#  Infinitesimal calculations in fundamental groups

Nir Gadish  Mathematics Department, University of Michigan<EMAIL_ADDRESS>,
Aydin Ozbek  Mathematics Department, University of Oregon
<EMAIL_ADDRESS>,  Dev Sinha  Mathematics Department, University of
Oregon<EMAIL_ADDRESS>and  Ben Walter  Mathematics Department, University of
the U.S. Virgin Islands<EMAIL_ADDRESS>
###### Abstract.

We show that Hopf invariants, defined by evaluation in Harrison cohomology of
the commutative cochains of a space, calculate the logarithm map from a
fundamental group to its Malcev Lie algebra. They thus present the zeroth
Harrison cohomology as a universal dual object to the Malcev Lie algebra. This
structural theorem supports explicit calculations in algebraic topology,
geometric topology and combinatorial group theory. In particular, we give the
first algorithm to determine whether a word raised to some power is a $k$-fold
nested product of commutators, in any group presented by generators and
relations.

To the memory and loved ones of the second author.

##  1\. Introduction

We fully develop the capacity of rational or real cochains to measure a
fundamental group, and thus any algebraically presented group, through a
variety of perspectives.

  * •

In algebraic topology, we establish that Harrison cohomology in degree zero
serves as a universal rational Lie dual of a group, equipped with an explicit
pairing. It thus provides “coordinates” for the infinitesimal approximation of
a group given by the logarithm map from a group to its Malcev Lie algebra.

  * •

Geometrically, intersection of a loop with codimension one subspaces of a
manifold, tracking the ordering of meeting times, can be used to define
functions on a fundamental group. The theory thus extends non-abelian
intersection theory [Hai85], as well as Cochran’s development of Milnor’s link
invariants [Coc90]. We initiate the art of finding suitable subspaces and
analyzing them through many examples in Section 5, with an emphasis on two-
complexes (thickened to four-manifolds) as well as Milnor invariants, which
will be fully developed in a sequel.

  * •

Our first full application is in combinatorial group theory. For algebraically
presented groups, the theory leads directly to letter-braiding and letter-
linking invariants, which generalize Magnus expansion and Fox calculus to all
groups. We determine an explicit set of functions on a presented group which
obstruct whether some power of a word lies in a desired $k$-fold commutator
subgroup. These functions are algorithmic, and have been implemented in SAGE
by the second author.

The appearance of Malcev Lie algebras as fundamental groups in rational
homotopy theory is classical [Qui69] (see [Iva22] for a recent review). But
standard approaches produce an abstractly presented Lie algebra, without
comparison to the group. They give no way to answer questions such as
determining Malcev filtration degree of specific group elements.

Our work produces an explicit pairing between a group $G$ and a central object
in one of the Quillen approaches to rational homotopy theory, namely the
zeroth Harrison cohomology of a space with that fundamental group. With this
pairing, one is able to compute the depth of some $\gamma\in G$ in the
rational lower central series of $G$. Classical algebraic presentation through
generators and relations are reflected in the topology of two-dimensional CW
complexes. Their analysis gives rise to a lifting criterion (Theorem 4.7),
which in turn gives rise to an algorithm which determines whether or not some
power of a word is a product of nested $k$-fold commutators. We introduce the
invariants which constitute the algorithm in Section 1.2 below.

While computational algebra is currently our most developed application, the
theory is ripe for a wide range of applications, as we address in Section
1.4.3. To study a group through these “coordinates”, one has freedom in
choosing a space which presents it as a fundamental group, a flavor of and
model for rational (or real) cochains, a presentation of Harrison cohomology,
and additional flexibility by considering maps between spaces. The resulting
”charts” can look different for two algebraic presentations of the same group
(see Section 5.1) and have a different flavor altogether in topological
examples (Sections 5.2 and 5.5). The main construction, that of the Hopf
pairing, is simple and natural enough that it should generalize to many
settings.

We elaborate on the three main perspectives for our results below. In Section
1.1, we share the framework for results which comes from algebraic topology.
In Section 1.2 we share our first applications, to combinatorial group theory.
In Section 1.3 we share basic motivating examples, which come from geometric
topology. We have written these independently, to be read in any order.
Readers are encouraged to explore their favorite “side of the
elephant”111https://www.flickr.com/photos/loxosceles/2334950809 first.

###### Contents

  1. 1 Introduction
    1. 1.1 Algebraic topology
    2. 1.2 Combinatorial group theory
    3. 1.3 Geometric topology
    4. 1.4 Relation to other work, technical setting, reading guidance, and future directions
      1. 1.4.1 Primary relation to other work
      2. 1.4.2 Technical setting and reading guidance
      3. 1.4.3 Future directions
  2. 2 Definitions
    1. 2.1 Eil trees and cofree Lie coalgebras
    2. 2.2 Harrison complex and Hopf invariants
  3. 3 Central Example: Handlebodies and linking of letters for free groups
    1. 3.1 First handlebody example
    2. 3.2 Equating topology and combinatorics of letter linking
    3. 3.3 Configuration braiding of trees and words
  4. 4 Structural theorems
    1. 4.1 Lie pairing
    2. 4.2 The lifting criterion
    3. 4.3 Letter linking algorithms
    4. 4.4 The Fundamental Theorem
  5. 5 Examples
    1. 5.1 Orientable surfaces and our favorite little two-complex
    2. 5.2 Milnor invariants for Borromean rings
    3. 5.3 Beyond letter linking
    4. 5.4 Commutators and Massey products
    5. 5.5 Pure braid groups
    6. 5.6 An infinite example
    7. 5.7 Beyond residual nilpotence using the product totalization
  6. A Bar and Harrison complexes
    1. A.1 Basics and connection with Sullivan models
    2. A.2 Explicit treatment of signs
  7. B Graphical (partial) models for groups through Thom Forms
    1. B.1 Thom forms
    2. B.2 Thickening two-complexes
    3. B.3 de Rham cochains associated to group generators and curves on disks

###  1.1. Algebraic topology

We first recall some basic facts about Malcev Lie algebras. The Baker-
Campbell-Hausdorff expansion for the logarithm of a product of exponentials in
a noncommutative algebra is

$\mathrm{BCH}(x,y)=\log(e^{x}e^{y})=x+y+\frac{1}{2}[x,y]+\frac{1}{12}([x,[x,y]]+[y,[y,x]])-\frac{1}{24}[x,[y,[x,y]]]+\cdots,$

where $[a,b]=ab-ba$.

The Malcev Lie algebra of a group $G$ is a complete Lie algebra
$\mathrm{Mal}({G})$ equipped with a function $\log\colon
G\to\mathrm{Mal}({G})$ such that

(1)
$\log(\gamma\cdot\gamma^{\prime})=\mathrm{BCH}(\log(\gamma),\log(\gamma^{\prime}))=\log(\gamma)+\log(\gamma^{\prime})+\frac{1}{2}[\log(\gamma),\log(\gamma^{\prime})]+\cdots$

satisfying the universal property that any function $G\to L$ satisfying (1)
into a nilpotent Lie algebra $L$ factors uniquely through a Lie algebra
homomorphism $\mathrm{Mal}({G})\to L$. This is the completed Lie algebra
topologically generated by the elements $\log(\gamma)$, subject only to the
relations coming from (1).

Quillen in [Qui69] presented $\mathrm{Mal}({G})$ as the set of primitive
elements in the group ring completed at the ideal generated by $(\gamma-1)$
for all $\gamma\in G$, with
$\log(\gamma)=\sum\frac{(-1)^{n+1}}{n}(\gamma-1)^{n}$.

In this paper we carefully develop and apply a dual notion to
$\mathrm{Mal}({G})$. With suitable finiteness, functions on a Lie algebra
inherit the dual structure of a Lie coalgebra. Such coalgebras can be briefly
understood as vector spaces $E$ equipped with a cobracket $]\cdot[:E\to
E\otimes E$ such that the dual space $E^{\\#}$ naturally forms a Lie algebra.
We develop cofree Lie coalgebras carefully in Section 2.

