Cohomogeneity one manifolds with singly generated rational cohomology

Abstract

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.We denote by QP n k any smooth, simply connected, closed manifold whose rational cohomology is isomorphic to the truncated polynomial algebra Q[x]/(x n+1 ) where the generator x has degree k. Note that such a manifold is a rational sphere or point if k is odd by the graded commutativity of the cup product. If k is even, then a QP n k has even dimension kn and positive Euler characteristic n + 1.Prototypical examples are simply connected, closed manifolds with the rational cohomology (equivalently, rational homotopy type) of a compact rank one symmetric space: a rational sphere is a QP 1 k , a rational CP n is a QP n 2 , a rational HP n is a QP n 4 , and a rational Cayley plane is a QP 2 8 . We call the parameters n and k standard if they correspond to a rank one symmetric space. The classification of parameters (n, k) for which a QP n k exists is reduced by way of the Barge-Sullivan rational realization theorem to a number theoretic problem (see Su [Su14]), but this problem is hard and there is no classification (cf. [FS16,KS]).This paper is motivated in part by the attempt to find highly symmetric models for QP n k with non-standard parameters. It is known that no such manifold admits a homogeneous or biquotient structure (see and Totaro [Tot02]). Our first theorem is a similar, negative result:

Manifolds that admit a double disk-bundle decomposition

“…In this way, it is a generalization of Lemma 6.3 of [29]. See [10,Prop. 2.7] for another related statement under different hypotheses.…”