{"paper":{"title":"The eta function and eta invariant of $\\mathbb{Z}_{2^r}$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ricardo A. Podest\\'a","submitted_at":"2014-07-28T15:50:29Z","abstract_excerpt":"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^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7454","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}