######  Definition 1.1.

A pairing $\langle-,-\rangle_{E}$ of a Lie coalgebra $(E,\,]\cdot[\;)$ with
the Malcev Lie algebra $\mathrm{Mal}({G})$ is a Lie pairing with $G$ if

$\bigl{\langle}\,]T[\,,a\otimes
b\bigr{\rangle}_{E}=\bigl{\langle}T,[a,b]\bigr{\rangle}_{E},$

where on the left $E^{\otimes 2}$ pairs with $\mathrm{Mal}({G})^{\otimes 2}$
by $\langle e\otimes e^{\prime},a\otimes a^{\prime}\rangle_{E}=\langle
e,a\rangle_{E}\cdot\langle e^{\prime},a^{\prime}\rangle_{E}$.

Such a pairing is universal if for any other Lie pairing
$\langle-,-\rangle_{E^{\prime}}$ there exists a unique homomorphism
$\varphi:E^{\prime}\to E$ of Lie coalgebras such that

$\bigl{\langle}\,{T},a\,\bigr{\rangle}_{E^{\prime}}=\bigl{\langle}\,\varphi(T),a\,\bigr{\rangle}_{E}.$

A Lie coalgebra equipped with a universal Lie pairing with $G$ is called the
Lie dual of $G$. It is unique up to unique isomorphism.

Lie pairings extend the notion of linear functionals on $G$ as follows. If
$T\in E$ satisfies $]T[=0$ then the pairing with $T$ automatically vanishes on
all commutators in $\mathrm{Mal}({G})$. It follows that

(2)
$\bigl{\langle}\,{T},\log(ab)\,\bigr{\rangle}_{E}=\bigl{\langle}\,{T},\log(a)+\log(b)\,\bigr{\rangle}_{E}=\bigl{\langle}\,{T},\log(a)\,\bigr{\rangle}_{E}+\bigl{\langle}\,{T},\log(b)\,\bigr{\rangle}_{E},$

so the Lie pairing with $T$ defines a rational functional on the
abelianization of $G$. The kernel of the cobracket of a Lie coalgebra is
called the weight zero subspace (see Proposition 2.7). By Equation (2) a Lie
pairing restricts to a bilinear pairing of between the weight zero subspace
and the abelianization of $G$. In other words, a Lie pairing naturally extends
homomorphisms from $G$ to $\mathbb{Q}$, and as we show below, the Lie dual of
$G$ must have weight zero subspace canonically isomorphic to
$G^{\\#}:=\text{Hom}_{Grp}(G,\mathbb{Q})$.

When $G^{\\#}$ is finite dimensional, a universal Lie pairing on $E$ defines
an isomorphism of Lie algebras $E^{\\#}\cong\mathrm{Mal}({G})$. But when
$G^{\\#}$ is infinite dimensional, the Lie dual is distinct from the standard
linear dual of $\mathrm{Mal}({G})$, which is no longer a Lie coalgebra.

Lie colagebras arise naturally in studying commutative differential graded
algebras, and in particular commutative cochain complexes. Given a commutative
augmented differential graded algebra $A^{*}$, Harrison constructed a cochain
complex $\mathcal{E}^{*}({A})$ computing the derived indecomposables (see
Definition 2.8 and Theorem A.4). Surprisingly, these naturally form a Lie
coalgebra, the central example for operadic Koszul-Moore duality [Sin13].
Rational homotopy theory relates the Harrison cohomology of a commutative
cochain model for a topological space $X$ with the linear dual of rational
homotopy groups of $X$, matching the Lie coalgebraic structures on both sides.

The third and fourth authors made this effective for higher homotopy groups in
the simply connected setting in [SW13]. The present work extends this to
fundamental groups, first showing in Proposition 2.11 that there exists a
canonical isomorphism $\int_{S^{1}}\colon
H^{0}_{\mathcal{E}}\big{(}{A^{*}(S^{1})}\big{)}\xrightarrow{\sim}\mathbb{Q}$.
This canonical isomorphism gives rise to the Hopf invariant pairing between
$H^{0}_{\mathcal{E}}\big{(}{A^{*}(X)}\big{)}$ and $\pi_{1}(X)$ via pull-back
and evaluation. The value of some Harrison cocycle $T$ on some $f:S^{1}\to X$
is given by $\int_{S^{1}}f^{*}T$. Our main structural result is the following.

######  Theorem 1.2.

Let $X$ be a pointed simplicial set or differentiable manifold and $A^{*}(X)$
the de Rham (PL or smooth, respectively) cochains on $X$. The Hopf invariant
pairing defined by

$\bigl{\langle}T,\,\log(\gamma)\bigr{\rangle}=\int_{S^{1}}\gamma^{*}T$

is a univeral Lie pairing of $H^{0}_{\mathcal{E}}\big{(}{A^{*}(X)}\big{)}$
with $\pi_{1}(X,*)$.

Thus the zeroth Harrison cohomology of $X$ is the natural Lie extension of the
evaluation of cohomology on homotopy, where $\pi_{1}(X)^{\\#}\cong H^{1}(X)$.
While conceptually satisfying, the true value of this theorem is that the
right hand side can be explicitly analyzed. Whenever a fundamental group is
tied to a space of interest, or such a space can be found, Harrison cohomology
provides a means for calculations in its Malcev Lie algebra. We provide
applications in algebra and geometric topology, as explained next.

###  1.2. Combinatorial group theory

Our framework leads to a remarkably elementary combinatorial approach to
understanding the radical of the lower central series of presented groups.

######  Definition 1.3.

A finite discrete function, or d-function, is a function from
$[n]=\\{1,2,\ldots,n\\}$ to an abelian group, here exclusively the additive
group of the rational numbers. If $f$ is a d-function, define $\int_{[n]}f$ to
be the sum $\Sigma_{i=1}^{n}\ f(i)$.

######  Definition 1.4.

Consider a word $w=s_{1}^{\epsilon_{1}}\ldots s_{n}^{\epsilon_{n}}$ where
$s_{i}$ are generators of a free group $F$ and $\epsilon_{i}=\pm 1$. If
$h\colon F\to\mathbb{Q}$ is a homomorphism let $w^{*}h\colon[n]\to\mathbb{Q}$
be the d-function $w^{*}h(i)=h(s_{i}^{\epsilon_{i}})$.

Thus $\int_{[n]}w^{*}h\;=h(w)$, the evaluation of a homomorphism. We construct
below a derived version of evaluation which we call letter braiding functions,
built from a discrete version of iterated integrals.

######  Definition 1.5.

Given $w$, a word of length $n$, for each $i$ let
$[i]_{w}=\\{1,2,\dots,i_{w}\\}$ where $i_{w}$ is either $(i-1)$ if
$\epsilon_{i}=1$ or else $i$ if $\epsilon_{i}=-1$. If $f$ and $g$ are two
d-functions with domain $[n]$, define
$\lfloor\lfloor\mathopen{}f\rfloor\rfloor{{}_{\\!{}_{w}}}$ to be the function
given by partial sums
$\lfloor\lfloor\mathopen{}f\rfloor\rfloor{{}_{\\!{}_{w}}}(i)=\int_{[i]_{w}}f$.
The braiding product of $f$ and $g$ is the point-wise product
$\lfloor\lfloor\mathopen{}f\rfloor\rfloor{{}_{\\!{}_{w}}}g$.

See Example 1.8 below. The braiding product terminology will be explained
below in Section 3.2, where we will see that the definition of $[i]_{w}$ is
dictated by topology.

######  Definition 1.6.

A formal braiding symbol in some set of formal variables, say
$X_{1},X_{2},\cdots$ is, inductively, a formal linear combination of
expressions each of which is either a variable or has the form
$\lfloor\lfloor\mathopen{}\alpha\rfloor\rfloor\beta$ or
$\beta\lfloor\lfloor\mathopen{}\alpha\rfloor\rfloor$, where $\alpha$ and
$\beta$ are symbols. The weight of a symbol is the largest number of formal
braiding products which occur in a term of the linear combination.

A braiding symbol is a pair $\sigma_{\vec{h}}$ where $\sigma$ is a formal
braiding symbol and $\vec{h}=h_{1},h_{2},\cdots$ are homomorphisms
$F\to\mathbb{Q}$.

In examples, we shorten notation by incorporating homomorphisms in the symbol
– for example $\lfloor\lfloor\mathopen{}h_{1}\rfloor\rfloor{h_{2}}$ stands for
the braiding symbol
$(\lfloor\lfloor\mathopen{}X_{1}\rfloor\rfloor{X_{2}})_{h_{1},h_{2}}$.

