#  Letter-braiding: a universal bridge between combinatorial group theory and
topology

Nir Gadish  Department of Mathematics, University of Michigan, Ann Arbor, MI
<EMAIL_ADDRESS>
###### Abstract.

We define invariants of words in arbitrary groups, measuring how letters in a
word are interleaving, perfectly detecting the dimension series of a group.
These are the letter-braiding invariants. On free groups, braiding invariants
coincide with coefficients in the Magnus expansion. In contrast with such
coefficients, our invariants are defined on all groups and over any PID. They
respect products in the group and are complete with respect to the dimension
series, so serve as the coefficients of a universal multiplicative finite-type
invariant, depending functorially on the group.

Letter-braiding invariants arise out of analyzing the bar construction on a
cochain model of a space with a prescribed fundamental group. This approach
specializes to simplicial presentations of a group as well as to more
geometric contexts, which we illustrate.

As an application, we define a Johnson style filtration and a Johnson
homomorphism on the automorphisms of any group, and use it to constrain
automorphisms of finite $p$-groups.

###### Key words and phrases:

letter-braiding numbers, Hopf invariants of words in groups, universal finite
type invariant, Magnus expansion for arbitrary groups, Johnson homomorphism,
automorphisms of finite p-groups

###### 2020 Mathematics Subject Classification:

57M05 20F34 55Q25 57K16 20J05 16S34 20-08 20F36

##  1\. Introduction

How can one detect whether a particular element in a group belongs to the
$k$-th term in the lower central series? In this work we describe a collection
of topologically motivated invariants of words in arbitrary groups, called
_letter-braiding invariants_. They are explicitly computable, from the
sequences of letters defining the word. The invariants distinguish group ring
elements over any PID up to powers of the augmentation ideal, and thus give
obstructions to being a $k$-fold commutator.

The story of these invariants starts with Monroe–Sinha [MS22], who based on
geometric constructions in rational homotopy theory observed that linking
numbers make sense for letters in words $w\in F_{n}$, the free group on $n$
generators. For example, the commutator $(xyx^{-1}y^{-1})$ has $x$’s linking
with $y$’s once. They define combinatorial linking of letters, with $k$-fold
linking invariants defined on the $k$-th term in the lower central series
$\gamma_{k}F_{n}$, and pairing perfectly with
$\nicefrac{{\gamma_{k}F_{n}}}{{\gamma_{k+1}F_{n}}}\otimes\mathbb{Q}$. They
furthermore show that, when defined, their linking invariants agree with
coefficients in the classical Magnus expansion of $F_{n}$ [Mag54].

Below, we give an “open-string” generalization of these letter-linking
invariants, which we therefore call _letter-braiding_ 111Recall that links are
closures of braids – the former consisting of closed strings while the latter
having open strands. Similarly, Monroe–Sinha’s letter-linking is defined when
all ‘strings of letters’ are closed, while we consider strands with ends..
These vastly extend the scope of Mornoe–Sinha’s letter-linking, as well as
Magnus’ expansion.

###  1.1. Main Results

We use constructions that are standard in algebraic topology. Let $\Gamma$ be
a group and let $X$ be any simplicial set whose fundamental group is $\Gamma$.
In applications, choices of $X$ can range from small combinatorial models to
functorial models to the singular simplicial set of a topological space. Let
$A$ be a PID.

The $A$-valued letter-braiding invariants of $\Gamma$ are parametrized by the
$0$-th (associative) Bar cohomology of $C^{*}_{\cup}(X;A)$, the simplicial
cochains on $X$ with the associative cup product and $A$-coefficients,

$\mathrm{H}^{0}_{Bar}(X;A):=\mathrm{H}^{0}\left(Bar(C^{*}_{\cup}(X;A))\right).$

See §2.1 for definitions.

Every $T\in\mathrm{H}^{0}_{Bar}(X;A)$ defines an $A$-valued invariant of
$w\in\Gamma$, as follows. Representing $w$ by a map $\tilde{w}\colon S^{1}\to
X$, it defines a pullback on Bar constructions

$\tilde{w}^{*}\colon\mathrm{H}^{0}_{Bar}(X;A)\to\mathrm{H}^{0}_{Bar}(S^{1};A).$

We give an explicit weight-reduction algorithm in Proposition 3.19, inspired
by Monroe–Sinha’s linking of letters [MS22], which defines an isomorphism
$L\colon\mathrm{H}^{0}_{Bar}(S^{1};A)\xrightarrow{\sim}A[t].$ Our letter-
braiding invariants are the composite $L\circ\tilde{w}^{*}$.

Thus $T\in\mathrm{H}^{0}_{Bar}(X;A)$ determines a polynomial invariant
$L_{T}(w)=L(\tilde{w}^{*}(T))\in A[t]$. The linear coefficient of the
polynomial $L_{T}(w)$ is the _letter-braiding number_ $\ell_{T}(w)$. This
collection of invariants $\mathrm{H}^{0}_{Bar}(X;A)$ depends only on the group
$\Gamma=\pi_{1}(X,*)$, as explained in §4, and so is denoted
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$.

Together, letter-braiding numbers assemble to an $A$-linear pairing with the
group ring $A[\Gamma]$, the _letter-braiding pairing_

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

When $\Gamma$ is presented by generators and relations, computing both
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$ and its pairing on words is given by
explicit, simple algorithms. And when $\Gamma$ is the fundamental group of
some manifold or variety, braiding invariants exhibit compelling geometric
interpretations.

The letter-braiding pairing matches the natural structures on $A[\Gamma]$ and
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$. Indeed, recall that $A[\Gamma]$ is a ring
under the convolution product, and it is furthermore filtered by powers of the
augmentation ideal $I=\ker(\epsilon\colon A[\Gamma]\to A)$. This filtered
algebra structure is functorial with respect to group homomorphisms.

Our fundamental result, proved in §5 and §6, is the following.

######  Theorem 1.1.

For any group $\Gamma$, its Bar cohomology $\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is
a filtered coalgebra, functorially in $\Gamma$. Under the letter-braiding
pairing, this structure is dual to the filtered algebra structure on
$A[\Gamma]$, as follows.

  1. (1)

Functoriality: for any group homomorphism $h\colon\Gamma_{1}\to\Gamma_{2}$,

$\langle h^{*}(T),w\rangle=\langle T,h_{*}(w)\rangle.$

  2. (2)

Product-coproduct duality222Thinking of letter-braiding numbers as
coefficients in a power series expansion, product–coproduct duality is
equivalent to the statement that this expansion is _multiplicative_. E.g. the
Kontsevich integral on braids is multiplicative, while the Mostovoy–Willerton
$\mathbb{Z}$-valued expansion is not. The $\mathbb{Z}$-valued braid invariant
produced by our procedure would be multiplicative, but it would take some work
to find its indeterminants – see Example 4.21 for coefficients of degree $\leq
3$.: defining $\langle T_{1}\otimes T_{2},x\otimes y\rangle=\langle
T_{1},x\rangle\langle T_{2},y\rangle$, the coproduct $\Delta$ on
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is dual to the convolution product on
$A[\Gamma]$,

$\langle\Delta T,x\otimes y\rangle=\langle T,x\cdot y\rangle.$

  3. (3)

Filtrations: let $I^{n}\leq A[\Gamma]$ denote the $n$-th power of the
augmentation ideal, then

$(I^{n})^{\perp}=\mathrm{H}^{0}_{Bar}(\Gamma;A)_{<n},$

the sub-coalgebra of weight $<n$ tensors.

  4. (4)

Completeness/Universal finite-type invariant: when $\Gamma$ is finitely
generated, the pairing defines an isomorphism of filtered coalgebras

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

and it is injective without assuming finite generation (see Remark 6.2).

  5. (5)

Extending Magnus: when $\Gamma$ is a free group, the letter-braiding pairing
coincides with coefficients in the classical Magnus expansion (see Example
1.2).

To elaborate point (4), recall that in braid theory, an invariant of the braid
group $PB_{n}$ has _finite type_ (it is a Vassiliev invariant) whenever it
vanishes on some power of $I\leq\mathbb{Z}[P_{n}]$ (see [MW02]). Extending
this to general groups, property (3) says that every
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ defines a finite type invariant
$\ell_{T}(\bullet)\colon\Gamma\to A$. Furthermore, a finite type invariant
$\Psi\colon\Gamma\to R$ is _universal_ if every finite type invariant factors
uniquely through $\Psi$. Property (4) says that every $A$-valued finite type
invariant is of the form $\ell_{T}(\bullet)$ for a unique
$T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ whenever $\Gamma$ is finitely generated,
exhibiting universality.

A generalization of Magnus expansion to all groups has until now been elusive,
and was only known in a small number of special classes of groups that are
close to free groups. Right-angled Artin groups (RAAGs) are partially free and
partially commutative, and a Magnus expansion was worked out by Duchamp–Krob
[DK92]. Braid groups are semi-direct products of free groups, and Magnus
expansions over $\mathbb{Z}$ were found by Mostovoy–Willerton [MW02] and
Papadima [Pap02]. Their expansions are not _multiplicative_ in that they do
not respect the product structure on braids. The best previous result along
these lines applying to general groups might still be that of Chen–Fox–Lyndon
[CFL58], which gave a word-rewriting algorithm to compute lower central series
of groups from a presentation. They build on Fox calculus as a dual basis for
the lower central series Lie algebra of free groups. Their algorithm produces
functionals on the lower central series subquotients of a group, but it is
inductive with equations that must be solved at each stage.

The construction in this paper explains why expansions for arbitrary groups
have been hard to find, by exhibiting the obstruction: as discussed in §4.1,
expansion coefficients of degree $k$ require knowledge of all Massey products
$\mu_{n}\colon\mathrm{H}^{1}(\Gamma;A)^{\otimes n}\to\mathrm{H}^{2}(\Gamma;A)$
of size $n\leq k$, and finding combinations of tensors such that all these
products vanish. RAAGs and braid groups happen to be formal over $\mathbb{Q}$,
i.e. their higher Massey products vanish, which is why their expansions were
easier to find.

We discuss and compare with other work throughout, but broadly speaking
highlight the following advantages of our approach.

  * •

General groups: every group $\Gamma$ can be probed by a collection of letter-
braiding invariants, which are well-defined even on non-free or infinitely
generated groups.

  * •

Globally defined: all letter-braiding invariants are well-defined on the
entire group, not just high-order commutators or elements of dimension
subgroups.

  * •

Cohomological interpretation: letter-braiding invariants are built from
combinations of classes in $\mathrm{H}^{1}(X;A)$, for any space with
$\pi_{1}(X,*)\cong\Gamma$, and thus carry topological meaning alongside their
combinatorial manifestation.

  * •

Coefficients in PIDs: invariants can be made to take values in any PID, and
thus detect torsion elements.

###### Contents

  1. 1 Introduction
    1. 1.1 Main Results
    2. 1.2 First examples, motivation and background
    3. 1.3 Current and planned applications
      1. 1.3.1 Combinatorial group theory
      2. 1.3.2 Geometric topology
      3. 1.3.3 Positive characteristic
      4. 1.3.4 Relative Massey products vs. relations
      5. 1.3.5 Algebraic models for spaces
      6. 1.3.6 Johnson homomorphisms
    4. 1.4 Related work
    5. 1.5 Acknowledgements
  2. 2 (Co)algebraic preliminaries
    1. 2.1 Bar construction
    2. 2.2 Group rings
    3. 2.3 Tensors and infinite cartesian powers
  3. 3 Free groups
    1. 3.1 Topological construction
    2. 3.2 Weight reduction algorithm
    3. 3.3 Properties of letter-braiding
  4. 4 General groups
    1. 4.1 Letter braiding and vanishing (Massey) products
    2. 4.2 Properties of letter-braiding
  5. 5 Group rings
    1. 5.1 Fox calculus and Magnus expansion
  6. 6 Completeness of letter-braiding
    1. 6.1 Introducing relations
    2. 6.2 Defect and obstruction to lifting
    3. 6.3 Consequences of the lifting criterion
    4. 6.4 Relative Massey products measure relations
  7. 7 Topological applications
    1. 7.1 Cochains determine fundamental group ring
    2. 7.2 Johnson homomorphisms
    3. 7.3 Massey products of finite groups

###  1.2. First examples, motivation and background

Letter-braiding invariants are motivated by intersection theory in the
geometry of loop spaces. If $M$ is a manifold, classes
$\alpha\in\mathrm{H}^{1}(M;\mathbb{Q})$ are represented by counting transverse
intersections of loops with real hypersurfaces in $M$. Given a sequence of
such hypersurfaces, one can also track the order in which intersections occur
and count series of intersections with prescribed orders – this is morally
what letter-braiding invariants are measuring. Such ordered intersection
counts are generally not homotopy invariant, as the hypersurfaces can
intersect. But one can find linear combinations of counts which are invariant
through analysis of such intersections, which encode cup products and Massey
products.

The apposite algebraic framework for this analysis is the classical bar
construction, which has been known to model cochains on (the completion of)
the loop space for over sixty years [Ada56, EM66, Dwy75a]. From this
perspective one is studying the fundamental group through the zeroth
cohomology of the loop space.

In this introduction we mention many classical results along these lines, but
few of those gave rise to invariants on groups that are explicit and
computable enough to, say, be calculated by a computer. Letter-braiding
invariants achieve this: they gives concrete and fast algorithms for obtaining
both $\mathrm{H}^{0}_{Bar}(\Gamma;A)$ and its letter-braiding pairing with
$\Gamma$, as currently being implemented on Sage by Aydin Ozbek.

######  Example 1.2.

The free group $F_{n}=\langle x_{1},\ldots,x_{n}\rangle$ is the fundamental
group of a wedge

$W_{n}=\bigvee_{i=1}^{n}S^{1}$

whose $1$-cochains are $C^{1}(W_{n};A)=A^{n}$ with trivial cup product. Its
letter-braiding invariants are thus parametrized by the tensor (co)algebra

$T(A^{n})=A\langle X_{1},\ldots,X_{n}\rangle$

of noncommutative polynomials on the standard basis elements
$X_{1},\ldots,X_{n}\in A^{n}$.

For $F_{2}=\langle x,y\rangle$, a visual depiction of the weight reduction
algorithm in Proposition 3.19 for the invariant $XYXX$ of the word
$w=[xy,x^{-2}]=xyx^{-1}x^{-1}y^{-1}x$ is

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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{aligned}\implies\ell_{XYXX}([xy,x^{-2}])=-1$

computing the higher braiding number “$x$’s after $y$’s after $x$’s after
$x$’s”, and detecting that $w$ is not a $4$-fold nested commutator (Example
3.20 gives details of the algorithm).

Theorem 5.9 shows that for every word $w\in F_{n}$, the letter-braiding number
is

$\ell_{X_{i_{1}}X_{i_{2}}\cdots
X_{i_{k}}}(w)=\epsilon\left(\frac{\partial}{\partial
x_{i_{1}}}\cdots\frac{\partial}{\partial x_{i_{k}}}w\right),$

where $\frac{\partial}{\partial x_{i}}$ is the Fox derivative of the group
ring $A[F_{n}]$ from [Fox53] and $\epsilon\colon A[F_{n}]\to A$ is the
standard augmentation map. Equivalently, the letter-braiding number labeled by
$X_{i_{1}}\cdots X_{i_{k}}$ coincides with the coefficient of $t_{i_{1}}\ldots
t_{i_{k}}$ in the classical Magnus expansion for $F_{n}$ (see [Mag54]). In
this way, letter-braiding invariants are a posteriori seen to be functorial
extensions of the classical Magnus expansion to arbitrary groups and
coefficients in any PID. At the same time, they are substantially easier to
calculate than either Fox or Magnus.

