The Extended Bloch Group and the Cheeger Chern Simons Class

Transcription

1 Geometr & Topolog (2007) The Extended Bloch Group and the Cheeger Chern Simons Class SEBASTIAN GOETTE CHRISTIAN K ZICKERT We present a formula for the full Cheeger Chern Simons class of the tautological flat complex vector bundle of rank 2 over BSL.2; ı /. This improves the formula b Dupont and Zickert [6], where the class is onl computed modulo 2 torsion. 57R20, G55 Introduction The Cheeger Chern Simons class c k is a natural refinement of the k th Chern class for complex vector bundles with connection, and it takes values in the ring of differential characters, see Chern Simons [2] and Cheeger Simons []. For a vector bundle with a flat connection, this class becomes an ordinar.2k / cohomolog class with coefficients in =.k/, where.k/ D.2i/ k. Let BSL.n; ı / denote the classifing space of the group SL.n; ı / with the discrete topolog. The universal Cheeger Chern Simons class c k 2H 2k.BSL.n; ı /; =.k// of the tautological flat complex vector bundle over BSL.n; ı / gives rise to the Borel regulator in algebraic K theor, and c 2 is also related to invariants of hperbolic 3 manifolds. One is interested in a combinatorial description of this class c k 2 H 2k.BSL.n; ı /; =.k//. Dupont derived an expression for c 2 2 H 3.BSL.2; ı /; =.2// modulo.2/ in [3]. A similar formula for Re c 3 is due to Goncharov [7]. The homolog of the classifing space of a discrete group is b definition the homolog of the group, and since =.2/ is divisible, we can regard c 2 as a homomorphism H 3.SL.2; //! =.2/. The natural map H 3.SL.2; //! H 3.PSL.2; // has cclic kernel of order 4, so we have a commutative diagram defining c 2 on H 3.PSL.2; //: H 3.SL.2; ı // c 2! =.2/ H 3.PSL.2; ı // c 2! = 2 Published: 2 August 2007 DOI: 0.240/gt

2 624 Sebastian Goette and Christian K Zickert An explicit formula for the lower map was obtained b Neumann [8], and extended b Dupont and Zickert [6] to a formula for the upper map. However, the formula given in [6] onl computes the image of c 2 in =2 2, thus onl computing c 2 up to 2 torsion (see [6, Remark 4.2] for a comment on the normalisation). In the present paper we extend their result obtaining a formula computing the full Cheeger Chern Simons class. In both [8] and [6] the formulas are obtained b factoring c 2 through a version of the extended Bloch group, an object defined b Neumann in [8]. There are two different versions of the extended Bloch group. One version, denoted B. / in Neumann s paper and B PSL. / later on in this note, is generated b smbols ŒzI p; q subject to a five term relation and a transfer relation. It is isomorphic to H 3.PSL.2; ı //. The other one, denoted EB. / in [8] and B SL. / in the following, is generated b smbols ŒzI 2p; 2q and onl subject to the five term relation. The latter version is called the more extended Bloch group, and conjectured to be isomorphic to H 3.SL.2; ı // in [8]. The role of the transfer relation is subtle and has caused some minor inaccuracies in [8] and [6], see Remark 3.3 below. [8, Proposition 8.2 and Corollar 8.3] are onl correct if we include the transfer relation. Proposition 8.2 has been used in the proof of [6, Proposition 4.5], so this result is also onl correct if we include the transfer relation. To the best of our knowledge these are the onl problems in [8] and [6]. We present corrections to these results as Theorem 3.4, Corollar 3.5 and Corollar 4.2, and Remark 4.3 below. This paper can thus be viewed as an erratum and a refinement of the papers [8] and [6]. To obtain the full class c 2, we construct a lift of the function LW B SL. /! =2 2 defined b Dupont and Zickert [6] with values in =4 2 D =.2/. Note that this lift is not compatible with the transfer relation, see Remark 3.3. This observation was the main motivation for the present note. The paper is organised as follows: In Section we recall the definition of the extended Bloch group. In Section 2 we define the modified extended Rogers dilogarithm. In Section 3 we work out some relations in the extended Bloch group, and in Section 4 we prove our main results that the extended Bloch group is isomorphic to H 3.SL.2; ı // as conjectured b Neumann [8], and that L computes the Cheeger Chern Simons class. In Section 5, we have added some more relations in P. / that might be of interest elsewhere. Acknowledgements We are grateful to J Dupont, G Kings and W Neumann for fruitful discussions and their interest in our work. The first named author was supported in part b DFG special