Braiding symbols give recipes for specifying d-functions associated to any
word.

######  Definition 1.7.

If $\sigma_{\vec{h}}$ is a braiding symbol and $w$ is a word of length $n$,
the associated d-function $w^{*}\sigma_{\vec{h}}$ is obtained by performing
the prescribed braiding products recursively

$w^{*}(\lfloor\lfloor\mathopen{}\alpha_{\vec{h}}\rfloor\rfloor\beta_{\vec{h}})=\lfloor\lfloor\mathopen{}w^{*}\alpha_{\vec{h}}\rfloor\rfloor{{}_{\\!{}_{w}}}w^{*}\beta_{\vec{h}},$

starting with $w^{*}(X_{i})_{\vec{h}}=w^{*}h_{i}$ as defined above, and
extending linearly for linear combinations. The value of the symbol on $w$ is
$\sigma_{\vec{h}}(w)=\int_{[n]}w^{*}\sigma_{\vec{h}}$. We call the assignment
$w\mapsto\sigma_{\vec{h}}(w)$ the letter braiding function associated to the
symbol $\sigma_{\vec{h}}$.

######  Example 1.8.

Suppose $w=bca^{{\text{-}}1}b^{{\text{-}}1}c^{{\text{-}}1}bb$ in $F=\langle
a,b,c\rangle$. Let $A$ be the indicator homomorphism which sends $a\mapsto 1$
and the other generators to zero; let $B$ and $C$ be defined similarly. The
table below gives a step-by-step computation of the braiding product
$\lfloor\lfloor\mathopen{}2A+B\rfloor\rfloor{{}_{\\!{}_{w}}}(-C)$. The caret
marks in the row for
$\lfloor\lfloor\mathopen{}2A+B\rfloor\rfloor{{}_{\\!{}_{w}}}$ indicate the
final term of the corresponding partial sum, dictated by $[i]_{w}$ ending at
either $(i-1)$ or $i$.

$w$ | $b$ | $c$ | $\ a^{{\text{-}}1}$ | $\ b^{{\text{-}}1}$ | $\,c^{{\text{-}}1}$ | $b$ | $b$
---|---|---|---|---|---|---|---
$2A+B$ | $1$ | $0$ | $-2$ | $-1$ | $0$ | $\;1$ | $\;1$
$\lfloor\lfloor\mathopen{}2A+B\rfloor\rfloor{{}_{\\!{}_{w}}}\\!$ | ${\hat{}}\,\,0\,\,$ | ${\hat{}}\,\,1\,\,$ | $\,{-1}\,\,{\hat{}}$ | $\,{-2}\,\,{\hat{}}$ | ${-2}\,\,{\hat{}}\,$ | ${\hat{}}\,{-2}\ $ | ${\hat{}}{-1}$
$-C$ | $0$ | $-1\ $ | $\phantom{-}0$ | $\phantom{-}0$ | $1$ | $\;0$ | $\;0$
$\lfloor\lfloor\mathopen{}2A+B\rfloor\rfloor{{}_{\\!{}_{w}}}(-C)$ | $0$ | $-1\ $ | $\phantom{-}0$ | $\phantom{-}0$ | $-2$ | $\;0$ | $\;0$

We deduce that $\lfloor\lfloor\mathopen{}2A+B\rfloor\rfloor(-C)(w)=-3$.

We omit the short, elementary proof of the following basic fact.

######  Proposition 1.9.

Letter braiding functions on words descend to well-defined functions on free
groups.

While letter braiding functions are elementary combinatorial objects, we will
connect them with larger frameworks from geometry and rational homotopy theory
under conditions we call the “eigen setting,” where multiple approaches of
invariant functions on groups coincide and have integer values. One such
theory consists of the subset of letter braiding functions with all proper
subsymbols vanishing (see Definition 3.1), which are called letter linking
functions.

Letter linking functions are the subject of [MS22], whose main result is that
letter linking functions associated to symbols decorated by indicator
homomorphisms – those which take some generator to one and all others to zero
– are sharp obstructions the lower central series of free groups. For example
$\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B(aba^{{{\text{-}}1}}b^{{{\text{-}}1}})=1$, as there is one $b$ in between an
$a-a^{-1}$ pair. Since the $a$’s and $b$’s have a non-zero “linking number”,
the word $aba^{{{\text{-}}1}}b^{{{\text{-}}1}}$ is not a two-fold commutator.

One of our main present results is to generalize this to the radical of the
lower central series for all groups. Let $q:F\to G$ present a group $G$ as a
quotient of a free group $F$, and say a letter linking function of $F$
descends to $G$ if the function is constant on cosets of $q$. Our Lifting
Theorem below (see Section 4.2) characterizes the letter linking functions
that descend to a quotient group. Clearly, such functions must vanish on
relations in $G$, but this is insufficient. Our theorem gives a complete
criterion using the language of Lie coalgebras, developed in Section 2. We
then have the following main theorem.

######  Theorem 1.10.

Let $w\in F$ be a word representing an element of $G$. Then some power of $w$
is a product of $k$-fold nested commutators in $G$ if and only if all weight
$k-1$ letter linking functions that descend to $G$ vanish on $w$.

When $k=1$ this simply states that the subgroup of $G$ on which all
homomorphisms to $\mathbb{Q}$ vanish is radical of the commutator subgroup
$[G,G]$. The $k>1$ cases are all new. Moreover, this along with the Lifting
Theorem then give rise to an algorithm to determine whether some power of a
word is a $k$-fold commutator, which has been implemented – see Section 4.3.

###  1.3. Geometric topology

The geometric genesis of our work is illustrated in an elementary analysis of
the Borromean rings, which are depicted in Figure 1(a).

(a) A presentation of the Borromean rings with Seifert surfaces for two
components.

(b) A cobordism between intersection points given by a homotopy

Figure 1.

Here two components have Seifert surfaces chosen, in red and blue. We record
intersections of the third component with those surfaces on the left edge of
the square in Figure 1(b), which represents the domain of that component.
These four intersections alternate between red and blue, with signs determined
by whether the third component is coming out of or into the page. These signs
alternate, and in fact the third component represents a simple commutator in
the fundamental group of the first two components.

The square in Figure 1(b) represents the behavior of such intersections under
an imagined isotopy of the third component. Assuming transversality
throughout, the preimage of the two Seifert surfaces would be curves, pictured
in red and blue, which are disjoint. Because the colors of the left ends of
these curves alternate, and the curves are disjoint, all four curves must have
endpoints on the right edge. So a homotopy of this component pulling it away
and resulting in the unlink is impossible. (The Isotopy Extension Theorem can
now be invoked to complete a proof that the Borromean rings are linked.)

Hopf invariants can measure elements of fundamental groups by tracking
intersections with codimension one submanifolds, for example these Seifert
surfaces. Cohomology does this though signed counts of intersection points.
The example above illustrates the use of finer combinatorial structure – the
interleaving of intersection times – in a case where the simple counts vanish.

We will focus further on the occurrence of blue points between cancelling
pairs of red points, the count of which is invariant in this situation and
also obstructs pulling away the third component. Similar counts are at the
heart of letter linking [MS22], as we introduced in Section 1.2 and will
connect to topology in Section 3.2. There are two directions of generalization
of such counts. One is that the idea can be iterated. Imagine for example
another unknot component, with its Seifert surface say colored green, so the
preimages of intersections with surfaces are colored red, blue or green. Then
counts of say some green point in between cancelling pairs of blue points
which themselves are between cancelling pairs of red points will also define a
homotopy invariant of the last component. Such counts will correspond to
higher weight Hopf invariants.

The second direction of generalization is to relax the disjointness of Seifert
surfaces needed for the argument above. Consider a Whitehead link, as pictured
in Figure 2(a), and imagine a third component linking with it. Since the red
and blue surfaces intersect, the order in which the third component intersects
them can change.

(a) A presentation of the Whitead link, a Seifert surface for each component,
and their intersection.

(b) Tracking what happens through a homotopy to intersections with Seifert
surfaces and a “correction” surface.

Figure 2.

