Homology operations on instantons
The moduli spaces, which arise naturally in the differential geometric formulation of the Yang-Mills theory, have been investigated in recent years by many people, using various techniques from algebraic geometry, complex manifold theory, global analysis and twistor theory. In particular, \textit{M. F. Atiyah} and \textit{J. D. Jones} [Commun. Math. Phys. 61, 97-118 (1978; Zbl 0387.55009)] gave the first homological information about the moduli spaces over the four-sphere for arbitrary instanton number k and they formulated the basic topological questions in this subject. In this paper the authors also investigate the possibility to use some homotopy theoretic techniques to study moduli spaces of instantons. More precisely, the authors show that the disjoint union of these moduli spaces (where the union is taken over all positive instanton numbers k) behaves homologically like a fourfold iterated loop space with associated iterated loop space operations. Then the authors use these operations to obtain new information about the homology of these moduli spaces. The key technical result consists in the proof that the moduli spaces admit loop sum maps *: \({\mathcal M}_ k\times {\mathcal M}_{\ell}\to {\mathcal M}_{k+\ell}\) that agree, up to homotopy, with the standard loop sum map on certain four-fold loop spaces. Then the authors restrict their attention to the case of the Lie group Sp(1) and when the instanton number is 1, and indentify the natural inclusion \({\mathcal M}_ 1\to {\mathcal C}_ 1\) with the classical J-homomorphism. This case has far-reaching consequences as seen in the computations given by the authors. The paper concludes with a list of open questions raised by the authors' computations and suggests directions for further study.
scientific article; zbMATH DE number 4090359