J. London Math. Soc. (2) 65 (2002) 757{768 Cf2002 London Mathematical Society DOI: 10.1112/S0024610702003125 A TOPOLOGICAL CRITERION FOR THE EXISTENCE OF

The following theorem is proved: if (M4n+1; g) is a complete Riemannian manifold and M is an oriented hypersurface partitioning M and with non-zero signature, then the spectrum of the Hodge{ deRham Laplacian is [0;1[. This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly L2 but not in L2) with the signature of . In this paper, we obtain the following result. Theorem 0.1. If (M4n+1; g) is a complete Riemannian manifold and M is an oriented hypersurface partitioning M and with non-zero signature, then the spectrum of the Hodge{deRham Laplacian is [0;1[