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 $\left(𝕃\right(V),d)$ 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...

Given a principal ideal domain $R$ of characteristic zero, containing $1/2$, and a connected differential non-negatively graded free finite type $R$-module $V$, we prove that the natural arrow $𝕃FH\left(V\right)\to FH𝕃\left(V\right)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow $UFH𝕃\left(V\right)\to FHU𝕃\left(V\right)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $𝕃$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also...

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...