Atiyah-Patodi-Singer \(\eta\)-invariant and invariants of finite degree (Q291164)

In the article under review, the author proves that for any additive subgroup \(A\) of \(\mathbb{R}\) that does not contain 1, then the eta-invariant \(\bmod\;A\), regarded as a functional from the set of all smooth compact oriented 3-dimensional submanifolds of \(\mathbb{R}^n\) without boundary to \(\mathbb{R}/A\), is not an invariant of finite degree. On the other hand, the author also proves that the eta-invariant mod \(A\), again regarded as a functional from the set of all smooth compact oriented 3-dimensional submanifolds of \(\mathbb{R}^n\) without boundary to \(\mathbb{R}/A\), is an invariant of degree 1 if \(\frac{1}{3}\in A\).