The author develops a completely new procedure to address the problem of studying the transversality properties of the compactification of the moduli space \(\overline{\mathcal M}\) of solutions of a nonlinear elliptic PDE \(F=0\) defined on a manifold \(X\). This new procedure is based on a notion of a virtual neighborhood of \(F\) which can be briefly explained as follows. Let \(({\mathcal B}, {\mathcal F}, F)\) be a triple, where \({\mathcal B}\) is a Banach manifold, \({\mathcal F}\) is a Banach bundle. Then an elliptic equation \(F\) is a section of a Banach bundle \(\mathcal F\). Assuming that \({\mathcal M}_F=F^{-1}(0)\) is compact (this assumption is false in general, but often it is true after a patching construction) one can construct an open finitely dimensional manifold \(U\) containing \(\overline{\mathcal M}\) and a bundle \(E\) over \(U\) with a section \(S\). Furthermore, \(S^{-1}(0)=\overline{\mathcal M}\). The author calls \((U,E,S)\) a virtual neighborhood of \(F\). The notion of a virtual neighborhood allows to reduce a difficult infinite dimensional problem \(({\mathcal B},{\mathcal F},F)\) to a classical finite dimensional problem \((U,E,S)\). Then, one can solve the infinite dimensional problem provided one can solve the corresponding finite dimensional problem. In this paper the author systematically uses a virtual neighborhood technique to study the resulting topological invariants in a variety of different contexts.
For the entire collection see [Zbl 0906.00020].