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We derive a formula for the $\bar\mu$-invariant of a Seifert fibered homologysphere in terms of the eta-invariant of its Dirac operator. As a consequence,we obtain a vanishing result for the index of certain Dirac operators onplumbed 4-manifolds bounding such spheres.

The $${\bar\mu}$$ -invariant of Seifert fibered homology spheres and the Dirac operator