5.6. We have (i) k H k = λ max ( H ) = λ max , λ min ( H ) = λ min ; (ii) k A k = max {| λ max − 1 | , | λ min − 1 |} ; (iii) k A k < 1 i f and only if λ max < 2 . Pr o of. U sing Lemma 5.5 we ha ve λ max ( H ( s )) ≤  1 − P n j =1 s j  λ max ( X ∗ 0 X 0 ) + P n j =1 s j λ max ( X ∗ j X j ) ≤ λ max and λ min ( H ( s )) ≥  1 − P n j =1 s j  λ min ( X ∗ 0 X 0 ) + P n j =1 s j λ min ( X ∗ j X j ) ≥ λ min for all s = ( s 1 , . . . , s n − 1 ) ∈ ∆ n − 1 . This prov es part (i). F or part (ii), w e ha ve k A k = sup s k H ( s ) − I k = sup s max i | λ i ( H ( s ) − I ) | = sup s ,i | λ i ( H ( s )) − 1 | . The supremum of t he distances b etw een a point and the p oints of a b ounded subset of R equals the distance to either the suprem um or the inﬁ m um of the set, k A k = max {| sup s ,i ( λ i ( H ( s )) ) − 1 | , | inf s ,i ( λ i ( H ( s )) ) − 1 |} = max {| λ max − 1 | , | λ min − 1 |} , using part (i). P art (iii) follo ws from (ii) observing that 0 < λ min ≤ λ max . 19 W e now explain how to guaran tee the condition k A k < 1 by rescaling the matrices X i . Let µ > 0. The b oundary of the tuple ( X 0 , . . . , X n , µ − 1 I ) is n X i =0 ( − 1) i ( X 0 , . . . , c X i , . . . , X n , µ − 1 I ) + ( − 1) n +1 ( X 0 , . . . , X n ) . Since Hamida’s function ϕ is a cocycle, it va nishes on boun d aries an d thus ϕ ( X 0 , . . . , X n ) = n X i =0 ( − 1) n + i ϕ ( X 0 , . . . , c X i , . . . , X n , µ − 1 I ) = n X i =0 ( − 1) n + i ϕ ( µX 0 , . . . , d µX i , . . . , µX n , I ) , the last equality coming from m ultiplying by a diagonal matrix with µ in the diagonal (Eq.(15) in Remark 5.3). W e now prov e that if µ is s mall enough, the matri x A associated to any of the tuples on the right-hand side satisﬁes k A k < 1 for the sp ectral norm. Hence th e val ue of Hamida’s function ϕ at ( X 0 , . . . , X n ) can b e computed as the alternating sum of th e v alues at tu ples satisfying th e h yp otheses of Theorem 5.4. Proposition 5.7. L et X 0 , . . . , X n ∈ GL N ( C ) and µ > 0 . Then ϕ ( X 0 , . . . , X n ) = n X i =0 ( − 1) n + i ϕ ( µX 0 , . . . , d µX i , . . . , µX n , I ) . L et λ max = max 0 ≤ j ≤ n λ max ( X ∗ j X j ) . I f 0 < µ < q 2 λ max then f or e ach tuple on the right-hand side, the asso ciate d matrix A satisﬁes k A k < 1 . Pr o of. W e are left with t he pro of of th e second statement. Let A b e the matrix asso- ciated to ( µX 0 , . . . , d µX i , . . . , µX n , I ) for some 0 ≤ i ≤ n . That is, A ( s ) = µ 2 X ∗ 0 X 0 − I + µ 2 i − 1 X j =1 s j  X ∗ j X j − X ∗ 0 X 0  + µ 2 n X j = i +1 s j − 1  X ∗ j X j − X ∗ 0 X 0  , for eac h s = ( s 1 , . . . , s n − 1 ) ∈ ∆ n − 1 . Let us write A ( s ) = H ( s ) − I . By Lemma 5.6(i) w e hav e λ max ( H ) = max j 6 = i λ max ( µ 2 X ∗ j X j ) ≤ µ 2 λ max < 2 . The result follo ws now from Lemma 5.6(iii). F or computational purp oses (cf. § A) w e will b e in terested in minimazing k A k . W e record the relev ant result here, foll ow ed by a remark. Lemma 5.8. L et X 0 , . . . , X n ∈ GL N ( C ) , λ max = max 0 ≤ j ≤ n λ max ( X ∗ j X j ) , λ min = min 0 ≤ j ≤ n λ min ( X ∗ j X j ) . Given µ > 0 write A µ for the matrix asso ciate d t o the tuple ( µX 0 , . . . , µX n ) . Then the function µ 7→ k A µ k r e aches a minimum value λ max − λ min λ max + λ min at µ = q 2 λ max + λ min . Pr o of. By Lemma 5.6(ii) k A µ k = max {| µ 2 λ max − 1 | , | µ 2 λ min − 1 |} . The maximum of the d istances to 1 reaches a minimum when b oth p oin ts are equidistant µ 2 λ max − 1 = 1 − µ 2 λ min ⇔ µ = r 2 λ max + λ min . F or this v alue of µ k A µ k = µ 2 λ max − 1 = 2 λ max λ max + λ min − 1 = λ max − λ min λ max + λ min . R emark 5.8 . Note that µ = q 2 λ max + λ min ≤ q 2 λ max and hence Prop osition 5.7 applies for this choice of scaling factor µ . Moreo ver, for all tu ples on the righ t-hand side excep t at most t wo , the associated matrix A will reac h its minim um norm. 20 5.4 The inﬁnite series for n = 3 In this section we simplify and rearrange th e form ula in Theorem 5.4 for n = 3; this is t he case relev ant for th e computation of V 1 ( F ). The resulting formula can b e im- plemented as a computer a lgorithm. The impatie nt reader ma y skip o v er t h e nex t calculations to Theorem 5.11. F or n = 3 Theorem 5.4 giv es the ex pression 3 X m 1 ,m 2 ≥ 0 ( − 1) m 1 + m 2 − 1 m 1 + m 2 + 4 m 1 X l 1 + l 2 ≤ m 1 + m 2 fact( l 1 , l 2 ) T r ( U ( m 1 , m 2 , l 1 , l 2 )) . (24) Recall from Lemma 5.2 that t h e matrix U ( m 1 , m 2 , l 1 , l 2 ) is a sum of words of the form s · V 1 1 . . . V 1 m 1 W 1 V 2 1 . . . V 2 m 2 W 2 . for s ∈ {± 1 } . W e exploit th e symmetry b et w een ( m 1 , m 2 ) and ( m 2 , m 1 ), and the inv ariance of the trace under cyclic p ermutations to simplify ( 24). W e will only need to consider t races of matrices of the form U 1 ω 1 U 2 ω 2 , and in this w a y w e can disregard the sign s of t h e permutation. Hence we d eﬁne: Deﬁnition 5.9. The matrix e U ( m 1 , m 2 , l 1 , l 2 ) is the sum of all the matrices of th e form U 1 V 1 1 . . . V 1 m 1 U 2 V 2 1 . . . V 2 m 2 (25) where V i j ∈ { U k : 0 ≤ k ≤ 2 } for all i, j and l k =   { ( i, j ) | V i j = U k }   for k = 1 , 2. T o ease notation let us write c = ( − 1) m 1 + m 2 − 1 m 1 + m 2 +4 whenever m 1 and m 2 are clear from t h e context. Lemma 5.10. X m 1 ,m 2 ≥ 0 c m 1 X | l |≤| m | fact( l ) T r ( U ( m , l )) = X m 1 ,m 2 ≥ 0 c ( m 2 − m 1 ) X | l |≤| m | fact( l ) T r( e U ( m , l )) . Pr o of. Fix m 1 , m 2 ≥ 0. Let ω 1 and ω 2 b e arbitrary words on letters U i , 0 ≤ i ≤ 2 of lenght m 1 respectively m 2 . On the LHS w e ha ve the four words ω 1 U 1 ω 2 U 2 , ω 1 U 2 ω 2 U 1 , ω 2 U 1 ω 1 U 2 and ω 2 U 2 ω 1 U 1 . The trace of t he ﬁrst and last, and of th e second and third are t h e same, say t and t ′ . Hence w e hav e the four summands c m 1 t , − c m 1 t ′ , c m 2 t ′ and − c m 2 t , times the factorial coeﬃcient ( l is the same for all w ords). S upp ose th at m 1 6 = m 2 . Then on the RHS w e hav e tw o w ords, U 1 ω 1 U 2 ω 2 and U 1 ω 2 U 2 ω 1 . By a cy clic p erm utation, t he traces are t ′ and t resp ectively . The correspond ing sum- mands are then c ( m 2 − m 1 ) t ′ and c ( m 1 − m 2 ) t , multiplied b y t he sa me factorial coeﬃcient. Finally , the case m 1 = m 2 giv es 0 on b oth sides. Now consider a matrix U 1 U i 1 . . . U i k with i j ∈ { 0 , 1 , 2 } an d k ≥ 1. W rite n i for the total num ber of letters equal to U i , i = 1 , 2 among the U i j . How many times do es this matrix ap p ear in a expression of the form (25 )? F or eac h i j = 2 it app ears in e U ( m 1 , m 2 , n 1 , n 2 − 1) for m 1 = j − 1 and m 2 = k − j , and coeﬃcient ( − 1) m 1 + m 2 − 1 m 1 + m 2 + 4 ( m 2 − m 1 ) fact ( l 1 , l 2 ) = ( − 1) k − 2 k + 3 ( k − 2 j + 1) fact( n 1 , n 2 − 1) . All in all we h a ve proven the follo wing. W rite χ a for the characteris tic function on an integ er a (so χ a ( a ) = 1 and χ a ( i ) = 0 if i 6 = a ). Theorem 5.11 (Inﬁnite series for n = 3) . L et X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ) and