A stable framework for proper and exterior homotopy and cohomology theories

Abstract

We develop a stable framework for exterior homotopy theory and apply it to proper homotopy theory. Starting from a known simplicial model structure on the category of exterior spaces we then pass to the proper, cellular model category of retractive exterior spaces over the ray, and establish the associated stable homotopy theory of exterior spectra. The resulting framework provides exterior analogues of several classical constructions in stable homotopy theory and sheds new light on proper homotopy theory, where the development of a satisfactory stable theory has been hindered by the lack of suitable categorical properties. As an application, we establish a Brown representability theorem for reduced exterior cohomology theories. This yields representability results for proper cohomology theories arising from the exterior setting, including cohomology theories with compact supports associated with classical generalized cohomology theories.