3 The Extended Bloch Group 625 programme Global Differential Geometr ; parts of this note were written while the first named author enjoed the hospitalit of the Chern Institute at Tianjin. The second named author was supported b Rejselegat for matematikere. The extended Bloch group We follow Neumann s description in [8] at the beginning of chapter 8. We first recall the construction of the classical Bloch group. Define a set of five term relations () FT D x; ; x ; =x = ; x ˇˇˇˇ x 2 n f0; g. n f0; g/ 5 : Consider the free Abelian groups generated b the elements of FT and n f0; g and the chain complex (2) ŒFT! Œ n f0; g! ^ with arrows defined on generators b.œz 0 ; : : : ; z 4 / D Œz 0 Œz C Œz 2 Œz 3 C Œz 4 ;.Œz / D z ^. z/ : Then P. /D Œ nf0; g = im is called the pre-bloch group, and the middle homolog of the complex (2) above is called the Bloch group B. /. Let denote the universal Abelian cover of n f0; g. To construct, we start with cut D n.. ; 0 [ Œ; //. For each x 2. ; 0/ [.; /, we add two boundar points x 0i WD lim x ti t&0 and put cut D cut [ fx 0i j x 2. ; 0/ [.; /g: We extend the principal branches of Log, Li 2 and Arg D Im Log to cut. In cut.2 / 2, identif (3).x C 0i; 2p; 2q/.x 0i; 2p C 2; 2q/ for all x 2. ; 0/, and.x C 0i; 2p; 2q/.x 0i; 2p; 2q C 2/ for all x 2.; / for all p, q 2, obtaining W! n f0; g. Equivalence classes will be denoted.xi 2p; 2q/ or simpl x. Note that we could have dropped the factors 2 above and worked in cut 2 instead, but we want to sta compatible with [6] and [8].

4 626 Sebastian Goette and Christian K Zickert As in [8], put FT C D.z 0 ; : : : ; z 4 / 2 FT ˇˇ Im z0 ; : : : ; Im z 4 > 0g: Then.z 0 ; : : : ; z 4 / 2 FT belongs to FT C if and onl if Im z > 0 and z 0 is in the interior of the Euclidean triangle spanned b 0, and z. Let FT denote the connected component of.ft/ 5 that contains FT C D..z 0 I 0; 0/; : : : ;.z 4 I 0; 0// ˇˇ.z0 ; : : : ; z 4 / 2 FT C : Also, note that the functions.zi 2p; 2q/ 7!.Log z C 2ip/ and.zi 2p; 2q/ 7!. Log. z/ C 2iq/ are holomorphic on. B [6], we can extend (2) to a chain complex (4) FT!! ^ with arrows defined on generators b.œ z 0 ; : : : ; z 4 / D Œ z 0 Œ z C Œ z 2 Œ z 3 C Œ z 4 ;.ŒzI 2p; 2q / D.Log z C 2ip/ ^. Log. z/ C 2iq/:. Definition The extended pre-bloch group P SL. / is defined as the quotient Œ = im, and the extended Bloch group B SL. / as the middle homolog of the complex (4). 2 The extended Rogers dilogarithm The classical dilogarithm is given b X z k Li 2.z/ D k 2 D kd Z z 0 log. t/ dt t for all z with jzj <. It extends to a multivalued function on with branch points at 0, and. Recall that the Rogers dilogarithm LW.0; /! is given b We extend L to a holomorphic function: L.x/ D Li 2.x/ C 2 log x log. x/ 2 6 : (5) LW cut.2 / 2! L.zI 2p; 2q/ D Li 2.z/ C 2 Log z C 2ip Log. z/ C 2iq 2 6 :