A = U 0 + s 1 U 1 + s 2 U 2 deﬁne d as in The or em 5.4 for n = 3 . Supp ose that k A k < 1 . Then Hamida’s f unction ϕ ( X 0 , X 1 , X 2 , X 3 ) e quals the lim it of the c onver gent series 3 ∞ X k =1 ( − 1) k k + 3 2 X i 1 ,...,i k =0 d fact( n 1 , n 2 − 1) T r ( U 1 U i 1 . . . U i k ) , (26) wher e n 1 , n 2 and d dep end on e ach tuple ( i 1 , . . . , i k ) as n 1 = P j χ 1 ( i j ) , n 2 = P j χ 2 ( i j ) and d = P j χ 2 ( i j ) ( k − 2 j + 1) = n 2 ( k + 1) − 2 P j χ 2 ( i j ) j , and fact( l 1 , l 2 ) is deﬁne d as i n L emma 5.3 for l 2 ≥ 0 and as 0 if l 2 = − 1 . 21 W e can now explain how to calculate ϕ ( X 0 , X 1 , X 2 , X 3 ) for arbitrary X i ∈ GL N ( C ). Firstly , use Equation (16) in Remark 5.3 to assume, without loss of generality , that X 0 = I . Secondly , by Proposition 5.7, for any µ > 0, ϕ ( X 0 , X 1 , X 2 , X 3 ) = − ϕ ( µX 1 , µX 2 , µX 3 , I ) + ϕ ( µX 0 , µX 2 , µX 3 , I ) − ϕ ( µX 0 , µX 1 , µX 3 , I ) + ϕ ( µX 0 , µX 1 , µX 2 , I ) . F or µ small enough (see Prop osition 5.7 or Remark 5.8) , each tuple on the right hand side will satisfy the conditions of Theorem 5.11. In addition, since X 0 = I , the last three terms satisfy t he hypoth esis of Lemma 5.12 b elow , and hen ce the Hamida fun ction on these tuples v anishes. Therefo re ϕ ( X 0 , X 1 , X 2 , X 3 ) = − ϕ ( µX 1 , µX 2 , µX 3 , I ) . Finally , w e can use the con verg ent series ( 26) to obtain an approximation of the val ue in the righ t-hand side. Lemma 5.12. If X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ) satisfy the hyp othesis of T he or em 5.11, and X ∗ 0 X 0 is c entr al (i e. a sc alar multiple of the i dentity) then ϕ ( X 0 , X 1 , X 2 , X 3 ) vanishes. Pr o of. This follo ws easily from Proposition A.5 (Ap p end ix A.4) and Theorem 5.11. All in all, w e hav e the follo wing. Theorem 5.13. L et X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ) arbitr ary. Deﬁne X ′ j = X j X − 1 0 for e ach 0 ≤ j ≤ 3 . L et λ max = max 0 ≤ j ≤ n λ max  ( X ′ j ) ∗ X ′ j  and 0 < µ < q 2 λ max . Then ϕ ( X 0 , X 1 , X 2 , X 3 ) = − ϕ ( µX ′ 1 , µX ′ 2 , µX ′ 3 , I ) , (27) and the tuple on the right-hand side satisﬁes the hyp othesis of The or em 5.11. Corollary 5.14. L et X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ) arbitr ary. If ( X 1 X − 1 0 ) ∗ X 1 X − 1 0 is c entr al, then ϕ ( X 0 , X 1 , X 2 , X 3 ) vanishes. Pr o of. U se (27) and Lemma 5.12. A Computational asp ects One adv an tage of the form ula in Theorem 5.11 for Hamida’s integral ev aluated at H 3 ( GL ( C )) is that it admits a straightf orw ard imp lementation as a computer algo- rithm. In this app endix we discuss det ails relev ant to an implementation of th e form ula in Theorem 5.11 or the computation of V 1 ( F ) for a cyclotomic ﬁeld F . The alg orithm to comput e H amida’s function takes as in p ut matrices X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ), p erforms the ‘homological trick’ of § 5.3 and outputs the par- tial sum of (26) up to a given k . These partial sums conv erge to H amida’s function ϕ ( X 0 , X 1 , X 2 , X 3 ), and we hav e an upp er b ound for th e error after k iterations (Ap- p endix A.3). T herefore, we can in principle compute the universal Borel class b 1 ev al- uated at H 3 ( GL N ( C )) (via Theorem 5.1) to any prescrib ed degree of accuracy . This, together with the inpu t c hain q Z 1 − Z 2 ( q ) (see § 2.3) allo w t he computation of V 1 ( F ) for any cy clotomic ﬁeld to any