Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The eta function and eta invariant of $\mathbb{Z}_{2^r}$-manifolds   We compute the eta function $\eta(s)$ and its corresponding $\eta$-invariant
for the Atiyah-Patodi-Singer operator $\mathcal{D}$ acting on an orientable
compact flat manifold of dimension $n =4h-1$, $h\ge 1$, and holonomy group
$F\simeq \mathbb{Z}_{2^r}$, $r\in \mathbb{N}$. We show that $\eta(s)$ is a
simple entire function times $L(s,\chi_4)$, the $L$-function associated to the
primitive Dirichlet character modulo 4. The $\eta$-invariant is 0 or equals
$\pm 2^k$ for some $k\ge 0$ depending on $r$ and $n$. Furthermore, we construct
an infinite family $\mathcal{F}$ of orientable $\mathbb{Z}_{2^r}$-manifolds
with $F\subset \mathrm{SO}(n,\mathbb{Z})$. For the manifolds $M\in \mathcal{F}$
we have $\eta(M)=-\tfrac{1}{2}|T|$, where $T$ is the torsion subgroup of
$H_1(M,\mathbb{Z})$, and that $\eta(M)$ determines the whole eta function
$\eta(s,M)$.

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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex