Homology Nilpotency in Rational Homotopy Theory: Cell Attachments, Retractive Towers, Divergence, and Stabilization

Abstract

Homology nilpotency of a simply connected minimal Sullivan algebra $M$ is the least integer $n$ for which $(M^+)^{n+1}$ is contained in an acyclic differential ideal. We develop lower- and upper-bound criteria, including methods for rational cell attachments, and construct an explicit finite-type minimal Sullivan algebra $M$ satisfying $nil_h(M)=3<Hnil(M)=4$. Complete strictly coordinated retractive towers identify the obstruction: they characterize homotopical nil-length while exposing the additional quotient defect measured by homology nilpotency. We then show that this rigid defect disappears after stabilization by degreewise finite wedges of simply connected rational spheres: $$ Hnil_s(X)=cat_o(X) $$ for every simply connected rational space $X$ of finite type.