requ ired precision. W e hav e included a complete description of the algorithm in App endix A .1. How ever, the computational complexity of this algorithm is exp onential on k (Ap- p endix A.2): t h e compu t ing time of t h e ( k + 1)th iteration is roughly 3 times that of the k t h iteration. The error after k iterations is b ounded by a constan t times k A k k k (see App endix A.3) so it will n ev ertheless con verge fast to zero, un less k A k is close t o one. Unfortunately , this seems to be the case in practice: for instance, more than 86% of th e terms in Z 1 | t = e 2 πi/ 3 hav e an associated matrix A of norm greater th an 0.9, even for the optimal c hoice of parameter µ (as in Lemma 5.8). A naive implementation with symbolic mathematical softw are (we u sed Maple [19]) is not very eﬃcie nt (it takes o n av erage 12 s p er term with k = 3). A numerica l implementa tion (we used Matlab [20]) i s muc h faster (it takes on a verag e 1.96 s p er term with k = 8) but it is still not fast en ough to guarantee a small error w ithin reasonable time, according to our error form ula. It is clear th at eith er a more eﬃcient implementa tion, or a tighter b ound on the error, or an alternative ‘homologica l trick’ is needed. I n an y case, t h e pu rpose of th e 22 present article is to describ e the theory b ehind a n ew approac h to c ompute Borel’s regulator, and it is ou r hope that more accurate implemen tations will b e built on these foundations. In App endix B we will d iscuss the results of a diﬀerent (C++) implemen tation of the same algorithm made by the 1 st author in the course of his doctoral thesis [7]. A.1 Complete algorithm Computation of ϕ ( X 0 , X 1 , X 2 X 3 ) for arbitrary X 0 , X 1 , X 2 , X 3 ∈ GL N ( C ) 1. Deﬁn e λ max = max 0 ≤ i ≤ 3 λ max ( X ∗ i X i ) , Y i = µX i X − 1 0 ( i = 1 , 2 , 3) , λ min = max 0 ≤ i ≤ 3 λ min ( X ∗ i X i ) , U 0 = Y ∗ 1 Y 1 − I , µ = p 2 / ( λ max + λ min ) , U i = Y ∗ i Y i − Y ∗ 1 Y 1 ( i = 2 , 3) . 2. F or eac h k max ≥ 1, consider the ﬁnite sum 3 k max X k =1 ( − 1) k k + 3 2 X i 1 ,...,i k =0 d ( i 1 , . . . , i k ) fact( n 1 , n 2 − 1) T r ( U 1 U i 1 . . . U i k ) , (28 ) with n 1 , n 2 , d and ‘fact’ deﬁned as in Theorem 5.11. Then (28) approximates ϕ ( Y 1 , Y 2 , Y 3 , I ) = − ϕ ( X 0 , X 1 , X 2 , X 3 ), with an error b ound ed abov e by N k U 1 kk U 2 k (1 − ρ ) 2 ρ k max (1 + k max (1 − ρ )) , where ρ = λ max − λ min λ max + λ min (Corollary A.4 and Lemma 5.6(ii)). Computation of b 1 ev aluated at a term in the inhomogeneous bar resolution Let [ g 1 | g 2 | g 3 ] ∈ B 3 (notation of § 2.2 ). By writing [ g 1 | g 2 | g 3 ] = ( I , g 1 , g 1 g 2 , g 1 g 2 g 3 ) and using Theorem 5.1 w e ha ve that the universal Borel class eval uated at [ g 1 | g 2 | g 3 ] is b 1 ([ g 1 | g 2 | g 3 ]) = 1 16 π i ϕ ( I , g ∗ 1 , ( g 1 g 2 ) ∗ , ( g 1 g 2 g 3 ) ∗ ) . W e can th en comput e the right-hand side by t he algorithm ab ov e. More generally , any elemen t z ∈ B 3 can b e written as a ﬁnite sum of elements as abov e. Computation of V 1 for a cyclotomic ﬁeld Let F = Q ( ξ ) b e a cy clotomic ﬁeld and suppose th at ξ = e 2 πi/q , for some integer q ≥ 3. Let Z 1 b e th e u nivers al c hain (recall we h a ve found an explicit represen tative 2 ) and deﬁn e Z 2 ( q ) as in Deﬁn ition 2.4. Let Z = q Z 1 − Z 2 ( q ) ∈ B 3 ( Z [ t, t − 1 ]). Let and ± v 1 , . . . , ± v l the u nits in Z /q Z . F or eac h 1 ≤ i, j ≤ l deﬁne Z ij by t h e substitut ion t = ξ v i v j in Z , Z ij = Z | t = ξ v i v j ∈ B 3 ( F ) . Then z ij = b 1 ( Z ij ) can b e ev aluated by our algorithm ab ov e. Finally , V 1 ( F ) equals the absolute v alue of the determinant of the matrix whose ( i, j ) -entry is z ij (Theorem 2.11). A.2 Complexit y The computational complexity of our prop osed algorithm (A p p en dix A.1) is exp onential on k : for each term in a homological cycle (for the c hain q Z 1 − Z 2 ( q ) w e hav e 6844 + 3 q terms) we compu te an instance of the inﬁnite series in Theorem 5.11 (cf. Theorem 5.13); for eac h series, w e ad d successiv e t erms until the error is smaller than the target error; ﬁnally , the k th summand of th e series in Theorem 5.11 inv olves cond u cting 3 k diﬀerent multipli cations of k + 1 matrices of size N (for the chain q Z 1 − Z 2 ( q ) we have N = 5). R emark A.1 . F or a general n > 3 o dd we exp ect an inﬁn ite series analogous to the one in Theore m 5 .11. Thus for ea ch term in a h omologi cal cycle we w ould ha v e n instances of t h e inﬁnite series, and in turn the k th summand in a series would inv olve n k multipli cations of k + 1 matrices of size typicall y 2 n + 1. 2 see ancillary ﬁle anc/univ ersalc hain.mat 23 A.3 Error b ound This section is devoted to b ounding the error in computing t he inﬁn ite series (26) after k iterations. The impatien t reader ma y refer to Proposition A.3 and Corollary A.4. Throughout this section, let k · k b e the spectral norm. Lemma A. 1. L et X b e a hermitian matrix of size N . Then | T r( X ) | ≤ N k X k . Pr o of. I f λ 1 , . . . , λ N are the eigen v alues of X then k X k = max i | λ i | an d hence | T r( X ) | =    X i λ i    ≤ N k X k . Lemma A.2. L et X b e a matrix o f size N of c ontinuous functions over a c om p act subset ∆ ⊂ R n . Supp ose that X ( s ) is hermiti an for e ach s ∈ ∆ and deﬁne k X k = max s ∈ ∆ k X ( s ) k . Then     T r Z ∆ X ( s ) d s     ≤ vol(∆) N k X k . Pr o of. By elementary properties of integration and Lemma A.1,     T r Z ∆ X ( s ) d s     =     Z ∆ T r ( X ( s )) d s     ≤ Z ∆ | T r ( X ( s )) | d s ≤ vol(∆) N k X k . Proposition A.3. L et a k b e the k th summand of the series in The or em 5.11 , and ke ep the same hyp othesis and notation. Then | a k | ≤ N k U 1 k k U 2 k k k A k k − 1 . Corollary A. 4. L et C = N k U 1 k k U 2 k , ρ = k A k < 1 and b k = k ρ k − 1 for e ach k ≥ 1 . Then      ∞ X k = l +1 a k      ≤ C ∞ X k = l +1 b k = C ρ l  1 + l (1 − ρ ) (1 − ρ ) 2  . Note that th e exp ression on the left-hand side is the absolute error of approximating P ∞ k =1 a k by P l k =1 a k , the outp ut of the algorithm after l iterations. The up p er b ound on the righ t-hand side conv erges t o zero as l → ∞ . R emark A.2 . Recall that Lemma 5.6(ii) allows us to have an explicit val ue for ρ = k A k . Pr o of of Pr op osition. In order to attain a neat u p per bound dep end ing on k A k , we retrace our steps in § 5.4 an d § 5.2 to write a k = X m 1 + m 2 = k − 1 ( − 1) k k + 3 m 1 T r Z ∆ 2 A m 1 dA A m 2 dA , since a k inv olv es all the w ords of total lenght k + 1. Recall that dA = U 1 ds 1 + U 2 ds 2 . Then T r Z ∆ 2 A m 1 dA A m 2 dA = T r Z ∆ 2 A m 1 U 1 A m 2 U 2 ds 1 ds 2 − T r Z ∆ 2 A m 1 U 2 A m 2 U 1 ds 1 ds 2 . By the triangle inequality and the Lemma A.2,     T r Z ∆ 2 A m 1 dA A m 2 dA     ≤ vol(∆ 2 ) N ( k A m 1 U 1 A m 2 U 2 k + k A m 1 U 2 A m 2 U 1 k ) ≤ N k U 1 k k U 2 k k A k k − 1 . There are k pairs of nonnegativ e in tegers ( m 1 , m 2 ) suc h th at m 1 + m 2 = k − 1. H ence | a