Lower central series quotients in terms of (co)homology

Question

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this answer

Denote by $(G_r)$ the lower central series of $G:G_0=G,G_{r+1}=[G,G_r]$. Can further quotients $G_r/G_{r+1}$ be expressed in terms of group (co)homology?

Moreover, if $G=\pi_1(X)$ where $X$ is a compact algebraic variety, quotients $G_r/G_{r+1}$ possess a Hodge structure introduced by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?

Answer 1

Regarding your question on Hodge structures on $\pi_1(X)$, what we know is that:

• for any topological space $X$ there exists a complex $A^*(X)$ equipped with a structure of commutative differential graded algebra that computes the singular cohomology of $X$: $$H^*(A^*(X))\cong H^*(X,\mathbb{Q})$$

• in rational homotopy theory "à la Sullivan" this complex determines when $X$ is nilpotent and of finite type the rational homtopy type of $X$ i.e. we can compute the rational homotopy groups of $X$ from $A^*(X)$. Otherwise in the non-nilpotent case we only get information on the lower central series of $\pi_1(X)$.

• when $X$ is complex algebraic variety (possibly singular and non-proper) we can put a mixed Hodge structure on $A^*(X)$ that induces the MHS on $H^*(X,\mathbb{Q})$ as described by Deligne in his foundational papers.

• we can extract from this Hodge structure on $A^*(X)$ a MHS on the lower central series of $\pi_1(X)$ and when $X$ is nilpotent of finite type we get a MHS on the higher homtopy groups of $X$.

• in general the MHS on $H^*(X,\mathbb{Q})$ does not determine the MHS on $\pi_1(X)$, you need to go to the model $A^*(X)$ together with its MHS. However when $X$ is said formal that is when $A^*(X)$ is quasi-isomorphic to $H^*(X,\mathbb{Q})$ as a commutative differential graded algebra, this MHS is determined by the MHS on $H^*(X,\mathbb{Q})$.

• Examples of formal complex algebraic varieties are given by smooth and proper ones.

Answer 2

I can provide the following results relating to the question about lower central series quotients and group cohomology.

Denote $gr(G;\mathbb{Q}):=\oplus_{i\geq 0}G_i/G_{i+1}\otimes_{\mathbb{Z}} \mathbb{Q}$ the associated graded Lie algebra of $G$.

In some specially cases, there is a close relationship between the lower central series quotients $G_r/G_{r+1}$ (away from torsion) and the group cohomology: $$H^*(G;\mathbb{Q})^!\cong Ext^*_U(\mathbb{Q},\mathbb{Q}).$$ Here, $H^*(G;\mathbb{Q})^!$ is the quadratic dual, and $U$ is the universal enveloping algebra of $gr(G;\mathbb{Q})$. For example, when $G$ is the classical pure braid groups $P_n$, the above isomorphism is valid. Another family of examples is the virtual pure braid groups $vP_n$ and their subgroups $vP_n^+$, which are also known as the quasitriangular group $QTr_n$ and triangular group $Tr_n$. See Paper of Bartholdi- Enriquez- Etingof- Rains, arXiv:math/0509661v6. In order to have the isomorphism, the following two conditions are necessary: the group $G$ is graded-formal(gr(G;\mathbb){Q}) is quadratic) and the cohomology algebra $H^*(G;\mathbb{Q})$ is Koszul.

Partial relation between the lower central series quotients and Massey products can be found in paper of Fenn and Sjerve: ''Massey products and lower central series of free groups''.

Relating question see: Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.

Answer 3

Lower central quotients can be extracted from group homology via spectral sequence built up from free simplicial resolution of a group. So, if your complex variety is aspherical, you probably know those Hodge structures because everything is as natural and functorial as it can be.

By a classical result of Magnus, for free groups we have isomorphism between free Lie ring on abelianisation $\mathcal LF_{ab} = \mathcal LH_1(F, \mathbb Z)$ and Magnus Lie ring $LG := \bigoplus L_nF = \bigoplus \gamma_i(F)/\gamma_{i+1}(F)$. In general, this morphism is just epi. Now take free simplicial resolution $F_{\bullet} \twoheadrightarrow G$. We can look at exact couple defined by exact sequences $L_nF \to F/\gamma_{n+1}F \to F/\gamma_{n}F$ and associate with it graded Lie ring spectral sequence converging to $L_n(\pi_0 (F)) = LG$. First sheet is given by $E^1_{p, q} = \pi_q (L_p F)$, and $s$-th differentials have degree $(s, -1)$. Actually, $E^1$ differs from free graded Lie ring on group homology only by torsion (we can check it introducing analogous s. s. for augmentation powers filtration on $\mathbb ZF$ and looking at morphism between them induced by $G \hookrightarrow \mathbb Z G$; rationally it's split injection by PBW, and it's known that $LG \otimes Q \cong \Delta_{\mathbb Q}G$), zeroth row is always free Lie on $G_{ab}$, first column is the shifted by 1 integral homology of $G$, and stripes below $k$ depend only on $H_{\leq k + 1}$.

Also, we instantly prove Stallings' result about maps inducing isomorphism on factors by $\gamma_i$ (for $G \xrightarrow{f} G'$ $G$ is para-$G'$ iff $H_1(f)$ iso and $H_2(f)$ is epi) just by checking differentials degree. Expression for $\gamma_1/\gamma_2$ comes from boundary morphism in this s. s.

(I don't remember reference for that stuff, but it's not hard to convert those speculations to actual proof. See also Cochran&Harvey, http://arxiv.org/pdf/math/0407203.pdf and Ellis, "Magnus-Witt type isomorphism for non-free groups".)