Letter-braiding can also be presented as a _discrete Chen integral_ , see
Corollary 3.25.

######  Example 1.3 (Torsion commutators).

Let $\Gamma=\mathscr{H}_{3}(\mathbb{F}_{2})$ be the Heisenberg group of
matrices

$\begin{pmatrix}1&X&Z\\\ 0&1&Y\\\
0&0&1\end{pmatrix}\in\operatorname{Gl}_{3}(\mathbb{F}_{2}).$

Then $X,Y,Z\colon\mathscr{H}_{3}(\mathbb{F}_{2})\to\mathbb{F}_{2}$ determine
cochains in $C^{1}(\mathscr{H}_{3}(\mathbb{F}_{2});\mathbb{F}_{2})$, with $X$
and $Y$ closed and $X\smile Y=dZ$. Thus
$XY+Z\in\mathrm{H}^{0}_{Bar}(\mathscr{H}_{3}(\mathbb{F}_{2});\mathbb{F}_{2})$
defines an invariant on $\mathscr{H}_{3}(\mathbb{F}_{2})$, and Example 5.11
shows that it pairs nontrivially with the commutator $E_{13}=[E_{12},E_{23}]$.

With this, letter-braiding has a similar flavor to Chen’s theory of iterated
integrals [Che71], but our invariants can detect torsion elements.

A direct predecessor of this work, Sinha–Walter [SW13] use the Bar
construction, and a variant for commutative algebras called the _Harrison
complex_ , to define define Hopf invariants of maps $\alpha\colon S^{n}\to X$
with $n\geq 2$ using higher linking numbers in spheres $S^{n}$. Their Hopf
invariants distinguish rational homotopy classes of maps into simply connected
spaces. The present paper grew out of a project which will define Hopf
invariants of maps $S^{1}\to X$ from linking in $S^{1}$, joint with A. Ozbek,
D. Sinha, and B. Walter [GOSW23]. That theory and the present one are
complementary, and some of the benefits of restricting focus to characteristic
zero, and making use of commutativity, are the following.

  * •

Hopf invariants detect and obstruct the Malcev Lie algebra of a group,
focusing directly on commutators, rather than the $\mathbb{Q}$-dimension
series. The Harrison complex is smaller, and is naturally a Lie coalgebra. As
such, it removes the redundancy on functionals on the group ring when
restricted to primitives, and computes a maximal Lie coalgebraic dual of the
Malcev Lie algebra of any $\Gamma$.

  * •

The theory is naturally geometric: de Rham forms supported near submanifolds
(a.k.a. Thom forms) realize integration and wedge products as intersection of
submanifolds. As such, letter-braiding theory is restated in terms of the
meeting times of a smooth loop with with various hypersurfaces, as described
above.

  * •

Hopf invariants specialize to Chen’s [Che71] theory of iterated integrals on
the one hand while on the other hand having a combinatorial formulation
relative to group presentations.

The resulting ‘commutative’ letter-braiding numbers differ subtly from ones
studied here based on simplicial cochains with their noncommutative cup
product. We prove that the former are obtained from the latter as the
invariants of $\log(w)=\sum_{n\geq 1}\frac{(-1)^{n+1}}{n}(w-1)^{n}$, as
expected from functionals on the Malcev Lie algebra.

Topological applications of this theory include the following (see [GOSW23]
for terminology).

######  Proposition 1.4.

Milnor invariants of links are computed from the intersections of a link
component with any collection of Seifert surfaces of the other components
along with further secondary surfaces: they are the letter-braiding numbers of
the intersection pattern.

Moreover, these invariants are defined past the weight which Milnor initially
considered, and for any element in the link group.

Here, unlike Milnor’s setup, Seifert surfaces are allowed to intersect, at the
cost of adding secondary surfaces. This additional freedom leads to
algorithmic calculations of Milnor invariants from a planar projection of the
link.

###  1.3. Current and planned applications

Since the invariants $T\in\mathrm{H}^{0}_{Bar}(\Gamma;A)$ are determined by
any space or simplicial set $X$ with $\pi_{1}(X,*)\cong\Gamma$, different
choices of model give invariants of distinct flavors, blending Algebra,
Combinatorics, Geometry and Topology.

####  1.3.1. Combinatorial group theory

A presentation $\Gamma=\langle S|R\rangle$ determines a $2$-complex $X_{S|R}$
whose $1$-cells are labeled by $S$ and $2$-cells come from relations $R$. Its
letter-braiding invariants are therefore presented in terms of the generators
and relation, as follows.

For any set $S$, let $F_{S}$ denote the free group generated by $S$. Its
letter-braiding invariants are computed from the wedge $X_{S}=\vee_{s\in
S}S^{1}$ and are simply the tensor (co)algebra

$T(A^{S})=\bigoplus_{k\geq 0}(A^{S})^{\otimes k}.$

As mentioned, letter-braiding numbers of words $w\in F_{S}$ are coefficients
of the classical Magnus expansion, as shown in Theorem 5.9. We thus think of
them as Fox’s differential operators: $f\in A^{S}\longleftrightarrow\sum_{s\in
S}f(s)\frac{\partial}{\partial s}$.

Now, the inclusion $X_{S}\hookrightarrow X_{S|R}$ identifies
$\mathrm{H}^{0}_{Bar}(\Gamma;A)$ as a sub-coalgebra of $T(A^{S})$, consisting
of possibly nonhomogeneous differential operators, so that the letter-braiding
numbers of $w\in\langle S|R\rangle$ are given by the Magnus expansion
coefficients of any lift $\tilde{w}\in F_{S}$.

But to yield well-defined invariants on $\langle S|R\rangle$, we may only
consider differential operators that happen to kill all relations $\langle
R\rangle\leq F_{S}$. Examples 4.5,4.6 discuss basic cases of relations
constraining the collection of invariants. In fact, we show that vanishing on
relations, suitably understood, is the only constraint.

######  Theorem 1.5.

For $\Gamma=\langle S|R\rangle$, the quotient map $p\colon
F_{S}\twoheadrightarrow\Gamma$ induces an inclusion

$p^{*}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)\hookrightarrow T(A^{S})$

whose image is the maximal sub-coalgebra of differential operators that vanish
on all $r\in R$.

In other words, a differential operator must vanish on all relations, and so
must all of its iterated coproducts. Vanishing of the operator alone does not
suffice. See Theorem 6.3 for a precise and applicable generalization of this
theorem. With it, one can compute the letter-braiding invariants of $\Gamma$
as follows: start with the linear differential operators that vanish on $r\in
R$; then find among their degree $2$ compositions those linear combinations
that continue to vanish on $r\in R$; and so on.

####  1.3.2. Geometric topology

Suppose $\Gamma$ arises as the fundamental group of a manifold $M$ rather than
from a presentation. Then its letter-braiding invariants are expressible in
geometric terms. Examples include surface groups, Artin groups, knot and link
groups and more general 3-manifold groups, and mapping-class groups (up to
finite index).

Lemma 4.11 recounts that $\mathrm{H}^{0}_{Bar}(M;A)$ is determined by sums of
tensors of cohomology classes $\mathrm{H}^{1}(M;A)^{\otimes k}$ all of whose
cup and higher Massey products vanish – see Examples 4.9,4.10. By Poincaré
duality, classes $\alpha\in\mathrm{H}^{1}(M;A)$ are represented by
intersection with a dual hypersurface $H_{\alpha}\subset M$. A word
$w\in\Gamma$ is represented by some loop in $M$, which intersects those
hypersurfaces at various times. The letter-braiding numbers of $w$ are then
determined by the interleaving of these intersections – see §3.2.

However, a different choice of representative loop for $w$ might meet the
hypersurfaces in a different order. The condition of vanishing cup and Massey
products thus turns out to be equivalent to the resulting letter-braiding
number being homotopy invariant. But since cup products are dual to
intersection, relations in $\pi_{1}(M,*)$ restrict the collections of
hypersurfaces that may define invariants, and therefore force nontrivial
intersection patterns of hypersurfaces.

For example, fundamental group of the Borromean ring complement is

$\Gamma_{3}=\langle
x,y,z\mid[x,[y^{-1},z]],[y,[z^{-1},x]],[z,[x^{-1},y]]\rangle.$

whose double-commutator relations are detected by the invariant $XYZ$ of
$F_{3}=\langle x,y,z\rangle$, reflecting a triple Massey product $\langle
X,Y,Z\rangle\neq 0$. Geometrically, $X,Y$ and $Z$ are dual to Seifert surfaces
of the three link components. Thus, the Massey product implies that no choice
of Seifert surfaces can be made disjoint.

####  1.3.3. Positive characteristic

Taking coefficients rings $A$ of positive charactertistic, letter-braiding can
detect torsion elements and commutators in finite groups. Recall that the
lower exponent $p$-central series of a group $\Gamma$ is the filtration
induced by $w\in F_{k}\Gamma\iff(w-1)\in I^{k}\leq\mathbb{F}_{p}[\Gamma]$. It
follows that $\mathbb{F}_{p}$-valued letter-braiding invariants obstruct this
$p$-central series. They are particularly informative when $\Gamma$ is a
finite $p$-group, since in this case the lower $p$-central series terminates
and, therefore, any two elements $w\neq w^{\prime}\in\Gamma$ can be
distinguished by letter-braiding invariants.

Conversely, the structure of $\mathbb{F}_{p}[\Gamma]$ provides information
about Massey products in $\mathbb{F}_{p}$-cohomology.

######  Example 1.6.

For $\Gamma=C_{p}$, the cyclic group of order $p$, the group ring is
$p$-dimensional, while
$\mathrm{H}^{1}(C_{p};\mathbb{F}_{p})=\mathbb{F}_{p}\cdot T$ generated by the
$\mathbb{F}_{p}$-group homomorphism $T(i)=i$. The braiding invariants for
$\Gamma$ are thus spanned by $T^{k}+(\text{lower weight correction terms})$
for $0\leq k\leq p-1$, see Example 7.10 for explicit cochain representatives
for which

$\langle T^{k}+\text{ (lower weights) },\,i\rangle=\frac{i^{k}}{k!}\mod{p}.$

That these already span the dual of $\mathbb{F}_{p}[C_{p}]$ means that $T^{p}$
does not extend to an invariant.

######  Corollary 1.7.

For any space $X$ with $\pi_{1}(X,*)\cong C_{p}$, the generator
$T\in\mathrm{H}^{1}(X;\mathbb{F}_{p})$ has vanishing cup and Massey powers up
to order $p-1$, but its $p$-th Massey power

$\langle\underbrace{T,\ldots,T}_{p\text{ times}}\rangle\neq 0.$

For $C_{p^{2}}$ the same $p$-fold Massey power is $0$, and $T^{p}$ detects the
nontrivial element $p\in C_{p^{2}}$.

See §7.3 for a generalization to all finite groups.

####  1.3.4. Relative Massey products vs. relations

The previous two subsections exhibit a dictionary between Magnus expansions of
relations in $\Gamma$ and Massey products on $\mathrm{H}^{1}(\Gamma;A)$. That
such a connection exists has been well-known, going back to Dwyer [Dwy75b] and
Fenn–Sjerve [FS87].

We complete this dictionary in §6.4 by defining _relative Massey products_.
Given a pair of spaces $(Y,X)$ there are higher product operations, that in
suggestive simple cases become

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

When $Y$ is obtained from $X$ by attaching ‘relation’ $2$-discs along a set
$R\subseteq\pi_{1}(X,*)$, the relative cohomology $\mathrm{H}^{2}(Y,X;A)\cong
A^{R}$ records functions on $R$, and Proposition 6.22 shows that the relative
Massey product $\mu_{n}$ computes the letter-braiding numbers of all $r\in R$.
Our setup thus vastly extends the picture painted by Fenn-Sjerve [FS87] in a
number of ways, listed at the end of §6.4.

####  1.3.5. Algebraic models for spaces

Mandell [Man06] showed that when $X$ is a nilpotent space of finite type, its
cochains $C^{*}(X;\mathbb{Z})$ with their $E_{\infty}$-algebra structure
determine the homotopy type of $X$, and in particular $\pi_{1}(X,*)$. More
constructively, building on Adams’ classical theorem [Ada56], Rivera–Zeinalian
[RZ18, Corollary 9.2] explains that the coBar construction on the singular
cochains $C_{*}^{Sing}(X)$ models the chains of the loop space $\Omega_{*}X$
and thus its $\mathrm{H}_{0}$ is computes the group ring
$\mathbb{Z}[\pi_{1}(X,*)]$. However, their proofs use rather abstract homotopy
theory, and do not provide any means for measuring words $w\in\pi_{1}(X,*)$.

In contrast, our proofs are essentially elementary, with algebraic rather than
homotopy theoretic flavor. Consequently, we get a simple and constructive
proof of the following.

######  Corollary 1.8 (Compare Mandell [Man06], and Dwyer [Dwy75a, Theorem
2.1]).

Let $(X,*)$ be a pointed connected simplicial set (or topological space) with
$\pi_{1}(X,*)$ finitely generated but perhaps infinitely related, and let $A$
be any PID. Then the filtered coalgebra of $A$-valued functionals of finite
type $\varinjlim\text{Hom}_{A}(A[\pi_{1}(X,*)]/I^{n},A)$ is determined by the
$2$-truncation of the augmented simplicial (or singular) cochain algebra
$C^{*\leq 2}_{\Delta}(X;A)$, via letter-braiding.

Compare this further to Rivera–Zeinalian [RZ21], who prove that the singular
chains of a connected space $X$ determine the entire fundamental group – no
nilpotency hypotheses necessary. However, one major obstacle in applying their
theory here is that one cannot simply pass to a small chain model for
calculations, as their construction is not quasi-isomorphism invariant
(instead, they use a finer notion of $\Omega$-equivalence).

####  1.3.6. Johnson homomorphisms

Work of Kawazumi [Kaw05] on Johnson filtrations indicates that Magnus
expansions define generalizations of the classical Johnson homomorphism of the
Torelli group. Since our setup is dual to Magnus expansions, and is defined
for general groups, it lets us define dual Johnson homomorphisms in general.

For any group $\Gamma$, the automorphism group $\mathrm{Aut}(\Gamma)$ is
filtered by subgroups $J^{A}(k)\leq\mathrm{Aut}(\Gamma)$ respectively fixing
$A[\Gamma]/I^{k+1}$ pointwise – a generalized Johnson filtration. Via the
letter-braiding pairing, one obtains a Johnson-type homomorphism on these
groups.

######  Theorem 1.9.

Denote $H=\mathrm{H}^{1}(\Gamma;A)$, and let $K_{n}\leq\mathrm{H}^{\otimes n}$
be the tensors on which the $n$-th Massey product is defined and vanishes.
Then for $\varphi\in J^{A}(k)$, the endomorphism $(T-\varphi^{*}(T))$ of
$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq k+1}$ determines an
$\mathrm{Aut}(\Gamma)$-equivariant ‘dual Johnson’ homomorphism

$J^{A}(k)\to\text{Hom}_{A}(K_{k+1},H),$

Furthermore, its kernel always contains $J^{A}(k+1)$, and is exactly this
subgroup whenever $\Gamma$ is finitely generated and $A[\Gamma]/I^{k+1}$ is
$A$-torsion free, e.g. when $A$ is a field.