5 The Extended Bloch Group Lemma The function L above induces a holomorphic function LW! =.2/ that satisfies the five term relation 4X. / k L. z k / D 0 for all. z 0 ; : : : ; z 4 / 2 FT. kd0 Note that L lifts Dupont and Zickert s function L in [6] from =2 2 to =4 2 D =.2/. Proof Because Li 2.z/ has no singularit at z D 0 and.zi p; q/ 7! Log z C 2ip is holomorphic on, the function L extends holomorphicall across. ; 0/.2 / 2 in. If we extend Li 2 and Log. Hence z/ to z 0i for z >, then Li 2.z C 0i/ D Li 2.z 0i/ C 2i Log z Log..z C 0i// D Log..z 0i// 2i: L.z C 0iI 2p; 2q/ D L.z 0iI 2p; 2q C 2/ C 4 2 p; so the extension L of L mod 4 2 is well-defined. B Neumann [8], we have a five term relation 4X. / k L. z k / D 0 2 kd0 for all. z 0 ; : : : ; z 4 / 2 FT C. Because FT is a connected complex manifold, the five term relation for L holds in =.2/ for all. z 0 ; : : : ; z 4 / 2 FT b analtic continuation. 2.2 Remark Along the commutator of a small loop around 0 and a small loop around in n f0; g, the holomorphic continuation of the Rogers dilogarithm changes b 4 2. This shows that we cannot lift L to a holomorphic function on with values in =A, with A.2/ a proper subgroup. 2.3 Corollar The function L induces homomorphisms LW P SL. /! =.2/ and LW B SL. /! =.2/:

6 628 Sebastian Goette and Christian K Zickert 3 Relations in the extended Bloch group Following Neumann [8], we find relations among elements of the extended pre-bloch group. We then parameterise the kernel of the forgetful maps P SL. /! P. / and B SL. /! B. / induced b the projection W!. In the following, we will identif z 2. ; 0/ [.; / with z C 0i. As explained b Dupont and Zickert in the appendix of [6]: n (6).FT C / \ FT D z 0 I 2p 0 ; 2q 0 ; z I 2p ; 2q ; z2 I 2.p p 0 /; 2q 2 ; z 3 I 2.p p 0 C q q 0 /; 2.q 2 q / ; z 4 I 2.q q 0 /; 2.q 2 q p 0 / o ˇ.z 0 ; : : : ; z 4 / 2 FT C and p 0 ; p ; q 0 ; q ; q 2 2 : For other choices of.z 0 ; : : : ; z 4 / 2 FT, a few of the p k, q k have to be adjusted b. Subtracting two instances of the five term relation and using () and (6), we obtain Neumann s ccle relation h i ŒxI 2p 0 ; 2q 0 2 ŒI 2p ; 2q 2 C x I 2p 2p 0 ; 2q 2 2 h i D ŒxI 2p 0 ; 2q 0 ŒI 2p ; 2q C x I 2p 2p 0 ; 2q 2 ; for all x, such that x; ; x ; =x = ; x 2 FT C. If we var x and continuousl, then some of the integers in this relation ma jump. Thus, we obtain (7) ŒxI 2p 0 ; 2q 0 2 ŒxI 2p 0 ; 2q 0 ŒI 2p ; 2q 2 C ŒI 2p ; 2q 8 x I 2p 2p 0 2; 2q 2 2 x I 2p if Arg Arg x ; 2p 0 2; 2q 2 h ˆ< D x I 2p 2p 0 ; 2q 2 2i h i x I 2p if < Arg Arg x ; 2p 0 ; 2q 2 h x I 2p 2p 0 C 2; 2q 2 2i ˆ: h x I 2p 2p 0 C 2; 2q 2 i if < Arg Arg x: Subtracting two instances of (7) gives (8) ŒxI 2p; 2.q / ŒxI 2p; 2q D ŒxI 2p; 2.q 0 / ŒxI 2p; 2q 0