But we can consider an auxiliary surface which bounds their intersection,
pictured as the purple segments in Figure 2(a), along with the part of one
component running from point $A$ to $D$ and the part of the other component
running from $B$ to $C$. If we track intersections of a third link component
both with the two Seifert surfaces and this auxiliary surface, say in purple,
we would see phenomena such as in Figure 2(b). Here red and blue preimages of
intersections cross, which happens where the Seifert surfaces intersect, but
their crossing point is a boundary point for the preimage curve for the purple
surface. So as the order of intersections with blue and red Seifert surfaces
changes – reading left to right the blue intersection goes from being before
red to after red — an intersection with the purple surface is created or
destroyed. Thus intersections with the purple surface can be used as
correction terms for finer counts of blue and red point which depend on order.

Another case worth considering is the Hopf link, for which no there can be no
such correction term. Indeed, we can recognize two Seifert surfaces as
generators of first cohomology of the complement of the Hopf link, and the
non-triviality of their cup product implies there is not a surface in the
complement which bounds their intersection (along with sub-curves of the
components).

In general the role of submanifolds and their intersections is formalized
using the language of Thom forms, as reviewed in Appendix B.1. Suppose $M$ is
a three-manifold such as a link complement, so surfaces have associated one-
forms. Wedge product corresponds to intersection, and the behavior of the
product $\Omega_{dR}^{1}(M)\otimes\Omega_{dR}^{1}(M)\to\Omega_{dR}^{2}(M)$
determines whether two surfaces can be used for “counting alternating
preimages”. If the product is zero at the cochain level, there is immediately
such an invariant. If it is homologous to zero then there is an invariant
involving a correction surface. And if the product is non-zero in cohomology
then there is no such invariant.

More generally, given a collection codimension one submanifolds one can count
preimages of them in prescribed orders. Such counts are generally not
invariant, as the codimension one submanifolds can intersect as the Seifert
surfaces for the Whithead link do. But one can find linear combinations which
are invariant through analysis of these intersections and their relations,
which encode cup products and Massey products. Remarkably, this story is
governed by a classical constructions from algebra and rational homotopy
theory: the bar complex and the Harrison complex. The Harrison theory
distinguishes fundamental group elements up to their images in the Malcev Lie
algebra.

###  1.4. Relation to other work, technical setting, reading guidance, and
future directions

####  1.4.1. Primary relation to other work

The gold standard for understanding fundamental groups explicitly through
cochains has been the theory of of Chen integrals [Che71]. Indeed, our
combinatorial theory of letter linking invariants could have been developed
through analysis of Chen integrals in the setting of handlebodies using Thom
forms for belt disks, as we use in Section 3.2.

In fact there are two other theories which can be defined through the
following formal framework to Hopf invariants. Let $C^{*}$ be a type of
cochains, such as singular or de Rham, and $F$ denote some homotopy functor on
the category of cochains. Then fix a map $\int_{S^{1}}$ from
$H^{*}(F(C^{*}(S^{1}))$ to some graded abelian group. For any $\gamma\in
H^{*}(F(C^{*}(X))$ we get a function on the fundamental group of any $X$
sending $f:S^{1}\to X$ to $\int_{S^{1}}f^{*}\gamma$. In Theorem 1.2 and again
in Definition 2.12 we define Hopf invariants by letting $C^{*}$ be any
commutative version of cochains and $F$ be the Harrison complex, in which case
$\int_{S^{1}}$ is a canonical isomorphism with $\mathbb{Q}$. When $C^{*}$ are
de Rham cochains and $F$ is the classical Bar complex (as in Appendix A.1),
Chen integrals can be used to compute the resulting invariants taking values
in $\mathbb{R}$. In the process of working on this project, the first author
realized that a distinct theory could be developed with $C^{*}$ being
simplicial cochains, with coefficients in any PID, and $F$ again the classical
Bar complex. That theory is called letter braiding and agrees in the algebraic
setting with the one given in Section 1.2, restricted to “nested” braiding
products, which in fact span all such products. As represented in Figure 3 all
of these theories produce the same set of integer-valued invariants in the
eigen-setting, as formulated for Hopf invariants in Definition 3.1.

Letter
Braiding

Letter
Linking

Hopf
Invariants

Chen-
Hain

Figure 3. Topological measures of the fundamental group. All produce letter
linking invariants in the eigen-setting

The three theories above are distinct. Hopf invariants give a perfect duality
with the Malcev Lie algebra, while Chen theory more naturally pairs with the
real group ring with its augmentation ideal filtration, and similarly for
letter braiding, which is defined over any PID. Calculations outside of the
eigen-setting differ even for the circle $\pi_{1}\cong\mathbb{Z}=\langle
a\rangle$ and first cohomology generated by a dual class $A$. The three
theories all have an invariant we call $(A|A)$ which agree in the eigen-
setting. The corresponding Hopf invariant is denoted $\lfloor A\rfloor A$, the
letter braiding invariant $\lfloor\lfloor\mathopen{}A\rfloor\rfloor A$ and the
Chen integral $\omega_{A}|\omega_{A}$ for $\omega_{A}$ a one-form representing
the class $A$. However, on the word $a^{2}$ the Hopf invariant $\lfloor
A\rfloor A$ will be $0$ because $\lfloor A\rfloor A$ is by definition zero in
the Harrison complex. The letter braiding invariant
$\lfloor\lfloor\mathopen{}A\rfloor\rfloor A$ is $1$, as can be checked from
Definition 1.7. Finally, the Chen integral of $\omega_{A}|\omega_{A}$ on
$a^{2}$ will be $\frac{1}{2}+1+\frac{1}{2}=2$ because there will be two “self-
interaction terms” on top of the interaction between the two occurrences of
$a$.

The other primary connection to highlight is work of the last two authors. In
[SW13] we show that the same simple formalism we use here, namely pull-back
and evaluation in the Harrison complex, gives complete rational functionals on
homotopy groups of simply connected spaces. Moreover, we show that the
resulting invariants are represented by “higher linking numbers, with
correction,” generalizing the classical Hopf invariant which measures a map
$S^{3}\to S^{2}$ through the linking number of preimages of two distinct
points in $S^{2}$. Then in [MS22], we develop letter linking invariants for
free groups purely combinatorially, establishing “predictions” of [SW13] for
maps $S^{1}\to\bigvee S^{1}$. We return to topology for our development of
letter linking for all groups here.

In recent work on a complementary approach, Rivera–Zeinalian [RZ18] showed
that a dual theory, the cobar construction on singular chains of a space $X$,
yields a chain model for the loop space $\Omega X$, without restrictive
hypotheses. In particular, they deduce that the $0$-th cobar homology of $X$
is the exactly group ring $\mathbb{Z}[\pi_{1}(X)]$ – no completion necessary.
While sharper, the difficulty of their theory in comparison to ours is that
theirs is not based on a quasi-isomorphism invariant functor, and for example
one cannot use a small simplicial chain model for $X$. More severely, the
cobar spectral sequence does not converge, for example when $\pi_{1}(X)$ is a
nonabelian simple group, and the natural filtrations are not exhaustive. Thus,
the sharp ability to capture the fundamental group does not translate as well
to calculations, as one might expect in light of constraints such as the
undecidability of the Word Problem.

Ultimately, by providing “coordinates” for the Malcev Lie algebra of any
group, Hopf invariants generalize the classical tools of Magnus expansions and
Fox derivatives. Moreover, the close relationship to standard constructions
such as bar complexes and Massey products means plenty of connection to
previous topological studies of groups. We defer to the paper of the first
author on letter braiding invariants [Gad23] for a thorough, though almost
certainly incomplete, discussion in its Introduction.

####  1.4.2. Technical setting and reading guidance

For accessibility for readers from a broad range of backgrounds, we have
chosen the most elementary developments available. These are grounded in
commutative cochain algebras, with primary versions being smooth or simplicial
de Rham theory. A reader with knowledge of the smooth theory, say from Bott
and Tu [BT82], is fully prepared. One facet of that theory which may require
review is that of Thom forms, which review and further develop in Appendix
B.1. The simplicial and smooth versions of de Rham theory are quasi-isomorphic
by [FHT15, Theorem 11.4] and we use both, for example using the smooth theory
in all of our examples while appealing to the simplicial theory for our
lifting criterion.

The main character in our story is the Harrison complex. Readers who are
familar with the bar complex could make a fruitful first reading by
“substituting” it for the Harrison complex throughout. Readers who are not
familiar with the bar complex are encouraged to read our short introduction to
it and the Harrison complex in Appendix A.