With $A=\mathbb{Z}$ and $\Gamma=F_{n}$ a free group, $K_{k}=H^{\otimes k}$ and
the integral Johnson filtration $J^{\mathbb{Z}}(k)\leq\mathrm{Aut}(F_{n})$ is
the classical one. In this case our dual Johnson homomorphism maps
$J(k)\leq\mathrm{Aut}(F_{n})$ to $(H^{*})^{\otimes k+1}\otimes H$ (compare
this with the Johnson homomorphism in [Kaw05, Theorem 6.1]). And when
$\Gamma=\pi_{1}(\Sigma_{g},*)$, the mapping class group
$\operatorname{Mod}(\Sigma_{g,1})$ acts on $\pi_{1}(\Sigma_{g},*)$, sending
the classical Johnson filtration into the our mod $A$ Johnson filtration. We
have not found tools for calculating the dual Johnson homomorphism in any
case, and leave it as an open problem to relate it with the classical ones.

Theorem 1.9 is proved in §7.2. With it, we deduce the following for any prime
number $p$.

######  Theorem 1.10.

If $\Gamma$ is a finite $p$-group, then
$J^{\mathbb{F}_{p}}(1)\leq\mathrm{Aut}(\Gamma)$ is a $p$-group. In particular,
if $\varphi\in\mathrm{Aut}(\Gamma)$ does not have $p$-power order, then
$\varphi$ acts nontrivially on the maximal elementary abelian quotient
$\Gamma^{ab}\otimes\mathbb{F}_{p}$.

###  1.4. Related work

A number of constructions of a similar flavor to ours appeared in the near and
far past. Most close in spirit are the Chen integrals [Che71], which attach
$\mathbb{R}$\- or $\mathbb{C}$-valued invariants to words $w\in\Gamma$ via
integration: if $M$ is a smooth manifold with $\pi_{1}(M,*)\cong\Gamma$ and
$\tilde{w}\colon[0,1]\to M$ is a loop representing the homotopy class of $w$,
Chen took

$\langle(\alpha_{1}|\ldots|\alpha_{n})\,,\,w\rangle=\int_{0\leq
t_{1}\leq\ldots\leq t_{n}\leq
1}\tilde{w}^{*}\alpha_{1}(t_{1})\wedge\ldots\wedge\tilde{w}^{*}\alpha_{n}(t_{n})$

for any tensor of de-Rham $1$-forms in the Bar construction
$\mathrm{H}^{0}(Bar(\Omega^{*}_{dR}(M))$. He further proved that this defines
an isomorphism with
$\varinjlim\text{Hom}_{\mathbb{R}}(\mathbb{R}[\Gamma]/I^{n},\mathbb{R})$. This
construction informed later integral invariants, such as the Kontsevich
integral, but it has not been made so explicit as to produce combinatorial
invariants of words in groups.

Our weight reduction algorithm (Proposition 3.19), in a special case, gives a
discrete analog of Chen’s iterated integral, see Corollary 3.25. With it, the
resulting invariants are made combinatorial, while also being defined over
general PIDs. In addition, weight reduction manifests substantial freedom to
choose the order in which $1$-forms are handled, of which Chen’s ‘time-
ordered’ formalism is but one: for most words, other orderings yield more
efficient calculations.

Our proof of the main technical theorem, Theorem 6.1, uses a spectral sequence
that is conceptually dual to that of Stallings’ in [Sta75, §3].

On the interaction between Massey products on $\mathrm{H}^{1}(\Gamma;R)$ and
relations in $\Gamma$, work of Dwyer [Dwy75b] interprets Massey systems as
homomorphisms from a group of upper triangular matrices $\Gamma\to U_{n}(R)$,
and Fenn–Sjerve [FS87] relate the evaluation of Massey products on
$\mathrm{H}^{2}(\Gamma;\mathbb{Z})$ with Magnus coefficients of relations.
Dually, Stavros–Levine [GL05] relate Massey products with lower central series
quotients of a group. They show that a homomorphism $F_{n}\to\Gamma$ that
induces an isomorphism on $H_{1}$, further induces an isomorphism modulo the
$k$-th step of the lower central series and a surjection on the $k+1$-st
quotient iff all Massey products of order $\leq k$ vanish. At the $k+1$-st
step, the difference between the two groups is measured by a homological dual
Massey product. Compare this with our setting, where we still consider
braiding invariants past the degree at which Massey products first appear.

Recently, Porter–Suciu [PS22, PS23] introduced an integral version of rational
homotopy theory for studying fundamental groups. Their construction makes
explicit use the Steenrod’s cup-1 operation, and requires a binomial structure
on the coefficient ring. With this, they develop a theory of $1$-minimal
models, that give explicit access to fundamental groups, including their
torsion, starting from a model for singular cochains over a binomial ring.
Around the same time, Horel [Hor22] found that binomial rings are a good setup
for integral homotopy theory, and exhibited a fully faithful functor between
nilpotent spaces and cosimplicial binomial rings. Our theory is not trying to
pick out $\pi_{1}$ from the group ring $A[\pi_{1}]$, since braiding invariants
are already defined at the group ring level, and therefore we can avoid the
cup-1 operations and binomial structures.

###  1.5. Acknowledgements

This work grew out of a closely related and forthcoming project, joint with
Aydin Ozbek, Ben Walter and Dev Sinha. I thank them for their insights,
questions, feedback and support. Special thanks to Dev Sinha for his careful
reading and detailed feedback on an early draft.

This paper was partially supported by the National Science Foundation under
Grant No. DMS-1929284 while the author was in residence at the Institute for
Computational and Experimental Research in Mathematics (ICERM) in Providence,
RI, during the Combinatorial Algebraic Geometry Reunion program.

##  2\. (Co)algebraic preliminaries

Let $A$ be a PID and recall the category of graded $A$-module with objects
$B=\bigoplus_{n\in\mathbb{Z}}B_{n}$ where every $B_{n}$ is an $A$-submodule.
The _shift_ of $B$ by $k$ is the graded modules $B[k]$ given by
$(B[k])_{n}:=B_{n+k}.$ Morphisms $f\colon B\to B^{\prime}$ are $A$-linear maps
that preserve grading. A homomorphism $f\colon B\to B^{\prime}[k]$ is said to
be a morphism of degree $k$

Also recall that tensor product of two graded modules has graded components

$(B\otimes
B^{\prime})_{n}=\bigoplus_{i\in\mathbb{Z}}B_{i}\otimes_{A}B^{\prime}_{n-i}.$

Throughout this paper we consider conilpotent associative coalgebras.

######  Definition 2.1.

A _differential graded coalgebra_ is a graded module $B$ equipped with a
coassociative comultiplication

$\Delta\colon B\to B\otimes B,\text{ a counit }\eta\colon B\to A$

and a coderivation $d\colon B\to B$ of degree $+1$ obeying the Koszul sign
convention (see [LV12, §1.5.3]). That is,

$(\Delta\otimes\operatorname{Id})\circ\Delta=(\operatorname{Id}\otimes\Delta)\circ\Delta,\quad(\operatorname{Id}\otimes\eta)\circ\Delta=(\eta\otimes\operatorname{Id})\circ\Delta=\operatorname{Id}\quad\text{
and }$

$\quad\Delta\circ
d=(d\otimes\operatorname{Id}+(-1)^{n}\operatorname{Id}\otimes d)\circ\Delta$
when evaluated on $B_{n}$.

A homomorphism of dg coalgebras is a morphism $h\colon B\to B^{\prime}$ such
that

$(h\otimes h)\circ\Delta_{B}=\Delta_{B^{\prime}}\circ
h,\quad\eta_{B^{\prime}}\circ h=\eta_{B},\quad\text{and}\quad h\circ
d_{B}=d_{B^{\prime}}\circ h.$

Denote iterated coproducts by $\Delta^{0}=\operatorname{Id}$ and

$\Delta^{n+1}=(\Delta\otimes\operatorname{Id}^{\otimes n})\circ\Delta^{n}.$

The ring $A$ itself has a tautological dg coalgebra structure, with
$\Delta\colon A\to A\otimes_{A}A$ the canonical isomorphism, $\eta\colon A\to
A$ the identity and $d=0$.

######  Definition 2.2.

An _augmentation_ of a dg coalgebra $B$ is a coalgebra homomorphism
$\epsilon\colon A\to B$. Denote $1\in B$ for the element $\epsilon(1)$. This
splits $B$ as $\overline{B}\oplus A\cdot 1$, and we call
$\overline{B}=\ker(\eta)$ the _reduced_ coalgebra of $B$. The _reduced
comultiplication_ $\overline{\Delta}\colon\overline{B}\to\overline{B}$ is

$\overline{\Delta}(b)=\Delta(b)-1\otimes b-b\otimes 1.$

Define

$\overline{\Delta}^{n+1}=(\overline{\Delta}\otimes\operatorname{Id}^{\otimes
n})\circ\overline{\Delta}^{n}.$

We say that $B$ is _conilpotent_ if for every $b\in B$ we have
$\overline{\Delta}^{N}(b)=0$ for some $N\geq 0$.

Filtrations of coagebras will play a significant role in this work. These are
most naturally understood through quotients rather than subspaces.

######  Definition 2.3.

Suppose $(B,\Delta,\eta,d)$ is a differential coalgebra. A _filtration_ on $B$
is a sequence of subcomplexes

$0=F_{-1}B\leq F_{0}B\leq F_{1}B\leq\ldots\leq F_{p}B\leq\ldots\leq B$

such that every quotient $B/F_{p}B$ is $A$-torsion free, and for all $i\leq p$
the comultiplication factors as indicated by

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{q_{p}}$$\textstyle{B\otimes
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{i}\otimes
q_{p-i}}$$\textstyle{\nicefrac{{B}}{{F_{p}B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists\Delta^{p}_{i}}$$\textstyle{\nicefrac{{B}}{{F_{i}B}}\otimes\nicefrac{{B}}{{F_{p-i}B}}.}$

Equivalently, $(q_{i}\otimes q_{p-i})\circ\Delta$ vanishes on $F_{p}B$.

The _associated graded_ coalgebra is

$\mathrm{gr}~{}{B}=\bigoplus\mathrm{gr}_{p}~{}{B}\text{ where
}\mathrm{gr}_{p}~{}{B}:=\ker\left(\nicefrac{{B}}{{F_{p-1}}}\twoheadrightarrow\nicefrac{{B}}{{F_{p}B}}\right)\cong\nicefrac{{F_{p}B}}{{F_{p-1}B}}$

equipped with the well-defined coproduct
$\Delta\colon\bigoplus_{p}\mathrm{gr}_{p}~{}B\to\bigoplus_{i\leq
p}\mathrm{gr}_{i}~{}B\otimes\mathrm{gr}_{p-i}~{}B$ consisting of

$\mathrm{gr}_{p}~{}B\subseteq\nicefrac{{B}}{{F_{p-1}B}}\xrightarrow{\oplus\Delta^{p-1}_{i}}\bigoplus_{i\leq
p-1}\nicefrac{{B}}{{F_{i}B}}\otimes\nicefrac{{B}}{{F_{p-1-i}B}},$

whose image lands in $\bigoplus_{i\leq
p}\mathrm{gr}_{i}~{}B\otimes\mathrm{gr}_{p-i}~{}B$.

######  Remark 2.4.

Since all $A$-modules in question are torsion free, the sequence

$0\to\mathrm{gr}_{i}~{}B\otimes\mathrm{gr}_{p-i}~{}B\to\nicefrac{{B}}{{F_{i}B}}\otimes\nicefrac{{B}}{{F_{p-i}B}}\to\left(\nicefrac{{B}}{{F_{i-1}B}}\otimes\nicefrac{{B}}{{F_{p-i}B}}\right)\oplus\left(\nicefrac{{B}}{{F_{i}B}}\otimes\nicefrac{{B}}{{F_{p-i-1}B}}\right)$

left-exact. Thus, a quick diagram chase shows that $\Delta^{p-1}_{i}$
restricts to
$\mathrm{gr}_{p}~{}B\to\mathrm{gr}_{i}~{}B\otimes\mathrm{gr}_{p-i}~{}B$,
giving a coassociative coproduct on $\mathrm{gr}~{}{B}$.

######  Lemma 2.5.

Suppose $B=(B,\Delta,\eta,d)$ is a dg-coalgebra let $\mathrm{H}^{*}(B)$ denote
its cohomology. Suppose that $\mathrm{H}^{0}(B)$ is torsion-free, then
$\mathrm{H}^{0}(B)$ has a natural coalgebra structure

$\mathrm{H}^{0}(B)\xrightarrow{\mathrm{H}^{0}(\Delta)}\mathrm{H}^{0}(B\otimes
B)\overset{\sim}{\longrightarrow}\mathrm{H}^{0}(B)\otimes\mathrm{H}^{0}(B)$

where the second arrow is inverse to the Künneth isomorphism.

Moreover, given a filtration $F_{\bullet}\leq B$ such that all
$\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p}B}})$ are torsion-free, the cohomology
coalgebra $\mathrm{H}^{0}(B)$ is naturally filtered by

$F_{p}\mathrm{H}^{0}(B):=\ker\left(\mathrm{H}^{0}(B)\to\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p}B}})\right).$

###### Proof.

The Künneth formula shows that whenever $\mathrm{H}^{0}(B)$ is torsion-free,
the natural homomorphism
$\mathrm{H}^{0}(B)\otimes\mathrm{H}^{0}(B)\to\mathrm{H}^{0}(B\otimes B)$ is
invertible. Its inverse gives a coproduct structure on $\mathrm{H}^{0}(B)$.
Coassociativity follows immediately from the naturality of the Künneth map.

Now, we claim that $F_{\bullet}\mathrm{H}^{0}(B)$ is a filtration of
coalgebras. Indeed, the quotients
$\nicefrac{{\mathrm{H}^{0}(B)}}{{F_{p}\mathrm{H}^{0}(B)}}$ are torsion-free as
submodules of $\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p}B}})$, and using the
Künneth formula again,

$\textstyle{\mathrm{H}^{0}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(B\otimes
B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{0}(B)\otimes\mathrm{H}^{0}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p}B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{H}^{0}(\Delta_{i}^{p})}$$\textstyle{\mathrm{H}^{0}(\nicefrac{{B}}{{F_{i}B}}\otimes\nicefrac{{B}}{{F_{p-i}B}})}$$\textstyle{\mathrm{H}^{0}(\nicefrac{{B}}{{F_{i}B}})\otimes\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p-i}B}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

so after inverting the Künneth isomorphisms we indeed find that the
comultiplication factors through $\mathrm{H}^{0}(\nicefrac{{B}}{{F_{p}B}})$,
as required. ∎

The filtered coalgebra structure on $\mathrm{H}^{0}(B)$ is obviously
functorial.

######  Definition 2.6.

Any augmented coalgebra $B$ with no $A$-torsion is equipped with the _weight
filtration_ as follows. Let $F_{p}B=\ker\left(\overline{\Delta}^{p}\colon B\to
B^{\otimes p+1}\right)$ where $\overline{\Delta}$ extends to $B$ via the
projection $B\to\overline{B}$ given by $b\mapsto b-\eta(b)\cdot 1$.
Coassociativity of $\overline{\Delta}$ implies that the comultiplication
factors through $\nicefrac{{B}}{{F_{p}B}}$, as required. Clearly, any
homomorphism of coalgebras preserves this filtration.

###  2.1. Bar construction

Coalgebras play a central role in our theory through the following
construction.