7 The Extended Bloch Group 629 for all x 2 cut. Similarl, one can prove (9) ŒxI 2.p /; 2q ŒxI 2p; 2q D ŒxI 2.p 0 /; 2q ŒxI 2p 0 ; 2q and (0) ŒxI 2.p C /; 2.q / ŒxI 2p; 2q D ŒxI 2.p 0 C /; 2.q 0 / ŒxI 2p 0 ; 2q 0 for all x 2 cut and all p, q, p 0, q 0 2 such that p 0 C q 0 D p C q. See Neumann [8] for a more geometric derivation of these relations. B [8, Lemma 7.3], we also have the relation () ŒzI 2p; 2q C Œ zi 2q; 2p D 2 2 I 0; 0 ; which is of course compatible with (8) (0). Let us define which is independent of q b (8). fzi 2pg D ŒzI 2p; 2q ŒzI 2p; 2.q / ; 3. Lemma For all z, w 2 cut with zw, we have the relation 8 ˆ< fzw C 0iI 2.p C r /g if Arg z C Arg w, fzi 2pg C fwi 2rg D fzw C 0iI 2.p C r/g if < Arg z C Arg w and ˆ: fzw C 0iI 2.p C r C /g if < Arg z C Arg w. Proof This is immediate from (7). 3.2 Lemma The element D fzi 2pg fzi 2.p /g 2 P SL. / is independent of z 2 n f0; g and p 2, and of order 2 in P SL. /. Proof Independence of p follows from (9) and independence of z is immediate from Lemma 3.. To prove that is of order two, use (8) and () to write D ŒzI 2; 0 ŒzI 2; 2 ŒzI 0; 0 C ŒzI 0; 2 D Œ zi 0; 2 C Œ zi 2; 2 C Œ zi 0; 0 Œ zi 2; 0 D f zi 0g f zi 2g D : To show that 0, we compute (2) L. / D L.zI 2; 2/ L.zI 2; 0/ L.zI 0; 2/ C L.zI 0; 0/ D 2 2 ; and in =.2/.

8 630 Sebastian Goette and Christian K Zickert 3.3 Remark Neumann has implicitl assumed that D 0 in [8, Section 8], and his conclusions have then been used b Dupont and Zickert in [6, Proposition 4.5]. More precisel, Neumann introduces a transfer relation ŒzI p; q C ŒzI p 0 ; q 0 D ŒzI p; q 0 C ŒzI p; q 0 for all p, q, p 0, q 0 2 in the definition of his extended Bloch group. The analogous relation in our context would read ŒzI 2p; 2q C ŒzI 2p 0 ; 2q 0 D ŒzI 2p; 2q 0 C ŒzI 2p; 2q 0 for all p, q, p 0, q 0 2 : In analog with [8, Proposition 7.2], one can show that the effect of the transfer relation above is equivalent to dividing b the subgroup of order two that is generated b. We have just computed L. / D =4 2 in (2). This explains that if one includes the transfer relation, then L is well-defined onl modulo 2 2 as in [6]. Assuming that D 0, one finds that fzi 2pg becomes independent of p. This allows the definition of a homomorphism! P SL. /=h i with z 7! fzi 0g, see [8, Proposition 8.2]. Starting from Neumann s map, we obtain a pullback square! P SL. /! P SL. /=h i: Here the left vertical arrow maps z to z 2, and is given b 8 0 if z D, ˆ< if z D, (3).z/ D fz 2 C 0iI 0g if Arg z 2 2 ˆ: ; 2 and z, and fz 2 C 0iI 2g if Arg z 2 ; 2 and z. 3.4 Theorem The map is a homomorphism, and the sequence 0!! P SL. /! P. /! 0 is exact and split, where the right arrow is induced b W! n f0; g. Proof First of all we note that b the definition of and Lemma 3.2, fz 2 I 2pg C D fz 2 I 2p C 2g D fz 2 I 2p 2g;