k | ≤ k 1 k + 3 k N k U 1 k k U 2 k k A k k − 1 . Observing that k k +3 ≤ 1 ﬁnishes the pro of. 24 A.4 V anishing of terms where X ∗ 0 X 0 is cen tral Recall that Theorem 5.13 and Corollary 5.14 depend on Lemma 5.12, whic h is in turn a consequence of the foll ow ing result. Proposition A.5. Ke ep the notation a nd hyp othesis of The or em 5.11. Le t k ≥ 1 , n 1 , n 2 ≥ 0 b e inte gers. If X ∗ 0 X 0 is c entr al then X ( i 1 ,...,i k ) ∈ I d ( i 1 , . . . , i k ) T r ( U 1 U i 1 . . . U i k ) = 0 , (29) wher e I denotes the set of al l se quenc es of length k which have n 1 terms e qual to 1 , n 2 terms e qual to 2 , and the r emaini ng terms e qual to 0 , and d : I → Z is deﬁne d as d ( i 1 , . . . , i k ) = n 2 ( k + 1) − 2 P i j =2 j . As a corollary , the Hamida function on the tuple will v anish: simply collate the t erms in the sum (26) with ﬁxed k , n 1 and n 2 . This prov es Lemma 5.12 . The rest of this section consists on a combinatorial pro of of Proposition A.5 . F or ev ery sequence ( i 1 , · · · i k ) ∈ I w e deﬁ ne T ( i 1 , · · · i k ) = (1 , i j 1 , i j 2 , · , i j n 1 + n 2 ), where the i j t form the subsequence of all n on-zero t erms, in the same order as they app ear in ( i 1 , · · · i k ). That is, th e op eration T ad d s a 1 to th e start of the sequence and remo ves all the 0’s. W e partition I into equiv alence classes by setting i ∼ i ′ precisely when T ( i ) is a cyclic p erm utation of T ( i ′ ). Our strategy in proving Proposition A.5 will b e to show that for an y equiv alence class J ⊂ I , we h a ve: X ( i 1 ,...,i k ) ∈ J d ( i 1 , . . . , i k ) T r ( U 1 U i 1 . . . U i k ) = 0 . (30) Lemma A. 6. The tr ac es T r ( U 1 U i 1 . . . U i k ) in the sum ab ove ar e al l e qual. Pr o of. The t race of a p roduct of matrices is inv arian t under cyclic p ermutations of the prod uct. F u rther, as U 0 is central, the trace is not eﬀected b y the exact position of the U 0 terms in the produ ct. Give n tw o traces T r( U 1 U i 1 . . . U i k ) and T r( U 1 U i ′ 1 . . . U i ′ k ) in t he ab o ve sum, w e may cycle the pro d uct of matrices in the ﬁrst one to get the the second one, if w e ignore the U 0 terms. Since b oth pro ducts hav e the same number k − n 1 − n 2 of U 0 terms, an d they are cen tral, both traces coincide. Therefore it then remains to pro ve the follo wing purely combinatorial result: X ( i 1 ,...,i k ) ∈ J d ( i 1 , . . . , i k ) = 0 . (31) W e ﬁrst examine d more closely: d ( i 1 , . . . , i k ) = n 2 ( k + 1) − 2 X i j =2 j = X i j =2 ( k + 1 − 2 j ) = X i j =2 k X x =1 σ j ( x ) (32) where σ j ( x ) =    − 1 if x < j, 0 if x = j, 1 if x > j. (F or the last equality in (32), note that P k x =1 σ j ( x ) = ( − 1)( j − 1) + 1( k − j ) = k − 2 j + 1.) Fix a sequence i ∈ J and let T ( i ) = ( a 1 , a 2 , · · · , a m ). Then m = n 1 + n 2 + 1 and each a t lies in { 1 , 2 } . Let s denote the smallest non - negativ e integer such that a t = a t + s for all t , where the indices are taken modulo m . That is, the order of the group of cyclic symmetries of T ( i ) is m /s . Now let C denote the set of m d imensional vectors with non-negative integer entries whic h sum to k − n 1 − n 2 . (Note that C is a ﬁnite set.) T o each integer l , 1 ≤ l ≤ m , such that a l = 1 and to each vector ~ c = ( c 1 , · · · , c m ) ∈ C , w e associate a sequ ence θ ( l , ~ c ) ∈ J constructed as follo ws: 1) Starting with the sequence ( a 1 , · · · , a m ), insert c t zeroes after a t , for eac h t . 