LOCALIZATION AND HOMOLOGICAL STABILITY OF CONFIGURATION SPACES

Abstract

In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW11] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold.