  We construct a variant of Floer homology groups and prove a gluing formula
for a variant of Donaldson invariants. As a corollary, the variant of Donaldson
invariants is non-trivial for connected sums of 4-manifolds which satisfy a
condition for Donaldson invariants. We also show a non-existence result of
compact, spin 4-manifolds with boundary some homology 3-spheres and with
certain intersection forms.
