such that the $r$-th differential of the above weight spectral sequence agrees
with $\mu_{r}$ as written in (11). For example, this is always possible when
$A$ is a field. More generally we have the following.

######  Proposition 4.13.

Let $(Y,*)$ be a finite $2$-complex with a single $0$-cell $*$, so that
$\mathrm{H}^{1}(Y;A)\hookrightarrow C^{1}(Y;A)$ is the submodule of closed
cochains. If this inclusion is split, then global Massey products in (12) are
defined and compute the differentials of the weight spectral sequence via
(11).

When $A=\mathbb{Z}$, the same holds even when $(Y,*)$ is countable and
$\mathrm{H}^{1}(Y;A)$ and $\mathrm{H}^{2}(Y;A)$ are countably generated. When
$A$ is a field, global Massey products exist with no hypotheses.

###### Proof.

Under the hypotheses, the complex $C^{*}(Y;A)$ is a complex of free
$A$-modules $C^{0}\xrightarrow{0}C^{1}\xrightarrow{d}C^{2}$ with $C^{0}=A\cdot
1$. There is already a projection $q\colon
C^{2}\twoheadrightarrow\mathrm{coker}(d)\cong\mathrm{H}^{2}$, so that a
splitting $p\colon C^{1}\to\mathrm{H}^{1}$ defines a quasi-isomorphism

$\textstyle{\big{(}A\cdot
1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{\phantom{\big{(}}C^{1}\phantom{\big{)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{p}$$\textstyle{C^{2}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\big{(}A\cdot
1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{\phantom{\big{(}}\mathrm{H}^{1}\phantom{\big{)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{\mathrm{H}^{2}\big{)}.}$

Since $C^{*}$ is a bounded-below complex of projectives, Theorem 2 of [Pet20]
and its following paragraph show that the cup product structure on $C^{*}$ can
be transferred to a square-zero coderivation on $Bar(\mathrm{H}^{*}(Y;A))$
making the induced map

$Bar(C^{*}(Y;A))\to Bar(\mathrm{H}^{*}(Y;A))$

into a quasi-isomorphism. Such a coderivation in degree $0$ has components
$\mu_{r}\colon\mathrm{H}^{1}(Y;A)^{\otimes r}\to\mathrm{H}^{2}(Y;A)$, extended
to $Bar^{0}(\mathrm{H}^{*}(Y;A))$ via the Leibnitz rule. Now, the two weight
spectral sequences are equal from their $E_{1}$-pages and on, but with
differentials of $Bar(\mathrm{H}^{*}(Y;A))$ restricted from (11), as claimed.

When $A=\mathbb{Z}$, countable generation of $\mathrm{H}^{1}$ and
$\mathrm{H}^{2}$ implies that there exists a quasi-isomorphic dg subalgebra
$\tilde{C}^{*}\leq C^{*}$ that is countably generated. Since every
$C^{i}\cong\mathbb{Z}^{\mathbb{N}}$ is a Baer–Specker group, Specker’s theorem
[Spe50] shows that the countably generated subgroup $\tilde{C}^{i}$ is free.
From this point, the same argument proceeds as above.

When $A$ is a field, every module is free and every injection is split, so the
above argument applies without further hypotheses. ∎

The weight spectral sequence gives an inductive process for calculating
invariants of high weight: if one has calculated all weight $<n$ braiding
invariants, then to get all invariants of weight $n$ it takes only to find
those tensor in $\mathrm{H}^{1}(\Gamma;A)^{\otimes n}$ with vanishing cup and
Massey products. Every such tensor extends to a weight $n$ braiding invariant,
and while the extension is not unique, any two extensions differ by a lower
weight invariant (which has already been found).

######  Example 4.14 (Surface groups).

Let $\Sigma_{g}$ be a closed orientable surface of genus $g$, with fundamental
group $\Gamma\cong\langle a_{1},\ldots,a_{g},b_{1},\ldots
b_{g}\mid[a_{1},b_{1}]\cdots[a_{g},b_{g}]\rangle$. Since
$H^{1}(\Sigma_{g};\mathbb{Z})=\mathbb{Z}^{2g}$ with the symplectic pairing
$\smile\colon\mathrm{H}^{1}\otimes\mathrm{H}^{1}\to\mathrm{H}^{2}\cong\mathbb{Z}$,
letter-braiding invariants of words in the surface group, in low weights, are
linear combinations of:

  * •

$A_{1},\ldots,A_{g},B_{1},\ldots,B_{g}\in H^{1}(\Sigma_{g};\mathbb{Z})$
counting signed occurrences of $a_{i}$ and $b_{i}$. These detect the image of
a word in the abelianization.

  * •

$A_{i}|A_{j},B_{i}|B_{j}$ for all $(i,j)$ and $A_{i}|B_{j},B_{i}|A_{j}$ for
all $i\neq j$. These count pairs of generators in a particular order, e.g.
$A_{i}|A_{j}$ is the signed count of times $a_{i}$ appears _before_ $a_{j}$.
They detect and distinguish between commutators $[a_{i},a_{j}],[b_{i},b_{j}]$
and $[a_{i},b_{j}]$.

  * •

$(A_{i}|B_{i}-A_{1}|B_{1})$ and $(B_{i}|A_{i}+A_{1}|B_{1})$ for all
$i=1,\ldots,g$. These linear combinations of counts of ordered pairs of
generators are unchanged by application of the surface relation, while the
individual count of pairs $A_{i}|B_{i}$ are not invariant.

  * •

$(A_{i}|B_{i}|A_{i})-(A_{1}|B_{1}|A_{i})-(A_{i}|B_{1}|A_{1})$ for all
$i=1,\ldots,g$ are invariants on double-commutators, detecting e.g. that
$[[a_{i},b_{i}],a_{i}]\neq 1$.

We give a full treatment across all weights in [GOSW23].

###  4.2. Properties of letter-braiding

Formal properties of letter-braiding polynomials proved in §3 continue to hold
for general groups. Most useful is the following.

######  Lemma 4.15 (Push-pull formula).

Let $h\colon\Gamma_{1}\to\Gamma_{2}$ be a group homomorphism, then for every
$w\in\Gamma_{1}$ and $T\in\mathrm{H}^{0}_{Bar}(\Gamma_{2};A)$,

(13) $L_{h^{*}(T)}(w)=L_{T}(h_{*}(w)\quad\text{ and
}\quad\ell_{h^{*}(T)}(w)=\ell_{T}(h(w)).$

###### Proof.

This is a direct application of Lemma 3.2 to the composition
$S^{1}\xrightarrow{w}B\Gamma_{1}\xrightarrow{h}B\Gamma_{2}$. ∎

In particular, the push-pull formula immediately reveals behavior with respect
to products.

######  Corollary 4.16 (Product formula).

Suppose $w_{1},w_{2}\in\Gamma$ are two elements, then

$\ell_{\bullet}({w_{1}\cdot
w_{2}})=\ell_{\bullet}({w_{1}})+\ell_{\bullet}({w_{2}})+(\ell_{\bullet}({w_{1}})\otimes\ell_{\bullet}({w_{2}}))\circ\overline{\Delta}.$

###### Proof.

Let $h\colon F_{\\{e,f\\}}\to\Gamma$ send $h(e)=w_{1}$ and $h(f)=w_{2}$. By
push-pull and Corollary 3.28,

$\ell_{T}(w_{1}w_{2})=\ell_{h^{*}(T)}(ef)=\ell_{h^{*}(T)}(e)+\ell_{h^{*}(T)}(f)+(\ell_{\bullet}(e)\otimes\ell_{\bullet}(f))_{\overline{\Delta}(h^{*}(T))}$

and since $h^{*}$ is a coalgebra homomorphism,
$\overline{\Delta}(h^{*}(T))=(h^{*}\otimes h^{*})\circ\overline{\Delta}(T)$.
Thus,

$\ell_{T}(w_{1}w_{2})=\ell_{h^{*}(T)}(e)+\ell_{h^{*}(T)}(f)+(\ell_{h^{*}(\bullet)}(e)\otimes\ell_{h^{*}(\bullet)}(f))_{\overline{\Delta}(T)},$

which proves the claim by another application of the push-pull formula. ∎

######  Corollary 4.17.

If $w\in\Gamma$ belongs to the $k$-th term of the lower central series,
$\gamma_{k}\Gamma$, and $T\in H^{0}(\Gamma;A)$ has weight $\leq n$, then

$\deg L_{T}(w)\leq\frac{n}{k}.$

In particular, $\ell_{T}(w)=0$ whenever $n<k$.

######  Definition 4.18.

A set of elements $s_{1},\ldots,s_{r}\in\Gamma$ determines a group
homomorphism $F_{r}\to\Gamma$ by $e_{i}\mapsto s_{i}$. Given a word $w\in
F_{r}$, we denote its image in $\Gamma$ by $w(s_{1},\ldots,s_{r})$. An element
of this form will be called _a word_ in $s_{1},\ldots,s_{r}$.

###### Proof of Corollary 4.17.

Suppose $w$ can be written as a $k$-fold commutator in the elements
$s_{1},\ldots,s_{r}\in\Gamma$. That is, there exists a word
$\tilde{w}\in\gamma_{k}F_{r}$ such that $\tilde{w}(s_{1},\ldots,s_{r})=w$.
Denote $h\colon F_{r}\to\Gamma$ the homomorphism with $e_{i}\mapsto s_{i}$, so
by push-pull

$L_{T}(w)=L_{h^{*}(T)}(\tilde{w}).$

But since $h^{*}$ preserves the weight filtration, the weight of $h^{*}(T)$ is
also $\leq n$, and by 4.20 it follows that $\deg L_{h^{*}(T)}(\tilde{w})\leq
n/k$. ∎

Since we already know how to calculate braiding numbers on free groups, it
remains to understand pullbacks
$h^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to\mathrm{H}^{0}_{Bar}(F_{S};A)$
for homomorphisms $h\colon F_{S}\to\Gamma$.

######  Theorem 4.19 (Composition formula).

Suppose $h\colon F_{S}\to\Gamma$ is a group homomorphism from a free group.
Then the induced coalgebra homomorphism

$h^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to\mathrm{H}^{0}_{Bar}(F_{S};A)\cong
T(A^{S})$

is determined by its weight $1$ component $\ell
h^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to A^{S}$, which is

$\ell h^{*}(T)\colon(s\mapsto\ell_{T}(h(s))).$

It follows that the letter-braiding numbers of a word
$w(s_{1},\ldots,s_{r})\in\Gamma$ are

$\ell_{T}(w(s_{1},\ldots,s_{r}))=\sum_{k\geq
1}\sum_{i_{1},\ldots,i_{k}}\ell_{S_{i_{1}}|\ldots|S_{i_{k}}}(w)\cdot(\ell_{\bullet}({h(s_{i_{1}})})\otimes\cdots\otimes\ell_{\bullet}({h(s_{i_{k}})}))_{\overline{\Delta}^{k-1}(T)}.$

###### Proof.

Theorem 3.8 above is the special case when $\Gamma$ is a free group, and the
same proof holds in the present generality after replacing $X_{S_{2}}$ by
$B\Gamma$. ∎

Application of the composition formula requires complete understanding of
$\ell_{\bullet}({h(s)})$ only for a particular set of generators. In the
following setup, this is obtained directly by weight reduction on leading
terms, without dealing with corrections coming from Massey products.

######  Corollary 4.20.

Consider $\Gamma=\pi_{1}(Y,*)$ and suppose
$w(s_{1},\ldots,s_{d})\in\gamma_{n}\Gamma$ is some $n$-fold commutator,
presented by word $w\in\gamma_{n}F_{S}$. Then on weight $\leq n$ invariants,
the letter-braiding number
$\ell_{\bullet}({w(s_{1},\ldots,s_{d})})\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to
A$ is computed via weight reduction of leading terms

$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
n}\xrightarrow{\operatorname{lead}}\mathrm{H}^{1}(\Gamma;A)^{\otimes
n}\xrightarrow{(\operatorname{res}|_{S}){}^{\otimes n}}(A^{S})^{\otimes
n}\xrightarrow{\ell(w)}A,$

where $\operatorname{res}|_{S}\colon\mathrm{H}^{1}(\Gamma;A)\to A^{S}$ is the
restriction of homomorphisms
$\mathrm{H}^{1}(\Gamma;A)\cong\text{Hom}_{grp}(\Gamma,A)$ to
$S\subseteq\Gamma$.

###### Proof.

By naturality of the weight spectral sequence in Lemma 4.11, there is a
commutative diagram

$\textstyle{\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}}$$\textstyle{\mathrm{H}^{1}(\Gamma;A)^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res|_{S}{}^{\otimes
n}}}$$\textstyle{\mathrm{H}^{0}_{Bar}(F_{S};A)_{\leq
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A^{S})^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ell(w)}$$\textstyle{A.}$

Since $w$ is an $n$-fold commutator, its letter-braiding numbers vanish on
weights $\leq n-1$, and thus the composition of the bottom two arrows
coincides with $\ell_{\bullet}(w)\colon\mathrm{H}^{0}_{Bar}(F_{S};A)\to A$ on
weights $\leq n$. The claim now follows form the commutativity of the diagram,
along with

$\ell_{T}(w(s_{1},\ldots,s_{d}))=\ell_{h^{*}(T)}(w).\qed$

######  Example 4.21 (Pure braid group).

The cohomology ring of Artin’s pure braid group $PB_{n}$ was calculated by
Arnol’d [Arn14] to be

$\mathrm{H}^{*}(PB_{n};A)\cong\frac{\Lambda^{*}_{A}(\omega_{ij}\mid 1\leq
i<j\leq
n)}{(\omega_{ij}\wedge\omega_{jk}+\omega_{jk}\wedge\omega_{ik}+\omega_{ik}\wedge\omega_{ij}\mid
1\leq i<j<k\leq n)},$

where $\omega_{ij}$ is the winding number of the $i$-th stand around the
$j$-th.