25 2) Cycle the resulting sequence of length k + 1 so that a l = 1 is the ﬁrst term. 3) Delete the ﬁrst term (whic h is a l = 1). Lemma A. 7. Every element of J c an b e c onstructe d in pr e cisely m/s ways as θ ( l, ~ c ) for an inte ger l with a l = 1 and a ve ctor ~ c ∈ C . Pr o of. Given a sequence j ∈ J , whic h we wish to constru ct by th e op eration abov e, we hav e a choice of m/s integ ers for l . An d for eac h such choice there is a uniqu e c hoice of vector ~ c ∈ C which results in j . Therefore w e can write X ( i 1 ,...,i k ) ∈ J d ( i 1 , . . . , i k ) = s m X ~ c ∈ C X a l =1 d ( θ ( l , ~ c )) ( 32 ) = s m X ~ c ∈ C X a l =1 X a j =2 k X x =1 σ b j ( x ) , (33) where in eac h case b j denotes the position of the term a j in the sequence θ ( l , ~ c ). Give n integers r, t, u mod ulo m , w e set: ǫ r tu =    − 1 if starting at r and rep eatedly adding 1 , one arrives at u b efore t, 0 if r, t, u are not pairwise distinct , 1 if starting at r and rep eatedly adding 1 , one arrive s at t b efore u. T o the sequence ( a 1 , · · · , a m ) we associate tw o m dimensional v ectors ~ e = ( e 1 , · · · , e m ) and ~ f = ( f 1 , · · · , f m ). They are deﬁned as e t = X a r =1 X a u =2 ǫ r tu , f t =  e t + n 2 a t = 1 , e t − ( n 1 + 1) a t = 2 . One ma y visualize these deﬁnitions by arranging th e a t round an oriented circle and noting that e t counts the arcs from a 1 to a 2 p assing through a t (with sign given by the d irection), and f t counts the arcs from a 1 to a 2 containing the arc b etw een a t and a t +1 (again with sign giv en by the direction). As alw a ys, when dealing with the a t , we regard the indices mod ulo m , and recall th at there is n 1 + 1 1’s and n 2 2’s in the sequence ( a 1 , . . . , a m ). Let e = P m t =1 e t and note that m X t =1 f t = e + ( n 1 + 1) n 2 − n 2 ( n 1 + 1) = e . Now by symmetry w e hav e P ~ c ∈ C ~ c = r (1 , · · · , 1), for so me in teger r . W e may separate the terms in t he ﬁnal sum in (33) into those where the x th term in θ ( l, ~ c ) is zero, and those where it is n ot: X ( i 1 ,...,i k ) ∈ J d ( i 1 , . . . , i k ) = s m X ~ c ∈ C X a l =1 X a j =2 k X x =1 σ b j ( x ) = s m X ~ c ∈ C m X t =1 ( c t f t + e t ) = s m ( r e + | C | e ) . Thus we ma y complete th e proof of (31) by sho wing t hat e = 0. That is: Lemma A. 8. Supp ose a ﬁnite numb er of r e d and blue dots ar e arr ange d r ound a cir cle. L et segment denote a p ortion of the cir cle b etwe en adjac ent dots. Consider the set of al l dir e cte d ar cs going f r om a blue dot to a r e d dot. Then the total numb er e of se gments tr averse d by the ar cs (c ounting wi th ne gative sign i f c ounter clo ckwise and p ositive sign if clo ckwise) is 0. (Note we hav e replaced 1’s and 2’s with b lue dots and red d ots respectively). Pr o of. First consider a diameter of the circle and supp ose th at all th e b lue d ots lie on one side of the diameter and all the red dots lie on the other side. Then by symmetry w e w ould hav e e = 0. It remains to show that e is constant under interc hanging an adjacen t red and blue dot. Let e 1 b e the v alue of e when the blue d ot is clo c kwise of the red d ot (state 1) and let e 2 b e the v alue of e when they are interc hanged, so the red dot is clockwis e of the blue dot (state 2). As b efore, let m denote t he t otal num ber of dots. 26 Then every dot oth er than the pair we are interc hanging, has tw o arcs joining it to one of the pair. Each of these arcs b