######  Definition 2.7 (Bar construction).

Let $(C^{*},d,\mu)$ be an augmented associative dg algebra, and for $c\in
C^{i}$ denote $|c|=i$. The augmentation ideal of $C$ is denoted by
$\overline{C}$ and its _desuspension_ is the shifted complex
$\mathbf{s}^{-1}\overline{C}:=\overline{C}[1]$, i.e.
$\mathbf{s}^{-1}\overline{C}^{i}=\overline{C}^{i+1}$. The _Bar construction_
of $C$, denoted $Bar(C)$, is the total complex of a bicomplex whose underlying
graded module is

$T(\mathbf{s}^{-1}\overline{C})=\bigoplus_{p=0}^{\infty}(\mathbf{s}^{-1}\overline{C})^{\otimes
p}.$

Tensors $(\mathbf{s}^{-1}\overline{C})^{\otimes p}$ are said to have _weight_
$p$, and the bigraded term $Bar^{-p,q}(C)$ consists of weight $p$ tensors of
degree $q$.

The ‘vertical’ differential $d_{1}$ of bidegree $(0,1)$ is the extension of
$d$ to a graded coderivation,

$d_{1}(c_{1}\otimes\ldots\otimes
c_{p})=-\sum_{i=1}^{p}(-1)^{(i-1)+|c_{0}|+\ldots+|c_{i-1}|}c_{0}\otimes\ldots\otimes
dc_{i}\otimes\ldots\otimes c_{p}.$

The ‘horizontal’ differential $d_{2}$ of degree $(1,0)$ is defined by
multiplying consecutive elements,

$d_{2}(c_{1}\otimes\ldots\otimes
c_{p})=-\sum_{i=1}^{p-1}(-1)^{i+|c_{1}|+\ldots+|c_{i}|}c_{1}\otimes\ldots\otimes\mu(c_{i},c_{i+1})\otimes\ldots\otimes
c_{p}.$

Both differentials satisfy the Koszul sign convention, and they anti-commute.
The (total) _Bar differential_ is $d_{Bar}=d_{1}+d_{2}$. Let
$\mathrm{H}^{*}_{Bar}(C)$ denote its cohomology, which we call the _Bar
cohomology of $C$_.

######  Definition 2.8 (Coalgebra structure).

The Bar construction is naturally a conilpotent dg coalgebra, with respect to
the deconcatenation coproduct

(2)
$\Delta(c_{1}|\ldots|c_{p})=\sum_{i=0}^{p}(c_{1}|\ldots|c_{i})\otimes(c_{i+1}|\ldots|c_{p}).$

where we replaced the tensor symbols by bars. The augmentation is given by the
$p=0$ summand $(\mathbf{s}^{-1}\overline{C})^{\otimes 0}=A$, so that

$\overline{Bar}(C)=\bigoplus_{p\geq 1}(\mathbf{s}^{-1}\overline{C})^{\otimes
p}$

with reduced coproduct omitting the $i=0$ and $p$ summands from (2).
Consequently, the weight of a tensor $x$ agrees with the order of vanishing of
$\overline{\Delta}^{n}(x)$.

######  Remark 2.9.

When $C$ is a commutative algebra, there is a further product structure on
$Bar(C)$, given by the shuffle product, making $Bar(C)$ into a Hopf algebra.
This structure is significant in the context of rational homotopy theory,
giving rise for example to Malcev Lie algebras of fundamental groups.

Bar cohomology is a quasi-isomorphism invariant of a dg algebra $C$, in the
following sense.

######  Lemma 2.10 ([LV12, Proposition 2.2.3]).

If $C\to C^{\prime}$ is a quasi-isomorphism of augmented dg algebras, and both
underlying $A$-modules are torsion-free, then the induced map

$Bar(C)\to Bar(C^{\prime})$

is a quasi-isomorphism.

###### Proof.

The argument of [LV12, Proposition 2.2.3] applies verbatim, with the
observation that torsion-free $A$-modules are flat. Recounting the argument,
one filters the bicomplexes $Bar(C)$ and $Bar(C^{\prime})$ by weight. Their
associated graded consist of tensor power complexes

$(C^{\otimes p},d_{1})\to((C^{\prime})^{\otimes p},d_{1}).$

Since tensor products of complexes of flat modules are quasi-ismorphism
invariant, the map induces an isomorphism of the $E^{1}$-pages of the
associated spectral sequence. It therefore induces an isomorphism on Bar
cohomology. ∎

######  Definition 2.11.

An augmented dg algebra $C^{*}$ is said to be _connected_ if it is quasi-
isomorphic to an augmented algebra $\tilde{C}$ with $\tilde{C}^{0}=A$.

######  Lemma 2.12.

Let $C^{*}$ be a dg algebra whose underlying $A$-module is torsion-free. Let
$Bar_{\leq p}(C)$ denote the sub-coalgebra of tensors of weight $\leq p$. Then
the quotients

$Bar_{>p}(C):=Bar(C)/Bar_{\leq p}(C)$

are torsion-free so $Bar_{\leq p}(C)$ defines a natural filtration on
$Bar(C)$.

If $C$ is connected, then $\mathrm{H}^{0}(Bar_{>p}(C))$ is torsion-free for
all $p\geq 0$, so that these induce a natural filtration on
$\mathrm{H}^{0}_{Bar}(C)$.

###### Proof.

The fact that $Bar_{>p}(C)$ is torsion-free is clear, and it gives a
filtration of $Bar(C)$ as a special case of the filtration of a conilpotent
coalgebra by weight.

To see that in the connected case $\mathrm{H}^{0}(Bar_{>p}(C))$ is torsion-
free, assume without loss of generality that $C^{0}=A$. Then the augmentation
ideal $\overline{C}$ is concentrated in positive degrees, and consequently
$Bar^{0}(C)$ is the lowest degree in which $Bar(C)$ is nontrivial. It follows
that $\mathrm{H}^{0}(Bar_{>p}(C))$ is a sub-coalgebra of $Bar_{>p}(C)$,
characterized by the vanishing of $d_{Bar}$, thus torsion-free. ∎

We summarize the main results of this section as follows.

######  Corollary 2.13.

The assignment $C^{*}\mapsto\mathrm{H}^{0}_{Bar}(C)$ is a functor which sends
connected dg algebras, whose underlying $A$-module is torsion-free, to
filtered coassociative coalgebras. Quasi-isomorphisms of algebras are sent to
isomorphism of coalgebras.

###  2.2. Group rings

Let $\Gamma$ be a group. We recall basic definitions.

######  Definition 2.14.

The _group ring_ $A[\Gamma]$ is the module of $A$-linear combinations of
elements $w\in\Gamma$. It forms an augmented (Hopf) algebra, with product

$m\colon A[\Gamma]\otimes A[\Gamma]\to A[\Gamma]$

extending $m(w_{1}\otimes w_{2})=w_{1}w_{2}$ to an $A$-linear map, with
coproduct $\Delta(w)=w\otimes w$, and with antipode $s(w)=w^{-1}$ for every
$w\in\Gamma$.

The augmentation $\epsilon\colon A[\Gamma]\to A$ sends every $w\in\Gamma$ to
$1$, so that

$\epsilon(\sum\lambda_{i}\cdot w_{i})=\sum\lambda_{i}.$

Let $I\lhd A[\Gamma]$ denote the augmentation ideal, which is the kernel of
$\epsilon$, generated by $(w-1)$ for $w\in\Gamma$. Then $A[\Gamma]$ is
filtered by powers $\ldots\leq I^{2}\leq I\leq A[\Gamma]$.

The association $\Gamma\mapsto A[\Gamma]$ is a functor, sending groups to
augmented (Hopf) algebras.

Recall the dimension series of a group.

######  Definition 2.15 ([Los60]).

Let $\Gamma$ be a group and $A$ any ring. The $A$-_dimension series_ of
$\Gamma$ modulo $A$ is the descending central series of subgroups

$\ldots\leq\gamma_{k}(\Gamma;A)\leq\ldots\leq\gamma_{2}(\Gamma;A)\leq\gamma_{1}(\Gamma;A)=\Gamma$

defined by

$w\in\gamma_{k}(\Gamma;A)\iff(w-1)\in I^{k}\subseteq A[\Gamma].$

Recall that the _lower central series_ of $\Gamma$, defined recursively as
$\gamma_{1}\Gamma=\Gamma$ and $\gamma_{k+1}\Gamma=[\gamma_{k}\Gamma,\Gamma]$,
is the minimal central series. It follows that
$\gamma_{k}\Gamma\subseteq\gamma_{k}(\Gamma;A)$ for every ring $A$.

######  Example 2.16.

For $A=\mathbb{Z}$, Passi [Pas68] described classes of groups for which
$\gamma_{k}\Gamma=\gamma_{k}(\Gamma;\mathbb{Z})$. However, starting with $k=4$
there exist groups for which $\gamma_{k}\Gamma\subsetneq\gamma_{k}(\Gamma;A)$,
as found by Rips [Rip72]. For $A=\mathbb{Q}$, the dimension series is
constructed recursively by the rule

$\gamma_{k+1}(\Gamma;\mathbb{Q})=\\{w\in\Gamma\mid\exists n\geq 1\text{ s.t.
}w^{n}\in[\gamma_{k}(\Gamma;\mathbb{Q}),\Gamma]\\},$

and for $A=\mathbb{F}_{p}$, the dimension series is the _lower exponent $p$
central series_ constructed recursively by

$\gamma_{k+1}(\Gamma;\mathbb{F}_{p})=[\gamma_{k}(\Gamma;\mathbb{F}_{p}),\Gamma]\cdot\gamma_{k}(\Gamma;\mathbb{F}_{p})^{p}.$

######  Definition 2.17.

Given an $A$-module $M$, let $M^{*}$ denote $\text{Hom}_{A}(M,A)$, the module
of $A$-linear functionals on $M$. For every $p\geq 1$ define an evaluation
pairing

$(M^{*})^{\otimes p}\otimes M^{\otimes p}\to
A\quad\text{via}\quad\langle\varphi_{1}\otimes\ldots\otimes\varphi_{p}\;,\;m_{1}\otimes\ldots\otimes
m_{p}\rangle=\prod_{i=1}^{p}\varphi_{i}(m_{i}).$

Given a function $f\colon\Gamma\to A$, its _linear extension_ to $A[\Gamma]$
is the $A$-linear map

$f\colon A[\Gamma]\to A\quad\text{defined by}\quad f\left(\sum\lambda_{i}\cdot
w_{i}\right)=\sum\lambda_{i}\cdot f(w_{i}),$

defining an isomorphism $A[\Gamma]^{*}\cong A^{\Gamma}$.

###  2.3. Tensors and infinite cartesian powers

The following technical fact will be crucial for constructing elements in
group rings that evaluate nontrivially on invariants of words.

######  Lemma 2.18 ([Goo72, Theorem 1(a)]).

Let $A$ be a commutative Noetherian ring. Then for any two sets $S$ and $T$,
the natural map evaluation $A^{S}\otimes A^{T}\to A^{S\times T}$ is injective.

###### Proof.

[Goo72, Theorem 1] shows that for any $A$-module $M$ the natural map

$M\otimes A^{T}\to M^{T}$

is injective. Taking $M=A^{S}$, the claim follows from the adjunction
$(A^{S})^{T}\cong A^{S\times T}$. ∎

##  3\. Free groups

We start by defining letter-braiding polynomials of words in free groups. In
§6.1 we will find which of these remain well-defined once relations have been
introduced. Our first approach in §3.1 is topological, with few calculations,
which are then the focus of §3.2.

###  3.1. Topological construction

The invariants we construct are grounded in the Bar construction for
simplicial cochains on simplicial sets.

######  Definition 3.1.

Given a pointed simplicial set $X_{\bullet}$, let $C^{*}_{\cup}(X;A)$ denote
its simplicial cochain complex with coefficients in $A$. This is the complex
of functions on nondegenerate simplices of $X$ (or dual to the normalized
chains). The cup product makes this into an associative algebra, and the
basepoint provides an augmentation. Denote

$Bar(X;A):=Bar(C^{*}_{\cup}(X;A)),$

and $\mathrm{H}^{*}_{Bar}(X;A)$ for its cohomology.

The following is a standard application of the Quillen adjunction between the
realization and singular functors and the quasi-isomorphism invariance of the
Bar construction.

######  Lemma 3.2.

The assignment $X\mapsto Bar(X;A)$ is a contravariant functor from simplicial
sets to filtered dg coalgebras.

A pointed continuous map on geometric realizations $g\colon|X|\to|Y|$ induces
a homomorphism of filtered coalgebras
$g^{*}\colon\mathrm{H}^{*}_{Bar}(Y;A)\to\mathrm{H}^{*}_{Bar}(X;A)$ that
furthermore satisfies

$(g_{2}\circ g_{1})^{*}=g_{1}^{*}\circ g_{2}^{*}\quad\text{ and }\quad f\sim
g\text{ homotopic }\implies f^{*}=g^{*}.$

If $g$ is the geometric realization of a simplicial map $|f|$, then
$f^{*}=g^{*}$ on Bar cohomology.

The definition of letter braiding requires calculations on simplicial models
for circles and wedges of circles.

######  Definition 3.3.

Let $\Delta[1]$ be the standard $1$-simplex, with endpoints
$\partial_{0}\Delta[1]$ and $\partial_{1}\Delta[1]$.

The _standard circle_ is the pointed simplcial set

$S^{1}_{\bullet}:=\Delta[1]/(\partial_{0}\Delta[1]\sim\partial_{1}\Delta[1]).$

Let $F_{S}$ be a free group on basis $S$. The _presentation complex of
$F_{S}$_ is the pointed simplicial set

$(X_{S})_{\bullet}=\bigvee_{s\in S}S^{1}_{\bullet},$

having one standard loop for every generator $s\in S$ of $F_{S}$.

The letter-braiding number is defined given the following choice of
isomorphism.

######  Lemma 3.4.

There is a canonical isomorphism of filtered augmented coalgebras

$\mathrm{H}^{*}_{Bar}(S^{1}_{\bullet};A)\cong A[t],$

with $|t|=0$, and $A[t]$ considered as the cofree coalgebra generated by $t$.

###### Proof.

Let $\tau\in C^{1}(S^{1}_{\bullet};A)$ be the cochain the assigns the value
$1$ to the unique $1$-simplex of $S^{1}$. The complex
$C^{*}_{\cup}(S^{1}_{\bullet};A)$ is the augmented algebra $A\cdot 1\oplus
A\tau$ with $d(\tau)=0=\tau^{2}$. Therefore, its Bar construction is the
cofree coalgebra

$Bar(A[\tau]/(\tau^{2}))=T(A\cdot\mathbf{s}^{-1}\tau)=\bigoplus_{p\geq
0}A(\mathbf{s}^{-1}\tau)^{\otimes p}$

with vanishing differentials. Denoting $t=\mathbf{s}^{-1}\tau$, this coalgebra
is polynomial in $t$. ∎

More generally, for the wedge of circles $X_{S}$, the cochain algebra
naturally consists of a unit along with

$C^{1}_{\cup}(X_{S};A)\cong A^{S},$

the $A$-valued functions on the set $S$ with trivial cup product. This is in
turn naturally interpreted as
$\mathrm{H}^{1}(X_{S};A)\cong\text{Hom}(F_{S},A)$. As in the last observation,
the Bar construction is concentrated in degree $0$, with vanishing
differentials. The Bar cohomology is thus natrually identified with the cofree
coalgebra

$\mathrm{H}^{*}_{Bar}(X_{S};A)\cong T(A^{S})=\bigoplus_{p\geq
0}\left(A^{S}\right)^{\otimes p}.$

######  Definition 3.5.

Recall that elements $w\in F_{S}$ are in bijection with based homotopy classes
of pointed maps $|S^{1}|\to|X_{S}|$. Abusing notation, let $w$ denote a
representative of the respective homotopy class.