The key example in this paper is that of handlebodies, given in Section 3.1.
We recommend a thorough understanding of how the Hopf invariant definition –
in a nutshell “pull-back, weight reduce and evaluate” – works in that example
to produce a letter-linking count. From there, a reader who prefers examples
could go right to Section 5.

The Harrison complex is central in the theory of $C_{\infty}$-algebras, and
our theory extends there. We leave it to experts to deduce the straightforward
generalization to that setting, though in Remark 4.4 we do outline progress
the second author made before his untimely passing.

####  1.4.3. Future directions

We share both our plans and possible areas of application more broadly.

  1. I.

Our first planned application is to Milnor invariants for links in in three-
manifolds, for which we give a taste in Sections 1.3 and 5.2. Stees [Ste24]
recently initiated the theory of Milnor invariants in arbitrary three-
manifolds from the completely different perspective of concordance with a
fixed representative “unlink.” We will proceed from a basic idea which is
central to our present work: measuring homotopy classes of curves through
patterns of intersections with Seifert surfaces and coboundings of their
intersections. Our current framework leads to broadening the notion of Seifert
surfaces to include possibly singular null-homologies between multiple link
components. We expect the framework developed in this paper, with that notion
of Seifert surface for input as cochains, to immediately yield a robust
theory.

  2. II.

A geometric reading of Theorem 1.2 using Thom forms, and analysis in the
Harrison spectral sequence (see Remark 2.14) shows that the rational
nilpotency class of a group is governed by Massey products of classes
represented by codimension one submanifolds of corresponding depth. That is,
intersections of fixed complexity are forced. It would be interesting to
connect with the substantial literature on nilpotency class and geometry.

  3. III.

The same techniques also lead to a geometric approach to the Johnson
filtration of mapping class groups. Consider a surface with one boundary
component for simplicity (but not necessity - see Section 5.1), and fix a
collection of disjoint curves $\alpha_{i}$ which represent first cohomology
(for example “belt disks” in a handlebody presentation). For a diffeomorphism
$f$ to be in Johnson filtration degree $<n$, there must exist a based loop $c$
such that the concatenated loop $c^{-1}*f(c)$ meets the curves $\alpha_{i}$
according to the following pattern:

     * •

In the zeroth degree we count intersections with the $\alpha_{i}$ (and they
must all vanish if $n>0$).

     * •

The first degree counts are intersections with some $\alpha_{i}$ which occur
between cancelling pairs of intersections with some $\alpha_{j}$ (and they
must vanish if $n>1$).

     * •

$\cdots$

     * •

The $n$th degree counts are that of intersections with some $\alpha_{k}$
between cancelling pairs of intersections in an $n-1$st count. This last count
must be nonzero.

Thus, for $f$ to be deep in the Johnson filtration forces some complexity of
$c^{-1}*f(c)$ which would be interesting to tie to other notions of complexity
of mapping classes.

  4. IV.

Our theory is in a sense dual to that of gropes [Tei02], which realize
iterated commutators through surfaces. Instead, we measure iterated
commutators through systems of codimension one submanifolds. Thus we can
imagine application in any area of geometric topology impacted by the
fundamental group, especially phenomena known to be tied to the lower central
series. The theory can also naturally be enriched by any structure compatible
with products and coboundary maps, such as Hodge structure [Hai85].

  5. V.

An important technical development for geometric application is the use of
geometric cochains as developed by Friedman, Medina and the third author
[FMMS22]. These are defined over the integers, and have a commutative product
given by intersection, but (then, of necessity) only have a partially defined
product. In that setting, the fundamental isomorphism
$\int_{S^{1}}:H_{\mathcal{E}}^{0}(S^{1}){\to}\mathbb{Q}$ does not hold. We
plan to explore the resulting differing theory, which conjecturally is closer
to that of letter braiding, in the course of our development of Milnor
invariants.

  6. VI.

In algebraic topology, our approach has an advantage over Chen integrals and
others in that it is fully integrated into a Quillen model for rational
homotopy theory. It would be helpful to more fully connect with recent
developments of Lie models based on the Lawrence-Sullivan interval [BFMT20],
which indeed use the Harrison complex at some key points in their development.
At a purely calculational level, there is a rich pairing between Lie algebras
and Lie coalgebras at play (see [SW11, Section 2]). And with our more explicit
connections to group theory, there could be an opportunity to transport
questions between domains, such as understanding analogues of rational
dichotomy for groups. (See [Hes07] for a lovely survey of rational dichotomy,
and rational homotopy theory more broadly.)

  7. VII.

A new tool which is helpful in exploring the interface of group theory and
homotopy theory is that of graphical models, used throughout Section 5 and
defined rigorously in Appendix B. Briefly, take a presentation of a group by
generators and relations, and use each relation to label the boundary of a
disk. That is, there will be one marked point on the boundary of the disk for
each letter in the relation, occurring in order. A graphical model is an
immersed graph in one such disk in which edges intersect transversely and have
endpoints either at marked points on the boundary or at intersections of other
edges. To such data we associate a differential graded commutative algebra
with partially defined product which maps to the de Rham cochains of a
manifold which realizes the group. In some cases, such as all those in Section
5, this partially defined subalgebra suffices for computations in our theory.

To our knowledge, these graphical partial models are new, and there are plenty
of open questions. For which groups can one find a graphical model which,
while being partially defined, suffices for calculations of the Lie dual? Can
de Rham theory be applied to the disk to make these models have a fully
defined product, for example making choices to understand products associated
to curves which share a vertex? Can group theoretic properties, such as
nilpotency class, be read off from a (partial) graphical model? For which
groups can finite models be found?

  8. VIII.

A main focus in the present work is computational efficacy. It is thus natural
to transfer constructions to computer code and implement as a library for a
mathematical computation package such as SageMath, as the second author did
for evaluating braiding invariants and finding bases of letter linking
invariants. This work is available as Python and SageMath jupyter notebooks
[Ozb23]. But there is still considerable room for optimizing algorithms, and
in particular working with Lie coalgebras. Previous work of the fourth author
[WS16] suggests that using specific bases for Lie coalgebras such as the star
basis may lead to quicker evaluation of cobrackets and computation of
pairings. Also the algorithm used to find letter linking invariants by [Ozb23]
relies heavily on the Baker–Campbell–Hausdorff product, which is much slower
than a native Lie coalgebraic algorithms would be.

  9. IX.

As we are not experts in algebra, we have less detailed broader speculation to
share. But one obvious line is to develop the theory beyond residual
nilpotence, using the completed Harrison complex as in Section 5.7. Another is
to more fully apply the theory over the integers or finite fields, using
either geometric cochains and the Harrison complex, as proposed in item V
above, or simplicial cochains which lead to letter braiding, as already
developed in [Gad23]. These theories shed light on group rings, rather than
the Malcev Lie algebra. The letter braiding setting, especially over finite
fields, is ripe for computer implementation as initiated for letter linking in
[Ozb23].

###### Contents

  1. 1 Introduction
    1. 1.1 Algebraic topology
    2. 1.2 Combinatorial group theory
    3. 1.3 Geometric topology
    4. 1.4 Relation to other work, technical setting, reading guidance, and future directions
      1. 1.4.1 Primary relation to other work
      2. 1.4.2 Technical setting and reading guidance
      3. 1.4.3 Future directions
  2. 2 Definitions
    1. 2.1 Eil trees and cofree Lie coalgebras
    2. 2.2 Harrison complex and Hopf invariants
  3. 3 Central Example: Handlebodies and linking of letters for free groups
    1. 3.1 First handlebody example
    2. 3.2 Equating topology and combinatorics of letter linking
    3. 3.3 Configuration braiding of trees and words
  4. 4 Structural theorems
    1. 4.1 Lie pairing
    2. 4.2 The lifting criterion
    3. 4.3 Letter linking algorithms
    4. 4.4 The Fundamental Theorem
  5. 5 Examples
    1. 5.1 Orientable surfaces and our favorite little two-complex
    2. 5.2 Milnor invariants for Borromean rings
    3. 5.3 Beyond letter linking
    4. 5.4 Commutators and Massey products
    5. 5.5 Pure braid groups
    6. 5.6 An infinite example
    7. 5.7 Beyond residual nilpotence using the product totalization
  6. A Bar and Harrison complexes
    1. A.1 Basics and connection with Sullivan models
    2. A.2 Explicit treatment of signs
  7. B Graphical (partial) models for groups through Thom Forms
    1. B.1 Thom forms
    2. B.2 Thickening two-complexes
    3. B.3 de Rham cochains associated to group generators and curves on disks

##  2\. Definitions

The Harrison complex presents the derived indecomposibles of an (augmented)
differential graded commutative algebra – see Appendix A.2. Our preferred
model for the Harrison complex is based on the operadic approach to Lie
coalegbras, and a model of the Lie co-operad arising from cohomology of
configuration spaces originating in [SW11].

