Geometric formality of rationally elliptic manifolds in small dimensions

TL;DR

This paper classifies certain low-dimensional rationally elliptic manifolds based on their rational cohomology and investigates their geometric formality, providing new classifications and counterexamples.

Contribution

It offers a classification of 5- and 6-dimensional rationally elliptic manifolds with small Betti numbers and examines geometric formality in these cases, including new examples and non-examples.

Findings

Six-dimensional formal rationally elliptic manifolds with second Betti number two are cohomologically equivalent to S^2×CP^2.

An infinite family of biquotients with second Betti number three are shown not to be geometrically formal.

Classification results for low-dimensional rationally elliptic manifolds based on Betti numbers.

Abstract

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rational cohomology $S^2\times {\mathbb C}P^2$. An infinite family of six-dimensional simply connected biquotients whose second Betti number is three, different from Totaro's biquotients, is considered and it is proved that none of biquotient from this family is geometrically formal.