The _letter-braiding polynomial_ of the element $w\in F_{S}$ is the
composition

$L_{\bullet}(w)\colon T(A^{S})\cong
H^{0}_{Bar}(X_{S};A)\overset{w^{*}}{\longrightarrow}\mathrm{H}^{0}_{Bar}(S^{1};A)\cong
A[t].$

The _letter-braiding number_ of $w$ is the linear coefficient of the
polynomial

$\ell_{\bullet}(w):=\frac{d}{dt}L_{\bullet}(w)|_{t=0}\colon
T(A^{S})\longrightarrow A$

Since induced maps on Bar cohomology depend only on homotopy class, the
polynomial $L(w)$ is independent of the choice of representative for $w$.

######  Example 3.6.

The most basic calculation is that of the braiding polynoimal for a generator
$s\in S$. In this case, given functionals $\alpha_{1},\ldots,\alpha_{n}\in
A^{S}$, we have

$L_{(\alpha_{1}|\ldots|\alpha_{n})}(s)=\alpha_{1}(s)\alpha_{2}(s)\ldots\alpha_{n}(s)\cdot
t^{n}\quad\text{ and }\quad L_{1}(s)=1.$

Indeed, let $i_{s}\colon S^{1}_{\bullet}\to(X_{S})_{\bullet}$ be the inclusion
of the edge labeled by $s$. On $1$-cochains it defines the pullback
$i_{s}^{*}\colon C^{1}(X_{S};A)\to C^{1}(S^{1};A)$ given by restriction
$A^{S}\to A^{\\{s\\}}$. That is, $i_{s}^{*}(\alpha)=\alpha(s)\cdot\tau$, where
$\tau\in A^{\\{s\\}}$ is the functional $s\mapsto 1$. Recalling that the
desuspension of $\tau$ in the Bar complex of $S^{1}_{\bullet}$ is $t$, the
general calculation follows.

The braiding number is therefore

$\ell_{(\alpha_{1}|\ldots|\alpha_{n})}(s)=\begin{cases}\alpha_{1}(s)&n=1\\\
0&\mathrm{otherwise}.\end{cases}$

The following is immediate from the fact that $A[t]$ is cofreely generated by
$t$, with $\Delta(t^{n})=\sum_{i=0}^{n}t^{i}\otimes t^{n-i}$.

######  Theorem 3.7.

The letter-braiding polynomial is a homomorphism of augmented filtered
coalgebras, meaning that $L_{(T+a\cdot T^{\prime})}(s)=L_{T}(w)+a\cdot
L_{T^{\prime}}(w)$ and if $\Delta(T)=\sum_{i}T_{1}^{i}\otimes T_{2}^{i}$ then

$\Delta L_{T}(w)=\sum_{i}L_{T_{1}^{i}}(w)\otimes
L_{T_{2}^{i}}(w)\quad\text{and}\quad L_{1}(w)=1.$

Furthermore, the braiding polynomial is determined by the braiding number, in
that

$L_{T}(w)=\eta(T)\cdot 1+\sum_{k=0}^{\infty}(\ell(w)^{\otimes
k+1})_{\overline{\Delta}^{k}(T)}\cdot t^{k+1}$

where $\overline{\Delta}^{k}$ is the iterated reduced coproduct
$(\overline{\Delta}\otimes\operatorname{Id}^{\otimes
k-1})\circ\ldots\circ(\overline{\Delta}\otimes\operatorname{Id})\circ\overline{\Delta}$,
and $(\ell(w)\otimes\ell(w))_{T_{1}\otimes
T_{2}}=\ell_{T_{1}}(w)\cdot\ell_{T_{2}}(w)$.

We next show that braiding number is functorial with respect to group
homomorphisms of free groups. On Bar cohomology

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

is the cofree coalgebra generated by functions $S\to A$. Coalgebra
homomorphisms $(B,\Delta)\to\mathrm{H}^{0}_{Bar}(X_{S};A)$ are in bijection
with linear maps $B\to A^{S}$. Thus to give such a homomorphism one must only
provide a function $f_{b}\colon s\mapsto f_{b}(s)\in A$ for every $b\in B$,
depending linearly on $b$.

######  Theorem 3.8.

Given a homomorphism of free groups $h\colon F_{S_{1}}\to F_{S_{2}}$, there is
a functorial pullback
$\mathrm{H}^{*}_{Bar}(X_{S_{2}};A)\to\mathrm{H}^{*}_{Bar}(X_{S_{1}};A)$
coinciding with the coalgebra homomorphism

$h^{*}\colon T(A^{S_{2}})\to T(A^{S_{1}})$

determined by the functions

$h^{*}(T)\colon s_{1}\mapsto\ell_{T}(h(s_{1})).$

In particular, these homomorphisms satisfy $(h\circ
h^{\prime})^{*}=(h^{\prime})^{*}\circ h^{*}$.

######  Example 3.9.

Let $S_{1}=\\{s\\}$ generate a cyclic group $F_{\\{s\\}}$. Then
$T(A^{\\{s\\}})\cong A[t]$ where $t$ corresponds to the functional $s\mapsto
1$. An element $w\in F_{S_{2}}$ corresponds to $h\colon F_{\\{s\\}}\to
F_{S_{2}}$, and the induced map $h^{*}\mapsto T(A^{S_{2}})\to A[t]$ is exactly
the letter-braiding polynomial $L_{\bullet}(w)$.

######  Example 3.10.

Let $S_{1}=\\{e_{1},e_{2}\\}$ generated a free group of rank two, so that
$A^{S_{1}}=A\langle E_{1},E_{2}\rangle$ generated by the ‘dual’ basis
$E_{i}(e_{j})=\delta_{ij}$. A homomorphism $h\colon F_{S_{1}}\to F_{S_{2}}$
corresponds to two elements $(w_{1},w_{2})=(h(e_{1}),h(e_{2}))$, and its
pullback is

(3) $\displaystyle\begin{split}h^{*}(T)=&\;\eta(T)\cdot 1+\ell_{T}(w_{1})\cdot
E_{1}+\ell_{T}(w_{2})\cdot E_{2}\\\
+\sum&\;\ell_{T_{1}^{i}}(w_{1})\ell_{T_{2}^{i}}(w_{2})\cdot
E_{1}|E_{2}+\ell_{T_{1}^{i}}(w_{2})\ell_{T_{2}^{i}}(w_{1})\cdot E_{2}|E_{1}\\\
+&\;\ell_{T_{1}^{i}}(w_{1})\ell_{T_{2}^{i}}(w_{1})\cdot
E_{1}|E_{1}+\ell_{T_{1}^{i}}(w_{2})\ell_{T_{2}^{i}}(w_{2})\cdot E_{2}|E_{2}\\\
+&\text{ weight }\geq 3\end{split}$

where $\overline{\Delta}(T)=\sum_{i}T_{1}^{i}\otimes T_{2}^{i}$.

Composing $h$ with the homomorphism $F_{\\{s\\}}\to F_{S_{1}}$ sending
$s\mapsto e_{1}e_{2}$, functoriality implies

$L_{T}(w_{1}\cdot w_{2})=\eta(T)\cdot 1+\ell_{T}(w_{1})\cdot
L_{E_{1}}(e_{1}e_{2})+\ell_{T}(w_{2})\cdot
L_{E_{2}}(e_{1}e_{2})+\Sigma\,\ell_{T_{1}^{i}}(w_{1})\ell_{T_{2}^{i}}(w_{2})\cdot
L_{E_{1}|E_{2}}(e_{1}e_{2})+\ldots$

thus computing the letter-braiding polynomial of a product $w_{1}\cdot w_{2}$
from polynomials of $F_{2}$.

###### Proof of Theorem 3.8.

A group homomorphism $h\colon F_{S_{1}}\to F_{S_{2}}$ corresponds to a unique
based homotopy class of pointed maps $h\colon|X_{S_{1}}|\to|X_{S_{2}}|$. We
show that the induced map $h^{*}$ on Bar cohomology has weight $1$ component
as stated.

For every $T\in\overline{T}(A^{S_{2}})$ let the projection of
$h^{*}_{\bullet}(T)$ onto this weight $1$ component be denoted $\ell
h^{*}_{\bullet}(T)\in A^{S_{1}}$. Fixing $s\in S_{1}$, we calculate $\ell
h^{*}_{s}(T)$ by considering the inclusion of the circle labeled by $s$

$i_{s}\colon S^{1}\hookrightarrow\bigvee_{s^{\prime}\in
S_{1}}S^{1}=X_{S_{1}}.$

On geometric realizations,

$|S^{1}|\overset{|i_{s}|}{\longrightarrow}|X_{S_{1}}|\overset{h}{\longrightarrow}|X_{S_{2}}|$

has composition representing the homotopy class of $h(s)\in F_{S_{2}}$. By
definition of the letter-braiding polynomial, this gives $(i_{s}^{*}\circ
h^{*})(T)=L_{h(s)}(T)$, having linear coefficient $\ell_{h(s)}(T)$. On the
other hand, as explained in Example 3.6,
$i_{s}^{*}(\alpha_{1}|\ldots|\alpha_{n})=\alpha_{1}(s)\ldots\alpha_{n}(s)\cdot
t^{n}$ and has vanishing linear coefficient unless $n=1$.

It follows that

$\ell_{h(s)}(T)=\ell(i_{s}^{*}\circ h^{*}(T))=\ell h_{s}^{*}(T),$

which proves the claimed description of $h^{*}$. Functoriality is automatic by
Lemma 3.2. ∎

###  3.2. Weight reduction algorithm

We give a combinatorial construction to calculate the letter-braiding
invariants of an element $w\in F_{S}$, Let $[n]:=\\{1,\ldots,n\\}$ and
$[n]_{*}:=\\{0,\ldots,n\\}$, considered modulo $n+1$.

######  Definition 3.11.

A _word_ in the generators of $F_{S}$ is a map $\tilde{w}\colon[n]\to S\cup
S^{-1}$. The _loop_ associated to $\tilde{w}$ is the simplicial circle

$S^{1}(\tilde{w})_{\bullet}=\left([n]_{*}\times\Delta[1]\right)/\sim$

where $\sim$ glues the endpoints of the intervals in order to get a circle,
but with orientation determined by the sign of the respective letter, as
indicated by the following examples:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{2}}$$\textstyle{3}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{3}^{-1}}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{4}^{-1}}$$\scriptstyle{s_{5}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{1}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{1}^{-1}}$$\scriptstyle{s_{2}}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{3}}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{4}}$$\textstyle{5}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{5}^{-1}}$

The first segment $0\to 1$ does not correspond to a letter and is always
oriented in increasing order. We refer to it as the _standard segment_.

Let $|s|=|s^{-1}|=s$ for every $s\in S$. There is a canonical simplicial map
$\tilde{w}$ from $S^{1}(\tilde{w})_{\bullet}$ to the wedge of circles
$(X_{S})_{\bullet}$, sending the $i$-th segment to the circle indexed by
$|\tilde{w}(i)|\in S$. The orientation of the segment matches that of the
target circle. The (unlabeled) standard segment is collapsed onto the
basepoint. We abuse notation333Maps from simplicial circles to $X_{S}$ are in
obvious bijection with words in $S\cup S^{-1}$. Here we insist that all such
maps start with a collapsed standard segment for reasons that will become
apparent next. and refer to this simplicial map by $\tilde{w}$.

For brevity, let $\mathrm{H}^{*}_{Bar}(\tilde{w};A)$ denote
$\mathrm{H}^{*}_{Bar}(S^{1}(\tilde{w});A)$.

######  Definition 3.12.

Let the _standard collapse_ $\operatorname{std}\colon S^{1}(\tilde{w})\to
S^{1}$ be the pointed simplicial map defined by collapsing all edges except
for the standard segment $0\to 1$, which is sent to the nondegenerate segment
of $S^{1}$. It induces an isomorphism of filtered coalgebras

$L^{-1}\colon{A[t]}=\mathrm{H}^{0}_{Bar}(S^{1};A)\to H^{0}_{Bar}(\tilde{w};A)$

whose inverse we denote by $L\colon H^{0}_{Bar}(\tilde{w};A)\to{A[t]}$.

These maps exist and are well-defined because the collapse map
$S^{1}(\tilde{w})\to S^{1}$ is a homotopy equivalence on geometric
realizations, and so induces an isomorphism on Bar cohomology.

######  Corollary 3.13.

Let $\tilde{w}\colon[n]\to S\cup S^{-1}$ be a word in the generators of
$F_{S}$, and let $w\in F_{S}$ denote the element it represents. The letter-
braiding polynomial of $w$ is given by

$L_{\bullet}(w)\colon{T}(A^{S})\overset{\tilde{w}^{*}}{\longrightarrow}\mathrm{H}^{0}_{Bar}(\tilde{w};A)\overset{L}{\longrightarrow}{A[t]}.$

###### Proof.

We have a commutative diagram of pointed simplicial sets

$\textstyle{S^{1}(\tilde{w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{std}}$$\scriptstyle{\sim}$$\scriptstyle{\tilde{w}}$$\textstyle{S^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{X_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Sing}(|S^{1}|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Sing}(w)}$$\textstyle{\operatorname{Sing}(|X_{S}|)}$

where the arrows decorated with $\sim$ induce quasi-isomorphisms on simplicial
cochain algebras. It follows that the two paths
$S^{1}(\tilde{w})\to\operatorname{Sing}(|X_{S}|)$ induce the same map on Bar
cohomology. By the construction of $w^{*}\colon{T}(A^{S})\to A[t]$ given in
Lemma 3.2, it agrees with $L_{\bullet}(w)$. ∎

We unpack the last corollary and give an explicit description of
$L_{\bullet}(w)$. We use the _sign function_ $(-1)^{\bullet}\colon S\cup
S^{-1}\to\\{\pm 1\\}$, defined by $(-1)^{s}=1$ and $(-1)^{s^{-1}}=-1$ for
every $s\in S$.

######  Definition 3.14.

Given a word $\tilde{w}\colon[n]\to S\cup S^{-1}$, let $C^{*}(\tilde{w};A)$ be
the following dg algebra.

$C^{0}(\tilde{w};A)=C^{1}(\tilde{w};A):=\\{a\colon[n]_{*}\to A\\}\text{ and
otherwise }C^{i}=0,$

with $d:C^{0}\to C^{1}$ given by $df_{i}:=(-1)^{\tilde{w}(i)}(f_{i+1}-f_{i})$
and $i+1$ is computed modulo $n$. Elements of $C^{0}(\tilde{w};A)$ are called
_functions_ and denoted by Latin letter, such $f$ and $g$. Elements of
$C^{1}(\tilde{w};A)$ are called _forms_ , and denoted by Greek letters.

The product is defined on functions by $(f\smile g)_{i}=f_{i}\cdot g_{i}$, and
between functions and forms as follows

(4) $\displaystyle\begin{aligned} \text{ if }\tilde{w}(i)\in S:\\\
(f\smile\alpha)_{i}&=f_{i}\cdot\alpha_{i}\\\ (\alpha\smile
f)_{i}&=f_{i+1}\cdot\alpha_{i}\end{aligned}\text{ and, }\begin{aligned} \text{
if }\tilde{w}(i)\in S^{-1}:\\\ (f\smile\alpha)_{i}&=f_{i+1}\cdot\alpha_{i}\\\
(\alpha\smile f)_{i}&=f_{i}\cdot\alpha_{i}\end{aligned}$

with $i+1$ computed modulo $n$, and the case $i=0$ satisfying the same
formulas as $\tilde{w}(i)\in S$. The augmentation is given by the evaluation
$f\longmapsto f_{0}$.

