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Locally symmetric spaces and K-theory of number fields

For a closed locally symmetric space M=\Gamma\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of R-rank one.

very natural way, an element in B (C), the Bloch invariant. By [28], this element does not depend on the chosen ideal triangulation.
A generalization to higher-dimensional hyperbolic manifolds was provided by Goncharov. He associated to an odd-dimensional hyperbolic manifold and flat bundles coming from the half-spinor representations, an element γ (M ) ∈ K * Q ⊗ Q, and proved its nontriviality by showing that application of the Borel regulator yields (a fixed multiple of) the volume.
It thus arises as a natural question, whether other locally symmetric spaces and different flat bundles give nontrivial elements in the K-theory of number fields (and eventually how much of algebraic K-theory in odd degrees can be represented by locally symmetric spaces and representations of their fundamental groups).
In section 2, we generalize the argument in [15] to the extent that, for a compact locally symmetric space M 2n−1 = Γ\G/K of noncompact type and a representation ρ : G → GL (N, C), nontriviality of the associated element γ (M ) ∈ K 2n−1 Q ⊗ Q is (independently of Γ) equivalent to nontriviality of the Borel class ρ * b 2n−1 .
While it does, in general, not work to associate elements in algebraic K-theory to flat bundles over manifolds with boundary, we show in section 4 that an element γ (M ) ∈ K * Q ⊗ Q can be associated to flat bundles over locally symmetric spaces of finite volume. ( [15] did this for hyperbolic manifolds and half-spinor representations, but implicitly assuming that ∂M be connected.) The results of section 2 (for closed manifolds) and section 4 (for cusped manifolds) are subsumed in the following theorem.
(In many cases one actually associates an element in K 2n−1 (F) ⊗ Q, for some number field F, see Theorem 2 in section 2.5.) In section 3, we work out the list of fundamental representations ρ : G → GL (N, C) for which ρ * b 2n−1 = 0 holds true. It is easy to prove that ρ * b 2n−1 = 0 is always true if 2n− 1 ≡ 3 mod 4. We work out, for which fundamental representations ρ * b 2n−1 = 0 holds if 2n − 1 ≡ 1 mod 4. (In [15] it was stated that the half-spinor representations would seem to be the only fundamental representations of Spin (2n − 1, 1) that yield nontrivial invariants of odd-dimensional hyperbolic manifolds. This is however not the case. Indeed, if 2n − 1 = dim (M ) ≡ 3 mod 4, then each irreducible representation of Spin (2n − 1, 1) yields nontrivial invariants.) The proof uses only standard Lie algebra and representation theory. The result reads as follows.
Theorem. The following is a complete list of irreducible symmetric spaces G/K of noncompact type and fundamental representations ρ : G → GL (N, C) with ρ * b 2n−1 = 0 for 2n − 1 := dim (G/K).
For hyperbolic manifolds and half-spinor representations, the construction of γ (M ) is due to Goncharov. (Though the proof in [15] implicitly assumes that ∂M be connected.) For hyperbolic 3-manifolds, another construction is due to Cisneros-Molina and Jones in [9]. (It was related in [9] to the construction of Neumann-Yang in [28].) The latter has the advantage that the number of boundary components does not impose technical problems, contrary to the group-homological approach in [15].
Our construction for closed locally symmetric spaces is a straightforward generalization of [15].
In the case of cusped locally symmetric spaces (with possibly more than one cusp) it would have seemed more natural to stick to the approach of Cisneros-Molina and Jones, and in fact this approach generalizes to locally symmetric spaces in a completely straightforward way (see section 4.1). However, we did not succeed to evaluate the Borel regulator (in order to discuss nontriviality and Q-independence of the obtained invariants) in this approach. On the other hand, Goncharov's approach, even in the case of only one cusp, uses very special properties of the spinor representation, which can not be generalized to other representations. Therefore, our argument is sort of a mixture of both approaches. On the one hand it is closer in spirit to the arguments of [15] (but with the cuspidal completion in section 4.2 memorizing the geometry of distinct cusps), on the other hand the argument in section 4.3 uses arguments from [9] to circumvent the very special group-homological arguments that were applied in [15] in the special setting of the half-spinor representations.
Of course, it should be interesting to relate the different constructions in a more direct way.

Preparations
The results of this section are fairly straightforward generalizations of the results in [15] from hyperbolic manifolds to locally symmetric spaces of noncompact type. We will define a notion of representations with nontrivial Borel class and will, mimicking the arguments in [15], show that representations with nontrivial Borel class give rise to nontrivial elements in algebraic K-theory of number fields. The problem of constructing representations with nontrivial Borel class will be tackled in the section 3.

Construction of elements in algebraic K-theory
In this paper, rings A will always be commutative rings with unit. (To be explicit: we will always consider subrings of C.) Assume that M is a closed, orientable n-manifold with Γ := π 1 M . Assume that we are given a ring A and a representation ρ : Γ → GL (A), where GL (A) denotes the increasing union of GL (N, A) over all N ∈ N.
We get an induced map Bρ : M → BGL (A) .
Throughout this paper BGL (A) (resp. BG for any Lie group) will mean the classifying space for G δ , that is the group G with the discrete topology. Thus π 1 BG = G. Quillen's plus construction (see [30]) provides us with a map (Bρ) If M happens to be a d-dimensional homology sphere, then M + is homotopy equivalent to S d and one gets a map which may be considered as representative of an element (Bρ) It was actually shown by Hausmann and Vogel (cf. [18] or [17]) that, for d ≥ 5 or d = 3, each element in K d (A) for a finitely generated commutative ring A can be constructed by some homology sphere M and some representation ρ.
If M is not a homology sphere, but a closed and oriented d-manifold, and A satisfies mild assumptions (see Section 2.4), e.g. for A = Q, then we will construct an element in K d (A) ⊗ Q, as follows. The continuous map Bρ induces a homomorphism Since M is a closed, oriented d-manifold, we have the fundamental class [M ] ∈ H d (M ; Q), which is the image of a generator of H d (M ; Z) under the change-of-rings homomorphism associated to the inclusion Z → Q. Let By the Milnor-Moore Theorem, the Hurewicz homomorphism gives, after tensoring with Q, an injective homomorphism Again by the Milnor-Moore Theorem, its image consists of the subgroup of primitive elements, which we denote by P H d (BGL (A) ; Q).
(If d is even and A is a ring of integers in any number field, then P H d (BGL (A) ; Q) = 0. Therefore one is only interested in the case that d is odd, d = 2n − 1.) Whenever we fix a projection pr d : In Section 2.4 we are going to show e.g. for A = Q (and also for many other rings) the projection pr d can be chosen such that the Borel regulators of h and pr d (h) agree for all h ∈ H d BGL Q ; Q . In particular, to check nontriviality of γ (M ) it will then suffice to apply the Borel regulator to (Bρ) * [M ].
If M is a (compact, orientable) manifold with nonempty boundary, then there is no general construction of an element in algebraic K-theory. However, we will show in section 4 that for finite-volume locally symmetric spaces one can generalize the above construction and again construct a canonical invariant γ (M ) ∈ K d Q ⊗ Q.
Volume class. We recall that the continuous cohomology H * c (G; R) of a Lie group G is defined as the homology of the complex for the continuous (with respect to the Lie group topology) G-invariant mappings from G * +1 to R and δ is the usual coboundary operator. In particular, the group cohomology of G is the continuous cohomology for G with the discrete topology. The comparison map comp : for f ∈ C k c (G). In particular, elements of H * c (G; R) can be evaluated on H * (BG; R).
Let M = G/K be a symmetric space of noncompact type. It is well-known ( [19], Ch.V, Thm.3.1) that M has nonpositive sectional curvature. G acts by isometries on M .
The volume class v d ∈ H d c (G; R) is defined as follows. Fix an arbitrary pointx ∈ M = G/K. Then we define a cocycle cν d ∈ C d (BG) by that is the signed volume of the straight simplex with verticesx 0 , g 1x0 , . . . , g 1 . . . g dx0 . (Note that in a simply connected space of nonpositive sectional curvature each ordered k +1-tuple of vertices (p 0 , . . . , p k ) determines a unique straight d-simplex str (p 0 , . . . , p k ).) By Stokes' Theorem we have Thus cν d is a simplicial cocycle on BG. (Its cohomology class does not depend onx, because anyx ∈ G/K can be mapped to any otherx 0 ∈ G/K by some g ∈ G and the action of G on G/K is homotopic to the identity and therefore preserves cohomology classes.) The corresponding cocycle ν d ∈ C d c (G; R) is given by the (clearly continuous) mapping