######  Corollary 4.22.

The tensors
$T_{ijk}=(\omega_{ij}|\omega_{jk}+\omega_{jk}|\omega_{ik}+\omega_{ik}|\omega_{ij})$
for $1\leq i<j<k\leq n$ define additive invariants on commutator braids
$\ell_{\bullet}(T_{ijk})\colon[PB_{n},PB_{n}]\to A$. They are computed on
basic commutators $[\sigma_{1},\sigma_{2}]$ by

$\ell_{\omega_{1}|\omega_{2}}([\sigma_{1},\sigma_{2}])=\omega_{1}(\sigma_{1})\omega_{2}(\sigma_{2})-\omega_{1}(\sigma_{2})\omega_{2}(\sigma_{1}).$

###### Proof.

This follows immediately from the last corollary and Proposition 3.23. ∎

Arnol’d shows, moreover, that when $A=\mathbb{Q}$ all Massey products vanish.
It follows that cup products are the only obstruction for higher weight
tensors $\mathrm{H}^{1}(PB_{n};\mathbb{Q})^{\otimes n}$ to define invariants.

######  Corollary 4.23.

Denoting $\omega_{ij}$ by $ij$, then for $i<j<k<l$ the sums of tensors

$\begin{pmatrix}ij|ik|ij&ij|ij|jk&ij|jk|ik\\\ jk|ij|ij&&jk|ik|ik\\\
ik|jk|ij&ik|ik|jk&ik|ij|ik\end{pmatrix}\text{and
}\begin{pmatrix}&ij|jl|jk&&&ij|kl|jl&ij|jk|kl\\\
&&jk|il|ik&jk|kl|il&&jk|ik|kl\\\ ik|kl|ij&&&&&ik|ij|kl\\\
il|ik|ij&il|ij|jk&il|jk|ik&&&\\\ &jl|il|jk&&jl|jk|il&&\\\
kl|il|ij&&&kl|jl|il&kl|ij|jl&\end{pmatrix}$

give well-defined additive invariants on double-commutators
$[[PB_{n},PB_{n}],PB_{n}]\to\mathbb{Z}$, where

$\displaystyle\ell_{\omega_{1}|\omega_{2}|\omega_{3}}([[\sigma_{1},\sigma_{2}],\sigma_{3}])=$
$\displaystyle\left[\omega_{1}(\sigma_{1})\omega_{2}(\sigma_{2})-\omega_{1}(\sigma_{2})\omega_{2}(\sigma_{1})\right]\omega_{3}(\sigma_{3})$
$\displaystyle-$
$\displaystyle\omega_{1}(\sigma_{3})\left[\omega_{2}(\sigma_{1})\omega_{3}(\sigma_{2})-\omega_{2}(\sigma_{2})\omega_{3}(\sigma_{1})\right].$

###### Proof.

The two grids of tensors have the property that within a column all tensors
have the same right-most factor, while the other two columns exhibit either
the Arnol’d relation or anti-symmetry of the cup product. It follows that the
contraction of the first two columns using the cup product gives $0$.

At the same time, within every row the left-most factor is the same while the
other two factors exhibit Arnol’d or anti-symmetry. Thus the contraction of
the last two columns also gives $0$. Together each of the two sums defines a
Bar cocycle.

Calculation of the braiding number of a double-commutator follows from the
same formula for the universal word $[[x,y],z]\in F_{3}$ and (5). ∎

As an application, let $\sigma_{ij}$ be the Artin generators of $PB_{n}$,
which satisfy $\omega_{ij}(\sigma_{kl})=\delta_{ik}\delta_{jl}$.

######  Corollary 4.24.

The double commutator braids $[[\sigma_{12},\sigma_{23}],\sigma_{12}]$ and
$[[\sigma_{12},\sigma_{23}],\sigma_{34}]$ are not triple-commutators in any
pure braid group $PB_{n}$ in which they are defined.

###### Proof.

Indeed, the former braid pairs nontrivially with the $3\times 3$ invariant of
Corollary 4.23 with $(i,j,k)=(1,2,3)$, while the latter braid pairs
nontrivially with the $6\times 6$ invariant with $(i,j,k,l)=(1,2,3,4)$. Any
triple commutator would give invariant $0$. ∎

We have computed a weight $4$ invariant that contains $1\cdot(12|23|34|23)$,
along with $80$ other tensor summands, which detects that
$[[\sigma_{12},\sigma_{23}],[\sigma_{23},\sigma_{34}]]\in PB_{n}$ is not a
$4$-fold commutator.

######  Problem 4.25.

Find a basis of leading terms of elements in
$\mathrm{H}^{0}_{Bar}(PB_{n};\mathbb{Q})$, perhaps as cord diagrams, and
determine its coproduct expansion.

Note that over $\mathbb{Z}$ the pure braid groups do have nontrivial Massey
products (see [Sal20]), and thus a-priori
$\mathrm{H}^{0}_{Bar}(PB_{n};\mathbb{Q})$ appears essentially larger than
$\mathrm{H}^{0}_{Bar}(PB_{n};\mathbb{Z})$. However, the above example shows
that the two coalgebras of invariants capture the same information.

######  Proposition 4.26.

Let $K$ be the fraction field of $A$. When $\Gamma$ is a finitely generated
group, the inclusion $\mathrm{H}_{Bar}^{0}(\Gamma;A)\otimes
K\hookrightarrow\mathrm{H}^{0}_{Bar}(\Gamma;K)$ is an isomorphism. Thus Massey
products on $\mathrm{H}^{1}(\Gamma;A)$ that are $A$-torsion do not essentially
restrict the coalgebra of braiding invariants: every invariant defined over
$K$ is realized by one over $A$ up to some scalar multiple.

###### Proof.

Since $\Gamma$ is finitely generated, there exists a simplicial set $Y$ with a
finite $1$-skeleton and $\pi_{1}(Y,*)\cong\Gamma$. It follows that

$C^{*}(Y;A)\otimes K\to C^{*}(Y;K)$

is an isomorphism in degrees $\leq 1$, so this map is $1$-connected. The same
argument as in the proof of Lemma 4.2 now shows that $Bar^{*}(Y;A)\otimes K\to
Bar^{*}(Y;K)$ is $0$-connected.

Since $K$ is flat over $A$, there is an isomorphism
$\mathrm{H}^{0}_{Bar}(Y;A)\otimes K\cong\mathrm{H}^{0}(Bar(Y;A)\otimes K)$. By
Lemma 4.2, the former is isomorphic to $\mathrm{H}^{0}_{Bar}(\Gamma;A)\otimes
K$ , while the latter is $\mathrm{H}^{0}_{Bar}(\Gamma;K)$ using the previous
paragraph. ∎

This comparison isomorphism no longer holds for infinitely generated groups.

######  Example 4.27.

Consider the group $F_{\infty}$ modulo relations
$x_{1}=(x_{2})^{2}=(x_{3})^{3}=\ldots$. Then
$\mathrm{H}^{1}(\Gamma;\mathbb{Z})=0$ but
$\mathrm{H}^{1}(\Gamma;\mathbb{Q})=\mathbb{Q}$. Therefore, letter-braiding
invariants over $\mathbb{Z}$ detect nothing, while over $\mathbb{Q}$ they
distinguish $x_{1}$ from $1$.

##  5\. Group rings

As discussed above, letter-braiding numbers define functionals
$\ell_{T}(\bullet)\colon\Gamma\to A$, for every
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$. These extend $A$-linearly to the group
ring $A[\Gamma]$.

######  Definition 5.1.

The letter-braiding pairing

$\langle-,\bullet\rangle\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\otimes
A[\Gamma]\to A$

is the $A$-bilinear map extending $\langle T,w\rangle=\ell_{T}(w)+\eta(T)$,
where $\eta\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to A$ is the counit. It
further extends to tensor powers as in §2.2.

The various properties proved in §4 are most naturally stated in terms of this
pairing, matching the functorial filtered coalgebra structure of
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$ with the filtered algebra structure of
$A[\Gamma]$, as stated in Theorem 1.1.

######  Remark 5.2 (Diagonal).

Recall that $A[\Gamma]$ is furthermore equipped with a diagonal coproduct
$\Delta(w)=w\otimes w$ for $w\in\Gamma$, making it into a Hopf algebra. In
contrast, the Bar construction of a commutative dg algebra has a shuffle
product, making it into a Hopf algebra as well. Chen [Che71] matches these
structures on $\mathbb{R}[\Gamma]$ and $\mathrm{H}^{0}_{Bar}(\Omega_{dR}(M))$
under his pairing by iterated integrals. However, since our cochain algebras
are rarely commutative, the shuffle product does not commute with $d_{Bar}$
and does not induce a Hopf algebra structure on
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$.

Instead, there is an $E_{\infty}$-algebra structure on cochains, as
instrumental in Mandell’s theory [Man06]. For our low dimensional
applications, only the first layer of this homotopy commutativity structure is
needed: the cup-1 operation, as central to Porter–Suciu work on 1-minimal
models [PS22].

######  Problem 5.3.

Describe the product structure on $\mathrm{H}^{0}_{Bar}(\Gamma;A)$ dual to the
coproduct on $A[\Gamma]$.

For the proof of Theorem 1.1, we first extend the product formula to group
rings.

######  Lemma 5.4.

Let $\epsilon\colon A[\Gamma]\to A$ denote the canonical augmentation
$\epsilon(\sum\lambda_{i}w_{i})=\sum\lambda_{i}$. Then for any $x,y\in
A[\Gamma]$ and $T\in\overline{\mathrm{H}^{0}_{Bar}(\Gamma;A)}$,

$\langle T,xy\rangle=\epsilon(y)\cdot\langle
T,x\rangle+\epsilon(x)\cdot\langle
T,y\rangle+\langle\overline{\Delta}(T),x\otimes y\rangle.$

###### Proof.

Let $x=\sum\lambda_{i}w_{i}$ and $y=\sum\mu_{j}w_{j}$. Then

$\langle
T,xy\rangle=\sum\lambda_{i}\mu_{j}\cdot\ell_{T}(w_{i}w_{j})=\sum\lambda_{i}\mu_{j}\cdot\left(\ell_{T}(w_{i})+\ell_{T}(w_{j})+(\ell_{\bullet}(w_{i})\otimes\ell_{\bullet}({w_{j}}))_{\overline{\Delta}(T)}\right).$

This is exactly $(\sum\mu_{i})\cdot\langle
T,x\rangle+(\sum\lambda_{i})\cdot\langle
T,y\rangle+\langle\overline{\Delta}T,x\otimes y\rangle$, as claimed. ∎

######  Definition 5.5.

For words $w_{0},\ldots,w_{n}\in\Gamma$, let the _multi-evaluation_

$\ell_{\bullet}(w_{0}|\ldots|w_{n})\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\to A$

be the composite
$(\ell_{\bullet}(w_{0})\otimes\ldots\otimes\ell_{\bullet}({w_{n}}))\circ\overline{\Delta}^{n}.$

######  Proposition 5.6.

Let $w_{0},\ldots,w_{n}\in\Gamma$ be any sequence of elements. The multi-
evaluation $\ell_{\bullet}({w_{0}|\ldots|w_{n}})$ satisfies the following.

  1. (1)

It is a linear combination of letter-braiding invariants of products,

$\ell_{\bullet}({w_{0}|\ldots|w_{n}})=\sum_{k=1}^{n}(-1)^{n-k}\sum_{0\leq
i_{1}<\ldots<i_{k}\leq n}\ell_{\bullet}({w_{i_{1}}\cdots w_{i_{k}}}).$

  2. (2)

In terms of the letter-braiding pairing,

$\ell_{T}({w_{0}|\ldots|w_{n}})=\langle
T\;,\;(w_{0}-1)\cdots(w_{n}-1)\rangle.$

  3. (3)

For every $0\leq i<n$,

$(\ell_{\bullet}({w_{0}|\ldots|w_{i}})\otimes\ell_{\bullet}({w_{i+1}|\ldots|w_{n}}))\circ\overline{\Delta}=\ell_{\bullet}({w_{0}|\ldots|w_{n}}).$

###### Proof.

The third statement follows immediately from coassociativity of $\Delta$.
Next, the first and the second equations are evidently equivalent, so let us
prove the latter by induction on $n$. Consider $x=(w_{0}-1)\cdots(w_{n}-1)\in
A[\Gamma]$ and assume that
$\ell_{\bullet}({w_{0}|\ldots|w_{n}})=\langle\bullet,x\rangle$. Using the
product formula for group rings,

$\langle\bullet,{x(w_{n+1}-1)}\rangle=\cancel{\epsilon(w_{n+1}-1)}\cdot\langle\bullet,x\rangle+\cancel{\epsilon(x)}\cdot\langle\bullet,w_{n+1}-1\rangle+\langle\overline{\Delta}(\bullet),x\otimes(w_{n+1}-1)\rangle,$
$=\langle\overline{\Delta}(\bullet),x\otimes w_{n+1}\rangle.$

The last expression is
$(\ell_{\bullet}({w_{0}|\ldots|w_{n}})\otimes\ell_{\bullet}{(w_{n+1})})\circ\overline{\Delta}$
and thus agrees with $\ell_{\bullet}({w_{0}|\ldots|w_{n+1}})$. ∎

######  Remark 5.7.

With Proposition 5.6 we can prove the recursive formula for invariants of
commutators.