###  2.1. Eil trees and cofree Lie coalgebras

To discuss the Lie cooperad using compact notation, we depart slightly from
[SW11] and use a new model via rooted trees. This approach is more in line
with the modern perspective of $\infty$-operads [HM22], and seamlessly
connects with letter braiding formalism.

######  Definition 2.1.

Write $\mathcal{T}\\!{\mathit{r}\/}_{*}(n)$ for the vector space generated by
rooted trees with vertices labeled uniquely by $\\{0,\dots,n\\}$.

Let $\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n)$, the
space of rooted Eil trees, be the quotient of
$\mathcal{T}\\!{\mathit{r}\/}_{*}(n)$ by the following local relations acting
at the root of a tree.

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In the diagrams above the root is circled, and $a$, $b$, $c$ stand for
vertices of a tree which could possibly have further branches (indicated by
the “whiskers”) not modified by these relations.

As emphasized by their name and notation, root change and root Arnold
relations only occur at the root of a tree. The root of a tree is moved via
the root change relation, so the Arnold identity can be applied anywhere, with
care taken with signs. We emphasize that $a$, $b$, $c$ above are vertices and
not subtrees.

The root gives a partial ordering on vertices which allows us to write trees
using compact notation.

######  Definition 2.2.

The tree symbol for a vertex-labeled rooted tree is defined as follows.

  * •

A tree with only one vertex has symbol given by the vertex’s label, $a$.

  * •

A corolla $\textstyle{\scriptstyle a_{1}}$$\textstyle{\scriptstyle
a_{2}}$$\textstyle{\scriptstyle\cdots}$$\textstyle{\scriptstyle
a_{n}}$$\textstyle{\scriptstyle b}$ (where $a_{i}$, $b$ are vertex labels, $b$
at the root) has symbol $\lfloor a_{1}\rfloor\lfloor
a_{2}\rfloor\\!\cdots\\!\lfloor a_{n}\rfloor\,b$.

  * •

More generally given a corolla $\textstyle{\scriptstyle
A_{1}}$$\textstyle{\scriptstyle
A_{2}}$$\textstyle{\scriptstyle\cdots}$$\textstyle{\scriptstyle
A_{n}}$$\textstyle{\scriptstyle b}$ where $b$ is the root vertex and
$A_{1},\dots,A_{n}$ are subtrees with symbols $\alpha_{1},\dots,\alpha_{n}$
respectively, the entire tree has symbol
$\lfloor\alpha_{1}\rfloor\lfloor\alpha_{2}\rfloor\\!\cdots\\!\lfloor\alpha_{n}\rfloor\,b$.

Similar to braiding symbols, the weight of a tree is its total number of
edges. This corresponds to the total number of $\lfloor,\rfloor\,$pairs in a
tree symbol. Note that $\mathcal{T}\\!{\mathit{r}\/}_{*}(n)$ are weight $n$
trees.

######  Example 2.3.

The tree $\textstyle{\scriptstyle a}$$\textstyle{\scriptstyle
b}$$\textstyle{\scriptstyle c}$ has symbol $\big{\lfloor}\lfloor
a\rfloor\,b\big{\rfloor}\,c$. The tree $\textstyle{\scriptstyle
a}$$\textstyle{\scriptstyle b}$$\textstyle{\scriptstyle
c}$$\textstyle{\scriptstyle d}$$\textstyle{\scriptstyle e}$ has symbol
$\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e$.

The definition above gives valid symbols for both planar and non-planar trees,
that is with or without commutativity at corollas. We employ non-planar trees,
and will implicitly impose commutativity for tree symbols so for example

$\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e=\big{\lfloor}\lfloor b\rfloor\lfloor
a\rfloor\,d\big{\rfloor}\big{\lfloor}c\big{\rfloor}\,e.$

For convenient notation we will also allow the unbracketed root vertex to be
written at arbitrary location, for example $\lfloor a\rfloor\lfloor
b\rfloor\,c=\lfloor a\rfloor\,c\,\lfloor b\rfloor\,=c\,\lfloor a\rfloor\lfloor
b\rfloor.$

Tree symbols are essentially the formal braiding symbols of Definition 1.6 but
with root change and root Arnold identities imposed. Letter braiding
invariants do not satisfy these identities universally but do satisfy them in
special cases. See Theorem 3.2.

######  Remark 2.4.

Symbols have their origin in [MS22, Section 3], where they are called “pre-
symbols” using parenthesis delimiters. In [MS22, Definition 3.1] the non-
parenthesized element was called the “free letter”; here it serves as the root
of its subtree. Aside from the new connection between rooted trees and
cooperads, the only substantial difference with [MS22] is that the current
theory allows repeated letters.

In [MS22, Definition 3.2] the correspondence between symbols and trees is made
via the containment partial ordering induced by nested parenthesizations in a
symbol. In our setting this becomes the canonical partial order on the
vertices of a rooted tree. This same equivalence between rooted trees and
partial orders also appears in [HM22, Lemma 3.2] in the context of dendroidal
objects and $\infty$-operads. The generating set of long graphs of [SW11, Fig
1] correspond to linear trees consisting of a single long branch extending
from the root, as in the second case of Example 2.3 above. In the notation of
[HM22] these are the total orders – the image of simplicial sets inside of
dendroidal sets. The important roles of rooted and other similar trees in
[HM22] as well as in our approach to Lie coalgebras and Harrison homology is
more than coincidental, and we hope to explore it further in future work.

Expanding the dictionary between symbols and trees, we let
$\lfloor\alpha\rfloor\beta$ denote a tree composed of disjoint subtrees $A$
and $B$ with symbols $\alpha$ and $\beta$, where $B$ is the subtree containing
the root and there is an edge connecting the root of $A$ to the root of $B$.
We say $A$ is the branch subtree and $B$ is the root subtree. For example
$\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e$ may be written as $\lfloor\alpha\rfloor\,\beta$
where $\alpha=\lfloor a\rfloor\lfloor b\rfloor\,d$ and $\beta=\lfloor
c\rfloor\,e$. With this notation, the root change and root Arnold relations in
$\mathcal{E}\\!{\mathit{i}l\/}(n)$ (Definition 2.1) are as follows.

(root change) $\displaystyle\qquad 0\ =\ \lfloor\alpha\rfloor\beta\ +\
\alpha\,\lfloor\beta\rfloor,$ (root Arnold) $\displaystyle\qquad 0\ =\
\lfloor\alpha\rfloor\lfloor\beta\rfloor\,\gamma\ +\
\lfloor\alpha\rfloor\,\beta\,\lfloor\gamma\rfloor\ +\
\alpha\,\lfloor\beta\rfloor\lfloor\gamma\rfloor$

where $\alpha$, $\beta$, $\gamma$ are expressions for subtrees. More
generally, included in the space generated by these relations are
$0=\sum_{i}\lfloor\alpha_{1}\rfloor\cdots{\alpha_{i}}\cdots\lfloor\alpha_{n}\rfloor$,
called the Leibniz relations in [MS22, Prop 3.10].

Branch and root subtrees also play an important role in the coproduct and Lie
cobracket. Translated to the realm of rooted trees, the edge cutting coproduct
of [SW11, Definition 2.6] becomes a tree pruning coproduct on
$\mathcal{T}\\!{\mathit{r}\/}_{*}$.

$\Delta T=\sum_{e\in E(T)}T_{l}^{\hat{e}}\otimes T_{r}^{\hat{e}}$

where the sum removes edges from $T$ one at a time, separating leaf and root
subtrees $T^{\hat{e}}_{l}$ and $T^{\hat{e}}_{r}$.

