$\eta$-invariant and flat vector bundles
  We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $\eta$ 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. 
Xiaonan MA and Weiping Zhang