###### Proof of Proposition 3.23.

In the group ring $A[F_{S}]$ there is an identity

$\displaystyle[w,s]-1$
$\displaystyle=wsw^{-1}s^{-1}-1=wsw^{-1}s^{-1}(1-s)+wsw^{-1}(1-w)+w(s-1)+(w-1)$
$\displaystyle=(w-1)(s-1)-([w,s]-1)(s-1)-(wsw^{-1}-1)(w-1).$

Therefore, by Proposition 5.6,

$\langle T,[w,s]\rangle=\ell_{T}(w|s)-\ell_{T}([w,s]|s)-\ell_{T}(wsw^{-1}|w).$

We note the following: $\ell_{T^{\prime}}(s)=0$ on any homogeneous tensor of
weight $\geq 2$, and $\ell_{T^{\prime}}([w,s])=0$ on any tensor of weight
$\leq n$. Similarly $\ell_{T^{\prime}}(w)=0$ on tensors of weight $\leq n-1$.
When $T=\alpha_{0}|\ldots|\alpha_{n}$, these observations leave only the terms

$\langle(\alpha_{0}|\ldots|\alpha_{n})\,,\,[w,s]\rangle=\ell_{\alpha_{0}|\ldots|\alpha_{n-1}}(w)\ell_{\alpha_{n}}(s)-\ell_{\alpha_{0}}(wsw^{-1})\ell_{\alpha_{1}|\ldots|\alpha_{n}}(w).$

But since $\ell_{\alpha_{0}}(\bullet)$ is an additive homomorphism, we have
$\ell_{\alpha_{0}}(wsw^{-1})=\ell_{\alpha_{0}}(s)$. ∎

###### Proof of Theorem 1.1.

The push-pull formula (13) shows that the pullback $h^{*}$ along a group
homomorphism is dual to the pushforward $h_{*}$. Now, observing the the
braiding pairing of the unit $1\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is $\langle
1,x\rangle=\epsilon(x)$ for every $x\in A[\Gamma]$, the product formula for
group rings (Lemma 5.4) becomes

$\displaystyle\langle\Delta(T),x\otimes y\rangle$ $\displaystyle=\langle
T\otimes 1+1\otimes T+\overline{\Delta}(T)\,,\,x\otimes y\rangle$
$\displaystyle=\langle T,x\rangle\cdot\epsilon(y)+\epsilon(x)\cdot\langle
T,y\rangle+\langle\overline{\Delta}(T),x\otimes y\rangle=\langle T,xy\rangle.$

Let $I\lhd A[\Gamma]$ denote the augmentation ideal. We prove that weight
$\leq n-1$ invariants vanish on $I^{n}$ by induction on $n$. The fact that the
only weight $0$ invariant up to scaling is $\langle 1,x\rangle=\epsilon(x)$
and $I=\ker(\epsilon)$ gives the base case. For the step, suppose $x\in I^{k}$
and $y\in I^{n-k}$ for $0<k<n$. Then

$\langle T,xy\rangle=\langle\Delta(T),x\otimes y\rangle=\sum_{i}\langle
T_{1}^{i},x\rangle\cdot\langle T_{2}^{i},y\rangle,$

where $\Delta(T)=\sum_{i}T_{1}^{i}\otimes T_{2}^{i}$. But for every $i$ the
weights of $T_{1}^{i}$ and $T_{2}^{i}$ must sum to at most $n-1$, so at either
the first has weight $<k$ or the second has weight $<n-k$. With the induction
hypothesis, it follows that every summand in the above sum vanishes.

Now, let $I\lhd A[\Gamma]$ be the augmentation ideal and let
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ have weight $<n$. we claim that
$\ell_{T}(\bullet)$ vanishes identically on $I^{n}$. By induction on $n$, if
$T$ has weight $0$ then it vanishes on $I$ by definition, since $\langle
1,x\rangle=\epsilon(x)$. Now let $T$ have weight $\leq n-1$ and suppose by
induction that $(I^{n-1})^{\perp}=\mathrm{H}^{0}_{Bar}(\Gamma;A)_{<n-1}$, then
if $x\in I^{n-1}$ and $y\in I$, we have

$\langle T,xy\rangle=\sum_{i}\langle T_{1}^{i},x\rangle\langle
T_{2}^{i},y\rangle$

where $\Delta(T)=\sum_{i}T_{1}^{i}\otimes T_{2}^{i}$. But since for every $i$,
the weights of $T_{1}^{i}$ and $T_{2}^{i}$ sum to $\leq n-1$, one of the
respective evaluations on $x$ and $y$ must vanish. Since $I^{n}$ is generated
by elements of the form $xy$, it follows that $\ell_{T}(\bullet)\equiv 0$ on
$I^{n}$.

Conversely, if $0\neq T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ has weight $n$, then
there exists some $x\in I^{n}$ such that $\langle T,x\rangle\neq 0$. Indeed,
let $S\subseteq\Gamma$ be a generating set, so that the restriction
$\mathrm{H}^{1}(\Gamma;A)\to A^{S}$ is injective, and consider the leading
term

$\operatorname{lead}(T)\in\mathrm{H}^{1}(\Gamma;A)^{\otimes n}.$

On tensor powers, restriction gives an injection
$\mathrm{H}^{1}(\Gamma;A)^{\otimes n}\hookrightarrow(A^{S})^{\otimes n}$, and
by Lemma 2.18 there is a further inclusion $(A^{S})^{\otimes n}\hookrightarrow
A^{S\times\cdots\times S}$. Therefore, if $T\neq 0$ then there exist
generators $(s_{1},\ldots,s_{n})$ such that

$\langle
T\,,\,(s_{1}-1)\cdots(s_{n}-1)\rangle=\langle\overline{\Delta}^{n}(T)\,,\,s_{1}\otimes\ldots\otimes
s_{n}\rangle=\langle\operatorname{lead}(T)\,,\,s_{1}\otimes\ldots\otimes
s_{n}\rangle\neq 0.$

This fact already shows that
$\mathrm{H}^{0}_{Bar}(\Gamma;A)\to\text{Hom}_{A}(A[\Gamma],A)$ is injective.
Completeness of letter-braiding is more difficult and will be the subject of
§6. ∎

######  Example 5.8.

We saw in Example 4.6 that in $F_{\\{e,f\\}}$ subject to the relation $ef=fe$,
all words are already distinguished by $E$ and $F$. Yet
$\mathrm{H}^{0}_{Bar}(\mathbb{Z}\langle e,f\rangle;\mathbb{Z})$ contains
further invariants, such as $E|E$ and $(E|F+F|E)$. Their role is now made
clear: the two elements $ef$ and $e+f$ in $A[\Gamma]$ can not be separated by
neither $E$ nor $F$, while

$\langle(E|F+F|E)\,,\,ef\rangle=1\quad\langle(E|F+F|E)\,,\,e+f\rangle=0.$

To wit, higher weight invariants in $\mathrm{H}^{0}_{Bar}(\Gamma;A)$
distinguish between group ring elements.

###  5.1. Fox calculus and Magnus expansion

For coefficients in $A=\mathbb{Z}$, the letter-braiding number is equivalent
to Fox calculus [Fox53] and dual to the Magnus expansion [Mag54], in the
following sense.

######  Theorem 5.9.

Let $F_{n}$ be the free group generated by $\\{x_{1},\ldots,x_{n}\\}$, and let
$X_{1},\ldots X_{n}\in\mathbb{Z}^{n}$ be the dual basis. Fix $w\in F_{n}$.
Then,

  * •

for every sequence of indices $(i_{1},\ldots i_{k})\in\\{1,\ldots,n\\}^{k}$
there is an equality

$\ell_{X_{i_{1}}|\ldots|X_{i_{k}}}(w)=\epsilon\left(\frac{\partial}{\partial
x_{i_{1}}}\ldots\frac{\partial}{\partial x_{i_{d}}}w\right)$

where $\frac{\partial}{\partial x_{i}}$ is the Fox derivative on
$\mathbb{Z}[F_{n}]$ associated to $x_{i}$, and
$\epsilon\colon\mathbb{Z}[F_{n}]\to\mathbb{Z}$ is the standard augmentation.

  * •

There is an equality of noncommutative power series

$M(w)=1+\sum_{1\leq i_{1},\ldots,i_{k}\leq
n}\ell_{X_{i_{1}}|\ldots|X_{i_{k}}}(w)t_{i_{1}}\cdots t_{i_{k}}$

where $M(-)$ is the Magnus expansion of [Mag54].

###### Proof.

On generators $x_{j}$, we computed in Example 3.6 that

$\ell_{X_{i_{1}}|\ldots|X_{i_{k}}}(x_{j})=\begin{cases}1&\text{ if
}X_{i_{1}}|\ldots|X_{i_{k}}=X_{j}\\\ 0&\mathrm{otherwise}.\end{cases}$

It follows that $\ell_{\bullet}(w)$ agrees with the augmentation of Fox
derivatives on generators of $F_{n}$.

Now, for products $w=u\cdot v$ [Fox53, Equation (3.2)] shows that Fox
derivatives satisfy the recurrence relation as given in Corollary 3.28 for
$\ell_{\bullet}(uv)$. Since the two agree on generators, they must be equal.

Characterization of the coefficients of Magnus’ expansion in terms of the Fox
derivatives is well-known, see e.g. [Fox53, Discussion following (3.4)]. ∎

######  Remark 5.10.

In light of the last theorem, our setup can be understood as an extension of
the Magnus expansion on free groups in the following ways.

  * •

The group $\Gamma$ needs not be free, where now the only well-defined Fox
derivatives are inhomogeneous linear combinations that exhibit vanishing cup
and Massey products.

  * •

The group $\Gamma$ need not be finitely generated.

  * •

The coefficient ring $A$ need not be $\mathbb{Z}$. E.g. it may be a field of
characteristic $p>0$.

  * •

The calculus is independent of a presentation for $\Gamma$. Instead, necessary
input is only a presentation of the cohomology $\mathrm{H}^{1}(\Gamma;A)$ with
its Massey products

$\mu_{n}\colon\mathrm{H}^{1}(\Gamma;A)^{\otimes
n}\to\mathrm{H}^{2}(\Gamma;A).$

Let us illustrate a major improvement afforded by allowing coefficients in
general PIDs. While rational homotopy theory gives insight into the ranks of
the lower central series quotients of a group, it can never detect words in
finite groups. More generally, any invariant theory with values in a torsion-
free ring $A$ will necessarily have $\mathrm{H}^{1}(\Gamma;A)=0$ for any
finite group $\Gamma$.

######  Example 5.11.

Take $A=\mathbb{F}_{2}$ and consider the Heisenberg group
$\Gamma=\mathscr{H}_{3}(\mathbb{F}_{2})$ of matrices

$\left\\{\begin{pmatrix}1&X&Z\\\ &1&Y\\\ &&1\end{pmatrix}\text{ s.t.
}X,Y,Z\in\mathbb{F}_{2}\right\\}\subseteq\operatorname{Gl}_{3}(\mathbb{F}_{2}).$

It has $8$ elements, a presentation $\langle x,y,z\mid
x^{2},y^{2},z^{2},[x,y]z^{-1}\rangle$, and
$\mathrm{H}^{1}(\Gamma;\mathbb{F}_{2})=\langle X,Y\rangle$ given by the sub-
diagonal entries.

On the group cochain level, the functions $X,Y$ and $Z$ are defined in
$C^{1}(B\Gamma;\mathbb{F}_{2})$, but $Z$ is not closed. Instead, $dZ=X\smile
Y$, so that the cup product vanishes on cohomology. Indeed,

$dZ(M,M^{\prime})=Z(MM^{\prime})-Z(M)-Z(M^{\prime})=X(M)Y(M^{\prime})=X\smile
Y(M,M^{\prime}).$

A well-defined invariant is therefore $(X|Y)+Z$, which is indeed unchanged by
application of any of the relations.

Now, $X$ and $Y$ detect the image of any group element in the abelianization.
But, more interestingly, the invariant $X|Y+Z$ detects the commutator
$E_{1,3}$.

More generally, finite $p$-groups are well-tailored for
$\mathbb{F}_{p}$-valued letter-braiding invariants.

##  6\. Completeness of letter-braiding

We next show that letter-braiding invariants furnish a _complete_ coalgebra of
functionals that are continuous with respect to the augmentation ideal
topology on the group ring. Recall that the group ring $A[\Gamma]$ is filtered
by powers of the augmentation ideal $I\lhd A[\Gamma]$. The $I$-adic completion
is inverse limit

$A[\Gamma]^{\wedge}_{I}=\varprojlim A[\Gamma]/I^{n}$

topologized so that all quotients $A[\Gamma]^{\wedge}_{I}\twoheadrightarrow
A[\Gamma]/I^{n}$ are continuous, when the target has the discrete topology. In
particular, a functional $\varphi\colon A[\Gamma]^{\wedge}_{I}\to A$ is
_continuous_ if and only if it vanishes on some power $I^{N}$.

######  Theorem 6.1 (Completeness of letter-braiding invariants).

Let $\Gamma$ be a finitely generated group. The letter-braiding pairing
defines a natural isomorphism of filtered coalgebras

$\mathrm{H}^{0}_{Bar}(\Gamma;A)\xrightarrow{\sim}\text{Hom}_{cts}(A[\Gamma]^{\wedge}_{I},A)$

where the $A[\Gamma]^{\wedge}_{I}$ is the $I$-adic completion of the group
ring equipped with the $I$-adic topology, and $\text{Hom}_{cts}(-,-)$ denotes
continuous linear functionals.

In other words, for every functional $\varphi\colon A[\Gamma]\to A$ that
vanishes on some power $I^{n}$, there exists a unique
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ of weight $\leq n$ such that

$\langle T,\bullet\rangle=\varphi(\bullet).$

######  Remark 6.2.

When the group $\Gamma$ is infinitely generated there is no hope that letter-
braiding would be an isomorphism in general. For example,
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is always a coalgebra, while duals of
infinite dimensional algebras are not – this is due to the failing of
$(V\otimes V)^{*}\cong V^{*}\otimes V^{*}$. We contend that in this case, the
coalgebra $\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is the better invariant of $\Gamma$
– Theorem 6.3 and Corollary 6.18 below detail exactly what it measures.

###  6.1. Introducing relations

Building towards a proof of Theorem 6.1, we quantify the effect of introducing
relations to a group. The main result of this subsection is that the letter-
braiding invariants of a quotient group $\Gamma_{0}/\langle R\rangle$ are
exactly those invariants on $\Gamma_{0}$ that vanish in a strong sense on all
$r\in R$.

Let $\Gamma_{0}$ be a group and $R\subseteq\Gamma_{0}$ a subset that will
serve as relations for $\Gamma:=\Gamma_{0}/\langle R\rangle$. The quotient map
$p\colon\Gamma_{0}\to\Gamma$ induces a pullback of filtered coalgebras

$p^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\longrightarrow\mathrm{H}^{0}_{Bar}(\Gamma_{0};A).$

######  Theorem 6.3 (Lifting criterion).

$p^{*}$ is an injection, identifying $\mathrm{H}^{0}_{Bar}(\Gamma;A)$ as a
subcoalgebra of $\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)$. Moreover, for
$T\in\overline{\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)}$, the following are
equivalent:

  1. (1)

$T$ is in the sub-coalgebra $p^{*}\mathrm{H}^{0}_{Bar}(\Gamma;A)$.

  2. (2)

One can write $\overline{\Delta}(T)=\sum_{i=1}^{m}T_{1}^{i}\otimes T_{2}^{i}$
with all $T_{1}^{i}$ and $T_{2}^{i}$ in the sub-coagelbra
$p^{*}\mathrm{H}^{0}_{Bar}(\Gamma;A)$, and moreover $\ell_{T}(r)=0$ for all
$r\in R$.

