Given a principal ideal domain $R$ of characteristic zero, containing 1/2, and a two-cone $X$ of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra $FH(\Omega X;R)$ to be isomorphic with the universal enveloping algebra of some $R$-free graded Lie algebra; as usual, $F$ stands for free part, $H$ for homology, and $\Omega$ for the Moore loop space functor.

Let $\left(𝕃\right(V),d)$ be a free graded connected differential Lie algebra over the field $\mathbb{Q}$ of rational numbers. An ideal $I$ in the Lie algebra $H\left(𝕃\right(V),d)$ is called if, for every cycle $\alpha \in 𝕃\left(V\right)$ such that $\left[\alpha \right]$ belongs to $I$, the kernel of the map $H\left(𝕃\right(V),d)\to H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$, $d\left(x\right)=\alpha$, is contained in $I$. We show that the center of $H\left(𝕃\right(V),d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$. We apply this computation for the determination of the rational homotopy Lie algebra $L_X={\pi }_{*}\left(\Omega X\right)\otimes \mathbb{Q}$ of a simply connected space $X$. We deduce...

Let $A={⨁}_k{A}_k$ and $B={⨁}_k{B}_k$ be graded Lie algebras whose grading is in $𝒵$ or ${𝒵}_2$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha$ and $\beta$ satisfy equations which look “derivatively knitted"; then $A\oplus B:={⨁}_{k,l}(A_k\oplus B_l)$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A{\oplus }_{(\alpha ,\beta )}B$. This graded Lie algebra is called the knit product of $A$ and $B$. The author investigates the general situation for any graded Lie subalgebras $A$ and...

We formulate first results of our larger project based on first fixing some easily accessible invariants of topological spaces (typically the cup product structure in low dimensions) and then studying the variations of more complex invariants such as ${\pi }_{*}\Omega S$ (the homotopy Lie algebra) or ${gr}^{*}{\pi }_{1}S$ (the graded Lie algebra associated to the lower central series of the fundamental group). We prove basic rigidity results and give also an application in low-dimensional topology.

Let M be a closed orientable manifold of dimension and ${𝒞}^{*}\left(M\right)$ be the usual cochain algebra on M with coefficients in a field. The Hochschild cohomology of M, $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is a graded commutative and associative algebra. The augmentation map $\epsilon :{𝒞}^{*}\left(M\right)\to 𝑘$ induces a morphism of algebras $I:H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))\to H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$, which is in general quite small. The algebra $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is expected to...