  We prove an existence theorem for gauge invariant $L^2$-normal neighborhoods
of the reduction loci in the space ${\cal A}_a(E)$ of oriented connections on a
fixed Hermitian 2-bundle $E$. We use this to obtain results on the topology of
the moduli space ${\cal B}_a(E)$ of (non-necessarily irreducible) oriented
connections, and to study the Donaldson $\mu$-classes globally around the
reduction loci. In this part of the article we use essentially the concept of
harmonic section in a sphere bundle with respect to an Euclidean connection.
  Second, we concentrate on moduli spaces of instantons on definite 4-manifolds
with arbitrary first Betti number. We prove strong generic regularity results
which imply (for bundles with "odd" first Chern class) the existence of a
connected, dense open set of "good" metrics for which all the reductions in the
Uhlenbeck compactification of the moduli space are simultaneously regular.
These results can be used to define new Donaldson type invariants for definite
4-manifolds. The idea behind this construction is to notice that, for a good
metric $g$, the geometry of the instanton moduli spaces around the reduction
loci is always the same, independently of the choice of $g$. The connectedness
of the space of good metrics is important, in order to prove that no
wall-crossing phenomena (jumps of invariants) occur. Moreover, we notice that,
for low instanton numbers, the corresponding moduli spaces are a priori compact
and contain no reductions at all so, in these cases, the existence of
well-defined Donaldson type invariants is obvious. The natural question is to
decide whether these new Donaldson type invariants yield essentially new
differential topological information on the base manifold have, or have a
purely topological nature.