  3. (3)

The group ring functional $\langle T,\bullet\rangle\colon A[\Gamma_{0}]\to A$
factors through the projection

$p_{*}\colon A[\Gamma_{0}]\twoheadrightarrow A[\Gamma].$

  4. (4)

For every sequence of group elements $w_{0},\ldots,w_{k}\in\Gamma_{0}$ such
that at least one of them belongs to $R$, the multi-evaluation

$\ell_{T}(w_{0}|\ldots|w_{n})=0.$

Injectivity of $p^{*}$ follows from the following more general fact.

######  Lemma 6.4.

Whenever $p\colon\Gamma_{1}\to\Gamma_{2}$ induces an injection
$p^{*}\colon\mathrm{H}^{1}(\Gamma_{2};A)\hookrightarrow\mathrm{H}^{1}(\Gamma_{1};A)$,
in particular when $p$ induces a surjection on abelianization
$\Gamma_{1}^{ab}\to\Gamma_{2}^{ab}$, the pullback

$p^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma_{2};A)\to\mathrm{H}^{0}_{Bar}(\Gamma_{1};A)$

is injective.

###### Proof.

Consider the induced map of the weight spectral sequences, as described in
Lemma 4.11. On their $E_{1}$-pages in total degree $0$ it consists of
injections

$p^{*}\colon\mathrm{H}^{1}(\Gamma_{2};A)^{\otimes
n}\hookrightarrow\mathrm{H}^{1}(\Gamma_{1};A)^{\otimes n}.$

But since the $E_{\infty}$-pages include inside the respective $E_{1}$-pages,
$p^{*}$ induces an injection on $E_{\infty}$. This already implies injectivity
on $\mathrm{H}^{0}_{Bar}(-;A)$. ∎

###### Proof of (1) $\implies$ (2).

If $T=p^{*}\tilde{T}$ for some $\tilde{T}\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$
and
$\overline{\Delta}(\tilde{T})=\sum\tilde{T}_{1}^{i}\otimes\tilde{T}_{2}^{i}$,
then

$\overline{\Delta}(T)=\overline{\Delta}(p^{*}\tilde{T})=\sum
p^{*}\tilde{T}_{1}^{i}\otimes p^{*}\tilde{T}_{2}^{i},$

so the components of the comultiplication indeed lift. By push-pull,
$\ell_{p^{*}\tilde{T}}(r)=\ell_{\tilde{T}}(p(r))=0$ since $r$ is in the kernel
of $p$. ∎

###### Proof of (2) $\implies$ (3):.

We must show that $\ell_{\bullet}(T)$ vanishes on the ideal

$I_{R}:=\ker\left(A[\Gamma_{0}]\twoheadrightarrow A[\Gamma]\right).$

Fox observed in [Fox53, §1] that the ideal $I_{R}$ is generated by the subset
$\\{r-1\\}_{r\in R}$. On every such generator, Condition (2) gives

$\langle T,r-1\rangle=\ell_{r}(T)-\ell_{1}(T)=0-0=0.$

To see that vanishing extends to any product with generators, we employ the
product formula for $A[\Gamma_{0}]$. Suppose that $x\in I_{R}$ was already
proven to be in the kernel of $\langle T,\bullet\rangle$ and $y\in
A[\Gamma_{0}]$ is any element. Then by the product formula

$\langle T,xy\rangle=\epsilon(y)\cdot\cancel{\langle
T,x\rangle}+\cancel{\epsilon(x)}\cdot\langle
T,y\rangle+\langle\overline{\Delta}T,x\otimes y\rangle,$

where $\langle T,x\rangle=0$ by assumption and $\epsilon(x)=0$ since $x$
belongs to the augmentation idea. But Condition 2 implies that
$\overline{\Delta}(T)=\sum T_{1}^{i}\otimes T_{2}^{i}$ where all terms
$T_{1}^{i}$ and $T_{2}^{i}$ are pulled back from $\Gamma=\Gamma_{0}/\langle
R\rangle$, so $\langle T_{1}^{i},x\rangle=0$ for all $i$. Together, this shows
that $\langle T,xy\rangle=0$.

The same proof shows that $yx$ also pairs trivially with $T$, where now we use
vanishing of $\langle T_{2}^{i},x\rangle$. With these, $\langle
T,\bullet\rangle$ vanishes on the two-sided ideal generated by
$\\{r-1\\}_{r\in R}$, thus factors through $A[\Gamma]$. ∎

###### Proof of (3) $\implies$ (4).

Assume that $\langle T,\bullet\rangle$ factors through $p_{*}\colon
A[\Gamma_{0}]\twoheadrightarrow A[\Gamma]$. We must show that
$\ell_{T}(w_{0}|\ldots|w_{n})=0$ whenever one or more of the $w_{i}$’s belongs
to $R$. Let $w_{0},\ldots,w_{n}$ be a sequence of group elements with
$w_{i}=r\in R$ for some $0\leq i\leq n$. Then by Proposition 5.6,

$\ell_{T}(w_{0}|\ldots|w_{n})=\langle T\,,\,(w_{0}-1)\cdots(w_{n}-1)\rangle.$

But the product $(w_{0}-1)\cdots(r-1)\cdots(w_{n}-1)$ is sent to $0\in
A[\Gamma]$ and so this multi-evaluation vanishes. ∎

The last step of the proof is the most elaborate, first requiring substantial
terminology.

###  6.2. Defect and obstruction to lifting

To complete the proof of the lifting criterion of Theorem 6.3 we show that if
$T$ does not lift to $\mathrm{H}^{0}_{Bar}(\Gamma;A)$, then there exists a
multi-evaluation containing $r\in R$ that does not vanish on $T$. To achieve
this we introduce a secondary “defect” filtration on
$\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)$.

In the construction below, a ‘space’ $Y$ can be taken to mean either a
topological space or a simplicial set. When working in the topological
category, all maps are continuous and $S^{1}(r)$ denotes a topological circle.
Otherwise, maps are simplicial and $S^{1}(r)$ is a subdivided simplicial
circle as defined in §3.2. Let $D^{2}(r)$ denote the reduced cone on
$S^{1}(r)$ – this is a disc with boundary $S^{1}(r)$.

######  Definition 6.5 (Presentation complex).

Let $(Y,*)$ be a pointed space and $R\subseteq\pi_{1}(Y,*)$ a subset of
elements, realized by pointed maps $\\{S^{1}(r)\to Y\\}_{r\in R}$. Then
$Y_{|R}$ is the reduced mapping cone

$Y_{|R}=\operatorname{Cone}\left(\vee r\colon\bigvee_{r\in R}S^{1}(r)\to
Y\right)=\left(\bigvee_{r\in R}D^{2}(r)\right)\coprod_{\vee_{r\in
R}S^{1}(r)}Y.$

Denote $\Gamma_{0}:=\pi_{1}(Y,*)$ and $\Gamma:=\pi_{1}(Y_{|R},*)$.

By Van-Kampen’s theorem, the natural inclusion $p\colon Y\hookrightarrow
Y_{|R}$ induces an isomorphism

$p_{*}\colon\pi_{1}(Y_{|R},*)\xrightarrow{\sim}\pi_{1}(Y,*)/\langle R\rangle.$

The long exact sequence of the pair $(Y_{|R},Y)$ gives an exact sequence

$0\to\mathrm{H}^{1}(\Gamma;A)\xrightarrow{p^{*}}\mathrm{H}^{1}(\Gamma_{0};A)\xrightarrow{d}A^{R}$

where $A^{R}$ is the relative cohomology
$\mathrm{H}^{2}(Y_{|R},Y;A)\cong\tilde{\mathrm{H}}^{2}(\bigvee_{r\in
R}S^{2};A)$.

######  Lemma 6.6.

Under the identification
$\mathrm{H}^{1}(\Gamma_{0};A)\cong\text{Hom}_{grp}(\Gamma_{0},A)$, the
connecting homomorphism $d$ is the restriction of functions $\Gamma_{0}\to A$
to $R\subseteq\Gamma_{0}$.

This exact sequence is the first instance of the lifting criterion: a
functional $\alpha\colon\Gamma_{0}\to A$ comes from $H^{1}(\Gamma;A)$ if and
only if it evaluated trivially on the relations. We will establish a vast
generalization of this for Bar cohomology.

Observe that at the cochain level we get a surjection of augmented dg algebras

$C^{*}(Y_{|R};A)\xrightarrow{p^{*}}C^{*}(Y;A)\to 0.$

Its kernel is $K=C^{*}(Y_{|R},Y;A)$, having $\mathrm{H}^{2}(K)\cong A^{R}$
detecting obstructions to lifting, with vanishing cohomology in lower degrees.

For simplicity of notation, let us consider this setup in the abstract.

######  Hypothesis 6.7.

Let $p\colon C\twoheadrightarrow C_{0}$ be a surjection of connected dg
algebras whose underlying $A$-modules are $A$-torsion free. Denote $K=\ker(p)$
and assume further that $\mathrm{H}^{1}(K)=0$.

######  Definition 6.8.

We next consider with complexes of bigraded modules, where the vertical
grading is the usual cohomological grading, and the new horizontal grading is
called _defect degree_. A plain (singly-graded) complex will be considered as
concentrated in defect degree $0$.

Let $\delta A$ denote the free module of bidegree $(-1,0)$. If $C^{*}$ is a
cochain complex concentrated in defect degree $0$, the tensor $(\delta
A)^{\otimes p}\otimes C^{*}$ is denoted $\delta^{p}C^{*}$ and is concentrated
in defect degree $-p$.

######  Theorem 6.9.

Let $p\colon C\twoheadrightarrow C_{0}$ be a surjection of dg algebras
satisfying Hypothesis 6.7. Then there exists a functorial ‘defect degree’
filtration $D_{\bullet}\mathrm{H}^{0}_{Bar}(C_{0})$ with the following
properties:

  * •

The induced map $p\colon\mathrm{H}^{0}_{Bar}(C)\to\mathrm{H}^{0}_{Bar}(C_{0})$
is an isomorphism onto defect degree zero

$H^{0}_{Bar}(C)\cong D_{0}H^{0}_{Bar}(C_{0}).$

  * •

The associated graded coalgebra
$\operatorname{gr}^{D}_{\bullet}\mathrm{H}^{0}_{Bar}(C_{0})$ is filtered by a
functorial secondary filtration $W_{\bullet}$, called _defect weight_ , such
that there is a natural injective homomorphism of bigraded coalgebras – the
‘defect map’ –

$\operatorname{def}\colon\operatorname{gr}^{W}\operatorname{gr}^{D}\mathrm{H}^{0}_{Bar}(C_{0})\hookrightarrow
T(\delta\mathrm{H}^{2}(K)\oplus\mathrm{H}^{1}(C))$

which sends defect weight $n$ and degree $p$ to tensors of weight $n$
containing exactly $p$ factors in $\delta\mathrm{H}^{2}(K)$.

  * •

On defect degree $0$, the defect weight filtration coincides with the usual
weight filtration on $\mathrm{H}^{0}_{Bar}(C)$, and the defect map on weight
$n$ is $\overline{\Delta}^{n-1}\colon\mathrm{H}^{0}_{Bar}(C)_{\leq
n}\to\mathrm{H}^{1}(\Gamma;A)^{\otimes n}$.

  * •

On weight $1$ classes
$\alpha\in\mathrm{H}^{1}(C_{0})\leq\mathrm{H}^{0}_{Bar}(C_{0})$ of defect
degree $1$, the defect map is the coboundary of the long exact sequence

$\mathrm{H}^{1}(C)\to\mathrm{H}^{1}(C_{0})\xrightarrow{d}\mathrm{H}^{2}(K).$

Lifting the defect weight filtration to $\mathrm{H}^{0}_{Bar}(C_{0})$, these
are refinements of $D_{\bullet}$ by

$D_{p-1}=W_{p-1,p}\subseteq W_{p,p}\subseteq W_{p+1,p}\subseteq\ldots\subseteq
W_{n,p}\subseteq\ldots\subseteq D_{p}$

so that the defect map in the last theorem is a collection of linear maps

$\operatorname{def}_{n,p}\colon W_{n,p}\longrightarrow
T\left(\delta\mathrm{H}^{2}(K)\oplus\mathrm{H}^{1}(C)\right)$

for every $n\geq p$, whose image consists of tensors of weight $n$ and exactly
$p$ many factors drawn from $\delta\mathrm{H}^{2}(K)$, and having

$\ker(\operatorname{def}_{n,p})=W_{n-1,p}.$

Comultiplication interacts with these subspaces by

(14) $\overline{\Delta}(W_{n,p})\subseteq\sum_{\begin{subarray}{c}m,q\\\ 0\leq
m-q\leq n-p\end{subarray}}W_{m,q}\otimes W_{n-m,p-q}+\sum_{0\leq i\leq
p-1}D_{i}\otimes D_{p-1-i}.$

The next construction is key to defining the defect filtrations.

######  Definition 6.10.

Let $p\colon C\twoheadrightarrow C_{0}$ be a surjection of connected dg
algebras, with kernel $K$. Treat the inclusion $i\colon K\hookrightarrow C$ as
a bicomplex

$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\delta
K^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\scriptstyle{-d}$$\textstyle{C^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{=:(C,K)^{*,*}}$$\textstyle{\delta
K^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{0}}$$\scriptstyle{-d}$$\textstyle{C^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$

where the horizontal grading is the _defect degree_.

Extend $p$ to a bigraded projection $p\colon(C,K)\longrightarrow C_{0}$ by
having it vanish of $\delta K$.

######  Lemma 6.11.

There is a natural bigraded differential algebra structure on $(C,K)$, making
the maps $C\hookrightarrow(C,K)\twoheadrightarrow C_{0}$ into bigraded
differential algebra homomorphisms, and the induced map
$\operatorname{tot}(C,K)\to C_{0}$ a quasi-isomorphism.

###### Proof.

Since $K\lhd C$ is an ideal, the product on $C$ induces maps

$\delta K\otimes C\to\delta K\quad\text{and}\quad C\otimes\delta K\cong\delta
C\otimes K\to\delta K,$

where we use the Koszul sign convention when moving $\delta$ past elements of
$C$:

$c\cdot(\delta k)=(-1)^{|c|}\delta(c\cdot k)$

for homogeneous $c\in C$. Along with the existing product $C\otimes C\to C$
and the zero product $\delta K\otimes\delta K\to 0$, these give a bigraded
algebra structure on $(C,K)$. Quick verification using the Koszul sign rule
shows that both differentials are indeed derivations.

