Stable splittings of configuration spaces of surfaces and related mapping spaces.

摘要

In this thesis, we study the stable homotopy theory of mapping spaces whose domains are surfaces. Classical results inextricably link this topic with the study of configuration spaces of surfaces. The main result is a stable splitting of these mapping spaces when the target is a sphere; that is, a wedge decomposition in the stable homotopy category. While this type of result is akin in spirit to the splittings of O nSigmanX due to James (for n = 1) and Snaith (for all n), the method of proof and type of results differ. By specializing to the case of surfaces, we obtain exact information about the homotopy type of the stable summands of the decomposition. These turn out to be constructed from Brown-Gitler spectra using the multiplicative structure of these spectra. As a consequence, we derive a complete calculation of the Steenrod operations on the cohomology of the function spaces of based maps from surfaces to spheres.