9 The Extended Bloch Group 63 which implies that.z/ C. / D. z/, and that fz 2 ; 2p C 4g D fz 2 I 2pg. B Lemma 3., we have.z/ C.w/ D.zw/ for almost all choices of z, w 2. The remaining cases are easil checked. The right arrow maps fzi 2pg D ŒzI 2p; 2 ŒzI 2p; 0 to Œz Œz D 0, so the sequence above is a chain complex. To prove injectivit of consider the composition L ı. For z 2 let p D 0 if Arg z 2 2 ; 2 and p D if Arg z 2 ; 2. Then and Log.z 2 C 0i/ C 2ip 2 Log z modulo 2i ; (4). L ı /.z/ D 2.Log z2 C 2ip/.Log. z 2 / C 2i/.Log z 2 C 2ip/ Log. and this even holds for z D b (2), (3). Hence is injective. z 2 / D 2i Log z 2 =.2/; It remains to show that im D ker. P SL. /! P. //. Relations (8) and (9) allow us to represent each generator of P SL. / as ŒzI 2p; 2q D pqœzi 2; 2 p.q /ŒzI 2; 0.p /qœzi 0; 2 C.p /.q /ŒzI 0; 0 ; see [8, Lemma 7.]. Using (), we see that the kernel of P SL. /! P. / is generated b elements of the form ŒzI 2p; 2q ŒzI 0; 0 D pqfzi 2g.p /qfzi 0g C pf zi 0g 2 im : B (4), a splitting of the sequence is given b the homomorphism exp ı L 2i W P SL. /! : 3.5 Corollar cf Neumann [8, Theorem 7.5] The sequence 0! = 7!.e2i /! B SL. /! B. /! 0 is exact and.2i/ 2 L.e 2i / D :

10 632 Sebastian Goette and Christian K Zickert 4 The Cheeger Chern Simons class and H 3.SL.2; ı // Recall that Dupont and Zickert have constructed a map W H 3.SL.2; ı //! B SL. / in [6, Section 3] without using the transfer relation. Following [6], we prove that L ı D c 2 2 =.2/ and conclude from this that is an isomorphism. Note that because =.2/ is divisible, there is a canonical isomorphism H 3 SL.2; ı /; =.2/ Š Hom H 3.SL.2; ı //; =.2/ : Let c 2 2 H 3.SL.2; ı /; =.2// denote the second Cheeger Chern Simons class of the tautological flat complex vector bundle of rank 2 over BSL.2; ı /. Here, we are using the same normalisation as Neumann [8]. Dupont and Zickert consider the class.2i/ 2 c 2 2 H 3.SL.2; ı /; = /, see [6, Remark 4.2]. 4. Theorem cf Dupont and Zickert [6, Theorem 4.] Under the isomorphism above, c 2 D L ı : Proof B [6, Theorem 4.], we have that 2c 2 D 2 L ı 2 Hom H 3.SL.2; ı //; =.2/ in our normalisation. Because H 3.SL.2; ı // is divisible, this implies our claim. 4.2 Corollar cf [6, Theorem 4.5] The map W H 3.SL.2; ı //! B SL. / is an isomorphism. Proof Consider the commutative diagram 0! =! H3.SL.2; ı //! B. /! 0 0! =.e2i /! B SL. /! B. /! 0: Exactness of the upper row has been established b Dupont and Sah in [5]. The lower row is just Corollar 3.5. Commutativit of the right hand square has been established in [6, Section 3]. Let 2, then implies that.2i/ 2.c 2 ı /. / D b Dupont [4, Theorem 0.2]. Theorem 4..2i/ 2 L. ı /. / D.2i/ 2.c 2 ı /. / D ;

11 The Extended Bloch Group 633 and. ı /. / 2 ker B SL. /! B. / b commutativit of the right hand square. On the other hand, Corollar 3.5 implies that.2i/ 2 L.e 2i / D ; and that L is injective on ker B SL. /! B. /. Thus the left hand square also commutes. Our claim now follows from the five-lemma. 4.3 Remark Let B PSL. / Š H 3.PSL.2; // denote Neumann s extended Bloch group in [8]. Then the diagram 0 0 =4 =4.i/ 0! =.e2i /! B SL. /! B. /! 0 4 b 0! =.e2i /! B PSL. /! B. /! commutes and has exact rows and columns. Here the map bw B SL. /! B PSL. / sends a generator Œz; 2p; 2q to the same generator in B PSL. /, and has been defined in [8, Proposition 7.4]. This is proved in analog with in [8, Corollar 8.3]. For example, commutativit of the lower left hand square follows from.b ı /.z/ D Œz 2 I 2p; 2 Œz 2 I 2p; 0 D 2 Œz 2 I 2p; Œz 2 I 2p; 0 D 2.z 2 / D 4.z/ 2 P PSL. /: This also shows that ker b is spanned b.i/ D f I 0g. 5 More relations in the extended pre-bloch group B Dupont and Sah [5], one has the relations Œz D D D D z z z Œ z D z z