With this product, $C\hookrightarrow(C,K)\twoheadrightarrow C_{0}$ are
homomorphisms of bigraded differential algebras. The short exact sequence of
complexes $0\to K\to C\to C_{0}\to 0$ shows that $\operatorname{tot}(C,K)\to
C_{0}$ is a quasi-isomorphism. ∎

Recalling that by Lemma 2.10 the Bar construction is a quasi-isomorphism
invariant of dg coalgebras, one can compute $\mathrm{H}^{0}_{Bar}(C_{0})$ with
the Bar construction of $(C,K)$. But since $(C,K)$ is bigraded, its Bar
construction is now triply graded.

######  Corollary 6.12.

Given a surjection of augmented dg algebras $C\twoheadrightarrow C_{0}$ whose
underlying $A$-modules are torsion free, $\mathrm{H}^{*}_{Bar}(C_{0})$ is
computed by the totalization of the tri-complex

$Bar(C,K)=T\left(\mathbf{s}^{-1}\delta
K\oplus\mathbf{s}^{-1}\overline{C}\right)$

with $d_{1}$ and $d_{2}$ induced by the internal differential $C$ and the
product structure, and an additional differential $d_{0}$ induced by the
inclusion $i\colon K\to C$. We call the three gradings: cohomological, weight
and defect degree. This tri-complex is functorial in commutative squares

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{C_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{C_{0}^{\prime}.}$

###### Proof.

Since $\operatorname{tot}(C,K)\to C_{0}$ is a quasi-isomrophism of dg algebras
whose terms are $A$-torsion free, Lemma 2.10 shows that the map
$\operatorname{tot}Bar(\operatorname{tot}(C,K))\to\operatorname{tot}Bar(C_{0})$
is a quasi-isomorphism. But the totalization of the bicomplex
$Bar(\operatorname{tot}(C,K))$ is natutrally isomorphic to totalization of the
tri-complex $Bar(C,K)$. Functoriality is clear by construction. ∎

######  Definition 6.13.

Let $C\twoheadrightarrow C_{0}$ now be a surjection of _connected_ dg algebras
whose underlying $A$-modules are torsion free. The _defect filtration_ on
$\mathrm{H}^{0}_{Bar}(C_{0})$ is the filtration induced by

$D_{p}Bar(C,K)=\text{subcomplex of terms with defect degree $\geq-p$}.$

This is the subcomplex in which at most $p$ tensor factors are taken from
$\delta K$. Denote the induced filtration on cohomology by
$D_{p}\mathrm{H}^{0}_{Bar}(C_{0})$.

###### Proof of Theorem 6.9.

Let us consider the spectral sequence associated with the defect filtration on
$\operatorname{tot}Bar(C,K)=T(\delta K\oplus\overline{C})$, suppressing the
weight desuspension from the notation. Its $E_{0}$-page consists of terms

$E_{0}^{-p,\bullet,\bullet}=D_{p}/D_{p-1}=\bigoplus_{i_{0},\ldots,i_{p}\geq
0}\overline{C}^{\otimes i_{0}}\otimes\delta K\otimes\overline{C}^{\otimes
i_{1}}\otimes\delta K\otimes\ldots\otimes\overline{C}^{\otimes
i_{p-1}}\otimes\delta K\otimes\overline{C}^{\otimes i_{p}}$

with differential given as the usual Bar differential $d_{1}+d_{2}$. In
particular, when $p=0$ one gets the usual total complex
$(\operatorname{tot}Bar(C),d_{Bar})$, so that the $p=0$ part of the
$E_{1}$-page is naturally isomorphic to $H^{*}_{Bar}(C)$.

Define the defect weight filtration on
$E_{0}^{\bullet,\bullet,\bullet}=(T(\delta K\oplus\overline{C}),d_{1}+d_{2})$
to be the filtration by tensor weight in $T(\delta K\oplus\overline{C})$ –
crucially, this filtration is distinct from the one induced from usual weight
filtration on $Bar(C,K)=(T(\delta K\oplus\overline{C}),d_{0}+d_{1}+d_{2})$,
see Remark 6.14. Let us examine the spectral sequence associated with defect
weight. To this end, fix $p\geq 0$ and let us denote
$B_{p}^{\bullet,\bullet}=(E_{0}^{-p,\bullet,\bullet},d_{1}+d_{2})$.

The associated graded for the defect weight filtration on
$B_{p}^{\bullet,\bullet}$ – an auxiliary $E^{\prime}_{0}$-page – consists of
the same terms but with only the internal differential
$E_{0}^{\prime}=(E_{0}^{-p,\bullet,\bullet},d_{1})$, so it is a direct sum of
tensor products of the complexes $(\overline{C},d)$ and $(\delta K,-d)$. Let
us apply the Künneth formula to the natural map

$\bigoplus_{i_{0},\ldots,i_{p}\geq 0}\mathrm{H}(\overline{C})^{\otimes
i_{0}}\otimes\delta\mathrm{H}(K)\otimes\mathrm{H}(\overline{C})^{\otimes
i_{1}}\otimes\delta\mathrm{H}(K)\otimes\ldots\otimes\mathrm{H}(\overline{C})^{\otimes
i_{p-1}}\otimes\delta\mathrm{H}(K)\otimes\mathrm{H}(\overline{C})^{\otimes
i_{p}}\to E_{1}^{\prime}.$

Recall our connectivity assumption, that $\overline{C}$ has cohomology only in
degrees $\geq 1$ and $\mathrm{H}(K)$ is concentrated in degrees $\geq 2$.
These imply that tensors of weight $n$ with $p$ many factors in $K$ only have
cohomology in cohomological degrees $\geq n+p$. Therefore, on tensors of
weight $n$ the Künneth map is an isomorphism in cohomological degree $\leq
n+p$, where in degrees $<n+p$ the $E_{1}^{\prime}$-page vanishes, and in
degree $n+p$

$\bigoplus_{i_{0}+\ldots+i_{p}=n-p}\mathrm{H}^{1}(\overline{C})^{\otimes
i_{0}}\otimes\delta\mathrm{H}^{2}(K)\otimes\mathrm{H}^{1}(\overline{C})^{\otimes
i_{1}}\otimes\ldots\otimes\delta\mathrm{H}^{2}(K)\otimes\mathrm{H}^{1}(\overline{C})^{\otimes
i_{p}}\cong(E_{1}^{\prime})^{-p,-n,n+p}.$

In terms of total degree in $E_{1}^{\prime}$, we conclude that
$E_{1}^{\prime}$ vanishes in total degree $<p$, and in total degree $p$ it is
the part of $T(\delta\mathrm{H}^{2}(K)\oplus\mathrm{H}^{1}(C))$ containing
exactly $p$ many factors in $\delta\mathrm{H}^{2}(K)$.

Diagrammatically, abbreviating $\mathrm{H}=\mathrm{H}^{1}(\overline{C})$ and
$V=\delta\mathrm{H}^{2}(K)$, the $E_{1}^{\prime}$-page takes the form

$\textstyle{0}$$\textstyle{\bigoplus V^{\otimes i}|\mathrm{H}|V^{\otimes
j}|\mathrm{H}|V^{\otimes
p-i-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\textstyle{\\#}$$\textstyle{\\#}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\bigoplus
V^{\otimes i}|\mathrm{H}|V^{\otimes
p-i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus
V^{\otimes i}|\delta\mathrm{H}^{3}(K)|V^{\otimes
p-i-1}}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{V^{\otimes
p}}$

from which we see that the entire auxiliary spectral sequence
$(E^{\prime}_{r},d^{\prime}_{r})$ vanishes total degree $<p$, and no
differentials can hit the terms in degree $p$. Thus there is an inclusion
$(E^{\prime}_{\infty})^{-q,p+q}\hookrightarrow(E^{\prime}_{1})^{-q,p+q}$, with
image characterized by the vanishing of all differentials $d^{\prime}_{r}$.

Recalling that the auxiliary spectral sequence computes the cohomology of
$\operatorname{tot}B_{p}=\operatorname{tot}E_{0}^{-p,\bullet,\bullet}$, which
is the $E_{1}$ page of the original spectral sequence, we conclude that
$\operatorname{tot}E_{1}^{-p,\bullet,\bullet}$ vanishes in total degree $<p$,
and in degree $p$ it has associated graded

$\bigoplus_{n=0}^{\infty}\operatorname{gr}^{F}E_{1}^{-p,-n,p+n}\hookrightarrow\bigoplus_{i_{0},\ldots,i_{p}\geq
0}\mathrm{H}^{1}(\overline{C})^{\otimes
i_{0}}\otimes\delta\mathrm{H}^{2}(K)\otimes\ldots\otimes
H^{1}(\overline{C})^{\otimes i_{p-1}}\otimes\delta\mathrm{H}^{2}(K)\otimes
H^{1}(\overline{C})^{\otimes i_{p}}.$

Summing over all $p\geq 0$ and observing that the defect weight filtration is
a filtration of coalgebras, the same argument performed in the category of
coalgebras shows that the inclusion

$\operatorname{tot}(E_{1}^{\bullet,\bullet,\bullet})^{0}=\bigoplus_{p\geq
0}\operatorname{gr}^{W}\operatorname{tot}(E_{1}^{-p,\bullet,\bullet})^{p}\hookrightarrow
T(\delta\mathrm{H}^{2}(K)\oplus\mathrm{H}^{1}(\overline{C}))$

is in fact a homomorphism, respecting defect degree and weight.

Return now to the original spectral sequence. As just proved, its
$E_{1}^{-p,\bullet}=\operatorname{tot}(E_{1}^{-p,\bullet,\bullet})$ term
begins as follows.

$\textstyle{0}$$\textstyle{E_{1}^{-2,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\textstyle{E_{1}^{-1,2}}$$\textstyle{\mathrm{H}^{2}_{Bar}(C)}$$\textstyle{0}$$\textstyle{0}$$\textstyle{E_{1}^{-1,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}_{Bar}(C)}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\mathrm{H}^{0}_{Bar}(C)}$

Since no differentials can hit the diagonal in total degree $0$, one obtains a
natural inclusion $E_{\infty}^{-p,p}\hookrightarrow E_{1}^{-p,p}$, whose image
is characterized by the vanishing of all higher differentials.

Summarizing, the defect degree filtration on $\operatorname{tot}Bar(C)$
defines $0\leq D_{0}\leq D_{1}\leq\ldots\leq H^{0}_{Bar}(C_{0})$, with
$D_{0}\cong H^{0}_{Bar}(C)$ and associated graded

$\operatorname{gr}^{D}H^{0}_{Bar}(D_{0})\cong\bigoplus_{p\geq
0}E_{\infty}^{-p,p}\hookrightarrow\bigoplus_{p\geq 0}E_{1}^{-p,p}$

as graded coalgebras. Furthermore, the $E_{1}$-coalgebra is filtered by the
defect weight $W_{\bullet}E_{1}$, so that the associated graded with respect
to $W_{\bullet}$ gives an inclusion of coalgebras

$\operatorname{def}\colon\operatorname{gr}^{W}E_{1}\hookrightarrow
T(\delta\mathrm{H}^{2}(K)\oplus\mathrm{H}^{1}(C))$

respecting both defect weight and degree. We call the last bigraded coalgebra
homomorphism the _defect map_. The restriction of the defect weight filtration
and the defect map to $E_{\infty}\leq E_{1}$ is the structure we set out to
construct.

Clearly, when $p=0$ the defect weight filtration is precisely the usual weight
filtration on $\mathrm{H}^{0}_{Bar}(C)$ and the defect map is the inclusion
recording the leading term as given by the $(n-1)$-fold reduced
comultiplication on $\mathrm{H}^{0}_{Bar}(C)_{\leq n}$.

Lastly, if $\alpha\in\mathrm{H}^{1}(C_{0})\leq\mathrm{H}^{0}_{Bar}(C_{0})$ has
defect degree $1$, then it is not the image of a cocycle in $C$. But $p\colon
C\twoheadrightarrow C_{0}$ is surjective, so there does exist a lift
$\tilde{\alpha}\in C^{1}$, which then necessarily has $0\neq
d\tilde{\alpha}\in K$. The sum

$\textstyle{\delta\cdot
d\tilde{\alpha}}$$\textstyle{+}$$\textstyle{\tilde{\alpha}}$

is a cocycle in $Bar(C,K)$ projecting to $\alpha$. Its image in
$\operatorname{gr}^{D}_{1}\mathrm{H}^{0}_{Bar}(C_{0})$ is therefore
$[d\tilde{\alpha}]$, which is then also its defect. But this construction is
the connecting homomorphism of the long exact sequence for $0\to K\to C\to
C_{0}\to 0$, as claimed. ∎

######  Remark 6.14.

An explicit description of the defect weight filtration on $Bar(C,K)$ is that

$W_{n,p}Bar(C,K)=\\{\text{highest defect summand has $\leq p$ many factors in
$\delta K$ and weight $\leq n$}\\}.$

This differs of the weight filtration, that bounds the weights of all tensors
in a cochain, and not just the top defect summand. The defect map returns the
cohomology class of this leading term of defect $p$ and weight $n$.

Shifting back to our topological setup of $C=C^{*}(Y_{|R};A)$ projecting onto
$C_{0}=C^{*}(Y;A)$, which satisfies Hypothesis 6.7 with
$\mathrm{H}^{2}(K)\cong A^{R}$ giving functions on relations, the defect map
is a bigraded coalgebra homomorphism

(15)
$\operatorname{def}\colon\operatorname{gr}^{W}\operatorname{gr}^{D}\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)\longrightarrow
T(\delta A^{R}\oplus\mathrm{H}^{1}(\Gamma;A)).$

We claim that this defect map records nonvanishing letter-braiding invariants
on relations $r\in R$.

######  Example 6.15 (The universal relation).

Consider the case of the free group on one generator $F_{\\{r\\}}$ to which we
introduce the relation $r$, making the group trivial. The associated
topological construction is the inclusion $S^{1}(r)\hookrightarrow D^{2}(r)$.

