The Omnibus Conjecture---disproved

TL;DR

This paper disproves the Omnibus Conjecture in Rational Homotopy Theory by providing counter-examples, showing that certain spaces can have infinite dimensional rational cohomology despite finite dimensional conditions.

Contribution

It presents counter-examples to the longstanding conjecture and explores related dual versions and special cases, advancing understanding in rational homotopy theory.

Findings

Counter-examples to the Omnibus Conjecture

Spaces with finite dimensional even-degree rational cohomology and spherical rational homology can have infinite dimensional rational cohomology

Discussion of dual versions and special cases of the conjecture

Abstract

We provide various counter-examples to the long-standing so-called "Omnibus Conjecture" in Rational Homotopy Theory. That is, we show that a space with finite dimensional even-degree rational cohomology and finite dimensional spherical rational homology may indeed have infinite dimensional rational cohomology. Moreover, we also discuss "dual" versions and special cases of the conjecture.