If the symbol of a tree is $\lfloor\alpha\rfloor\,\beta$, then pruning yields
trees with symbols $\overline{\alpha}$ and $\overline{\beta}$, where the
closure symbol indicates that labels have been shifted downwards to be
consecutive integers starting at 0, while perserving relative ordering. For
example $\big{\lfloor}\;\underbrace{\lfloor 4\rfloor\lfloor
0\rfloor\,2}_{\alpha}\;\big{\rfloor}\\!\underbrace{\big{\lfloor}3\big{\rfloor}1\,}_{\beta}\longmapsto\lfloor
2\rfloor\lfloor 0\rfloor 1\otimes\lfloor 1\rfloor 0.$ Writing general tree
pruning via symbols extends this example by accounting for corolla
commutativity and further nesting.

######  Definition 2.5.

Let

$\Delta\psi=\sum_{\psi=\cdots\lfloor\alpha\rfloor\cdots}\overline{\alpha}\otimes\overline{\psi^{\widehat{\alpha}}}$

where the sum ranges over all ways of picking a subsymbol $\alpha$ of $\psi$
and the symbol $\psi^{\widehat{\alpha}}$ is the result of removing the
subsymbol $\lfloor\alpha\rfloor\,$ from $\psi$.

The anti-commutative Lie cobracket is $]\psi[=\Delta\psi-\tau\Delta\psi$,
where $\tau$ is the twist map $\tau(a\otimes b)=b\otimes a$.

In [SW11] we construct Lie coalgebras using oriented graphs instead of rooted
trees – see Appendix A. In the appendix we explain that these two models are
naturally isomorphic, but we prefer the one presented here. Moreover, in
[SW11] we show that the kernel of the cobracket in weight greater than zero is
precisely the expressions generated by arrow reversing (root change) and
Arnold identities. Cobracket thus descends to the quotient
$\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n)$, yielding
a model for the Lie cooperad. The perfect pairing between
$\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n)$ and
$\mathcal{L}\\!{\mathit{i}e\/}(n)$ is combinatorially rich - see [SW11,
Definition 2.11] for its description using graphs. We use symbols and their
cobracket to construct (conilpotent) cofree Lie coalgebras and then the Lie
coalgebraic bar construction, which underlies the Harrison complex.

######  Definition 2.6.

Let $W$ be a vector space. Define the (conilpotent) cofree Lie coalgebra on
$W$ to be the vector space

${{\mathbb{E}}}(W)\ \cong\
\bigoplus_{n}\raisebox{3.0pt}{$\bigl{(}\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n)\,\otimes\,W^{\otimes(n+1)}\bigr{)}$}\\!\raisebox{-2.0pt}{$\diagup$}\\!\raisebox{-4.0pt}{$\mathcal{S}_{n+1}$},$

with cobracket operation induced by the cobracket on
$\mathcal{T}\\!{\mathit{r}\/}_{*}(n)$ and deconcatenation on
$W^{\otimes(n+1)}$. Here, the symmetric group $\mathcal{S}_{n+1}$ acts on
$\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n)$ through
its action on $\mathcal{T}\\!{\mathit{r}\/}_{*}(n)$ by permuting labels
$\\{0,\dots,n\\}$ and on $W^{\otimes(n+1)}$ by reordering (with Koszul signs
if $W$ is graded).

The index of the summand in the decomposition of ${{\mathbb{E}}}(W)$ above is
called the weight. Let ${{\mathbb{E}}}^{\leq k}(W)$ denote the subcoalgebra
given by restricting to weights less than or equal to $k$.

From [SW11, Proposition 3.18], we have the following, which dualizes our
understanding of free $k$-step nilpotent Lie algebras as either “$k$ or less
nested brackets of generators” or “Lie algebras modulo $(k+1)$ iterated
brackets”.

######  Proposition 2.7.

${{\mathbb{E}}}^{\leq
k}(W)=\mathrm{ker}\bigl{(}\,]\cdot[^{\circ(k+1)}:{{\mathbb{E}}}W\to({{\mathbb{E}}}W)^{\otimes(k+2)}\bigr{)}$
is the cofree $(k+1)$-conilpotent Lie coalgebra on $W$.

As we did for braiding symbols, we write elements in the $n$-th summand of
${{\mathbb{E}}}(W)$ using the elements in $W^{\otimes(n+1)}$ as labels of
vertices within a tree symbol. For example

$\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e\ =\
\big{\lfloor}0\big{\rfloor}\big{\lfloor}\lfloor 1\rfloor\lfloor
2\rfloor\,3\big{\rfloor}\,4\;\bigotimes\;\bigl{(}c\otimes a\otimes b\otimes
d\otimes e\bigr{)}.$

We give detailed treatment, with examples, of the Koszul signs for cobracket
as well as for the differentials in the Harrison complex in Appendix A.

###  2.2. Harrison complex and Hopf invariants

With cochain algebras in mind, we focus on differential graded-commutative,
augmented, unital algebras $(A,\ \mu_{A},\ d_{A})$, with augmentation ideal
$\bar{A}$ so that $A=k\cdot 1\oplus\bar{A}$ with $k=\mathbb{Q}$ or
$\mathbb{R}$. Key examples are commutative algebras of differential forms on
simplicial sets and manifolds, augmented by restriction to a basepoint.

Recall that the desuspension of a cochain complex $(C^{\bullet},d)$ is the
shifted complex $s^{{{\text{-}}1}}C^{\bullet}:=C^{(\bullet-1)}$ with the same
differential.

######  Definition 2.8.

The Harrison complex is the complex with underlying graded vector space
${{\mathbb{E}}}(s^{{{\text{-}}1}}\\!\bar{A})$, the cofree Lie coalgebbra on
the desuspension of the augmentation ideal, with its natural cohomological
grading coming from tensor powers of $s^{{\text{-}}1}\\!\bar{A}$.

Its differential is the sum of two terms
$d=d_{{{\mathbb{E}}}(s^{{{\text{-}}1}}\\!\bar{A})}+d_{\mu}$. The first is the
differential extended from that of $A$ to its tensor powers by the Leibniz
rule, and second is built from the operation of “contracting edges and
multiplying labels” as follows. For rooted trees with a single edge define
$d_{\mu}\bigl{(}\lfloor
s^{{{\text{-}}1}}a\rfloor\,s^{{{\text{-}}1}}b\bigr{)}=(-1)^{|a|}\,s^{{{\text{-}}1}}(ab)$.
As shown in [SW11, Proposition 4.7], this freely extends to a differential
$d_{\mu}$ on all of ${{\mathbb{E}}}(s^{-1}\bar{A})$.

Equivalently, the Harrison complex is the totalization of a second-quadrant
bicomplex whose horizontal grading is by tensor degree in
${{\mathbb{E}}}(s^{-1}\bar{A})$ (the number of $s^{{{\text{-}}1}}$ symbols in
an expression) and vertical grading by cohomological degree inherited from $A$
(ignoring the desuspensions). The differential
$d_{{{\mathbb{E}}}s^{{\text{-}}1}\\!\bar{A}}$ has bidegree $(0,1)$ and will be
called the vertical differential, while $d_{\mu}$, which is defined through
edge contraction and multiplication, has bidegree $(1,0)$ and will be called
the horizontal differential. That is

$\mathcal{E}(A)={\rm Tot}_{\oplus}\bigl{(}{{\mathbb{E}}}(s^{-1}\bar{A}),\
d_{{{\mathbb{E}}}s^{{\text{-}}1}\\!\bar{A}},\ d_{\mu}\bigr{)},$

from which a natural spectral sequence can be constructed, as discussed in
Remark 2.14.

In general, the maps above should involve Koszul signs tracking the
interaction of differentials with different gradings. We give details in
Appendix A.2, as these are needed in particular to integrate this theory with
the action of the fundamental group on higher homotopy groups. In our setting,
desuspended one-cochains are in degree zero, in which case these Koszul signs
all vanish.