Since $\mathrm{H}^{1}(D^{2})=0$, the defect map takes values in the coalgebra
$T(\delta\mathrm{H}^{2}(D^{2},S^{1}))$, in which weight and defect degree
coincide. It then follows that the defect weight filtration is trivial in that
it matches the defect degree. Since $\mathrm{H}^{2}(D^{2},S^{1})=A^{\\{r\\}}$
is generated by $(\tau\colon r\mapsto 1)$ and the connecting homomorphism
$\mathrm{H}^{1}(S^{1})\to\mathrm{H}^{2}(D^{2},S^{1})$ sends the generator
$t\in\mathrm{H}^{1}(S^{1})$ to $\tau$, the defect filtration splits and is the
evident isomorphism $\mathrm{H}^{0}_{Bar}(S^{1};A)\cong A[t]\to
A[\delta\tau]$, with $t^{n}$ having defect degree $n$.

This gives an interpretation of defect: it records the letter-braiding
invariants of the relation $r$. The fact that every nontrivial braiding
invariant is nonzero on $r$ reflects the fact that none are pulled back from
the (trivial) quotient group.

This interpretation will be made precise in general. Indeed, suppressing
$\delta$ from the notation, the defect map takes values in
$T(A^{R}\oplus\mathrm{H}^{1}(\Gamma;A))$, but

$\bigoplus_{n\geq 0}(A^{R}\oplus\mathrm{H}^{1}(\Gamma;A))^{\otimes
n}\hookrightarrow\bigoplus_{n\geq 0}(A^{R}\oplus A^{\Gamma})^{\otimes
n}\hookrightarrow\bigoplus_{n\geq 0}A^{(R\coprod\Gamma)^{n}}$

defining $A$-valued $n$-ary functions on the relations $R$ and the quotient
group $\Gamma$. We lift these to functions $\Gamma_{0}\to A$ by setting for
every $w\in\Gamma_{0}$

(16) $f\in A^{\Gamma}\rightsquigarrow\langle
f,w\rangle=f(p(w))\quad\text{and}\quad g\in A^{R}\rightsquigarrow\langle
g,w\rangle=\begin{cases}g(r)&w=r\in R\\\ 0&\mathrm{otherwise}.\end{cases}$

######  Proposition 6.16 (Defect evaluation).

Consider the defect map of $\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)$ obtained from
the surjection of dg algebras $C^{*}(Y_{|R};A)\twoheadrightarrow C^{*}(Y;A)$.
If $T\in\overline{\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)}$ has defect degree $p$
and defect weight $n$, then for every sequence of group elements
$w_{1},\ldots,w_{m}$ such that $q\geq p$ of them that are relations $r_{i}\in
R$,

(17)
$\ell_{T}(w_{1}|\ldots|w_{n})=\begin{cases}\langle\operatorname{def}_{n,p}(T)\,,\,w_{1}\otimes\ldots\otimes
w_{n}\rangle&p=q\text{ and }n=m\\\ 0&p<q\text{ or }n<m,\end{cases}$

where
$\langle-,\bullet\rangle\colon(A^{R}\oplus\mathrm{H}^{1}(\Gamma;A))\otimes
A[\Gamma_{0}]\to A$ extends the evaluations in (16). We allow $m<n$ but make
no claims about its evaluation when $p=q$.

###### Proof.

Lets first consider the case $p=0$. In this case $T$ is pulled back from some
$\tilde{T}\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$. By push-pull (13)

$\ell_{T}(w_{1}|\ldots|w_{m})=(\ell_{\bullet}(w_{1})\otimes\ldots\otimes\ell_{\bullet}(w_{m}))_{\overline{\Delta}^{m-1}(p^{*}\tilde{T})}=(\ell_{\bullet}(p(w_{1}))\otimes\ldots\otimes\ell_{\bullet}(p(w_{m})))_{\overline{\Delta}^{m-1}(\tilde{T})}$

so since $\operatorname{def}_{n,0}(T)=\overline{\Delta}^{n-1}(\tilde{T})$, the
two evaluations in (17) coincide when $m=n$. In particular, when a positive
number of $w_{i}$ belong to $R$, some of these evaluations vanish and thus so
does their product. Also, when $m>n$ we have $\overline{\Delta}^{m-1}(T)=0$,
as needed for (17) to hold. This concludes the proof for all cases
$(n,p)=(n,0)$, and we subsequently assume that $p\geq 1$.

Now, for vanishing in the case $q>p$, assume by induction on $p$ that for all
$p^{\prime}<p$, whenever $T^{\prime}$ has defect degree $\leq p^{\prime}$ and
$w_{1},\ldots,w_{m^{\prime}}\in\Gamma_{0}$ contains $q^{\prime}>p^{\prime}$
many relations $r_{i}\in R$ then

$\ell_{T^{\prime}}(w_{1}|\ldots|w_{m^{\prime}})=0.$

Let $T$ have defect degree $p$ and $w_{1},\ldots,w_{m}\in\Gamma_{0}$ contain
$q>p$ many relations. Say $w_{i}=r$ is the first relation, then Proposition
5.6 shows

$\ell_{T}(w_{1}|\ldots|r|w_{i+1}|\ldots|w_{m})=(\ell_{\bullet}(w_{1}|\ldots|r)\otimes\ell_{\bullet}(w_{i+1}|\ldots|w_{m}))_{\overline{\Delta}(T)}.$

By the first case considered, $\ell_{\bullet}(w_{1}|\ldots|r)$ vanishes on
defect degree $0$, so the only nontrivial contributions to this evaluations
come from summands of $\overline{\Delta}(T)$ in which the left-hand factor has
defect degree $>0$. But since defect degree is a coalgebra filtration, all
those summands must have right-hand factor of defect degree $<p$. Since there
are $q-1\geq p$ many relations among $w_{i+1},\ldots,w_{m}$, the induction
hypothesis implies that those evaluations vanish as well.

We thus proved that the multi-evaluation vanishes whenever the number of
relations in the evaluation is greater than the defect degree. In particular,
if $w_{1},\ldots,w_{n}$ contains $p$ many relations, then
$\ell_{\bullet}(w_{1}|\ldots|w_{n})$ vanishes on $D_{p-1}$ and factors through
the associated graded

$\ell_{\bullet}(w_{1}|\ldots|w_{n})\colon\operatorname{gr}^{D}_{p}\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)\to
A.$

Similarly,
$(\ell_{\bullet}(w_{1}|\ldots|w_{i})\otimes\ell_{\bullet}(w_{i+1}|\ldots|w_{m}))\colon\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)\otimes\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)\to
A$ vanishes on all tensors $D_{j}\otimes D_{k}$ with $j+k<p$. Thus the
relation
$(\ell_{\bullet}(w_{1}|\ldots|w_{i})\otimes\ell_{\bullet}(w_{i+1}|\ldots|w_{n}))_{\overline{\Delta}(\bullet)}=\ell_{\bullet}(w_{1}|\ldots|w_{n})$
descends to the associated graded quotients as well.

We next prove that evaluations vanish when $m>n$. First, suppose $T$ has
defect degree $p$ and defect weight $n=1$. Then
$(\operatorname{def}\otimes\operatorname{def})\circ\overline{\Delta}(T)=\overline{\Delta}(\operatorname{def}(T))=0$,
so by injectivity of the defect map, it must be that $\overline{\Delta}(T)$
has total defect degree $<p$. But then

$\ell_{T}(w_{1}|\ldots|w_{m})=(\ell_{\bullet}(w_{1})\otimes\ell_{\bullet}(w_{2}|\ldots|w_{m}))_{\overline{\Delta}(T)}=0$

from the previous discussion. The same argument now shows the case of general
$m>n\geq 1$ by induction on $n$.

We are left with the cases $m=n$ and $q=p$, where we want to show the defect
map determines the multi-evaluation. Firstly, the vanishing of
$\ell_{\bullet}(w_{1}|\ldots|w_{n})$ on defect weight
$W_{n-1}\operatorname{gr}^{D}_{p}\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)$ shows
that these evaluations factor through the associated graded

$\ell_{\bullet}(w_{1}|\ldots|w_{n})\colon\operatorname{gr}^{W}_{n}\operatorname{gr}^{D}_{p}\mathrm{H}^{0}_{Bar}(\Gamma_{0})\to
A$

on which the defect map is defined and injective. We prove (17) by double
induction on $(n,p)$. The cases with $(n,0)$ were already proved at the top of
this proof.

For $(n,p)=(1,1)$, we must calculate $\ell_{T}(r)$ for $r\in R$. The relation
is witnessed topologically by a map of pairs
$r\colon(D^{2}(r),S^{1}(r))\to(Y_{|R},Y)$. This map induces a commutative
square of cochain algebras

$\textstyle{C^{*}(Y_{|R};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\textstyle{C^{*}(Y;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\textstyle{C^{*}(D^{2}(r);A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{*}(S^{1}(r);A)}$

showing that the pullback
$r^{*}\colon\mathrm{H}^{0}_{Bar}(Y;A)\to\mathrm{H}^{0}_{Bar}(S^{1}(r);A)$
respects the respective defect structures. For the pair $(S^{1}\hookrightarrow
D^{2})$ we already computed the defect map in Example 6.15, so we are left
with a commutative diagram

$\textstyle{\operatorname{gr}^{W}_{1}\operatorname{gr}^{D}\mathrm{H}^{0}_{Bar}(Y;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\scriptstyle{\operatorname{def}_{1,1}}$$\textstyle{\operatorname{gr}^{W}_{1}\operatorname{gr}^{D}\mathrm{H}^{0}_{Bar}(S^{1};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{def}_{1,1}}$$\scriptstyle{\cong}$$\textstyle{At\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{A}$$\textstyle{\mathrm{H}^{2}(Y_{|R},Y;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r^{*}}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{2}(D^{2},S^{1};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\textstyle{A\tau}$$\textstyle{A^{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}_{r}}$$\textstyle{A}$

The composition the top row is the letter-braiding number $\ell_{\bullet}(r)$,
while the bottom arrow identifies this number with the evaluation of functions
$A^{R}$ at $r$. This proves the claim when $(n,p)=(1,1)$.

Next, suppose by induction that (17) holds for all $(n^{\prime},p^{\prime})$
such that $n^{\prime}<n$ or $p^{\prime}<p$. Then
$\ell_{T}(w_{1}|\ldots|w_{n})=(\ell_{\bullet}(w_{1})\otimes\ell_{\bullet}(w_{2}|\ldots|w_{n}))_{\overline{\Delta}(T)}$
and $\overline{\Delta}(T)$ is a sum of tensors $T_{1}\otimes T_{2}$, each of
which either have total defect degree $<p$, or it has total defect degree $p$
but respective defect weights $n_{1}$ and $n_{2}$ such that $n_{1}+n_{2}=n$.
In the former case the evaluation vanishes since there are $p$ relations in
the list, while in the latter case the induction hypothesis gives

$\ell_{T_{1}}(w_{1})\otimes\ell_{T_{2}}(w_{2}|\ldots|w_{n})=\langle\operatorname{def}(T_{1})\otimes\operatorname{def}(T_{2})\,,\,w_{1}\otimes\ldots\otimes
w_{n}\rangle$

Equality in (17) thus holds since the defect map is a coalgebra homomorphism.
∎

With this we can complete the proof of our lifting criterion.

###### Proof of Theorem 6.3 (4)$\implies$(1).

Suppose that $T\in\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)$ is not pulled back from
$\Gamma$. Then by Theorem 6.9 it has defect degree $p>0$. Let $n$ denote the
defect weight of $T$ modulo $D_{p-1}$. Then,

$0\neq\operatorname{def}_{n,p}(T)\in(A^{R}\oplus\mathrm{H}^{1}(\Gamma;A))^{\otimes
n}$

a linear combination of tensors with exactly $p$ factors taken from $A^{R}$.
As discussed in this subsection, the tensor power above injects into the space
of $n$-ary functions on $R\coprod\Gamma$, and it does not vanish when
evaluated on some sequence $w_{1},\ldots,w_{n}$ with $p$ many entries in $R$.
Now, the evaluation formula (17) shows that

$\ell_{T}(w_{1}|\ldots|w_{n})=\langle\operatorname{def}_{n,p}(T)\,,\,w_{1}\otimes\ldots\otimes
w_{n}\rangle\neq 0.\qed$

###  6.3. Consequences of the lifting criterion

There is a nice categorical restatement of the lifting criterion of Theorem
6.3.

######  Corollary 6.17 (Lifting criterion v2).

Let $p\colon\Gamma_{0}\twoheadrightarrow\Gamma$ be a surjective group
homomorphism, then the commutative diagram of inclusions induced by the
letter-braiding pairing

$\textstyle{\mathrm{H}^{0}_{Bar}(\Gamma;A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}}$$\textstyle{A[\Gamma]^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{*}}$$\textstyle{\mathrm{H}^{0}_{Bar}(\Gamma_{0};A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A[\Gamma_{0}]^{*}}$

is Cartesian. That is, a linear functional on $A[\Gamma]$ is realized by
letter-braiding invariants on $\Gamma$ iff its pullback to $A[\Gamma_{0}]$ is
realized by letter-braiding invariants on $\Gamma_{0}$.

###### Proof.

This is immediate from the implication (1)$\iff$(3) of Theorem 6.3. ∎

In the special case of a presentation $\Gamma=\langle S|R\rangle$, one can be
more concrete.

######  Corollary 6.18.

Let $\Gamma=\langle S|R\rangle$ be a group presentation (possibly infinite).
Then the on Bar cohomology of $\Gamma$ is the maximal sub-coalgebra

$\mathrm{H}^{0}_{Bar}(\Gamma;A)\leq T(A^{S})$

of possibly nonhomogeneous tensors of forms in $A^{S}$, characterized by the
property that its elements have vanishing braiding numbers of all relations
$r\in R$.

###### Proof.

The image of $\mathrm{H}^{0}_{Bar}(\Gamma;A)\hookrightarrow T(A^{S})$ is
clearly a sub-coalgebra and all its elements have vanishing letter-braiding
numbers on $r\in R$. To see that it is the maximal such, if $H\leq T(A^{S})$
is a coalgebra with this property, then every $T\in H$ satisfies
$\ell_{T}(w_{1}|\ldots|w_{n})=0$ whenever $w_{1},\ldots,w_{n}\in F_{S}$ is a
sequence of elements at least one of which is an element in $R$. But the
lifting criterion now shows that $T$ comes from
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$. ∎

With these facts establishes, we can prove the letter-braiding invariants are
complete.

