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

Abstract

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag structures can be recovered by a fundamental class construction.

Invariants preserved by mutation

“…If int (M ) ∼ = Γ\G/K is a Q-rank 1 locally symmetric space, then by [7,Lemma 6.2] composition of (Ψ M ) * with the isomorphism H * (| BΓ comp |) ∼ = H simp * (BΓ comp ) yields the inverse of the Eilenberg-MacLane-isomorphism:…”

“…It is proved in the appendix of [2] that the Bloch-Wigner morphism sends the P SL(2, C)-fundamental class of a closed hyperbolic 3-manifold to its Bloch invariant β(M ). The corresponding result for cusped hyperbolic 3-manifolds as well as for the generalized Bloch invariant of (closed or R-rank 1) locally symmetric spaces has been proved in [10, Theorem 1], see also [7,Definition 8.1]…”

“…Bloch invariant of CR structures ( [3]). If M is a finite-volume hyperbolic manifold and ρ : π 1 M → SU (2, 1) a reductive representation, then [7,Lemma 10.3] implies that the Bloch invariant…”

“…In [9] we gave a topological proof of Ruberman's theorem using the fundamental class construction. In [7] we used an analogous argument to prove that also the volume of flag structures is preserved under mutation. The aim of this paper is to prove in a general setting that G-fundamental classes and hence various geometric invariants are preserved under generalized mutation.For a closed, orientable d-manifold M and a representation ρ : π 1 M → G one has the naturally associated G-fundamental class (Bρ) * [M ] ∈ H d (BG).…”