Geodesic
Atiyah--Patodi--Singer $\eta$-invariant and invariants of finite degree
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 163-193

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We consider the problem of computing the degree of invariants of the form $\eta\bmod A$, where $\eta$ is the Atiyah–Patodi–Singer invariant considered on smooth compact oriented three-dimensional submanifolds of $\mathbb R^n$ and $A$ is an additive subgroup of $\mathbb R$. We use the functional definition of invariants of finite degree. (A similar approach is used in the paper “Quadratic property of the rational semicharacteristic” by S. S. Podkorytov.) The main results are as follows. If $1\notin A$, the degree is infinite. If $\frac13\in A$, the degree equals one. 
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     author = {A. N. Trefilov},
     title = {Atiyah--Patodi--Singer $\eta$-invariant and invariants of finite degree},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {163--193},
     publisher = {mathdoc},
     volume = {415},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a15/}
}
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A. N. Trefilov. Atiyah--Patodi--Singer $\eta$-invariant and invariants of finite degree. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 163-193. http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a15/