The following is immediate from definitions.

######  Lemma 3.15.

$C^{*}(\tilde{w};A)$ is isomorphic to $C^{*}_{\cup}(S^{1}(\tilde{w});A)$ with
its standard cup product.

The combinatorial model of the letter-braiding polynomial using
$C^{*}(\tilde{w};A)$ supports an explicit algorithm for calculation, distinct
from Fox calculus, using weight reduction as introduced by Monroe–Sinha
[MS22].

We build forms from the following special forms in $C^{1}(\tilde{w};A)$.

######  Definition 3.16.

For $0\leq i\leq n$, the _indicator form_ $\delta_{i}\colon[n]_{*}\to A$ is

$\delta_{ij}=\begin{cases}1&i=j\\\ 0&\mathrm{otherwise},\end{cases}$

where we also denote $\delta_{0}$ by $t$. The _orientation form_ $dx\in
C^{1}(w;A)$ is

$dx_{i}:=\begin{cases}+1&w(i)\in S\\\ -1&w(i)\in S^{-1}\\\ 0&i=0.\end{cases}$

Given a function $f\in[n]\to A$, let $fdx$ denote the form

$fdx_{i}:=f_{i}\cdot dx_{i}\quad\text{with}\quad fdx_{0}=0.$

Our definition of $fdx$ is _not_ an instance of the cup product on
$C^{*}(w;A)$.

######  Definition 3.17.

For every $1\leq i\leq j\leq n+1$, the _integral_ of $fdx\in
C^{1}(\tilde{w};A)$ is

$\int_{i}^{j}fdx:=f(i)+f(i+1)+\ldots+f(j-1),$

and $\int_{w}fdx=f(1)+\ldots+f(n)$.

The _cobounding_ of a form $fdx\in C^{1}(\tilde{w};A)$ is the function
$d^{-1}(fdx)\in C^{0}(\tilde{w};A)$ defined by

$d^{-1}(fdx)_{j}:=\int_{1}^{j}fdx-\int_{w}fdx=-\left(f(j)+f(j+1)+\ldots+f(n)\right)$

and $d^{-1}(fdx)_{0}=0$.

The following facts are easy to check.

######  Lemma 3.18.

Any form $\alpha\in C^{1}(\tilde{w};A)$ can be written uniquely as

$\alpha=fdx-a_{0}\cdot\delta_{0}.$

Given a function $F\in C^{0}(\tilde{w};A)$, its differential is

$dF=F^{\prime}dx-F_{1}\cdot\delta_{0},$

where $(F^{\prime})_{i}=F_{i+1}-F_{i}$. In particular, the cobounding of $fdx$
belongs to the augmentation ideal of $C^{*}(\tilde{w};A)$ and satisfies

$d(d^{-1}(fdx))=fdx-\int_{w}fdx\cdot\delta_{0}.$

Thus $fdx$ is exact if and only if $\int_{w}fdx=0$.

######  Proposition 3.19 (0-linking and weight reduction).

Every element $T\in Bar^{0}(\tilde{w};A)=T(C^{1}(\tilde{w};A))$ is
cohomologous to a polynomial in $\delta_{0}$. The isomorphism

$L\colon\mathrm{H}^{0}_{Bar}(\tilde{w};A)\to{A[t]}$

defining the letter-braiding polynomial sends a polynomial in $\delta_{0}$ to
one in $t$ by substitution $\delta_{0}\mapsto t$. In the notation below, we
identify $\delta_{0}$ with $t$.

The polynomial can be computed via following weight-reduction procedure. Given
a tensor $(\alpha_{1}|\ldots|\alpha_{n})$ such that every $\alpha_{i}$ is
either $t$ or $fdx$, let its _modified weight_ be $n-\\#(t\text{ entries})$.

  1. (1)

If the modified weight is $0$, then the tensor is already a polynomial in $t$,
and we halt. Otherwise, pick one $\alpha_{k}=fdx$ (preferably one with
$\int_{w}fdx=0)$ and proceed.

  2. (2)

The tensor $(\alpha_{1}|\ldots|fdx|\ldots|\alpha_{n})$ is cohomologous to

$\displaystyle\begin{split}\left(\int_{w}fdx\right)\cdot&(\alpha_{1}|\ldots|\alpha_{k-1}|t|\alpha_{k+1}|\ldots|\alpha_{n})\\\
-&(\alpha_{1}|\ldots|\alpha_{k-2}|(\alpha_{k-1}\smile
d^{-1}(fdx))|\alpha_{k+1}|\alpha_{k+2}|\ldots|\alpha_{n})\\\
+&(\alpha_{1}|\ldots|\alpha_{k-2}|\alpha_{k-1}|(d^{-1}(fdx)\smile\alpha_{k+1})|\alpha_{k+2}|\ldots|\alpha_{n}).\end{split}$

(By convention, if $(fdx)$ occurs in one of the tensor’s extremal terms, omit
the undefined cup product.) These cup products are given by the following
rules:

$\displaystyle t\smile d^{-1}(fdx)=-(\int_{w}fdx)\cdot t\quad$
$\displaystyle\text{ and }\quad d^{-1}(fdx)\smile t=0$ $\displaystyle
gdx\smile d^{-1}(fdx)=-(\int_{\partial_{1}(\bullet)}^{n+1}fdx)g\cdot dx\quad$
$\displaystyle\text{ and }\quad d^{-1}(fdx)\smile
gdx=-(\int_{\partial_{0}(\bullet)}^{n+1}fdx)g\cdot dx,$

where $\partial_{0}(\bullet)$ (or $\partial_{1}(\bullet)$) is the function on
segments that returns the initial vertex (or terminal vertex), and we define
$\int_{0}^{n+1}fdx=0$.

  3. (3)

The summands obtained in Step (2) have strictly smaller modified weight than
the original. Collect like terms and return to Step (1) for each resulting
summand.

We refer to this weight-reduction procedure as _$0$ -linking_, since general
linking of submanifolds is obtained by cobounding and intersecting

$\operatorname{linking}(M_{1},M_{2})=\partial^{-1}(M_{1})\cap M_{2},$

which is Poincaré dual to computing $d^{-1}(-)$ followed by the cup product of
cochains. This manifests above with linking of $0$-chains (dual to $1$-forms)
in a $1$-manifold (the circle $S^{1}(w)$).

######  Example 3.20.

Example 1.2 depicts the higher $0$-linking of letters in $[xy,x^{-2}]\in
F_{\\{x,y\\}}$. The picture describes the weight reduction $X\smile
d^{-1}(Y\smile d^{-1}(X\smile d^{-1}(X)))$, where the drawn intervals at every
level depict the graph of the cobounding function and the dots record cup
products: a solid line represents addition of $+1$ to a function while a
dotted line represents addition of $-1$; similarly, a solid dot represents a
mass of $+1$ on an interval, while an empty circle represents mass $-1$. E.g.
at the first step, the function $d^{-1}X$ takes value $+1$ over the 2nd
segment and $-1$ on the last three. Its cup product with $X$ assigns mass $+1$
to the 3rd segment and $-1$ to the last.

A special case of these higher linking invariants of letters appeared recently
in Monroe–Sinha [MS22]. The weight-reduction algorithm given above is based on
their work and is also the starting point in the sibling paper to this one
based on the Harrison complex [GOSW23]. This algorithm gives an alternative to
Fox calculus, that is both faster and simpler to carry out by hand.

###### Proof.

The isomorphism $L^{-1}\colon{A[t]}\to\mathrm{H}^{0}_{Bar}(\tilde{w};A)$ is
induced by the simplicial map $S^{1}(\tilde{w})\to S^{1}$ collapsing all but
the standard $0$-th segment. On cochains, this induces the map
$t\mapsto\delta_{0}$. It follows that any cocycle in
$Bar^{0}(\tilde{w};A)={T}(C^{1}(\tilde{w};A))$ is cohomologous to a polynomial
in $\delta_{0}$, obtained by the substitution $t\leftrightarrow\delta_{0}$.

To compute this polynomial, one can follow the steps described in the
proposition. The coboundary

$\displaystyle
d_{Bar}(\alpha_{1}|\ldots|\alpha_{k-1}|d^{-1}(fdx)|\alpha_{k+1}|\ldots|\alpha_{n})=-(\alpha_{1}|\ldots|\bigg{[}fdx-(\int_{w}fdx)\cdot\delta_{0}\bigg{]}|\ldots|\alpha_{n})$
$\displaystyle-(\alpha_{1}|\ldots|\alpha_{k-1}\smile
d^{-1}(fdx)|\alpha_{k+1}|\ldots|\alpha_{n})+(\alpha_{1}|\ldots|\alpha_{k-1}|d^{-1}(fdx)\smile\alpha_{k+1}|\ldots|\alpha_{n})$

shows that $(\alpha_{1}|\ldots|fdx|\ldots|\alpha_{n})$ and the expression in
Step (2) are cohomologous. The cup products described in Step (3) follow
directly from Definition 3.14. ∎

With this, one gets an explicit procedure for computing letter-braiding
polynomials.

######  Corollary 3.21.

Given $(\alpha_{1}|\ldots|\alpha_{n})\in(A^{S})^{\otimes n}$, the letter-
braiding polynomial of $w\in F_{S}$ is obtained by applying the weight-
reduction procedure of Proposition 3.19 to the tensor

$(\tilde{w}^{*}(\alpha_{1})|\ldots|\tilde{w}^{*}(\alpha_{n}))\in
C^{1}(\tilde{w};A)^{\otimes n}$

where $\tilde{w}^{*}(\alpha)$ is the composition
$[n]\overset{\tilde{w}}{\to}S\cup S^{-1}\overset{\alpha}{\to}A$ extended by
zero to $0\in[n]_{*}$.

###### Proof.

In Corollary 3.13 we saw that $L_{\bullet}(w)$ is obtained as the composition
$L\circ w^{*}$. The weight-reduction procedure gives a formula for $L$, while
$w^{*}$ is induced by $S^{1}(w)\to X_{S}$ which pulls back 1-cochains
$\alpha\in A^{S}$ as claimed. ∎

Monroe–Sinha [MS22] consider the above weight-reduction procedure in the
special case of tensors $\tilde{w}^{*}(\alpha_{1}|\ldots|\alpha_{n})$ for a
word $w\in F_{S}$ in the $n$-th step of the lower central series. In that
case, they shows that all integrals $\int_{w}fdx$ vanish in every iteration of
the weight-reduction process. As a result, their braiding polynomials never
involve $\delta_{0}$ until the very last step, resulting in a linear
polynomial, as echoed in Corollary 3.30 below.

######  Example 3.22.

For tensors of weight $1$, letter-braiding factors through the abelianization:
suppose $\alpha\in
A^{S}\cong\mathrm{H}^{1}(F_{S};A)\cong\text{Hom}_{grp}(F_{S},A)$

$L_{\alpha}(w)=\alpha(w)\cdot t.$

Indeed, writing the pullback as $\tilde{w}^{*}(\alpha)=fdx$, Proposition 3.19
shows that

$\tilde{w}^{*}(\alpha)\sim(\int_{w}fdx)\cdot t$

and one can check that the integral agrees with $\alpha(w)$.

Next consider the universal commutator $[x,y]=xyx^{-1}y^{-1}$ in $F_{2}$.
Using the weight reduction procedure, its letter-braiding number on
$(\alpha_{1}|\alpha_{2})\in(A^{\\{x,y\\}})^{\otimes 2}$ is

$\ell_{\alpha_{1}|\alpha_{2}}([x,y])=\alpha_{1}(x)\alpha_{2}(y)-\alpha_{1}(y)\alpha_{2}(x)$

as indicated by the following weight-reduction diagram.

${}^{\phantom{-1}}x^{\phantom{-1}}$${}^{\phantom{-1}}y^{\phantom{-1}}$$x^{{-1}}$$y^{{-1}}$$+\alpha_{1}(x)$$+\alpha_{1}(y)$$-\alpha_{1}(x)$$-\alpha_{1}(y)$$\bullet$$+\alpha_{2}(y)$$\bullet$$\circ$$\bullet$$\bullet$$-\alpha_{2}(x)$$\circ$$\bullet$$\circ$$\bullet$

More generally, let $\gamma_{n}F_{S}$ be the $n$-th step of the lower central
series of $F_{S}$, recalled in §2.2.

######  Proposition 3.23 (Invariants of commutators).

For a nested commutator $w\in\gamma_{n}F_{S}$ and any generator $s\in S$, the
commutator $[w,s]$ has invariants

(5)
$\ell_{\alpha_{0}|\alpha_{1}|\ldots|\alpha_{n}}([w,s])=\ell_{\alpha_{0}|\ldots|\alpha_{n-1}}(w)\cdot\alpha_{n}(s)-\alpha_{0}(s)\cdot\ell_{\alpha_{1}|\ldots|\alpha_{n}}(w).$

The proof is deferred to Remark 5.7 below.

Weight reduction can be treated as a recursive procedure. The $k$-th step of
the weight-reduction process shows that any tensor
$(\alpha_{1}|\ldots|\alpha_{n})\in Bar^{0}(\tilde{w};A)$ is cohomolgous to a
linear combination of tensors of the form
$\alpha_{1}|\ldots|\alpha_{n-k}|t^{d}$ for $d\leq k$. Suppose we already
completed a weight-reduction calculation for $\alpha_{1}|\ldots|\alpha_{n-k}$,
then we can use the following.

######  Lemma 3.24.

If $T\in Bar^{0}(\tilde{w};A)$ has weight reduction $P(t)\in A[t]$, then
$T|t^{d}$ has weight reduction $P(t)\cdot t^{d}$.

###### Proof.

If $T=P(t)+d_{Bar}(\eta)$ for some polynomial $P(t)\in{A[t]}$ and for $\eta\in
Bar^{-1}(w;A)$, then we claim that

$T|\underbrace{t|\ldots|t}_{d\text{ times}}=P(t)\cdot
t^{d}+d_{Bar}(\eta|\underbrace{t|\ldots|t}_{d\text{ times}}).$

Indeed, since every $f\in\overline{C}^{0}(\tilde{w};A)$ satisfies $f(0)=0$, it
follows that $f\smile t=0$. Thus,

$d_{Bar}(\eta|\underbrace{t|\ldots|t}_{d\text{
times}})=d_{Bar}(\eta)|\underbrace{t|\ldots|t}_{d\text{ times}}$

demonstrating the claimed equality. ∎

One particular choice of weight reduction gives rise to the following discrete
analog of Chen’s iterated integrals in [Che71].

######  Corollary 3.25 (Letter braiding as iterated sum).

For functions $\alpha_{1},\ldots,\alpha_{r}\in A^{S}$, the letter-braiding
number of a word $\tilde{w}\colon[n]\to S\cup S^{-1}$ is

(6)
$\ell_{\alpha_{1}|\ldots|\alpha_{r}}(w)=\sum_{i_{1}<_{w}i_{2}<_{w}\ldots<_{w}i_{r}}\alpha_{1}(\tilde{w}(i_{1}))\ldots\alpha_{r}(\tilde{w}(i_{r}))$

where $i<_{w}j$ is the relations $i<j$ when $\tilde{w}(i)\in S$, and $i\leq j$
when $\tilde{w}(i)\in S^{-1}$.