Proof:
Let π : G/K = M → M be the covering map. Fix a point x 0 ∈ M and a liftx 0 ∈ M . Let C str,x0 * (M ) be the chain complex of straight simplices with all vertices in x 0 . Let w 0 , . . . , w k be the vertices of the standard simplex ∆ k . Each σ ∈ C str,x0 k (M ) is a continuous map σ : ∆ k → M which (by asphericity of M ) is uniquely determined by the homotopy classes of the sub-1-simplices for Γ = π 1 (M, x 0 ), where the bijection maps σ to (1, γ 1 , . . . , γ d ). It follows from the definition of the boundary operator on BΓ that this bijection is a chain map, thus an isomorphism of chain complexes.
Moreover, inclusion C str,x0 * (M ) → C * (M ) induces an isomorphism in homology. Indeed, each cycle in C * (M ) can first be homotoped such that all vertices are in x 0 , and then be straightened (by induction on dimension of subsimplices, depending on a chosen order of vertices). Straightening a simplex σ with straight boundary means to chose the unique geodesic simplex which is homotopic rel. ∂ to σ. (In particular, its edges represent the same elements of π 1 (M, x 0 ) as the corresponding edges of σ. ) It is well-known 1 that straightening all simplices of a cycle yields a cycle in the same homology class 2 .
So, let r i=1 a i σ i be a representative of the fundamental class. (One may choose e.g. a triangulation σ 1 + . . . . (Possibly after straightening some simplices overlap, so we do not get a triangulation. However, it will be sufficient to have a fundamental cycle consisting of geodesic simplices.) By Stokes Theorem, vol

Borel classes
Let M = G/K be a symmetric space of noncompact type. Then G is a semisimple, noncompact Lie group and K is a maximal compact subgroup.
1 This is proved in [2], Lemma C.4.3, for hyperbolic manifolds. Word by word the same proof works if M is any Riemannian manifold of nonpositive sectional curvature, in particular if M is a locally symmetric space of noncompact type. 2 This construction uses that M is a manifold of nonpositive sectional curvature. That assumption might actually be avoided, since it was shown in [13] that, for each aspherical space M with fundamental group Γ, there is a subcomplex K (M ) := EM C simp * Let g be the Lie algebra of G, and k ⊂ g be the Lie algebra of K. There is the Cartan decomposition g = k ⊕ p with [k, k] ⊂ k, k, p ⊂ p, p, p ⊂ k. It is a well-known fact that the Killing form B (X, Y ) = T r (ad (X) • ad (Y )) is negatively definite on k and positively definite on p.
The dual symmetric space is G u /K, where G u is the simply connected Lie group with Lie algebra g u = k ⊕ ip ⊂ g ⊗ C. The Killing form on g u is negatively definite, thus G u /K is a compact symmetric space.
The Lie algebra cohomology H * g is the cohomology of the complex Λ * g, d with . . ,X j , . . . , X n . The relative Lie algebra cohomology H * g, k is the cohomology of the subcomplex C * g, k , d with C * g, k = φ ∈ Λ * g : X φ = 0, ad (X) φ = 0 ∀X ∈ k . If G/K is a symmetric space of noncompact type, and G u /K its compact dual, then there is an obvious isomorphism H * g, k = H * g u , k , dual to the obvious linear map k ⊕ ip → k ⊕ p. Moreover, H * g, k is the cohomology of the complex of G-invariant differential forms on G/K. Since G u is compact and connected, there is an isomorphism (defined by averaging) Dualizing representations. Let ρ : G → GL (N, C) be a representation. ρ can be conjugated such that K is mapped to U (N ). We will henceforth always assume that ρ has been fixed such that ρ sends K to U (N ). the associated Lie-algebra homomorphism, and, with g = k ⊕ p, g u := k ⊕ ip, the induced homomorphism on k ⊕ ip. The corresponding Lie group homomorphism will be called the dual homomorphism to ρ.
Here g u , k and ip are to be understood as subsets of the complexification g ⊗ C, and G u is the simply connected Lie group with Lie algebra g u . In particular, the complexification of gl N C is isomorphic to gl N C ⊕ gl N C, and ip ≃ u (N ) in this case. We emphasize that ρ u sends K to the first factor of U (N ) × U (N ), and not to the diagonal subgroup as has been claimed in [15].
Van Est Theorem. The van Est theorem states that there is a natural isomorphism If ρ : G → GL (N, C) is a representation, sending K to U (N ), then we conclude that there exists the following commutative diagram, where all vertical arrows are isomorphisms Thus the volume class [v d ] is the (up to multiplication by real numbers unique) nontrivial continuous cohomology class in degree d. Proof: According to [10], Prop. 1.5, v d corresponds to the class of the volume form in H d g, k . It is obvious that the isomorphism H d g, k ≃ H d i g u , k maps the volume form of G/K to the volume form of G u /K. QED

