Rational analogs of projective planes (Q395751)

In [\textit{D. Sullivan}, Publ. Math., Inst. Hautes Étud. Sci. 47, 269--331 (1977; Zbl 0374.57002)] Sullivan proved that, ``for spaces with nilpotent homotopy systems, there is a perfect replica in terms of nilpotent minimal dga's of the rational homotopy theory.'' In this paradigm, a natural question is the realization of a rational cohomology algebra, \(H\) (with \(H^1=0\)) together with prescribed Pontryagin classes, by a simply connected, closed, smooth manifold. A first obvious requirement is that \(H\) satisfies an \(n\)-dimensional Poincaré duality (with \(n>5\)). If the dimension \(n\) is different of \(4k\) for some integer \(k\), this realization can be done without any restriction. If \(n=4k\), there is a solution with possibly one singular point. For removing this singular point, there are necessary and sufficient conditions (see Sullivan [loc. cit.] and \textit{J. Barge} [Ann. Sci. Éc. Norm. Supér. (4) 9, 469--501 (1976; Zbl 0348.57016)] involving Pontryagin numbers (they must be integers and satisfy a Riemann-Roch relation) and the signature (it has to verify the Hirzebruch signature formula).NEWLINENEWLINEIn the paper under review, the author gives first a variant of the proof of Barge-Sullivan's theorem. The main goal is the application of this realization theorem to cohomology algebras of the form, \({\mathbb Q}[a]/a^3\), with \(a\) of even degree, \(p\). For \(p=2\), 4 or 8, the projective planes of complex, quaternionic or octonionic numbers give solutions. We know also that, for a realization of \({\mathbb Z}[a]/a^3\), they are the only ones, but what is the situation inside the rational homotopy type? The author proves that the next smallest dimension where a rational analog of projective plane exists is for \(p=16\) (i.e., \(n=32\)) and there are infinitely many homeomorphism types of such manifolds. From the realization theorem, the proof consists of resolutions of systems of Diophantine equations.