Tautological rings of fake quaternionic spaces

Abstract

The tautological ring $R^{\ast }(M)$ of a smooth manifold $M$ is the ring of characteristic classes generated by the Miller-Morita-Mumford classes, and is often more accessible than the ring of all characteristic classes of smooth $M$-bundles. In this paper, we show that the Krull dimension of the tautological ring vanishes for almost all manifolds homotopy equivalent to $\mathbb{H}{P}^2$ through a combination of new methods in rational homotopy theory developed by Alexander Berglund and the family signature theorem.