$η$-invariant and flat vector bundles

We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.

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