Borel classes
Let G be a compact Lie group. Let I k S (G) resp. I k A (G) be the ad-invariant symmetric resp. antisymmetric multilinear k-forms on g. We have the isomorphism I k A (G) → H k (G; R). Moreover, we remind that there is the Chern-Weil isomorphism I k S (G) → H 2k (BG; R), where in this section (contrary to the remainder of the paper) BG means the classifying space for G with its Lie group topology. In particular, if G = U (N ), then the invariant polynomial 1 (2πi) n T r A n+1 is mapped to C n ∈ H 2n (BU (N ) ; Z) , the n-th component of the universal Chern character.
There is a fibration G → EG → BG and an associated 'transgression map' τ which maps a subspace of H 2n−1 (G; Z) (the so-called transgressive elements) to a quotient of H 2n (BG; Z), cf. [4], p.410. If G = U (N ), then, according to Borel ([4], p.412), one has The classes b 2n−1 are called Borel classes. By definition, Borel classes exist only in odd degrees. (We will not distinguish between b 2n−1 and its image in H 2n−1 (U (N ) ; R).) According to Cartan ([8]), there is a homomorphism whose image corresponds (after the isomorphism I 2n−1 precisely to the transgressive elements. It is explicitly given by It will be clear from the context whether we consider the Borel classes as elements of H * (u (N )) ≃ H * (U (N ) ; R) or as the (under the van Est isomorphism) corresponding elements of H * c (GL (N, C) ; R). Stabilization H * (U (N + 1) ; R) → H * (U (N ) ; R) preserves b 2n−1 , thus b 2n−1 may also be considered as an element of H 2n−1 (U ; R) ∼ = H 2n−1 c (GL (C) ; R).
We consider K 2n−1 (C) ⊗ Q as a subgroup of H 2n−1 (BGL (C) ; Q), as in section 2.1. The Borel regulator (We will show in section 2.4 that there is a projection pr 2n−1 : If A ⊂ C is a subring, then inclusion induces a homomorphism K 2n−1 (A) ⊗ Q → K 2n−1 (C) ⊗ Q, thus the Borel regulator also defines a homomorphism Borel class of representations. Definition 2. Let M = G/K be a symmetric space of noncompact type of odd dimension d = 2n − 1. We say that a (continuous) representation ρ : , and the latter holds if and only if Proof: The first equivalence follows from naturality of the van Est isomorphism. The second equivalence follows from Then there exists a projection Proof: such that the following two conditions are satisfied: • There exists a subset J ⊂ I such that {e j : j ∈ J} is a basis of D.
We use this basis to define the projection pr 2n−1 by To decide whether the Borel class is indecomposable we apply 3 Borel's computation of K-theory of integer rings in number fields in [5].
Let O F be the ring of integers in a number field F , which has r 1 real and 2r 2 complex embeddings. Borel proves that the Borel regulator, applied to the different embeddings of SL (O F ), yields an isomorphism between P H 2n−1 (BSL (O F ; Z)) and Z r1+r2 resp. Z r2 if n is even resp. odd.
Since decomposable cohomology classes vanish on primitive homology classes, this implies in particular: If A = O F for a number field F, then the Borel class b 2n−1 is not decomposable for even n If moreover F is not totally real, then the Borel class b 2n−1 is not decomposable for all n.
In particular, we can apply Lemma 2 to A = O F .
If A 1 ⊂ A 2 ⊂ C are subrings and the Borel class is not decomposable for A 1 , then of course it is also not decomposable for A 2 . Thus we can actually apply Lemma 2 to all rings A with O F ⊂ A ⊂ C. In particular to A = Q or A = C: There exists a projection 3 We remark that in the already interesting case A = C one can prove indecomposability of the Borel class without using Borel's K-theory computation.
By Borel's Theorem (see [5], Thm.9.6.), for any arithmetic subgroup Γ ⊂ SL N, C we have an isomorphism j * : This isomorphism is constructed via the van Est isomorphism, that is by integration of forms over simplices. In particular, if h ∈ H * BSL N, C ; Q and i : Γ → SL N, C is the inclusion, then j * i * h maps to h under the canonical homomorphism from continuous to ordinary group cohomology.

Compact locally symmetric spaces and K-theory
In this subsection, we finally show that to each representation of nontrivial Borel class, and each compact, oriented, locally symmetric space of noncompact type we can find a nontrivial element in K * Q ⊗ Q.
Theorem 2. For each symmetric space G/K of noncompact type and odd dimension d = 2n − 1, and to each representation ρ : G → GL (N, C) with ρ * b 2n−1 = 0, there exists a constant c ρ = 0, such that the following holds: to each compact, oriented, locally symmetric space M = Γ\G/K, with ρ (Γ) ⊂ GL (N, A) for a subring A ⊂ C satisfying the conclusion of Lemma 2, there exists an element By Lemma 2 we have a projection pr 2n−1 . Thus, as in section 2.1., we can consider Since the Borel regulator is defined by pairing with b 2n−1 , and by Lemma 2, we get where the last equality is true by Theorem 1. QED Corollary 3. For each symmetric space G/K of noncompact type and odd dimension d = 2n − 1, and to each representation ρ : G → GL (N, C) with ρ * b 2n−1 = 0, there exists a constant c ρ = 0, such that the following holds: to each compact, oriented, locally symmetric space M = Γ\G/K there exists an element Proof: G is a linear semisimple Lie group without compact factors. dim (G/K) = 2n−1 implies that G is not locally isomorphic to SL (2, R), because dim (SL (2, R) /SO (2)) = 2. Thus G satisfies the assumptions of Weil's rigidity theorem, which implies that there exists some g ∈ G with gΓg −1 ∈ G Q . Thus, upon replacing Γ by Each irreducible representation ρ : G → GL (N, C) is isomorphic to a representation ρ ′ such that G Q is mapped to GL N, Q . This follows from the classification of irreducible representations of Lie groups, see [14].
Moreover, by Corollary 2, A = Q satisfies the conclusion of Lemma 2. Thus we can apply Theorem 2. QED Then compact, oriented, locally symmetric spaces Γ\G/K of rationally independent volumes yield rationally independent elements in K 2n−1 Q ⊗ Q.
Remark: In [15] it was claimed that for (2n-1)-dimensional compact hyperbolic manifolds one can construct an element However, since ρ * b 2n−1 is an integer cohomology class, c ρ is rational if and only if v 2n−1 is a rational cohomology class, and this is equivalent Since, conjecturally, all hyperbolic manifolds have irrational volumes, one can probably not get rid of the factor c ρ in Theorem 2.
In conclusion, we are left with the problem of finding representations of nontrivial Borel class, which will be solved in section 3.
The Matthey-Pitsch-Scherer construction. The following construction gives a somewhat stronger invariant under the assumption that M is stably parallelizable. Assume that M d → R n is an embedding with trivial normal bundle νM . Let U be a regular neighborhood. Then there is the composition giving an element γ (M ) ∈ π s d (M ). By [26], if M is a closed hyperbolic 3-manifold and ρ : M → BSL is the map given by the stable trivialization, then ρ * (γ (M )) is the Bloch invariant.
An analogous construction works for locally symmetric spaces, as long as they are stably parallelizable.
It is known by a Theorem of Deligne and Sullivan that each hyperbolic manifold M admits a finite covering M which is stably parallelizable. Let k be the degree of this covering. Then, rationally, we can define γ (M ) := 1 k γ M ∈ π s d (M ) ⊗ Q, and thus get a finer invariant which gives back γ (M ) ∈ K d (C) ⊗ Q. We will not pursue further that approach in this paper.

