On the omnibus conjecture (Q1971021)

This paper concerns the so-called ``omnibus conjecture'': Suppose \(X\) is a 1-connected finite type CW complex and \(\mathbb{F}_p=\mathbb{Q}\) if \(p=0\), \(\mathbb{F}_p= \mathbb{Z}_{p\mathbb{Z}}\) if \(p\) is odd; if \(\dim H^{\text{pair}}(X; \mathbb{F}_p)< \infty\) and \(H^*_* (\Omega X;\mathbb{F}_p)\) is an elliptic Hopf algebra, then \(H^* (X;\mathbb{F}_p)\) satisfies Poincaré duality. The author proves that the conjecture is true in two cases: when \(X\) is a \(p\)-formal space when \(p\) is an odd, and when \(p=0\) and \(X\) is the toal space of a fibration with basis a product \(\pi K(\mathbb{Q},2n_i)\) and fibre a rational elliptic space.