The weak form of Hirzebruch’s prize question via rational surgery

Abstract

We present a relatively elementary construction of a spin manifold with vanishing first rational Pontryagin class satisfying the conditions of Hirzebruch’s prize question, using a modification of Sullivan’s theorem for the realization of rational homotopy types by closed smooth manifolds. As such this is an alternative to the solutions of the problem given by Hopkins–Mahowald, though without the guarantee of the constructed manifold admitting a string structure. We present a particular solution which is rationally 7 connected with eighth Betti number equal to one; our approach yields many other solutions with complete knowledge of their rational homotopy type.