12 634 Sebastian Goette and Christian K Zickert in the pre-bloch group P. /. If we interpret z as the cross-ratio of a generic configuration of four points in P, then these relations sa that up to orientation, the order of the points is not important. Note that Im c 2 is alread well-defined on B. /. Similar relations hold in Neumann s extended pre-bloch group P PSL. / b [8, Proposition 3.]. As a consequence, unordered oriented simplices are also sufficient to compute Re c 2 up to some finite ambiguit. Unfortunatel, these relations become 4 more complicated in P SL. /. Let p z denote the standard fourth root of z with Arg 4p z 2 4 ; Proposition Let Im z > 0. Then () z I 2p; 2p C 2q (2) (3) (4) (5) Proof I 2p 2qI 2p z I 2p C 2q; 2q D z z I 2q; 2p 2q z The involutions D ŒzI 2p; 2q C i p 4p z ; D ŒzI 2p; 2q e i 2. 6p/ 4p z ; ŒzI 2p; 2q C e i 2.C6q/ p 4 z ; D ŒzI 2p; 2q Œ zi 2q; 2p D ŒzI 2p; 2q C e i 2.2C6q/ p 4 z ; e i 2 ŒzI 2p; 2q 7! Œ zi 2q; 2p and ŒzI 2p; 2q 7! : z I 2pI 2p C 2q generate an action of S.3/ on. Using Im z > 0 and Im z > 0, it is now eas to see that the five relations above follow from () and (5). Note that b [5], both relations are true modulo ker P SL. /! P. / D im. Because the Rogers dilogarithm L is injective on im b (4), it suffices to check both relations after appling L. This can be done using some elementar facts about the classical dilogarithm, and is thus left to the reader. 5.2 Remark The relations in [8] look somewhat nicer, since the correction term does not involve the variable z. This is possible because in Neumann s definition of P PSL. /, odd integers are allowed, so that one can consider the involution ŒzI p; q 7! Œ=zI p; C p C q. Following [8], the various terms on the left hand side of the equations in Proposition 5. correspond to flattenings of a given oriented simplex with different orderings of vertices.

13 The Extended Bloch Group 635 Thus the proposition seems to indicate that it is not enough to consider unordered oriented simplices if one wants to compute the full class c 2. For higher classes c k, not man formulas are available. The onl formula known to the authors is Goncharov s formula for Re c 3 in [7], which uses unordered oriented simplices. References [] J Cheeger, J Simons, Differential characters and geometric invariants, from: Geometr and topolog (College Park, MD, 983/84), Lecture Notes in Mathematics 67, Springer, Berlin (985) MR [2] S S Chern, J Simons, Characteristic forms and geometric invariants, Ann. of Math..2/ 99 (974) MR [3] J L Dupont, The dilogarithm as a characteristic class for flat bundles, J. Pure Appl. Algebra 44 (-3) (987) MR8850 [4] J L Dupont, Scissors congruences, group homolog and characteristic classes, Nankai Tracts in Mathematics, World Scientific Publishing Co., River Edge, NJ (200) MR [5] J L Dupont, C H Sah, Scissors congruences II, J. Pure Appl. Algebra 25 (982) MR [6] J L Dupont, C K Zickert, A dilogarithmic formula for the Cheeger Chern Simons class, Geom. Topol. 0 (2006) MR [7] A B Goncharov, Geometr of configurations, pollogarithms, and motivic cohomolog, Adv. Math. 4 (995) MR [8] W D Neumann, Extended Bloch group and the Cheeger Chern Simons class, Geom. Topol. 8 (2004) MR Mathematisches Institut, Universität Freiburg, Eckerstr., 7904 Freiburg, German Department of Mathematics, Columbia Universit, New York NY 0027, USA sebastian.goette@math.uni-freiburg.de, zickert@math.columbia.edu Proposed: Robion Kirb Received: 5 June 2007 Seconded: Shigeuki Morita, Joan Birman Revised: Jul 2007