In particular, when the $\alpha_{j}$’s are indicator forms, this sum counts
the number of occurrences of a sequence of generators, in order, within a
word. Both the signs and the “edge cases” specified by $<_{w}$ are critical
for invariance. For example, in $F_{\\{x,y,z\\}}$ the invariant
$\ell_{X|Y|Z}(w)$ counts the number of times the generator $z$ follows $y$
which in turn follows $x$ in the word $w$ (see Example 1.2).

###### Proof.

Firstly, observe that for any $\alpha\in A^{S}$, writing
$\tilde{w}^{*}\alpha=fdx$ we have

$\tilde{w}^{*}(\alpha)_{i}=\alpha(\tilde{w}(i))=f(i).$

With this, the corollary follows from the next calculation of the linear
coefficient of the weight-reduced polynomial: if
$(w^{*}(\alpha_{1})|\ldots|w^{*}(\alpha_{n}))\sim\ell\cdot t+t^{2}P(t)$, then

(7) $\ell=\sum_{\begin{subarray}{c}1\leq i_{1}\leq i_{2}\leq\ldots\leq
i_{r}\leq n\\\ \text{s.t. }\forall j:\,\partial_{1}(i_{j})\leq
i_{j+1}\end{subarray}}f_{1}(i_{1})f_{2}(i_{2})\ldots f_{r}(i_{r}),$

where $\partial_{1}(i)$ is the terminal vertex of the $i$-th edge in
$S^{1}(\tilde{w})$. Indeed, the condition $\partial_{1}(i)\leq j$ is
equivalent to $i<_{w}j$.

The equality is proved by induction on $r$, with $r=1$ being the equality in
Example 3.22. Now, denote the iterated sum in (7) by
$\int_{w}(f_{1}dx|\ldots|f_{r}dx)$ and assume by induction that for all
$(f_{1},\ldots,f_{r})$

$(f_{1}dx|\ldots|f_{r}dx)\sim t^{2}\cdot
P(t)+t\cdot\int_{w}(f_{1}dx|\ldots|f_{r}dx),$

where $\sim$ denotes the relation of differing by coboundary. Then for $r+1$
we have

$f_{1}dx|\ldots|f_{r}dx|f_{r+1}dx\sim\left(\int_{w}f_{r+1}dx\right)\cdot(f_{1}dx|\ldots|f_{r}dx|t)-(f_{1}dx|\ldots|f_{r}dx\smile
d^{-1}(f_{r+1}dx)),$

Lemma 3.24 shows that the first term has polynomial invariant divisible by
$t^{2}$ and thus does not contribute to the linear coefficient. And by the cup
product formulas in Proposition 3.19,

$f_{r}dx\smile
d^{-1}(f_{r+1}dx)=-\bigg{(}\sum_{i_{r+1}\geq\partial_{1}(\bullet)}f_{r+1}(i_{r+1})\bigg{)}f_{r}(\bullet)dx.$

The inductive hypothesis now implies that the linear coefficient is

$\sum_{\begin{subarray}{c}1\leq i_{1}\leq\ldots\leq i_{r}\leq n\\\ \forall
j:\,\partial_{1}(i_{j})\leq i_{j+1}\end{subarray}}f_{1}(i_{1})\ldots
f_{r-1}(i_{r-1})\cdot\bigg{(}f_{r}(i_{r})\cdot\sum_{\begin{subarray}{c}i_{r}\leq
i_{r+1}\\\ \partial_{1}(i_{r})\leq
i_{r+1}\end{subarray}}f_{r+1}(i_{r+1})\bigg{)}.\qed$

###  3.3. Properties of letter-braiding

The following example specializes the iterated sum formula to the case of the
universal 2-letter word.

######  Example 3.26 (The universal product).

Let $\\{e_{1},e_{2}\\}$ generate a free group of rank $2$. We compute the
letter-braiding invariants of the word $\tilde{w}=e_{1}e_{2}$. Denote the dual
basis to $e_{1}$ and $e_{2}$ in $A^{\\{e_{1},e_{2}\\}}$ by $\langle
E_{1},E_{2}\rangle$, that is, $E_{1}(e_{1})=E_{2}(e_{2})=1$ and
$E_{1}(e_{2})=E_{2}(e_{1})=0$. Then ${T}(A^{\\{e_{1},e_{2}\\}})\cong A\langle
E_{1},E_{2}\rangle$ is the ring of noncommutative polynomials in $E_{1}$ and
$E_{2}$.

The simplicial map corresponding to the word $e_{1}e_{2}$ is

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.87032pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{e_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
12.75pt\raise-5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$\scriptstyle{1}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
46.87032pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{e_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
47.75pt\raise-5.25555pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$\scriptstyle{2}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.5pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
64.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
30.25pt\raise-18.74445pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
2.25586pt\raise-2.99405pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces\end{split}\quad\xrightarrow{e_{1}e_{2}}\quad\underbrace{S^{1}}_{e_{1}}\vee\underbrace{S^{1}}_{e_{2}}\;=\;X_{\\{e_{1},e_{2}\\}}$

collapsing the $0$-th edge to the basepoint. Its pullback
$(e_{1}e_{2})^{*}\colon C^{1}(X_{\\{e_{1},e_{2}\\}};A)\to C^{1}(e_{1}e_{2};A)$
is sends $E_{1}$ to the indicator form $\delta_{1}$ and $E_{2}$ to the
indicator $\delta_{2}$.

######  Proposition 3.27.

Let $T\in A\langle E_{1},E_{2}\rangle$ be any noncommutative monomial in
$E_{1}$ and $E_{2}$. The letter-braiding number of the word $E_{1}E_{2}\in
F_{\\{e_{1},e_{2}\\}}$ on $T$ is

$\ell_{T}(e_{1}e_{2})=\begin{cases}1&\text{ if }T=E_{1},E_{2}\text{ or
}E_{1}|E_{2}\\\ 0&\mathrm{otherwise}.\end{cases}$

###### Proof.

By the iterated sum formula in Corollary 3.25,

$\ell_{E_{j_{1}}|\ldots|E_{j_{n}}}(e_{1}e_{2})=\sum_{1\leq
i_{1}<\ldots<i_{n}\leq 2}E_{j_{1}}(e_{i_{1}})\ldots E_{j_{n}}(e_{i_{n}}).$

If $n>2$ then this sum is empty, and the invariant vanishes. When $n=2$ the
invariant is $E_{j_{1}}(e_{1})E_{j_{2}}(e_{2})$, which vanishes unless
$(E_{j_{1}}|E_{j_{2}})=(E_{1}|E_{2})$ whose invariant is $1$. The remaining
case of $n=1$ is clear. ∎

We can now calculate the letter-braiding number of a product.

######  Corollary 3.28.

If $w_{1},w_{2}\in F_{S}$ are two elements, and $T\in\overline{T}(A^{S})$ in
the augmentation ideal satisfies
$\overline{\Delta}(T)=\sum_{i}T_{1}^{i}\otimes T_{2}^{i}$, then

(8) $\ell_{T}(w_{1}\cdot
w_{2})=\ell_{T}(w_{1})+\ell_{T}(w_{2})+\sum_{i}\ell_{T_{1}^{i}}(w_{1})\cdot\ell_{T_{2}^{i}}(w_{2}).$

###### Proof.

Define a group homomorphism $h\colon F_{\\{e_{1},e_{2}\\}}\to F_{S}$ by
$e_{i}\mapsto w_{i}$. Now, the composition

$\textstyle{F_{\\{s\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{1}\cdot
e_{2}}$$\textstyle{F_{\\{e_{1},e_{2}\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{F_{S}}$

sends $s$ to $w_{1}\cdot w_{2}$. Consider the functorial pullback maps defined
in Theorem 3.8.

On the one hand, Example 3.9 shows that the pullback along the composition is
the letter-braiding polynomial $L_{T}(w_{1}\cdot w_{2})$. On the other hand,
Example 3.10 gives

$h^{*}(T)=\ell_{T}(w_{1})\cdot E_{1}+\ell_{T}(w_{2})\cdot
E_{2}+\sum_{i}\ell_{T_{1}^{i}}(w_{1})\ell_{T_{2}^{i}}(w_{2})\cdot
E_{1}|E_{2}+\text{ all other monomials}.$

Applying Example 3.9 again, now to $(e_{1}e_{2})^{*}$, gives the letter-
braiding polyonmial of the word $e_{1}e_{2}$ on any noncommutative monomial in
$E_{1}$ and $E_{2}$. But these were computed in Example 3.26, and the braiding
numbers $\ell_{\bullet}(e_{1}e_{2})$ vanish on all monomials other than the
ones listed in the last equation. Substituting the nonzero braiding numbers
proves (8). ∎

######  Example 3.29.

The product formula (8) gives $\ell_{T}(s^{-1})$, for inverses of generators.
Indeed, if $T=\alpha_{1}|\ldots|\alpha_{n}$ then

$0=\ell_{T}(ss^{-1})=\ell_{T}(s)+\ell_{T}(s^{-1})+\sum_{i=1}^{n-1}\ell_{\alpha_{1}|\ldots|\alpha_{i}}(s)\ell_{\alpha_{i+1}|\ldots|\alpha_{n}}(s^{-1})$

and since $\ell_{\alpha_{1}|\ldots|\alpha_{i}}(s)$ vanishes unless $i=1$, we
get

$\ell_{\alpha_{1}|\ldots|\alpha_{n}}(s^{-1})=-\alpha_{1}(s)\ell_{\alpha_{2}|\ldots|\alpha_{n}}(s^{-1}).$

It thus follows that

(9)
$\ell_{\alpha_{1}|\ldots|\alpha_{n}}(s^{-1})=(-1)^{n}\alpha_{1}(s)\ldots\alpha_{n}(s).$

More generally, one can check that for any $w\in F_{S}$, its inverse has

(10) $\ell_{T}(w^{-1})=L_{T}(w)|_{t=-1},$

the evaluation the braiding polynomial or $w$ at $(-1)$.

We will see in §5 that the product formula (8) is equivalent to the statement
that the (unreduced) coproduct $\Delta$ on $\mathrm{H}^{0}_{Bar}(F_{S};A)$ is
dual to the convolution product of the group ring $A[F_{S}]$.

######  Corollary 3.30.

Let $\gamma_{k}F_{S}$ be the $k$-th term in the lower central series. If
$w\in\gamma_{k}F_{S}$, and $T\in\overline{T}(A^{S})$ is a reduced tensor of
weight $n$, then the letter-braiding polynomial satisfies

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

In particular,

$L_{T}(w)=\begin{cases}\ell_{T}(w)\cdot t&\text{ if }n<2k\\\ 0&\text{ if
}n<k.\end{cases}$

###### Proof.

For $T$ of weight $1$ the claim is obvious, since all
$\alpha\in\mathrm{H}^{1}(F_{S};A)$ vanish on $[F_{S},F_{S}]$.

Now, assume by induction on $n$ that any tensor $T^{\prime}$ of weight
$n^{\prime}<n$ has $\deg L_{T^{\prime}}(w)\leq n^{\prime}/k$, so in particular
if $L_{T^{\prime}}(w)\neq 0$ then $n^{\prime}\geq k$. Now, if $T$ has weight
$n$ and $\deg L_{T}(w)=d$, then $\overline{\Delta}^{d}L_{T}(w)\neq 0$. But
since $L_{\bullet}(w)$ is a coalgebra homomorphism, this means that

$\overline{\Delta}^{d}(T)=\sum_{i}T_{1}^{i}\otimes\ldots\otimes
T_{d}^{i}\implies
0\neq\overline{\Delta}^{d}(L_{T}(w))=\sum_{i}L_{T_{1}^{i}}(w)\otimes\ldots\otimes
L_{T_{d}^{i}}(w).$

Fix a summand $T_{1}^{i}\otimes\ldots\otimes T_{d}^{i}$ on which
$L_{\bullet}(w)$ does not vanish. Since all $T_{j}^{i}$ are tensors of weight
$<n$, the induction hypothesis implies that they must each have weight $\geq
k$. It follows that $n\geq d\cdot k$. ∎

######  Corollary 3.31.

If $T\in\overline{T}(A^{S})$ is a reduced tensor of weight $<2k$, then the
function $w\mapsto\ell_{T}(w)$ is a group homomorphism $\gamma_{k}F_{S}\to A$.

###### Proof.

Using the product formula $\ell_{T}(w_{1}\cdot
w_{2})=\ell_{T}(w_{1})+\ell_{T}(w_{2})+\sum_{i}\ell_{T_{1}^{i}}(w_{1})\cdot\ell_{T_{2}^{i}}(w_{2})$,
it suffices to show that $\ell_{T_{1}^{i}}(w_{1})\ell_{T_{2}^{i}}(w_{2})=0$
for every $i$. This follows since their weights satisfy
$\operatorname{wt}(T_{1})+\operatorname{wt}(T_{2})\leq\operatorname{wt}(T)<2k$,
so one of the weights must be smaller than $k$ and thus vanish on
$\gamma_{k}F_{S}$. ∎

In fact, over $\mathbb{Q}$, the results in [MS22] imply that every
homomorphism $\gamma_{k}F_{S}/\gamma_{2k}F_{S}\to\mathbb{Q}$ is obtained as
the letter-braiding invariant for a unique $T\in\overline{T}(\mathbb{Q}^{S})$
of weight $<2k$.

##  4\. General groups

We extend letter-braiding invariants to general groups, first recalling
definitions. Define the fundamental group of a pointed simplicial set
$(Y_{\bullet},*)$ as that of its geometric realization. Let
$\pi_{1}(Y,*)=\Gamma$. Letter-braiding invariants of $\Gamma$ are assigned to
the coalgebra $\mathrm{H}^{0}_{Bar}(Y;A)$. Every $w\in\Gamma$ determines a
continuous map $w\colon|S^{1}|\to|Y|$, well-defined up to homotopy, and thus
induces a pullback

$w^{*}\colon\mathrm{H}^{0}_{Bar}(Y;A)\to\mathrm{H}^{0}_{Bar}(S^{1};A).$

Using the canonical isomorphism $L\colon\mathrm{H}^{0}_{Bar}(S^{1};A)\to
A[t]$, we define the following.

######  Definition 4.1.

Fix $T\in\mathrm{H}^{0}_{Bar}(Y;A)$. For every $w\in\Gamma$ the _letter-
braiding polynomial_ of $w$ associated to $T$ is

$L_{T}(w)=L(w^{*}(T))\in A[t].$

The _letter-braiding number_ $\ell_{T}(w)\in A$ is the linear coefficient of
$L_{T}(w)$.

By relating these letter-braiding polynomials to the ones in §3, we will get
formulas for computing them and obstructions for their existence.

The following claim shows that the particular space that realizes a group
$\Gamma$ is its fundamental group is immaterial. Consequently, any such space
$(Y,*)$ gives a distinct viewpoint on the braiding invariants of words in
$\Gamma$, in some cases by a geometric description of the ring
$\mathrm{H}^{*}(Y;A)$.

######  Lemma 4.2.

If $(Y,*)$ and $(Y^{\prime},*^{\prime})$ are connected pointed simplicial sets
and $f\colon(|Y|,*)\to(|Y^{\prime}|,*^{\prime})$ is a pointed map of their
realizations that induces an isomorphism on $\pi_{1}$, then the pullback

$f^{*}\colon\mathrm{H}^{0}_{Bar}(Y^{\prime};A)\to\mathrm{H}^{0}_{Bar}(Y;A)$

given by Lemma 3.2 is an isomorphism.