###### Proof of Theorem 6.1.

For a finitely generated free group $F_{r}$ generated by
$\\{x_{1},\ldots,x_{r}\\}$, we claim that the truncated group ring
$A[F_{r}]/I^{n}$ is the free $A$-module generated by products

$(x_{i_{1}}-1)(x_{i_{2}}-1)\cdots(x_{i_{d}}-1)$

for sequences $(i_{1},\ldots,i_{d})$ of length $0\leq d<n$. Indeed, every
generator $x_{i}$ can be written as $1+(x_{i}-1)$, its inverse is
$x_{i}^{-1}=1-(x_{i}-1)+(x_{i}-1)^{2}-\ldots+(-1)^{n-1}(x_{i}-1)^{n-1}$, and
every element in the free group can be written as a product of $x_{i}^{\pm 1}$
so expressible as a linear combination of shifted monomials. Freeness will
follow from the following observation.

Let $X_{i}\colon F_{r}\to A$ be the linear functional that counts instances of
$x_{i}$ with sign, so that $\mathrm{H}^{1}(F_{r};A)=\langle
X_{1},\ldots,X_{r}\rangle$. Then the letter-braiding invariants of $F_{r}$ are
labeled by the tensor coalgebra $T(X_{1},\ldots,X_{r})$ of noncommutative
polynomials in the $X_{i}$’s. Proposition 5.6

$\langle
X_{i_{1}}|\ldots|X_{i_{d}}\,,\,(x_{j_{1}}-1)\cdots(x_{j_{d}}-1)\rangle=\ell_{x_{j_{1}}}(X_{i_{1}})\ldots\ell_{x_{j_{d}}}(X_{i_{d}})=\delta_{(i_{1},\ldots,i_{d}),(j_{1},\ldots,j_{d})}$

gives a dual basis to degree $d$ monomials, and vanishes on all higher
degrees. By induction on degree, one can find appropriate linear combinations
of weight $\leq n$ that define a dual basis to the monomials. This proves that
letter-braiding $\mathrm{H}^{0}_{Bar}(F_{r};A)_{\leq
n}\hookrightarrow(A[F_{r}]/I^{n})^{*}$ is surjective and thus an isomorphism
(it also gives a recipe for constructing the Taylor-Maclaurin expansion of
every group ring element, i.e. writing it as a linear combination of
monomials).

Now, let $F_{n}\twoheadrightarrow\Gamma$ be a presentation of some group. Then
the Cartesian square in Corollary 6.17 shows that
$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq n}\to(A[\Gamma]/I^{n})^{*}$ is also an
isomorphism. ∎

###  6.4. Relative Massey products measure relations

In Lemma 4.11 we saw that the obstructions to having a tensor
$T(\mathrm{H}^{1}(\Gamma;A))$ define a letter-braiding invariant of $\Gamma$
is that all its cup and Massey products must vanish. In the last section we
gave an alternative characterization as having vanishing defect, as measured
by evaluations on relations. The relationship between Massey products and
relations in a group has a long history, going back to the work of Dwyer
[Dwy75b] and Fenn–Sjerve [FS87]), but they only cover a somewhat narrow set of
examples: ones in which the Massey product of cohomology classes
$\langle\alpha_{1},\ldots,\alpha_{n}\rangle$ is defined, and it pairs with
relations that are commutators.

However, even the most elementary examples we considered above, of surface
groups and braid groups (Examples 4.14,4.21), exhibit interesting invariants
in which one must consider a linear combination of tensors from
$\mathrm{H}^{1}(\Gamma;A)$ to detect various group elements. In this section
we expand the scope of Dwyer’s analysis, giving a full picture of how Massey
products coincide with letter-braiding numbers of relations, including non-
commutators, via a _relative Massey product_.

Let $\Gamma=\langle S|R\rangle$ be a group presentation. Applying the
construction $Y\mapsto Y_{|R}$ from §6.1 to the wedge of circles $X_{S}$ from
§3, we obtain a presentation $2$-complex $X_{S|R}$ whose fundamental group is
$\Gamma$. It contains a subcomplex $p\colon X_{S}\hookrightarrow X_{S|R}$,
inducing the surjection $F_{S}\twoheadrightarrow\Gamma$. Observe that the
$1$-simplices of $X_{S}$ are in bijection with the set of generators
$S\subseteq\Gamma$, while $X_{S|R}$ has additional edges coming from the
relations. $1$-cochains are $A$-valued functions on those edges.

######  Observation 6.19.

Restricting $1$-cochains of $X_{S|R}$ to the generators $S\subseteq\Gamma$
gives a surjection of coalgebras $p^{*}\colon Bar(X_{S|R};A)\twoheadrightarrow
Bar(X_{S};A)=T(A^{S})$, where the target is the coalgebra of letter-braiding
invariants on $F_{S}$ from §3. So every letter-braiding invariant of $F_{S}$
lifts to a Bar cochain of $\Gamma$ by extending the functions from $X_{S}$ to
the interiors of relation discs $D^{2}(r)\subseteq X_{S|R}$. But those
extensions will typically not be $d_{Bar}$-closed: their coboundary involves a
$2$-cochain of $X_{S|R}$, detecting some of the relations. Let us quantify the
obstruction to closed lifting.

Consider then the short exact sequence

(18) $0\to(K)_{Bar}\to Bar(X_{S|R};A)\to T(A^{S})\to 0$

where $(K)_{Bar}$ is a coalgebraic coideal. In analogy with relative cochains
$C^{*}(X_{S|R},X_{S};A)$, denote this coideal by $Bar(X_{S|R},X_{S};A)$. More
generally,

######  Definition 6.20.

When $p\colon X\hookrightarrow Y$ is a subcomplex inclusion, denote

$Bar(Y,X;A)=\ker\left(p^{*}\colon Bar(Y;A)\twoheadrightarrow Bar(X;A)\right),$

the coideal cogenerated by $K=C^{*}(Y,X;A)$ in $T(\overline{C^{*}(Y;A)})$. Its
cohomology will be denoted $\mathrm{H}^{*}_{Bar}(Y,X;A)$.

An alternative equivalent definition proceeds by recalling that a surjection
$Bar(C)\twoheadrightarrow Bar(C_{0})$ is quasi-isomorphic to the injection
$Bar(C)\hookrightarrow Bar(C,K)$. Then $Bar(Y,X;A)$ could be defined as
$\mathrm{coker}(Bar(C)\hookrightarrow Bar(C,K))$ – the quotient by the defect
$0$ subcomplex – up to a degree shift.

######  Proposition 6.21.

An inclusion $X\hookrightarrow Y$ defines an exact sequence

$\mathrm{H}^{0}_{Bar}(Y;A)\to\mathrm{H}^{0}_{Bar}(X;A)\xrightarrow{\mu}\mathrm{H}^{1}_{Bar}(Y,X;A),$

with $\mu(-)$ measuring the obstruction of lifting Bar cocycles from $X$ to
$Y$. We call $\mu(-)$ the _relative Massey product_ , for reasons explained
below.

In particular, a tensor $T\in T(A^{S})$ extends to a closed cochain in
$Bar^{0}(X_{S|R};A)$ if and only if its relative Massey product
$\mu(T)\in\mathrm{H}_{Bar}^{1}(X_{S|R},X_{S};A)$ vanishes.

###### Proof.

The snake lemma applied to

$0\to Bar(Y,X;A)\to Bar(Y;A)\to Bar(X;A)\to 0$

gives the claimed exact sequence. It follows that $T\in T(A^{S})$ lifts to a
class $\tilde{T}\in\mathrm{H}^{0}_{Bar}(X_{S|R};A)$ if and only if $\mu(T)=0$.
∎

For $T\in\mathrm{H}^{0}_{Bar}(X;A)$ its the relative Massey product is
computed by picking any lift $\tilde{T}\in Bar^{0}(Y;A)$ and computing its Bar
differential:

$\mu(T)=[d_{Bar}(\tilde{T})]\in\mathrm{H}^{1}_{Bar}(Y,X;A).$

When $\mu(T)$ has weight $1$, it is a relative cohomology class
$\mu(T)\in\mathrm{H}^{2}(Y,X;A)$ whose image in the absolute cohomology
$\mathrm{H}^{2}(Y;A)$ is the usual Massey product of the lift $\tilde{T}$, as
explained in Proposition 4.13. Thus $\mu(T)$ is a refinement of the Massey
product that remembers that $T$ came from $X$.

To see how the relative Massey product detects relations in a group, return to
the special case $(X_{S|R},X_{S})$. Then $\mu(T)$ takes values in
$\mathrm{H}^{2}(X_{S|R},X_{S};A)\cong A^{R}$, and is therefore exactly a
function on relations.

######  Proposition 6.22 (Relative Massey = letter-braiding of relations).

Suppose $T\in T(A^{S})$ is a letter-braiding invariant of $F_{S}$ whose
relative Massey product has weight $1$, that is
$\mu(T)\in\mathrm{H}^{2}(X_{S|R},X_{S};A)\cong A^{R}$. Then for every relation
$r\in R$,

(19) $\ell_{T}(r)=\langle\mu(T),[D^{2}(r)]\rangle,$

where $[D^{2}(r)]\in H_{2}(X_{S|R},X_{S};A)$ is the relative homology class of
the relation disc of $r$.

###### Proof.

This is essentially a special case of Proposition 6.16. With the
interpretation of $\mathrm{H}^{*}_{Bar}(Y,X;A)$ using the defect filtration as
in the paragraph preceding Proposition 6.21, the condition on $\mu(T)$ is
equivalent to saying that $T$ has defect $1$. The proof of (17) here works,
using the case of the universal relation $(D^{2},S^{1})$: the relative Massey
product in the exact sequence

$\cancel{\mathrm{H}^{0}_{Bar}(D^{2};A)}\to\mathrm{H}^{0}_{Bar}(S^{1};A)\xrightarrow{\mu}\mathrm{H}^{1}_{Bar}(D^{2},S^{1};A)\to\cancel{\mathrm{H}^{1}_{Bar}(D^{2};A)}$

is an isomorphism, identifying the letter-braiding invariant with the
evaluation on the relative fundamental class
$[D^{2}]\in\mathrm{H}^{2}(D^{2},S^{1};A)$. ∎

The absolute version of this proposition follows as a special case, and it
appears e.g. in [FS87, Main theorem]. They evaluate the absolute Massey
product $\mu_{n}\colon\mathrm{H}^{1}(\Gamma;\mathbb{Z})^{\otimes
n}\to\mathrm{H}^{2}(\Gamma;\mathbb{Z})$ on $H_{2}(\Gamma)$, which by Hopf’s
formula is generated by $R\cap[\Gamma,\Gamma]$ – relations that happen to also
be commutators. These correspond to disc classes $[D^{2}(r)]$ that come from
$\mathrm{H}^{2}(X_{S|R};\mathbb{Z})$ as they have vanishing boundary.
Fenn–Sjerve show that the evaluation coincides with Magnus expansion
coefficients, which by Theorem 5.9 is the same as $\ell_{T}(r)$. Proposition
6.22 therefore subsumes their result and extend it in the following
directions:

  * •

While older work considers Massey products
$\langle\alpha_{1},\ldots,\alpha_{n}\rangle$, which are
$\mu_{n}(\alpha_{1}|\ldots|\alpha_{n})$ on rank $1$ tensors, many groups
require considering linear combinations of tensors, which our theory
accommodates – see Examples 4.14,4.21.

  * •

Our generalization sees the entire set of relations $R$, and also considers
products of cochains $C^{1}(\Gamma;A)^{\otimes n}$ that are no a priori
closed. In fact, the pairing of $\alpha\in C^{1}(\Gamma;A)$ with a non-
commutator $r\in R$ is exactly the obstruction to $\alpha$ being closed, with
(19) specializing to

$\ell_{\alpha}(r)=d\alpha([D^{2}(r)])$

so $d\alpha=0$ iff $\ell_{\alpha}(r)=0$ for all $r\in R$.

  * •

While we only discussed $\mu(T)$ of weight $1$ above, higher weights appear
and are significant obstructions to lifting as well. This is necessary because
$T(A^{S})$ is a coalgebra, and one might attempt to lift braiding invariants
$T$ that split into several factors with nontrivial relative Massey products.
This situation is quantified by the defect map and Proposition 6.16.

##  7\. Topological applications

So far, topology provided us with invariants of groups. In the other
direction, we get a hands-on proof the following.

###  7.1. Cochains determine fundamental group ring

######  Theorem 7.1.

Let $X$ be any connected pointed CW-complex (or simplicial set) with a
finitely generated fundamental group. Then with coefficients in any PID, the
$2$-truncation of its singular (or simiplicial) cochain algebra $C^{*\leq
2}_{\cup}(X;A)$, along with its augmentation, determines the completed group
ring $A[\pi_{1}(X,*)]_{I}^{\wedge}$ explicitly via letter-braiding invariants.

###### Proof.

Given a filtered coalgebra, its dual is naturally a completed algebra, and
this operation is inverse to the continuous dual (see e.g. [SS15, Subsection
(2.5.3)]). In light of Theorem 6.1, it follows that
$A[\pi_{1}(X,*)]_{I}^{\wedge}$ is isomorphic to the complete algebra of
$A$-linear functionals on $\mathrm{H}^{0}_{Bar}(X;A)$, where their pairing is
the letter-braiding pairing. But the latter coalgebra is computed from
$C^{*}_{\cup}(X;A)$ in degrees $\leq 2$ along with its augmentation coming
form evaluation at the basepoint. ∎

To get ones hands on the (completion of the) group $\pi_{1}(X,*)$ itself,
additional structure is needed: the diagonal $X\to X\times X$, inducing the
Hopf algebra structure on $A[\pi_{1}(X,*)]$. This involves the
$E_{\infty}$-algebra structure on $C^{*}(X;A)$, and is not present in our
discussion.

###  7.2. Johnson homomorphisms

Let us construct the Johnson homomorphism mentioned in Theorem 1.9. Fix a
group $\Gamma$ and let $A$ be a PID. Denote
$\mathrm{H}=\mathrm{H}^{1}(\Gamma;A)$.

