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For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes $$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$ as defined in http://www.jstor.org/stable/1971013

It is claimed at several places in the literature that $\hat{c}_k(E_\rho)$ is constant on components of the representation variety. (For example in Reznikov's proof of the Bloch conjecture http://arxiv.org/pdf/dg-ga/9407007.pdf where this fact is called rigidity of secondary characteristic classes and is attributed to the above-linked paper of Chern and Simons. However I was not able to find such a statement in that paper.)

Question: is there are a reference for this fact, or is there some well-known fact from which this local constantness follows?

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  • $\begingroup$ I think, flat connections are the critical points of the CS functional on the space of all connections on the bundle $E$. This will imply local constancy. $\endgroup$
    – Misha
    Sep 26, 2013 at 19:53
  • $\begingroup$ It should be noted that $\hat c_1$ is not rigid. For example, all bundles with connection on $S^1$ are flat, and $\hat c_1$ is related to the determinant of holonomy, which varies continously over all of $\mathbb C^\times$. $\endgroup$
    – Sebastian Goette
    Oct 6, 2015 at 12:12

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To answer my own question: the wanted rigidity follows from Theorem 3.4 in http://arxiv.org/pdf/math/9904131.pdf together with the main result from P. Ntolo, Homologie de Leibniz d'algébres de Lie semi-simples, Comptes Rendus Acad. Sci. 318 (1994)

UPDATE: the details of Reznikov‘s original argument have been worked out in a recent paper of Pitsch and Porti: https://arxiv.org/pdf/1704.01321.pdf

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