Spin structures and spectra of Z 2 k -manifolds

Abstract

We give necessary and sufficient conditions for the existence of pin± and spin structures on Riemannian manifolds with holonomy group ℤ2 k . For any n≥4 (resp. n≥6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M 1 , M 2 , non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M 1 admits a pin ± (resp. spin) structure whereas M 2 does not.