# Is there a simplicial volume definition of Chern Simons invariants?

Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the fundamental class $[M]\in H_3(M,\mathbb Z)$, which is a homotopy invariant.

Now the hyperbolic volume and Chern-Simons invariant $M$ are connected by the following definition: $$i(\operatorname{Vol}(M)+i\operatorname{CS}(M))=\frac 12\int_M\operatorname{tr}(A\wedge dA+\frac 32A\wedge A\wedge A)\in\mathbb C/4\pi^2\mathbb Z$$ where $A$ is any flat connection on the trivial principal $\operatorname{SL}(2,\mathbb C)$-bundle over $M$ whose monodromy is the isomorphism $\pi_1(M)=\Gamma$. This corresponds to a particularly natural homomorphism (based on a dilogarithm) in $H_3(\operatorname{SL}(2,\mathbb C),\mathbb Z)\to\mathbb C/4\pi^2\mathbb Z$ (see work of Neumann and Zickert).

This close connection between the two invariants $\operatorname{Vol}(M)$ and $\operatorname{CS}(M)$ motivates the following question:

Is there a definition of $\operatorname{CS}(M)$ within the framework of simplicial volume?

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