Locally symmetric spaces andK–theory of number fields

Abstract

For a closed locally symmetric space M = Γ\G/K and a representation ρ : G → GL (N, C) we consider the push-forward of the fundamental class in H * BGL Q and a related invariant in K * Q ⊗ Q. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of R-rank one.(C) is, up to torsion, isomorphic to the Bloch *will be called the Eilenberg-MacLane map. The chain homotopy inverse is given by the composition of str with the chain isomorphism Φ, thusThe geometric realization | BΓ | of BΓ in the sense of [22] is aspherical by [20, p.128]. Given a manifold M and an isomorphism I : π 1 M ∼ = Γ, there is an up to homotopy unique continuous mapping h M : M →| BΓ | which induces I on π 1 , see [20, p.177]. The map h M (rather its homotopy class) is called the classifying map for π 1 M . If M is aspherical and has the homotopy type of a CW-complex then h M is a homotopy equivalence, and h M * : H * (M ; Z) → H * (| BΓ |; Z) is the composition of EM −1 with the isomorphism i * : H simp * (BΓ; Z) → H * (| BΓ |; Z) that is induced by the inclusion i of the simplicial into the singular chain complex.

Homological and Bloch invariants for $\mathbb Q$-rank one spaces and flag structures

“…Obviously, Ψ • Φ • i = id and thus we have an injective homomorphism Remark. In the R-rank 1 case, the geodesic straightening map str * : [21]). However, in the higher rank case, the straightening map str * : C x 0 * (M ) → C str,x 0 * (M ) is not well defined as we mentioned in Section 4.…”

“…In [21] the third-named author generalized Goncharov's (first) construction to finite-volume locally symmetric spaces of noncompact type M = Γ\G/K which are either closed or of R-rank one. To each representation ρ : G → SL(n, C) the construction yields an element in H * (SL(n, Q)) or, after a suitable projection, an element in P H * (GL(Q)) ∼ = K * (Q) ⊗ Q such that application of the Borel class (to either of the two elements) yields a multiple c ρ Vol(M ) of the volume Vol (M ).…”

“…The factor c ρ depends only on ρ, in particular one can recover the volume if c ρ = 0. Moreover, [21,Section 3] provides a complete list of fundamentals representations ρ : G → SL(n, C) with c ρ = 0, for example one has c ρ = 0 whenever dim (G/K) ≡ 3 mod 4 and ρ = id. However, in the noncompact case, the only R-rank one examples with c ρ = 0 were (odd-dimensional) real-hyperbolic manifolds.…”

“…. , s. Make z a chain z 0 in C x 0 d (M ) via the chain homotopy C * (M ) → C x 0 * (M ) in the proof of[21, Lemma 8]. Note that z 0 is obtained by adding several 1-dimensional paths to z and hence algvol(z 0 ) = algvol(z) = Vol(M ).…”

Invariants preserved by mutation

“…Several invariants can be derived from this element.Volume of locally symmetric spaces. If ρ : Γ → G is the inclusion of a Q-rank 1 lattice and int(M ) = Γ\G/K the locally symmetric space, then by[8, Section 4.2.3] there is the extended volume cocycle cν d defined by cν d (g 1 , . .…”