Examples
Compact examples can e.g. be obtained by Borel's construction of arithmetic hyperbolic manifolds using quadratic forms.
Let u be an algebraic integer such that all roots of its minimal polynomial have multiplicity 1 and are real and negative (except u). Assume moreover that (0, . . . , 0) is the only integer solution of It is isomorphic to a discrete cocompact subgroup of SO (2n − 1, 1). By Selberg's lemma, it contains a torsionfree cocompact subgroup Γ ⊂ SO (2n − 1, 1; Z [u]). With the computations in section 3 below one concludes: If n is even, then the compact manifold M := Γ\H 2n−1 (and, for example, a half-spinor representation) gives a nontrivial element 3 Existence of representations of nontrivial Borel class

Invariant polynomials
Then a representation ρ : G → GL (N, C) has nonvanishing Borel class if and only if T r (D e ρ (t) n ) = 0 for some t ∈ t.
Proof: As in section 2.3, we consider the dual representation We know that ρ has nonvanishing Borel class if and only if The projection p : because a left inverse to p * is given by averaging differential forms over the compact group K. Hence, ρ * u b 2n−1 = 0 if and only if its image in H 2n−1 (G u ) does not vanish. The latter equals (π 2 ρ u ) be Cartan's homomorphism (see section 2.3). According to [8], the image of R are the transgressive elements and one has In view of naturality of the transgression map, C n = τ (b 2n−1 ) implies that Moreover, R • τ = id implies injectivity of τ , hence (π 2 ρ u ) * C n = 0 is also a necessary condition for (π 2 ρ u ) is not trivial on g u . Let t u be the Cartan subalgebra of g u , which corresponds to t under the canonical bijection k ⊕ p ≃ k ⊕ ip. There is an action of the Weyl group W on t u , we denote its space of invariant polynomials by S W * (t u ). By a theorem of Chevalley (see [6]), restriction induces an isomorphism In particular, it suffices to check that T r (π 2 Dρ u (.) n ) is not trivial on t u . We note that the Cartan algebra t can be conjugated into a subspace of p. Since the conclusion of Lemma 2 is invariant under conjugation, we can without loss of generality assume that t ⊂ p and thus t u ⊂ ip. This implies that, for t ∈ t u , Dρ u (t) belongs to the second factor of u (N ) ⊕ u (N ), and thus π 2 Dρ u (t) = Dρ u (t) on t u . Finally we note that, for t ∈ p, T r (Dρ (t) n ) and T r (Dρ u (it) n ) coincide up to a power of i. The claim follows. QED Example: Spinor representations. We consider the half-spinor representations of so (m − 1, 1) because their Borel classes (for m = 2n even) have been computed in [15] and the computation given there appears not to be correct. (The point of confusion seems to be that [15] computes a supertrace rather than a trace. This seems to be related to the wrong assertion that K = U (N ) embeds as a diagonal subgroup into G u = U (N )×U (N ). However, K corresponds to the first factor of U (N ) × U (N ), see section 2.3.) For simplicity, we consider the complexified representation of so (m − 1, 1) ⊗ C ∼ = so (m, C). We use the description of the half-spinor representations as it can be found in [14].
Cl (Q) acts on Λ * W as follows: e i sends v ∈ W to e i ∧ v and e n+i sends v ∈ W to 2v − 2Q (v, e n+i ) e i , for i = 1, . . . , n. (This follows from the proof of [14], Lemma 20.9.) This action extends in the obvious way to an action of Cl (Q) on Λ * W . In particular 1 2 (e i e n+i − 1) acts by sending v ∈ W to e i ∧ v − 1 2 v. Thus it maps e i to 1 2 e i and e j to − 1 2 e j for j = i. This action gives rise to an isomorphism Cl (Q) even ∼ = End (Λ even W ) ⊕ End Λ odd W (see [14], p.305). The induced actions of so (Q) on Λ even W resp. Λ odd W are the positive resp. negative half-spinor representations. We will denote them by S + and S − .
For the positive half-spinor representation (and any l) we have In particular, we have T r S + (A i ) l < 0 for l odd and T r S + (A i ) l > 0 for l even.
For the negative half-spinor representation we have In particular, we have T r S − (A i ) l < 0 for l odd and T r S − (A i ) l > 0 for l even. It follows from the proof of [14], Lemma 20.16., that Cl (Q) acts on Λ * W as follows: the action of e i and e n+i , for i = 1, . . . , n is defined as in the case n even, and e 2n−1 acts as multiplication by 1 on Λ even W and as multiplication by -1 on Λ odd W . In particular, we have again that 1 2 (e i e n+i − 1) acts by sending e i to 1 2 e i and e j to − 1 2 e j for j = i. This action gives rise to an isomorphism Cl (Q) even ∼ = End (ΛW ) (see [14], p.306). The induced action of so (Q) on ΛW is the spinor representation S.
As a Cartan-algebra we choose the algebra of diagonal matrices Let {A i : i = 1, . . . , n} be a basis, where Thus we have (for any l) In particular, we have T r S (A i ) l < 0 for l odd and T r S (A i ) l > 0 for l even.

Borel class of Lie algebra representations
Let g be a semisimple Lie algebra and R g its (real) representation ring, with addition ⊕ and multiplication ⊗. Let t be a Cartan subalgebra of g.
In this section we consider, for n ∈ N, the map given by b 2n−1 (π) (t) = T r (π (t) n ) .

