2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59362The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P. Gilkey. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented. This result answers the question of P. Gilkey (1989) concerning the existence of even-order operators on odd-dimensional manifolds with nontrivial fractional part of eta-invariant.24 pages, 1 figure; final version; see http://www.kluweronline.com/issn/0001-4346/contentsK-Theory and HomologyAnalysis of PDEsAlgebraic TopologyDifferential GeometryOperator AlgebrasSpectral Theory58J28, 19L64 (Primary); 58J22, 19K56, 58J40 (Secondary)text