Equivariant Floer theory and gluing Donaldson polynomials
Let \({\mathcal M}(P)\) be a moduli space of anti-self-dual connections on an \(SU(2)\)-bundle \(P\) over a closed, oriented, Riemannian 4-manifold \(X=X_1\cup_Y X_2\). Then even if \({\mathcal M}(P)\) consists entirely of irreducible connections and is smooth for this reason, this need not hold for moduli spaces associated to the \(X_i\), \(i=1,2\). Indeed, pasting reducible instantons on \(X_2\) to irreducible ones on \(X_1\) always gives irreducibles on \(X\). The target of the authors of this paper is to deal with reducible instantons on \(X_2\) through the use of equivariant cohomology. This means creating Floer groups for more general 3-manifolds \(Y\), relative Donaldson invariants for 4-manifolds \(X_i\) cobounding \(Y\) and a pairing on the groups which recovers the invariant for the closed 4-manifolds \(X\). The authors discuss finite dimensional techniques from equivariant topology and Morse theory, some of which appear to be novel. They review the relevant gauge theory on cylinders and on 4-manifolds with cylindrical ends, including good classes of perturbations. The equivariant Floer (co)homology groups, their pairing and module structure, which is followed by some examples of these groups for space forms, are defined. It is shown that these groups are independent of the perturbation, provided it is chosen from a natural class. Then the authors show that invariants for closed four-manifolds can be extracted from the invariants for \(X_i\), and they deduce a vanishing theorem for polynomial invariants.
scientific article; zbMATH DE number 863699