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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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1 Answer 1

If you use eta invariant in place of Chern-Simons invariant, there is almost such a definition, at least in a closely related context. If we restrict to surface bundles over the circle with fiber of fixed genus, then the eta invariant of a fibered 3-manifold can be thought of as a certain kind of class function on the mapping class group. Such eta invariants exist for many different kinds of (unitary) representations (of subgroups of mapping class groups), and under suitable circumstances (see e.g. http://arxiv.org/abs/1003.4977) the functions they define on (subgroups of) mapping class groups are examples of what are known as homogeneous quasimorphisms.

Quasimorphisms arise in the theory of bounded cohomology; there is a duality theory relating them to a (relative) 2-dimensional Gromov norm, called stable commutator length. Gromov norm here is just a synonym for simplicial volume, as in your question.

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