###### Proof.

By quasi-isomorphism invariance of $\mathrm{H}^{0}_{Bar}(-;A)$ and Lemma 3.2,
it is sufficient to prove the claim for a simplicial map
$(Y,*)\to(Y^{\prime},*^{\prime})$. Let $Z$ be the mapping cone of $f$. By Van
Kampen’s theorem $Z$ is simply connected, and thus the long exact sequence in
cohomology has

$\cancel{\tilde{\mathrm{H}}^{0}(Z;A)}\overset{d}{\to}\mathrm{H}^{1}(Y^{\prime};A)\overset{f^{*}}{\to}\mathrm{H}^{1}(Y;A)\to\cancel{\mathrm{H}^{1}(Z;A)}\overset{d}{\to}\mathrm{H}^{2}(Y^{\prime};A)\overset{f^{*}}{\to}\mathrm{H}^{2}(Y;A),$

so that $f^{*}\colon\mathrm{H}^{*}(Y^{\prime};A)\to\mathrm{H}^{*}(Y;A)$ is an
isomorphism when $*\leq 1$ and injective when $*=2$. At the cochain level,
this means that the pullback $f^{*}\colon C^{*}(Y^{\prime};A)\to C^{*}(Y;A)$
is $1$-connected, so that after desuspension it becomes $0$-connected on
$\mathbf{s}^{-1}{\overline{C}(-;A)}$.

Since the complexes here consist of torsion-free and thus flat $A$-modules, it
follows that on every tensor power

$(f^{*})^{\otimes p}\colon\mathbf{s}^{-1}{\overline{C}(Y^{\prime};A)}^{\otimes
p}\to\mathbf{s}^{-1}{\overline{C}(Y;A)}^{\otimes p}$

is also $0$-connected. Passing to Bar constructions, we imitate the proof that
$Bar(-)$ is a quasi-isomorphism invariant (Lemma 2.10) and consider the
spectral sequence of the filtration by weight. Its $E_{0}$ page is simply the
direct sum of complexes $\mathbf{s}^{-1}{\overline{C}(-;A)}^{\otimes p}$, and
thus $f$ induces a $0$-connected map on the $E_{0}$-page. Since this
connectivity property passes through the pages of the spectral sequence, it
follows that the map

$f^{*}\colon Bar(Y^{\prime};A)\to Bar(Y;A)$

is already $0$-connected. It therefore induces an isomorphism on
$\mathrm{H}^{0}_{Bar}(-;A)$ and an injection on $\mathrm{H}^{1}_{Bar}(-;A)$. ∎

In the present work, our preferred choice of simplicial set with fundamental
group $\Gamma$ the standard simplicial model for the classifying space.

######  Definition 4.3.

Recall, the simplicial Bar construction of a group $\Gamma$ is the simpilicial
set

$B\Gamma_{\bullet}\colon[p]\mapsto\Gamma^{p}$

with face maps $\partial_{0}$ and $\partial_{p}$ forgetting the respective
extremal coordinate, internal face maps given by multiplication in $\Gamma$ of
adjacent terms, and degeneracies inserting copies of the unit. Its fundamental
group is canonically isomorphic to $\Gamma$. Denote

$Bar(\Gamma;A):=Bar(B\Gamma;A),\text{ and
}\mathrm{H}^{*}_{Bar}(\Gamma;A)=\mathrm{H}^{*}(Bar(B\Gamma;A)).$

The functoriality of the construction $\Gamma\mapsto B\Gamma$ implies the
following.

######  Lemma 4.4.

The assignment $\Gamma\mapsto Bar(\Gamma;A)$ is a contravariant functor from
the category of groups to the category of bigraded dg-coalgebras. Thus
$\Gamma\mapsto\mathrm{H}^{0}_{Bar}(\Gamma;A)$ is a contravariant functor
taking values in the category of filtered coalgebras.

Moreover, any connected $(Y,*)$ admits a natural isomorphism

$\mathrm{H}^{0}_{Bar}(\pi_{1}(Y,*);A)\overset{\sim}{\to}\mathrm{H}^{0}_{Bar}(Y;A).$

###### Proof.

Recalling that $|B\Gamma_{\bullet}|$ is a $K(\Gamma,1)$ space, homotopy
classes of pointed maps $(X,*)\to|B\Gamma_{\bullet}|$ are in bijection with
group homomorphisms $\pi_{1}(X,*)\to\Gamma$. In particular, for any
topological space $(X,*)$ there exists a pointed map $i\colon
X\to|B(\pi_{1}(X,*))_{\bullet}|$, unique up to homotopy, inducing the identity
on $\pi_{1}$. Applied to $X=|Y|$, Lemma 3.2 gives a canonical pullback

$i^{*}\colon\mathrm{H}^{0}_{Bar}(\pi_{1}(Y,*);A)\to\mathrm{H}^{0}_{Bar}(Y;A),$

which by Lemma 4.2 is an isomorphism. Functoriality and naturality are clear.
∎

###  4.1. Letter braiding and vanishing (Massey) products

The spectral sequence of a bicomplex applied to the Bar complex of $\Gamma$
gives a first (upper bound) approximation to $H^{0}_{Bar}(\Gamma;A)$: the
tensor (co)algebra on $H^{1}(\Gamma;A)=\text{Hom}_{grp}(\Gamma,A)$, which is
sharp when $\Gamma$ is free as we saw in §3. The second order approximation is
the Bar cohomology of the ring $H^{*}(B\Gamma;A)$. We see immediately that in
weight two, $H^{0}_{Bar}(\Gamma;A)$ is computed by the kernel of cup product
$H^{1}(B\Gamma;A)\otimes H^{1}(B\Gamma;A)\to H^{2}(B\Gamma;A)$

In general constructing letter-braiding invariants for $\Gamma$ relies on
understanding the cohomology ring, including Massey products, of $B\Gamma$ or
any other $X$ with $\pi_{1}(X,*)\cong\Gamma$ in degrees one and two. We will
observe that introducing relations to a group simultaneously gives
obstructions for braiding invariants to be well-defined as well as creates
Massey products in this cohomology ring, both encoded by $d_{Bar}\colon
Bar^{0}\to Bar^{1}$.

######  Example 4.5.

Let $F_{\\{e,f\\}}$ be the free group on two generators, with letter-braiding
invariants indexed by noncommutative polynomials $\mathbb{Z}\langle
E,F\rangle$. For example, $\ell_{E}(-)$ counts the signed number of $e$’s in
words.

Now, introduce the relation $e=f$, so a letter $e$ may be replaced by an $f$.
With this, the functional $\ell_{E}(-)$ is no longer well-defined, e.g.
$eefe=fffe$ under this relation, but

$\ell_{E}(eefe)=3\neq 1=\ell_{E}(fffe).$

So $E$ and $F$ are lost as invariants of the group. However, $E+F$ still
defines an invariant.

At the level of the Bar complex, the effect of introducing the relation is to
add a $2$-cell $D^{2}_{e^{-1}f}$, so that $dE=-dF=[D^{2}_{e^{-1}f}]$ are no
longer closed. But their sum $E+F$ is closed, and represents a braiding
invariant.

######  Example 4.6.

A more interesting example is obtained by introducing the relation $ef=fe$ to
$F_{\\{e,f\\}}$, modeled by a $2$-torus. Now, $E$ and $F$ still define
invariants of words, but $(E|F)$ does not: it counts occurrences of $e$
_before_ $f$, which changes upon application of the relation.

Accordingly, $(E|F)$ is not a Bar cocycle, because $E\smile F\neq 0$ in
cohomology. Anti-symmetry of the cup product means that $(E|F+F|E)$ is an
invariant.

######  Remark 4.7.

The loss of the invariant $(E|F)$ in the last example is reflected by the fact
that in the abelian group $F_{\\{e,f\\}}/\langle[e,f]\rangle$, any two
distinct elements are already distinguished by their evaluation on $E$ and
$F$, so no additional invariants are needed. Still, higher weight invariants
exist and serve to distinguish linear combinations of elements in the group
ring, as discussed in Example 5.8.

On the cohomological side, the Bar cohomology features cochains with vanishing
products, as explained next.

######  Example 4.8.

Let $Y$ be any pointed simplicial set. Given $1$-cochains
$\alpha_{1},\ldots,\alpha_{n}\in C^{1}(Y;A)$, lets examine when the tensor
$T=(\alpha_{1}|\ldots|\alpha_{n})\in Bar^{0}(Y;A)$ is a cocycle. The
differential in the Bar construction is $d_{Bar}=d_{1}+d_{2}$, with $d_{1}$
internal to $C^{*}(Y;A)$ and $d_{2}$ given by cup products of cochains. So
$d_{Bar}T=0$ iff

  * •

every $\alpha_{i}$ is closed in $C^{1}(Y;A)$, and

  * •

every cup product $\alpha_{i}\smile\alpha_{i+1}$ vanishes in $C^{2}(Y;A)$.

When these conditions are satisfied, $T$ represents a class in
$\mathrm{H}^{0}_{Bar}(Y;A)$.

Now, let $w\colon S^{1}(w)\to Y$ be a simplicial map representing some
$w\in\pi_{1}(Y,*)$. Then

$w^{*}(T)=w^{*}(\alpha_{1})|\ldots|w^{*}(\alpha_{n})\in Bar^{0}(w;A)$

can be weight-reduced to a polynomial $L_{T}(w)\in{A[t]}$, thus giving the
letter-braiding polynomial of $w$. Most importantly, due to relations in
$\pi_{1}(Y,*)$ there might be different words representing the same group
element. But, Lemma 3.2 guarantees that the letter-braiding polynomial only
depends on the pointed homotopy class $S^{1}\to Y$, and is thus an invariant
of $w$.

To better understand the condition $d_{Bar}(T)=0$, consider the following two
examples.

######  Example 4.9.

Suppose $\alpha,\beta\in C^{1}(Y;A)$ are $1$-cocycles whose cup product
vanishes in cohomology. Then $d_{Bar}(\alpha|\beta)=-\alpha\smile\beta=d\eta$
for some $\eta\in C^{1}(Y;A)$ but might not be $0$. However, the in-
homogeneous linear combination $(\alpha|\beta)+\eta$ satisfies,

$d_{Bar}\colon(\alpha|\beta)+\eta\longmapsto-\alpha\smile\beta-d\eta=0$

giving a Bar $0$-cocycle of weight $2$. We thus get a well-defined letter-
braiding invariant, by weight reducing
$(w^{*}(\alpha)|w^{*}(\beta))+w^{*}(\eta)\in Bar^{0}(w;A)$, while neither of
the summands is by itself a homotopy invariant.

Summarising, cup products need only vanish at the level of cohomology to give
invariants, but one must then include a correction term of lower weight
‘witnessing’ this.

######  Example 4.10.

To construct a weight $3$ Bar cocycle, consider closed $1$-cochains
$\alpha,\beta,\gamma\in C^{1}(Y;A)$ such that both $\alpha\smile\beta=d\eta$
and $\beta\smile\gamma=d\zeta$ vanish in cohomology. Then

$d_{Bar}(\alpha|\beta|\gamma-\eta|\gamma-\alpha|\zeta)=(\eta\smile\gamma+\alpha\smile\zeta)$

might still be nonzero. This last expression represents the _triple Massey
product_ $\langle\alpha,\beta,\gamma\rangle$, and it is a $2$-cocycle in
$C^{2}(Y;A)$. If it vanishes in cohomology, then a secondary correction term
$\tau$ exists, making the inhomogeneous sum
$(\alpha|\beta|\gamma-\eta|\gamma-\alpha|\zeta+\tau)$ into a Bar cocycle and
defining a letter-braiding invariant.

More generally, tensors $(\alpha_{1}|\ldots|\alpha_{n})$ extend to Bar
cocycles iff all their consecutive cup and higher Massey products vanish in
cohomology. For general linear combinations of tensors, this is all best
housed in the spectral sequence we discussed above, given formally as follows.

######  Lemma 4.11 (Weight spectral sequence).

Suppose $(Y,*)$ is connected. There is a natural second quadrant spectral
sequence converging to $\mathrm{H}^{*}_{Bar}(Y,A)$, which in total degree $0$
admits a natural inclusion of graded coalgebras

$E_{r}^{-\bullet,\bullet}\hookrightarrow
T(\mathrm{H}^{1}(Y;A))\quad\text{where}\quad
E_{r}^{-p,p}\hookrightarrow\mathrm{H}^{1}(Y;A)^{\otimes p}$

on every page. On the $E_{1}$-page this inclusion is an equality, and the
differential given by the cup product in cohomology

$[\alpha_{1}]\otimes\ldots\otimes[\alpha_{p}]\mapsto-\sum_{i=1}^{p-1}[\alpha_{1}]\otimes\ldots\otimes[\alpha_{i}\smile\alpha_{i+1}]\otimes\ldots\otimes[\alpha_{p}].$

Later differentials are given by higher Massey products, so that cycles in
$E_{r}^{-p,p}$ are tensors of weight $p$ in $\mathrm{H}^{1}(Y;A)$ with
vanishing Massey products of size $\leq r$. For example444In practice, it is
not likely that pure tensors have well-defined Massey products, and only
linear combinations of them survive to the $E_{r}$-page of the spectral
sequence. However, see Proposition 4.13.,

(11)
$d_{r}\colon[\alpha_{1}]\otimes\ldots\otimes[\alpha_{p}]\mapsto-\sum_{i=1}^{p-r}[\alpha_{1}]\otimes\ldots\otimes\langle\alpha_{i},\ldots,\alpha_{i+r}\rangle\otimes\ldots\otimes[\alpha_{p}],$

This spectral sequence is natural in pointed simplicial maps
$(Y,*)\to(Y^{\prime},*^{\prime})$.

###### Proof.

The spectral sequence comes from the filtration of $Bar^{*}(Y;A)$ by weight,
which is clearly natural in maps of pointed simplicial sets. The $E_{0}$-page
of this spectral sequence consists of columns $\overline{C}^{*}(Y;A)^{\otimes
p}$ shifted down $p$ degrees, with the usual tensor product differential.
Since $\tilde{\mathrm{H}}^{0}(Y;A)=0$, the Künneth spectral sequence shows
that in degree $<p$ this tensor product complex has vanishing cohomology, and
in degree $p$ its cohomology is $\cong\mathrm{H}^{1}(Y;A)^{\otimes p}$.

It follows that $E_{1}^{-p,q}$ vanishes in negative total degree, and
$E_{1}^{-p,p}\cong\mathrm{H}^{1}(Y;A)^{\otimes p}$, as claimed. In particular,
no differentials land in total degree $0$, and thus all pages of the spectral
sequence have

$E_{r}^{-p,p}\hookrightarrow E_{1}^{-p,p}$

characterized by the vanishing of the first $r$ differentials. The cup product
on cohomology and higher Massey products coincide with the differentials of
this spectral sequence essentially by their definition, see [May69]. ∎

######  Corollary 4.12.

For every weight $n\geq 1$ there is a natural ‘leading term’ map

$\operatorname{lead}\colon\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq
n}\to\mathrm{H}^{1}(\Gamma;A)^{\otimes n}$

with image all tensors with vanishing cup and Massey products, and kernel
$\mathrm{H}^{0}_{Bar}(\Gamma;A)_{\leq n-1}$.

In many contexts, the celebrated Homotopy Transfer Theorem [Kad80] gives
globally defined Massey products

(12) $\mu_{r}\colon\mathrm{H}^{1}(Y;A)^{\otimes
r}\to\mathrm{H}^{2}(Y;A)\quad\forall r\geq 2$