It is obvious that
holds for representations π 1 , π 2 . Therefore b 2n−1 is uniquely determined by its values for irreducible representations. Moreover, for representations π 1 , π 2 .
Complex-linear representations. First we consider complex simple Lie algebras g and the ring R C g ⊂ R g of their C-linear representations.
Let V = C l+1 be the standard representation, with basis e 1 , . . . , e l+1 . Then with A k the induced representation on Λ k V , cf. [14]. In particular, irreducible representations occur as representations of dominant weight in tnesor products of the fundamental representations A 1 , . . . , A l . We compute b 2n−1 on the fundamental representations A k , k = 1, . . . , l.
A basis of Λ k V is given by As Cartan-subalgebra we may choose the diagonal matrices For n = 1, the sum is a multiple of h 1 + . . . + h l+1 = 0. For l = k = 1 and n odd, we have that b n (A 1 ) = h n 1 + h n 2 is a multiple of h 1 + h 2 = 0. In all other cases, i.e. for l ≥ 2, n > 1 or l = k = 1, n even, the sum is not divisible by h 1 +. . .+h l+1 and thus not trivial. This is obvious for even n. In the case of odd n > 1 and l ≥ 2, it follows for example from the com- Thus, fundamental representations have nontrivial b 2n−1 , for all l ≥ 2, n > 1 or l = k = 1, n even. g = sp (l, C).
Let V = C 2l be the standard representation. Then with B k the induced representation on Λ k V . We compute b 2n−1 on the fundamenta representations B k , k = 1, . . . , l. Let {e 1 , . . . , e l , f 1 , . . . , f l } be a basis of C 2l for the standard representation.
A basis of Λ k V is given by If n is even, we clearly get a nonvanishing polynomial without cancellations. If n is odd, then the permutation, which transposes i r and j r simultaneously for all r, multiplies the sum by −1, but on the other hand preserves the sum. Thus b 2n−1 (B k ) = 0 if n is odd.
Let V = C 2l+1 be the standard representation. Then with C k the induced representation on Λ k V , and S the spin representation. As Cartan-subalgebra we may choose the diagonal matrices diag (h 1 , . . . , h l , −h 1 , . . . , −h l , 0) .
Then the computation of b 2n−1 on C k is exactly the same as for sp (l, C), in particular b 2n−1 (C k ) = 0 for n even and b 2n−1 (C k ) = 0 for n odd. Moreover, we have computed in section 3.1 that b 2n−1 (S) = 0 for all n. g = so (2l, C).
Let V = C 2l be the standard representation. Then with D k the induced representation on Λ k V , and S ± the half-spinor representations. As Cartan-subalgebra we may choose the diagonal matrices diag (h 1 , . . . , h l , −h 1 , . . . , −h l ) .
Again the same computation as for sp (l, C) shows that b 2n−1 (D k ) = 0 for n even and b 2n−1 (D k ) = 0 for n odd. Moreover, we have computed in section 3.1 that b 2n−1 (S ± ) = 0 for all n.
Exceptional Lie groups.
We will see in the next section that we will only be interested in Lie groups which admit a symmetric space of odd dimension. The only exceptional Lie group admitting an odd-dimensional symmetric space is E 7 with dim (E 7 /E 7 (R)) = 163 = 2.82 − 1. The fact that 163 ≡ 3 mod 4, i.e. n = 82 even, implies automatically that ρ * b 163 = 0 holds for each irreducible representation ρ. For completeness we also show, at least for a specific representation, that ρ * b 2n−1 = 0 holds for each n > 1. Namely, we consider the representation ρ : E 7 → GL (56, C), which has been constructed in [1], Corollary 8.2, and we are going to show that this representation satisfies ρ * b 2n−1 = 0 for each n > 1, in particular for n = 82.
For even n, we know that ρ * b 2n−1 = 0. If n is odd then, for the derivative π 1 of the standard representation λ 1 of SU (2) we have T r (π 1 (h) n ) = 0, whenever h ∈ t ∩ su (2) belongs to the Cartan-subalgebra of su (2) , because the latter are the diagonal 2x2matrices of trace 0. Thus the first direct summand λ 1 12 ⊗ λ 1 does not contribute to T r (π (h) n ). Hence , for h = (h spin , h su ) ∈ t ⊂ spin (12) ⊕ su (2), we have T r (π (h) n ) = T r (S − (h spin ) n ). But the nontriviality of the latter has already been shown in section 3.1.
Noncomplex Lie algebras. Let π : g → gl (N, C) be an R-linear representation of a simple Lie-algebra g which is not a complex Lie algebra. Then g ⊗ C is a simple complex Lie algebra and π is the restriction of some C-linear representation g ⊗C → gl (N, C). Let t be a Cartan subalgebra of g. Then it is obvious that an element t ∈ t ⊗ C with Real representations of complex Lie algebras. If g is a simple complex Lie algebra, then each R-linear representation π : g → gl (N, C) is of the form π = π 1 ⊗ π 2 for C-linear representations π 1 , π 2 . We have In particular, real representations with nontrivial b 2n−1 can only exist if there are complex representations of nontrivial b 2n−1 .
Proposition 1. Let ρ : G → GL (N, C) be a representation of a Lie group G, and π : g → gl (N, C) the associated Lie algebra representation π = D e ρ. Then ρ has nontrivial Borel class if and only if π has nontrivial Borel class.
Proof: This is precisely the statement of Lemma 3. QED We use the classification of symmetric spaces as it can be read off table 4 in [29]. Of course, we are only interested in symmetric spaces of odd dimension. A simple inspection shows that all odd-dimensional irreducible symmetric spaces of noncompact type are given by the following list: First we note that for n even all representations satisfy ρ * b 2n−1 = 0. This applies to locally symmetric spaces of dimension ≡ 3 mod 4. In the above list this are the following symmetric spaces: We now analyze the irreducible locally symmetric spaces of dimension ≡ 1 mod 4.
For those locally symmetric spaces, whose corresponding Lie algebra g is not a complex Lie algebra (this concerns the first 3 cases), we can, as observed at the end of section 3.2, directly apply the results for the complexifications. Thus we get : -for SL l (R) /SO l , l ≡ 3, 4 mod 8, every fundamental representation yields a nontrivial element, -for SL 2l (H) /Sp l , l ≡ 0 mod 4, every fundamental representation yields a nontrivial element, -for Spin p,q / (Spin p × Spin q ) , p, q ≡ 1 mod 2, p ≡ q mod 4, the positive and negative half-spinor representations are the only fundamental representations yielding nontrivial elements.
For those locally symmetric spaces whose corresponding Lie algebra g is a complex Lie algebra, we use the fact that each real representation is of the form ρ 1 ⊗ ρ 2 . We get: -for SO l (C) /SO l , l ≡ 3 mod 4, the spinor representation and its conjugate are the only fundamental representations yielding nontrivial elements, -for Sp l (C) /Sp l , l ≡ 1 mod 4, no fundamental representation yields nontrivial elements.
Example (Goncharov): Hyperbolic space H n is the symmetric space H n = Spin n,1 / (Spin n × Spin 1 ) .
Let n be odd. It was shown in [15] that the positive and negative half-spinor representation have nontrivial Borel class. The question was raised ( [15], p.587) whether these are the only fundamental representations of Spin n,1 with this property. As a special case of the above results we see that for n ≡ 3 mod 4 each irreducible representation has nontrivial Borel class, but for n ≡ 1 mod 4 the positive and half-negative spinor representation are the only fundamental representations with this property.
On the other hand, if n = 3, then the invariants coming from different irreducible representations, albeit distinct and nontrivial, all are rational multiples of each other. This will follow from the computation in Section 3.4.

Some clues on computation
So far we have only discussed how to decide whether ρ * b 2n−1 = 0, which is in view of Lemma 3 easier than computing ρ * b 2n−1 . The aim of this subsection is only to give some clues to the computation of ρ * b 2n−1 . Its results are not needed throughout the paper, except for the explicit values of the Borel regulator in section 5.
For each Lie-algebra-cocycle P ∈ C n g u , k , we denote by ω P ∈ Ω n (G u /K) the corresponding G u -invariant differential form. Then we have the following obvious observation. ([ω P ] denotes the cohomology class of ω P , and [G u /K] v ∈ H n (G u /K, R) denotes the dual of the fundamental class [G u /K]. The Riemannian metric is given by −B, that is the negative of the Killing form.) Lemma 4. : Let X 1 , . . . , X n be an orthonormal basis for ip with respect to -B. Then, for each P ∈ I n g u , k , we have Corollary 5. : [ω P ] = 0 iff P (X 1 , . . . , X n ) = 0 for some (hence any) basis of ip.

Example: Hyperbolic 3-manifolds.
Let G = SL (2, C). Then An ON-basis of ip (with respect to the Killing form) is given by 1 Thus, for each representation ρ : Sl (2, C) → GL (m + 1, C) with associated Lie algebra representation Π : sl (2, C) → M at (m + 1, C) we have By the classification of irreducible representations of sl (2, C), each m + 1-dimensional irreducible representation is equivalent to π m given by In particular, T r π m (X) 2 = T r π m (Y ) 2 and we conclude If m = 1, we get One should note that the hyperbolic metric is given by a half of the negative of the Killing form. Thus an orthonormal basis of p = T [e] H 3 with respect to the hyperbolic metric is given by It follows that the Borel regulator is 1 8π 2 times the hyperbolic volume.
(There seem to be different normalizations of the Borel regulator in the literature. [11] computes the Borel regulator to be 1 4π 2 times the hyperbolic volume, while [28] defines the imaginary part of the Borel regulator to be 1 2π 2 times the hyperbolic volume.) In [15] it was stated that the half-spinor representations seem to be the only fundamental representations of Spin (d, 1) that yield nontrivial invariants of odd-dimensional hyperbolic manifolds. This is however not the case.
We will use the convention that, for A ∈ {H, X, Y } if A 1 is defined (in a given basis), then A 2 is obtained via the base change e 1 → e 2 , e 2 → e 3 , e 3 → e 1 and A 3 is obtained via the base change e 1 → e 3 , e 3 → e 2 , e 2 → e 1 .
We have [H 1 , = iY 3 and more relations are obtained out of these ones by base changes.
A basis of ip is given by H 1 , H 2 , X 1 , X 2 , X 3 . The formula for ρ * b 5 (H 1 , H 2 , X 1 , X 2 , X 3 ) contains 120 summands. ( We note that H 1 , H 2 , X 1 , X 2 , X 3 are pairwise orthogonal and have norm 2 √ 3. Dividing each of them by 2 √ 3 gives an ON-basis, on which evaluation of ρ * b 5 gives 4 The cusped case

Preparations
Let G be a connected, semisimple Lie group with maximal compact subgroup K. Thus G/K is a symmetric space of noncompact type. In this section we will assume that G/K is a symmetric space of rank one. Let ρ : G → GL (N, C) be a representation. We assume that ρ maps K to U (N ), which can be achieved upon conjugation. We note that connected, semisimple Lie groups are perfect, hence ρ has image in SL (N, C) and maps K to SU (N ).

Negative curvature and visibility manifolds
If M = Γ\G/K is a locally symmetric space of rank one, then its sectional curvature sec is bounded between two negative constants, after scaling with a constant factor one has −4 ≤ sec ≤ −1.
In particular, by [12], page 440, the universal covering M is a 'visibility manifold' in the sense of [12]. By [12], Theorem 3.1, this implies that each end of int (M ) = M − ∂M has a neighborhood E homeomorphic to U c /P c , where c ∈ ∂ ∞ int (M ), U c is a horoball centered at c and P c ⊂ Γ is a discrete group of parabolic isometries fixing c.
In the proof of Proposition 2 and Theorem 3 we will use that π 1 ∂ i M → π 1 M is injective for each path-component ∂ i M of ∂M . We are going to explain how this fact follows from well-known properties of negatively curved manifolds. Proof: M = M /Γ for a group of isometries Γ ⊂ Isom M with Γ ≃ π 1 M . By the proof of [12], Theorem 3.1, we have that each end of M is of the form N ×(0, ∞) and that N × {0} = L c /P c for a simply connected horosphere L c and a subgroup P c ⊂ Γ that acts freely and properly discontinuously on L c .
In particular, the boundary component ∂ i M that corresponds to this end is homeomorphic to N and its fundamental group is isomorphic to P c . We have N = L c ⊂ M , which implies that π 1 N → π 1 M is injective because, by covering theory, N resp. M are the set of homotopy classes (rel. {0, 1}) of paths γ : [0, 1] → N resp. M with γ (0) = x 0 , for a fixed point x 0 ∈ N ⊂ M . QED Moreover M and all ∂ i M are aspherical by the Cartan-Hadamard Theorem resp. by [12].

