Homology of moduli spaces of instantons on ALE spaces. I
In earlier works the author and P. B. Kronheimer introduced a new class of hyper-Kähler manifolds which arise as moduli spaces of anti-self- dual connections on a certain class of 4-dimensional noncompact manifolds, the ALE spaces. An ALE space is diffeomorphic to the minimal resolution of the simple singularity \(C^ 2/ \Gamma\) for a finite subgroup \(\Gamma\) of SU(2). The author in a previous paper studied the geometry of the moduli space and showed that, under a topological condition on the vector bundle, its middle cohomology group is isomorphic to a weight space of an irreducible finite-dimensional representation of a simple Lie algebra. The author's goal in this paper is to compute the homology of the moduli spaces by using the moment map for the \(S^ 1\)-action as a Morse function. In this paper the case where \(\Gamma\) is a cyclic group (i.e., the base ALE space is of type \(A_ n\)) is studied. A main result in this paper is the following: If \(\mathfrak M\) is the moduli space of anti-self- dual connections on a vector bundle over the ALE space of type \(A_ n\), and if the hyper-Kähler metric on the moduli space is complete, then the space has a nondegenerate Morse function \(F\) which has only critical points of even index. In particular, the homology of \(\mathfrak M\) has no torsion and vanishes in odd degrees, and every component of \(\mathfrak M\) is simply connected.
scientific article; zbMATH DE number 663767