Tautological rings of fake quaternionic spaces

Abstract The tautological ring $$R^*(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 introduce a new method to obtain upper bounds on the Krull dimension of $$R^*(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for manifolds homotopy equivalent to a fixed, simply connected Poincaré duality space by using recent progress in rational homotopy theory and the family signature theorem. In particular, we show that the Krull dimension of the tautological ring vanishes for almost all manifolds homotopy equivalent to $$\mathbb {H}P^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> .