Relative classifying spaces.
First, we briefly discuss the approach via relative classifying spaces, which works exactly as in [9]. Let M be a compact d-manifold with boundary such that int (M ) = M − ∂M admits a locally symmetric Riemannian metric of finite volume. Let M + be the quotient space obtained by identifying points in respectively each boundary component. In particular H d (M + ) has a fundamental class.
(Remark: In [9], M + is the one-point compactification. This is not homeomorphic to our M + , but has the same homology in degrees ≥ 2.) Let P ⊂ G and B ⊂ SL (N ; C) be maximal unipotent subgroups, such that ρ : (G, K) → (SL (N, C) , SU (N )) sends P to B. To a rank one locally symmetric space Γ\G/K of finite volume and dimension n, we can consider G/K ∪ C, where C denotes the set of parabolic fixed points in ∂ ∞ G/K and get as in [9], section 3, an (up to Γ-equivariant homotopy unique) Γ-equivariant map F (P )) .
The quotient of G/K∪C by Γ is homeomorphic to M + . In particular, H d (Γ\ (G/K ∪ C)) has a fundamental class, and we get as in [9] an element α (M ) ∈ H d (B (G, F (P ))) , which by the representation ρ is pushed forward to an element ρ * (α (M )) ∈ H d (B (SL (N, C) , F (B))) .
In the case of hyperbolic 3-manifolds, Cisneros-Molina and Jones ([9]) lifted this invariant to K (C) ⊗ Q, and proved its nontriviality by relating it to the Bloch invariant. We describe here how to do a very similar construction for arbitrary locally symmetric spaces. (Unfortunately we did not succeed to compute the Borel regulator of this invariant. This is the reason why we will actually pursue another approach, using relative group homology and closer in spirit to [15], in the remainder of this section. The construction is however included at this point because its main step, Lemma 4, will be crucial for the proof of Proposition 2.)

Generalized Cisneros-Molina-Jones construction
Let M be an aspherical d-manifold with aspherical boundary, F ⊂ C a subring and ρ : π 1 M → SL (F) a representation 4
By assumption, ρ maps π 1 ∂ 1 M into some maximal unipotent subgroup B ⊂ SL (n, F) of parabolic elements. B is conjugate to B 0 , the group of upper triangular matrices with all diagonal entries equal to 1. By [30], Lemma 4.2.3, there exists a homomorphism Π : B 0 → St (N, F) with ΦΠ = id. Applying conjugations and composing with ρ, we get a homomorphism τ : ∂ 1 M is aspherical, hence τ is induced by some continuous mapping g 0 : ∂ 1 M → F , and the diagram commutes up to some homotopy H t . This construction can be repeated for all connected components ∂ 1 M, . . . , ∂ s M of ∂M . For each r = 1, . . . , s we get a continuous map g r : ∂ r M → F such that jg r ∼ (Bρ) i r . Altogether, we get a continuous map g : ∂M → F such that jg is homotopic to (Bρ) i.
By [9], Lemma 8.1. this implies the existence of the desired map R. QED Hence one obtains an element Q * R * [M + ] ∈ H d (BSL (F) ; Q). Unfortunately we did not succeed to prove its nontriviality, i.e. to evaluate the Borel regulator. Therefore we will in the remainder of this section pursue a different approach, closer in spirit to [15], but surrounding the problem that ∂M may be disconnected. We mention that another "basis-point independent" approach might use multicomplexes in the sense of Gromov, but also here we were not able to compute the Borel regulator in the case that there are 3 or more boundary components. Also, in the case of hyperbolic 3-manifolds, yet another approach is due to Neumann-Yang [28]. It should be interesting to generalize and compare the different constructions.
For hyperbolic 3-manifolds of finite volume, Zickert has given in [31] a direct construction of a fundamental class [M, ∂M ] ∈ H 3 (SL (2, C) , B 0 ), even in the case of possibly disconnected boundary.

Cuspidal completion
We start with a notational remark. Let X be a topological space and A 1 , . . . , A s ⊂ X a set of (not necessarily disjoint) subspaces. There is a (not necessarily injective) continuous mapping i : A 1∪ . . .∪A s → X from the disjoint union A 1∪ . . .∪A s to X. We define the disjoint cone to be the pushout of the diagram If X is a CW-complex and A 1 , . . . , A s are disjoint sub-CW-complexes, then clearly A special case is that of a compact manifold M with disconnected boundary ∂M , consisting of path-components ∂ 1 M∪ . . .∪∂ s M . Then DCone (∪ s i=1 ∂ i M → M ) is the space M + from the Cisneros-Molina-Jones construction. (In this case, the union of components is a disjoint union. Nonetheless DCone is different from Cone.) Another special case is the cuspidal completion of a classifiying space, whose geometry will be described in this section. The point of the construction is that it may remember the geometry of the cusps of locally symmetric spaces. Thus it serves as a technical device to handle the cusped case.

Construction of BG comp :
We recall from the beginning of section 2.2 that BG is the simplicial complex realizing the bar construction. Thus its n-simplices are of the form (1, g 1 , . . . , g n ) with g 1 , . . . , g n ∈ G.
Definition 4. Let G/K be a symmetric space of noncompact type. We define the cuspidal completion BG comp of BG to be the pushout of the following diagram: In other words, BG comp is the mapping cone 5 Cone ∪ c∈∂∞G/K BG → BG , but with the union∪ c∈∂∞G/K BG to be understood as a disjoint union.
Notation: The cone point of Cone (BG) corresponding to c ∈ ∂ ∞ G/K will also be denoted c.
For the remainder of this section we assume somex ∈ G/K to be fixed.
Thus we obtain a well-defined simplex-wise Riemannian metric on the push-out. In particular, we have a simplex-wise local isometry The local isometry p defines a volume form p * dvol on Int (BG comp ). (Recall that a differential form ω on a simplicial complex consists of a smooth form ω σ on every simpex σ such that ω σ | τ = ω τ whenever τ ⊂ σ is a subsimplex.) By Thom's analogue of the deRham Theorem one has H k dR (BG comp ) = H k simp (BG comp ), where H * dR is the cohomology of the complex of differential forms. The isomorphism is given by integration of differential forms over simplices. In particular, p * dvol gives a simplicial cohomology class v d ∈ H d simp (BG comp ) represented by the cocycle ∆ → ∆ p * dvol. (This is defined because ideal d-simplices in a d-dimensional symmetric space G/K of noncompact type have finite volume, see e.g. [24], which provides even a uniform bound.) Recall that in section 2.3 we defined, for d = 2n−1 odd, the Borel class b d ∈ H d c (GL (C) ; R) which, of course, can also be considered as a class b d ∈ H d (BSL (N, C) ; R). Proposition 2. Let M be a compact, oriented manifold with boundary components ∂ 1 M, . . . , ∂ s M such that Int (M ) = Γ\G/K is a finite-volume locally symmetric space of noncompact type. Assume that, for some subring F ⊂ C, we have an inclusion

Proof:
First we notice that it suffices to prove that (Bρ . . , s. Indeed, if z is a relative cycle that represents [M, ∂M ], and for i = 1, . . . , s we have chains z i ∈ C d (BΓ i ) is a cycle, whose image in C d (DCone (∪ s i=1 BΓ i → BSL (F))) again represents ( for i = 1, . . . , s. QED Remark: For the special case of hyperbolic manifolds and half-spinor representations, Proposition 2 was proved in [15], Theorem 2.12. The proof in [15] uses very special properties of the half-spinor representations and seems not to generalize to other representations.

Straightening of interior and ideal simplices
Definition 5. For a manifold M with boundary components ∂ 1 M, . . . , ∂ s M , we denote Since int (M ) − ∪ s i=1 E i is homeomorphic to M , this means that we have a projection is said to be straight if some (hence any) lift to M ∪ ∂ ∞ M is straight. Recall that DCone (∪ s i=1 BΓ i → BΓ) is a simplicial complex as defined in the footnote in section 4.2.
Lemma 7. Let M be a manifold with boundary ∂M with a Riemannian metric such that int (M ) is a visibility manifold of nonpositive curvature. Let x 0 ∈ M , Γ := π 1 (M, x 0 ) and Γ i := π 1 (∂ i M, x i ) for the boundary components ∂ i M of ∂M and some x i ∈ ∂ i M . Fix (using some path from x 0 to x i ) isomorphisms of Γ i with subgroups of Γ. Then there is an isomorphism of chain complexes Proof: int (M ) is a simply connected manifold of nonpositive curvature. Hence a straight (possibly ideal) simplex is uniquely determined by its ordered set of vertices. Since each straight simplex in M can be lifted to a straight simplex in M , this implies: -each interior simplex σ ∈ C str,x0 k (M ) is uniquely determined by the homotopy classes of the k edges γ j between its vertices σ (v j−1 ) and σ (v j ) for j = 1, . . . , k. That is, the interior simplices generate a subcomplex inĈ str,x0 * (M ) that is isomorphic, as a chain complex, to C simp * (BΓ). The bijection Φ 1 is given by where γ i ∈ Γ = π 1 (M, x 0 ) is the homotopy class of the edge from σ (v j−1 ) to σ (v j ), for j = 1, . . . , k. It is easy to check that the bijection Φ 1 is a chain map.
-each ideal simplex σ ∈ C str,x0 k (M ) is uniquely determined by the choice of a boundary component ∂ i M (which corresponds to the Γ-orbit Γc i of a parabolic fixed point c i ) and by the homotopy classes of interior edges from σ (v j−1 ) to σ (v j ) for j = 1, . . . , k − 1. By definition, the interior edges belong to Γ i . Thus, for i = 1, . . . , s, ideal simplices with ideal vertex at Γc i generate a subcomplex inĈ str,x0 * (M ) that is isomorphic to C simp * (Cone (BΓ i )) ⊂ C simp * (Cone (BΓ i → BΓ)). The bijection Φ 2 is given by where γ j ∈ Γ i ⊂ Γ = π 1 (M, x 0 ) is the homotopy class of the edge from σ (v j−1 ) to σ (v j ), for j = 1, . . . , k − 1. It is easy to check that the bijection Φ 2 is a chain map.
The two chain isomorphisms Φ 1 and Φ 2 are compatible: if σ is an ideal k-simplex, then ∂ k σ is an interior simplex and we have by construction: Thus Φ 1 and Φ 2 define a chain isomorphism Proof: Denote d = 2n − 1.

Evaluation of Borel regulator
G is a linear semisimple Lie group without compact factors, not locally isomorphic to SL (2, R). By Weil rigidity we can assume (upon conjugation) that Γ ⊂ G Q . For a semisimple Lie group G, each representation ρ : G → GL (N, C) is isomorphic to a representation which maps G (A) to SL (N, A). (This can be read off the classification of representations of semisimple Lie groups, see [14].) By Corollary 2, A = Q satisfies the assumptions of Lemma 2. Thus a) is a consequence of b). We are going to prove b). Let . . , C s } to be zero). We homotope the cycle r i=1 τ i + p j=1 κ j into a cycle r i=1 τ ′ i + p j=1 κ ′ j such that all interior vertices are homotoped into x 0 , all ideal vertices remain fixed during the homotopy, and all edges between interior vertices of κ ′ j are homotoped into loops representing elements of Γ i . That is τ ′ 1 , . . . , τ ′ r , κ ′ 1 , . . . , κ ′ p ∈ C x0 * (M ). G/K has negative sectional curvature. In particular it is a visibility manifold and we have the subcomplex C str,x0 * (M ) ⊂ C x0 * (M ) as defined by Definition 6. By [12], Theorem 3.1, some neighborhood of C l , for l ∈ {1, . . . , s}, is homeomorphic to U c l /P c l , where c l ∈ ∂ ∞ G/K, U c l is a horoball centered at c l and P c l ⊂ Γ is a discrete group of parabolic isometries fixing c l . In particular, each ideal simplex with an ideal vertex at C l lifts to an ideal simplex in G/K ∪ ∂ ∞ G/K with an ideal vertex at c l .
Let w 0 , . . . , w d be the vertices of the standard simplex ∆ d . By Lemma 7, the isomorphism C str,x0 *

Examples from hyperbolic manifolds
The case of hyperbolic 3-manifolds has been discussed to some extent in [28].
If M is any hyperbolic 3-manifold of finite volume, then π 1 M can be conjugated to be contained in SL (2, F ), where F is an at most quadratic extension of the trace field ( [25]), thus one gets an element in K 3 (F ) ⊗ Q. In [28], section 9, some examples of this construction are given. (The discussion in [28] is about elements in B (F ) ⊗ Q for the Bloch group B (F ), but of course analogously one is getting elements in K 3 (F ) ⊗ Q associated to the respective manifolds.) For example (cf. [28], section 9.4.) for any number field F with just one complex place there exists a hyperbolic 3-manifold of finite volume, such that its invariant trace field equals F . The associated γ (M ) gives a nontrivial element, and actually a generator, in K 3 (F ) ⊗ Q.

Representation varieties
Let M be a compact d-manifold with boundary. If ψ : π 1 M → G is a homomorphism, one can construct a ψ-equivariant map f : M → G/K which is unique up to homotopy. In particular, vol (ψ) := F f * dvol, for a fundamental domain F ⊂ M , is well-defined. If ψ preserves parabolics, i.e. ψ (π 1 ∂ i M ) is unipotent for all components ∂ i M ⊂ ∂M , then literally the same argument as in the proof of Theorem 3 shows ∈ H d DCone ∪ s i=1 BΓ i → BG ; Z is constant. We note that the map is however not constant on the variety of parabolics-preserving representations. This follows, for example, from the volume rigidity theorem (which for hyperbolic manifolds has been proved by Thurston and Dunfield and in the higher rank case is a consequence of Margulis superrigidity theorem) which states that elements of the component of Rep (π 1 M, G) that contains the discrete representation are the only representations of maximal volume.