  The 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.
