The Aˆ-genus and symmetry of the Dirac spectrum on Riemannian product manifolds

It is well known [M.F. Atiyah, V.K. Patodi, I.M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975) 43–69] that the spectrum of the Dirac operator on a closed Riemannian spin manifold MnMn of dimension n,n≢3mod4, is always symmetric (with respect to zero). In the paper we prove, over odd-dimensional Riemannian product manifolds M12p×M22q+1 with 2p+2q+1≡3mod4, that if the Aˆ-genus of M12p vanishes, Aˆ(M1)=0, then the Dirac spectrum of M12p×M22q+1 is symmetric. We point out that if the Dirac spectrum of M22q+1 is not symmetric, then Aˆ(M1)=0 is a necessary condition for the spectral symmetry on M12p×M22q+1.