Unbounded Nilpotency Gaps on Rationally Elliptic Manifolds

Abstract

We construct simply connected, closed smooth manifolds whose rational homotopy types are elliptic and whose homology nilpotency exceeds rational category by arbitrarily large amounts. This separates short models from short quotients of minimal models. The construction starts from an explicit family of minimal Sullivan algebras with finite-dimensional cohomology. A finite quotient establishes Poincaré duality and gives an upper bound for homology nilpotency, while a closed orientation functional computes rational category. Two forced primitives yield the lower bound. An even regrading separates the relevant weights, extending this obstruction to every graded acyclic differential ideal. Rational surgery realizes the resulting models in their formal dimensions, with vanishing rational Pontryagin classes and arbitrarily large rational connectivity. The ratio of homology nilpotency to rational category tends to four thirds. Finally, under an external spherical stabilization statement, attaching a single cone can increase homology nilpotency by an arbitrarily large amount while leaving rational category unchanged.