######  Definition 7.2.

The _Johnson filtration modulo $A$_ of $\mathrm{Aut}(\Gamma)$ consists of
subgroups

$\ldots\leq J^{A}(k)\leq J^{A}(k-1)\leq\ldots\leq
J^{A}(0)=\mathrm{Aut}(\Gamma)$

where $J^{A}(k)$ is the pointwise stabilizer of the quotient ring
$A[\Gamma]/I^{k+1}$.

Since $\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k}$ injects into the functionals
$\text{Hom}_{A}(A[\Gamma]/I^{k+1},A)$, it follows that any $\varphi\in
J^{A}(k)$ defines a pullback
$\varphi^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
k+1}\to\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k+1}$ fixing
$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k}$. Therefore, for every $\varphi\in
J^{A}(k)$ the assignment

$\tau_{\varphi}\colon T\longmapsto T-\varphi^{*}(T)$

induces is an $A$-linear map

$\tau_{\varphi}\colon\nicefrac{{\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
k+1}}}{{\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
k}}}\to\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k+1},$

and by the weight spectral sequence (Lemma 4.11), its domain is naturally
isomorphic to $K_{k+1}\leq\mathrm{H}^{1}(\Gamma;A)^{\otimes k+1}$ – the locus
of definition and vanishing of the $(k+1)$-st Massey product.

######  Proposition 7.3.

The image of $\tau_{\varphi}$ has weight $1$, and is therefore an element of
$\mathrm{H}^{1}(\Gamma;A)$.

###### Proof.

It suffices to prove that $T-\varphi^{*}(T)$ is killed by the reduced
coproduct $\overline{\Delta}$.

Suppose $\overline{\Delta}(T)=\sum T_{1}^{i}\otimes T_{2}^{i}$, then

$\overline{\Delta}(T-\varphi^{*}(T))=\sum T_{1}^{i}\otimes
T_{2}^{i}-\varphi^{*}(T_{1}^{i})\otimes\varphi^{*}(T_{2}^{i})=\sum(T_{1}^{i}-\varphi^{*}(T_{1}^{i}))\otimes
T_{2}^{i}+\varphi^{*}(T_{1}^{i})\otimes(T_{2}-\varphi^{*}(T_{2}^{i})).$

But since both $T_{1}^{i}$ and $T_{2}^{i}$ have weight $\leq k$, they are
fixed by $\varphi^{*}$ and the coproduct vanishes. ∎

######  Proposition 7.4.

For $k\geq 1$, the assignment $\varphi\mapsto\tau_{\varphi}$ is a group
homomorphism $J^{A}(k)\to\text{Hom}_{A}(K_{k+1},H)$.

###### Proof.

Let $\varphi,\psi\in J^{A}(k)$ be two automorphisms of $\Gamma$. Then

$\tau_{\varphi\psi}=T-(\varphi\psi)^{*}(T)=T-\psi^{*}\varphi^{*}(T)=T-\psi^{*}(T)+\psi^{*}(T-\varphi^{*}(T)).$

But since $T-\varphi^{*}(T)$ has weight $1$, it is fixed by $\psi^{*}$ and we
get $\tau_{\varphi\psi}=\tau_{\psi}+\tau_{\varphi}$. ∎

######  Proposition 7.5.

The Johnson homomorphism $J^{A}(k)\to\text{Hom}_{A}(K_{k+1},\mathrm{H})$ is
$\mathrm{Aut}(\Gamma)$-equivariant, where the action on $J^{A}(k)$ is by
conjugation.

###### Proof.

For any $\chi\in\mathrm{Aut}(\Gamma)$,

$\tau_{\chi\varphi\chi^{-1}}(T)=T-(\chi^{-1})^{*}\varphi^{*}\chi^{*}(T)=(\chi^{-1})^{*}(\chi^{*}(T)-\varphi^{*}(\chi^{*}(T)))=(\chi^{-1})^{*}\tau_{\varphi}(\chi^{*}(T))$

which is exactly the action of $\chi$ on $\text{Hom}_{A}(K_{k+1},\mathrm{H})$.
∎

######  Proposition 7.6.

The kernel of the Johnson homomorphism
$J^{A}(k)\to\text{Hom}_{A}(K_{k+1},\mathrm{H})$ contains $J^{A}(k+1)$. When
$\Gamma$ is finitely generated and $A[\Gamma]/I^{k+1}$ is $A$-torsion free
(e.g. when $A$ is a field), this is the entire kernel.

###### Proof.

Any $\varphi\in J^{A}(k+1)$ stabilizes all of
$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k+1}$ and thus $\tau_{\varphi}=0$.
Conversely, if $\varphi\in J^{A}(k)$ has $\tau_{\varphi}=0$ then
$T=\varphi^{*}(T)$ for all $T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k}$. We
claim that, for $\Gamma$ finitely generated with $A[\Gamma]/I^{k+1}$ is
$A$-torsion-free, $\varphi$ fixes all of $A[\Gamma]/I^{k+1}$.

Indeed, since $\Gamma$ is finitely generated, the $A$-module
$A[\Gamma]/I^{k+1}$ is also finitely generated and we assume it to be torsion-
free, thus it is in fact free. If $x\neq\varphi_{*}(x)\in A[\Gamma]/I^{k+1}$,
then there exists a functional $f\colon A[\Gamma]/I^{k+1}\to A$ such that
$f(x-\varphi(x))\neq 0$. But, by completeness of letter-braiding, there exists
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k}$ such that $\langle
T,\bullet\rangle=f$. Now,

$f(x)=\langle T,x\rangle=\langle\varphi^{*}(T),x\rangle=\langle
T,\varphi_{*}(x)\rangle=f(\varphi_{*}(x))$

contradicting our choice of $f$. It therefore must be that $\varphi_{*}=Id$ on
$A[\Gamma]/I^{k+1}$. ∎

Together, these propositions prove Theorem 1.9. We apply this theory to finite
$p$-groups.

######  Lemma 7.7.

Let $p$ be a prime number and $\Gamma$ a finite $p$-group. Then for every
$k\geq 1$, the quotient
$\nicefrac{{J^{\mathbb{F}_{p}}(k)}}{{J^{\mathbb{F}_{p}}(k+1)}}$ is an
elementary abelian $p$-group.

###### Proof.

$\Gamma$ is finitely generated and $\mathbb{F}_{p}$ is a field, so the Johnson
homomorphism gives an embedding

$\nicefrac{{J^{\mathbb{F}_{p}}(k)}}{{J^{\mathbb{F}_{p}}(k+1)}}\hookrightarrow\text{Hom}_{\mathbb{F}_{p}}(K_{k+1},H)$

whose domain is an $\mathbb{F}_{p}$-vector space. It follows that the quotient
is elementary abelian. ∎

######  Lemma 7.8.

For $\Gamma$ a finite $p$-group, there exists $k\geq 1$ such that
$J^{\mathbb{F}_{p}}(k+1)=\\{1\\}\leq\mathrm{Aut}(\Gamma)$. The same holds for
any $p$-nilpotent group, i.e. one for which the lower-exponent $p$-central
series terminates, with $k$ being the $p$-nilpotency class.

###### Proof.

Fix $x\in\Gamma$ and recall that $(x-1)\in I^{k}\lhd\mathbb{F}_{p}[\Gamma]$ if
and only $x$ belongs to $\gamma_{k}(\Gamma;\mathbb{F}_{p})$, the $k$-th term
in the lower-exponent $p$-central series for $\Gamma$ (see 2.2).

Suppose $\gamma_{k+1}(\Gamma;\mathbb{F}_{p})=\\{1\\}$. Then if
$\varphi\in\mathrm{Aut}(\Gamma)$ acts trivially on
$\mathbb{F}_{p}[\Gamma]/I^{k+1}$, it follows that for all $x\in\Gamma$,

$(\varphi(x)-x)\in I^{k+1}\implies(\varphi(x)x^{-1}-1)\in
I^{k+1}\implies\varphi(x)x^{-1}\in\gamma_{k+1}(\Gamma;\mathbb{F}_{p})=\\{1\\}$

so $\varphi(x)=x$ and thus $\varphi=1\in\mathrm{Aut}(\Gamma)$. ∎

With this, we complete the proof of Theorem 1.10.

###### Proof of Theorem 1.10.

The mod $p$ Johnson filtration

$\\{1\\}\leq J^{\mathbb{F}_{p}}(k)\leq\ldots\leq J^{\mathbb{F}_{p}}(1)\leq
J^{\mathbb{F}_{p}}(0)=\mathrm{Aut}(\Gamma)$

has finite length and all consecutive quotients below $J^{\mathbb{F}_{p}}(1)$
are elementary abelian $p$-groups. It follows that $J^{\mathbb{F}_{p}}(1)$ is
itself a finite $p$-group.

In particular, it follows that any $\varphi\in\mathrm{Aut}(\Gamma)$ that does
not have $p$-power order cannot belong to $J^{\mathbb{F}_{p}}(1)$, implying
that it acts nontrivially on $\mathbb{F}_{p}[\Gamma]/I^{2}$. But then, by the
same argument as in the proof of the last lemma, there exists some
$x\in\Gamma$ such that
$\varphi(x)x^{-1}\notin\gamma_{2}(\Gamma;\mathbb{F}_{p})=[\Gamma,\Gamma]\Gamma^{p}$,
so $\varphi(x)\neq x$ in the quotient $\Gamma^{ab}\otimes\mathbb{F}_{p}$. ∎

###  7.3. Massey products of finite groups

For finite groups, our theory leads to the following fact about Massey
products in $\mathrm{H}^{1}(-;\mathbb{F}_{p})$.

######  Theorem 7.9.

Let $\Gamma$ be a finite group, and $p$ any prime number such that
$\mathrm{H}^{1}(\Gamma;\mathbb{F}_{p})\neq 0$. Then
$\mathrm{H}^{2}(\Gamma;\mathbb{F}_{p})\neq 0$ and nontivial Massey products
$\mu_{k}\colon\mathrm{H}^{1}(\Gamma;\mathbb{F}_{p})^{\otimes
k}\to\mathrm{H}^{2}(\Gamma;\mathbb{F}_{p})$ exist.

Quantitatively, denote
$b_{1}=\dim_{\mathbb{F}_{p}}\mathrm{H}^{1}(\Gamma;\mathbb{F}_{p})$, if $k\geq
2$ is the length shortest nontrivial Massey product

$0\neq\langle\alpha_{1},\ldots,\alpha_{k}\rangle\in\mathrm{H}^{2}(\Gamma;\mathbb{F}_{p}),$

which is necessarily well-defined with no indeterminacy, then

(20) $k<\begin{cases}|\Gamma|+1&b_{1}=1\\\
\log_{b_{1}}|\Gamma|+2&b_{1}>1.\end{cases}$

###### Proof.

Since $\mathbb{F}_{p}[\Gamma]$ is an $\mathbb{F}_{p}$-vector space of
dimension $|\Gamma|$, and the letter-braiding invariants
$\mathrm{H}^{0}_{Bar}(\Gamma;\mathbb{F}_{p})$ inject into its linear dual,
they must also have dimension $\leq|\Gamma|$. Now, suppose $k\geq 2$ is the
smallest length of a nontrivial Massey product in
$\mathrm{H}^{1}(\Gamma;\mathbb{F}_{p})$. Then by the weight spectral sequence,

$\mathrm{gr}~{}\mathrm{H}^{0}_{Bar}(\Gamma;\mathbb{F}_{p})_{\leq
k-1}\cong\mathbb{F}_{p}\oplus\mathrm{H}^{1}\oplus(\mathrm{H}^{1})^{\otimes
2}\oplus\ldots\oplus(\mathrm{H}^{1})^{\otimes k-1},$

so in particular
$|\Gamma|\geq\dim_{\mathbb{F}_{p}}\mathrm{H}^{0}_{Bar}(\Gamma;\mathbb{F}_{p})\geq
1+b_{1}+b_{1}^{2}+\ldots+b_{1}^{k-1}$. The claimed inequalities for $k$
follow. ∎

######  Example 7.10 (Cyclic $p$-groups).

Let $\Gamma=C_{p}$ by the cyclic group of prime order $p$ and take
$A=\mathbb{F}_{p}$. Then $C^{1}(BC_{p};\mathbb{F}_{p})=\mathbb{F}_{p}^{C_{p}}$
is spanned by the power functions $m_{k}\colon C_{p}\to\mathbb{F}_{p}$,

$m_{k}(i)=i^{k},$

for $0\leq k\leq p-1$. Only $m_{1}$ is a group homomorphism, and accordingly,
$H^{1}(C_{p};\mathbb{F}_{p})=\mathbb{F}_{p}\cdot m_{1}$.

One can check directly from the definitions that the Massey products of
$m_{1}$ with itself are

$\langle\underbrace{m_{1},\ldots,m_{1}}_{k\text{
times}}\rangle=\frac{1}{k!}\sum_{j=1}^{k-1}\frac{k!}{j!(k-j)!}m_{j}\smile
m_{k-j}=d\left(\frac{1}{k!}m_{k}\right)$

so vanish in cohomology. They thus define letter-braiding invariants

$I_{k}:=\frac{1}{k!}\sum_{\begin{subarray}{c}i_{1}+\ldots+i_{r}=k\\\ \forall
j,\,i_{j}>0\end{subarray}}\frac{k!}{i_{1}!\cdots
i_{r}!}(m_{i_{1}}|m_{i_{2}}|\ldots|m_{i_{r}})=(\underbrace{m_{1}|\ldots|m_{1}}_{k\text{
times}})+\text{(lower weight terms)}.$

This invariant pairs with $C_{p}$ as

$\langle I_{k},i\rangle=\frac{i^{k}}{k!}\mod{p},$

which by the multinomial theorem is independent of a presentation of $i$ as
$a_{1}+\ldots+a_{n}$.

As for the $p$-th Massey power of $m_{1}$, observe that $m_{p}=(-)^{p}$ is
already a group homomorphism555In fact $m_{p}=m_{1}$ by Fermat’s little
theorem, i.e. $dm_{p}=0$, allowing the $p$-th Massey power of $m_{1}$ to be
nontrivial.

Similar analysis for the group $C_{p^{2}}$ shows that now the $p$-th Massey
power of $m_{1}$ is $d\gamma_{1}$ for the divided power operation

$\gamma_{1}(i)=\frac{i^{p}-i}{p}\mod p.$

It then follows that the weight $p$ invariant of $C_{p^{2}}$ is exactly the
function $\gamma_{1}$, which detects $p$.

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