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I am learning the APS's papers. What I only know is that the relative $\eta$ invariant is a homotopy invariant.

The reduced $\bar\eta_D=\frac12(\eta_D(0)+\ker(D))\bmod\mathbb Z$, for a self adjiont operator $D$ on an closed odd dimensional manifold, depends smoothly on the defining data.

Q What does "depends smoothly" mean? Is it a topological invariant?

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    I assume that this means the reduced eta invariant depends smoothly on the choices made to define the operator $D$. Often these choices might involve geometric data such as a Riemannian metric, vector bundle metric, and/or a connection. But the reduced eta invariant is _not_ a topological invariant, i.e., it depends on these choices in general. Only the relative eta invariant is, as you mentioned, a topological invariant. – 2017-02-09
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    Literally, it means the reduced eta invariant varies smoothly, but in Bismut and Cheeger's paper(eta invariant and adiabatic limits) Thm 2.7 says that the reduced eta invariant is " independent on the choice of metric ". So, I am a little confused . – 2017-02-09
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    I am looking at Bismut-Cheeger now, although I am not familiar enough with superconnections to comment with any certainty. On page 41 (above Theorem 2.7), they define this reduced eta invariant $\tilde{\eta}(D^\xi)$ as a _difference_ of reduced eta invariants (mod $\mathbb{Z}$). So is it possible that this reduced eta invariant is similar to (or possibly generalizes) the relative eta invariant of A-P-S, which is also a difference in eta invariants? – 2017-02-09

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