Engel elements in the homotopy Lie algebra

Separated Lie Models and the Homotopy Lie Algebra

A simply connected topological space X has homotopy Lie algebra ( X) Q. Following Quillen, there is a connected di erential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X , and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property we call separated. The homo...

Engel-5 Lie Algebras

Free Cell Attachments and the Rational Homotopy Lie Algebra

Given a space X let LX denote its rational homotopy Lie algebra π∗(ΩX) ⊗ Q. A cell attachment f : ∨iSi → X is said to be free if the Lie ideal in LX generated by f is a free Lie algebra. This condition is shown to be general in the following sense. Given a space X with rational cone length N , then X is rationally homotopy equivalent to a space constructed using at most N + 1 free cell attachme...

On the Homotopy Lie Algebra of an Arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra gA is defined to be the Lie algebra of primitives of the Yoneda algebra, ExtA(k, k). Under certain homological assumptions on A and its quadratic closure, we express gA as a semi-direct product of the well-understood holonomy Lie algebra hA with a certain hA-module. This allows us to compute the h...

Invariant elements in the dual Steenrod algebra

‎In this paper‎, ‎we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${mathcal{A}_p}^*$ under the conjugation map $chi$ and give bounds on the dimensions of $(chi-1)({mathcal{A}_p}^*)_d$‎, ‎where $({mathcal{A}_p}^*)_d$ is the dimension of ${mathcal{A}_p}^*$ in degree $d$‎.