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We prove an existence theorem for gauge invariant $L^2$-normal neighborhoodsof the reduction loci in the space ${\cal A}_a(E)$ of oriented connections on afixed Hermitian 2-bundle $E$. We use this to obtain results on the topology ofthe moduli space ${\cal B}_a(E)$ of (non-necessarily irreducible) orientedconnections, and to study the Donaldson $\mu$-classes globally around thereduction loci. In this part of the article we use essentially the concept ofharmonic section in a sphere bundle with respect to an Euclidean connection.  Second, we concentrate on moduli spaces of instantons on definite 4-manifoldswith arbitrary first Betti number. We prove strong generic regularity resultswhich imply (for bundles with "odd" first Chern class) the existence of aconnected, dense open set of "good" metrics for which all the reductions in theUhlenbeck compactification of the moduli space are simultaneously regular.These results can be used to define new Donaldson type invariants for definite4-manifolds. The idea behind this construction is to notice that, for a goodmetric $g$, the geometry of the instanton moduli spaces around the reductionloci is always the same, independently of the choice of $g$. The connectednessof the space of good metrics is important, in order to prove that nowall-crossing phenomena (jumps of invariants) occur. Moreover, we notice that,for low instanton numbers, the corresponding moduli spaces are a priori compactand contain no reductions at all so, in these cases, the existence ofwell-defined Donaldson type invariants is obvious. The natural question is todecide whether these new Donaldson type invariants yield essentially newdifferential topological information on the base manifold have, or have apurely topological nature.

Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds