  We show that for a suitable class of ``Dirac-like'' operators there holds a
Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are
closed Riemannian manifolds of dimension $n\ge 3$ together with such operators,
then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that
the spectrum of its associated operator is close to the disjoint union of the
spectra of the two original operators. As an application, we show that in
dimension $n\equiv 3$ mod 4 harmonic spinors for the Dirac operator of a spin,
$\Spinc$, or $\Spinh$ manifold are not topologically obstructed.

Differential Geometry;High Energy Physics - Theory;Differential Geometry