ecomes one segment larger going from state 1 to state 2 (b earing in mind that we are counting segmen ts with sign). Th us e 2 has an extra 2( m − 2) more than e 1 coming from these arcs. The contri butions from th e arcs not invol ving either of the interc hanged pair are not eﬀected. Thus the only other consideration is the tw o arcs joining th e pair. In state 1 t hey contribute − 1 and m − 1 t o the sum. In state 2 th ey contribute 1 and 1 − m to the sum. Thus we ha ve e 2 = e 1 + 2( m − 2) − ( m − 2) + (2 − m ) = e 1 , as required. B A n umerical computation of V 1 ( F ) In the course of his doctoral stu dies [7], the 1 st author carried out h is o wn C++ implementa tion of an algorithm b ased on Theorem 5.11 to compute V 1 ( F ) in the case where F is the cyclotomic ﬁ eld Q ( ω ) with ω = e 2 πi 3 . W e will now b rieﬂy discuss the numerical results of [7] based on our theory . There is just one pair of conjugate embeddings ψ 1 , ψ 1 : F ֒ → C , and one pair of conjugate primitive 3 rd roots of unity , ω , ω 2 ∈ F . Thus V 1 ( F ) is the absolute v alue of the determinant of the one by one matrix whose entry (see § 2.3) is: L 11 = b 1 ([(3 Z 1 − Z 2 (3)) | t = ω ]) /i . This has 6844 terms coming from Z 1 and 9 terms coming from Z 2 (3). In § 5.1 of [7], it is explained that the program ﬁrst remov ed terms containing rep eated matrices, as they d o not contribute to L 11 , and combined terms with the same matrices in p erm uted order (see Remark 5.2). This left 3450 distinct terms. The series form ula of Theorem 5.11 was app lied to t h ese t erms. A fter running for 12 hours on a 2.4GHz computer the estimated v alue of V 1 ( F ), up to tw o decimal points, w as V 1 ( F ) = 0 . 02 . Assuming that the error is in fact no larger than 0.01, w e w ould kn o w th at V 1 ( F ) 6 = 0 and, from Theorem 2.10 , that R 1 ( F ) = V 1 ( F ) / ind( F ) with V 1 ( F ) lyin g in the range [0 . 01 , 0 . 03], and ind( F ) ∈ Z . F or t his num b er ﬁeld the Borel regulator R 1 ( F ) is known to b e − 2 k 3 ζ ∗ F ( − 1) for some k ∈ Z , and ζ ∗ F ( − 1) = − 0 . 02692 . . . (see § 5.2 in [7] for details). If ind( F ) = 1, t hen we would h ave R 1 ( F ) = V 1 ( F ), and th e ca lculation w ould suggest that k = − 2, as this w ould give R 1 ( F ) = 3 4 ζ ∗ F ( − 1) = 0 . 02019 . . . . H ow ever, even assuming that the error is bou n ded by 0.01, th is is still just big enough to allo w k = − 3. Of course reducing the size of the error would elimi nate th is possibility . This w ould require a faster implemen tation than the one in [7]. Ac kno wledgemen ts W e w ould like to w armly th ank H erbert Gangl for his helpful adv ice and detailed com- ments on a previous version of our pap er. W e w ould also lik e to thank MathOverﬂo w user ‘Ralph’ for answering a q uestion ab out the Hu rewicz homomorphism, Rob Sno c ken and Christopher V oll for explaining v arious aspects of Ded ekind zeta funct ions to us, Bernhard K oeck for useful discussions on K -theory , Matthias Flac h for his comment on the orders of some K-th eory groups and the Algebra and Numb er Theory Group at Universit y College Dublin, in particular Rob ert Osburn, for their in vitation to discuss this w ork. Finally w e would like to th an k EPSRC for fun ding this research, Sa jeda Mannan for facilitating a research visit, and the universities of S heﬃeld and Southampton for supp orting this w ork. References [1] D. Benois and T. 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