Recall the bar complex $\mathcal{B}(A)$, defined similarly and more readily as
the total complex of the (coassociative) cofree tensor coalgebra
$T(s^{{\text{-}}1}\bar{A})$ with first differential
$d_{s^{{\text{-}}1}\\!\bar{A}}$ and second differential given by summing over
the ways to multiply pairs of consecutive terms of a tensor – see Definition
A.1. Harrison’s original definition of $\mathcal{E}$ for commutative algebras
in [Har62] was as a quotient of the bar complex modulo shuffles, in particular
quotienting by cycles of the form $\bigl{(}a|b-b|a\bigr{)}$, which lie in the
kernel of $\mu_{A}$ tautologically by commutativity. Our presentation of
Harrison’s complex as part of an explicit operadic framework which is more
flexible, and it pairs naturally with Lie algebraic constructions. There is a
surjection from the classical bar complex mapping tensors to linear trees (the
“long graphs” of [SW11])

$s^{{\text{-}}1}a_{0}|s^{{\text{-}}1}a_{1}|\cdots|s^{{\text{-}}1}a_{n}\longmapsto\Big{\lfloor}\\!\cdots\\!\big{\lfloor}\lfloor
s^{{\text{-}}1}a_{0}\rfloor
s^{{\text{-}}1}a_{1}\big{\rfloor}\cdots\Big{\rfloor}s^{{\text{-}}1}a_{n}.$

Because linear trees span $\mathcal{E}\\!{\mathit{i}l\/}(n)$, the bar classes
above indeed span ${{\mathbb{E}}}(s^{{\text{-}}1}\bar{A})$.

In the setting of $1$-connected differential graded commutative algebras we
show in [SW11, Section 4.1] that the functor $\mathcal{E}$ is part of a
Quillen adjoint pair. In Proposition 2.10 of [BFMT20], the authors show that
even without any connectivity hypotheses, $\mathcal{E}$ is quasi-isomorphism
invariant, which leads to the following.

######  Definition 2.9.

Let $X$ be a connected topological space. Define the Harrison cohomology of
$X$, denoted $H_{\mathcal{E}}^{*}(X)$, as the cohomology of
$\mathcal{E}(C^{*}(X))$ for any augmented commutative cochain algebra
$C^{*}(X)$ quasi-isomorphic as differential graded associative algebras to the
singular cochains of $X$.

######  Example 2.10.

Because a wedge of circles is formal, we may replace cochains by cohomology,
which in addition to having zero differential has trivial product. Thus the
zeroth Harrison cohomology of a wedge of circles is canonically isomorphic to
the cofree Lie coalgebra on the first cohomology:

$H_{\mathcal{E}}^{0}\bigl{(}\bigvee_{s\in
S}S^{1}\bigr{)}\cong{{\mathbb{E}}}\bigl{(}\mathbb{Q}^{S}\bigr{)}.$

Harrison cohomology provides a dual to $\pi_{1}(X)$ because the two admit a
natural pairing, which we now define. The definition works equally well for
general homotopy groups, building on the following.

######  Proposition 2.11.

For every $n\geq 1$ there is a canonical isomorphism
$H^{n-1}_{\mathcal{E}}(S^{n})\cong
H^{n}(S^{n})\xrightarrow[\int_{S^{n}}]{\cong}\mathbb{Q}$.

###### Proof.

The Harrison spectral sequence reads

$E_{1}^{p,q}=\mathcal{E}(s^{{\text{-}}1}\bar{H}^{*}(X)),$

the cofree Lie coalgebra on $s^{{\text{-}}1}\bar{H}^{*}(X)$. Since
$\bar{H}^{*}(S^{n})$ is one dimensional, the spectral sequence collapses, with
either a sole class in bi-degree $(-1,n)$ canonically identified with
$H^{n}(S^{n})$ when $n$ is odd or both that class and another one in bi-degree
$(-2,2n)$ when $n$ is even. Evaluation on the fundamental class defines the
claimed isomorphism. ∎

######  Definition 2.12.

Let $\gamma\in H^{0}_{\mathcal{E}}(X)$ and $f:S^{1}\to X$ a pointed map
representing $[f]\in\pi_{1}(X,*)$. The Hopf invariant associated to $\gamma$
is the function $h_{\gamma}\colon[f]\mapsto\int_{S^{1}}f^{*}\gamma$.

Equivalently, the Hopf invariant defines a pairing
$H^{0}_{\mathcal{E}}(X)\times\pi_{1}(X)\to\mathbb{Q}$ by

$\langle\gamma,[f]\rangle=\int_{S^{1}}f^{*}(\gamma).$

In practical terms, to perform this evaluation we pull back $\gamma$ to the
Harrison complex of $S^{1}$, find a homologous cocycle in weight zero (which
must exist by Proposition 2.11) and then evaluate that on the fundamental
class of $S^{1}$, typically by counting points or by integration. The process
of finding a homologous class in weight zero is called weight reduction, often
performed inductively. For weight zero cocycles, Hopf invariants simply
evaluate cohomology on the fundamental group. Evaulation of higher weights can
be viewed as derived versions of this.

Hopf invariants are equally effective regardless of which space is used with a
given fundamental group.

######  Lemma 2.13.

If $f\colon X^{\prime}\to X$ is a map of pointed connected spaces inducing an
isomorohism on $\pi_{1}$, then

$H^{0}_{\mathcal{E}}(f)\colon H^{0}_{\mathcal{E}}(X^{\prime})\to
H^{0}_{\mathcal{E}}(X)$

is an isomorphism.

###### Proof.

Let $f^{*}\colon C^{*}(X^{\prime})\to C^{*}(X)$ be the induced map of Sullivan
minimal models for $X$ and $X^{\prime}$. Then the hypothesis implies it is
$1$-connected, i.e. inducing an isomorphism on cohomology in degrees $\leq 1$
and an injection in degree $2$. It follows that its desuspension
$s^{{{\text{-}}1}}f^{*}$ is $0$-connected. By the Künneth isomorphism and
exactlness of $\mathcal{S}_{n}$-coinvariants over $\mathbb{Q}$, the induced
maps

$\raisebox{3.0pt}{$\bigl{(}\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n-1)\otimes
s^{{{\text{-}}1}}\bar{C}^{*}(X^{\prime})^{\otimes
n}\bigr{)}$}\\!\raisebox{-2.0pt}{$\diagup$}\\!\raisebox{-4.0pt}{$\mathcal{S}_{n}$}\to\raisebox{3.0pt}{$\bigl{(}\mathcal{E}\\!{\mathit{i}l\/}{\mathcal{T}\\!{\mathit{r}\/}_{*}}(n-1)\otimes
s^{{{\text{-}}1}}\bar{C}^{*}(X)^{\otimes
n}\bigr{)}$}\\!\raisebox{-2.0pt}{$\diagup$}\\!\raisebox{-4.0pt}{$\mathcal{S}_{n}$}$

are similarly $0$-connected for all $n\geq 1$. From here, a standard spectral
sequence argument shows that
$\mathcal{E}(f^{*})\colon\mathcal{E}(C^{*}(X^{\prime}))\to\mathcal{E}(C^{*}(X))$
is $0$-connected, and in particular induces an isomorphism on
$H^{0}_{\mathcal{E}}$. ∎

We will take full advantage of the freedom to choose $X$ with fundamental
group $G$. In some cases, there are good manifold models, in others we can
directly analyze two-complexes, as we do in Section 5 using Appendix B.1. And
if for example we take $X$ as the simplicial model for $BG$ and use PL-forms
then we immediately get functorality with respect to group homomorphisms.

######  Remark 2.14.

The central role of $H^{0}_{\mathcal{E}}(X)$ in our theory motivates its
calculation. A standard tool is the spectral sequence of a bicomplex, which
leads to the Harrison spectral sequence. (See Section A.1 for an introduction
to the corresponding spectral sequence for the Bar complex.) In degree zero,
the $E_{1}$-page of the Harrison spectral sequence is identified with the
cofree Lie coalgebra on the first cohomology of $X$, namely
${{\mathbb{E}}}(\pi_{1}(X)^{\\#})$ cogenerated by the linear dual
$\pi_{1}(X)^{\\#}=\text{Hom}_{Grp}(\pi_{1}(X),\mathbb{Q})$.

Since the cohomology of the augmentation ideal $\bar{C}^{*}(X)$ for cochains
of a connected space vanishes in degree zero, the Harrison $E_{1}$-page above
vanishes in negative total degree and thus no differentials land in total
degree zero. So it is the vanishing of differentials out of
${{\mathbb{E}}}(\pi_{1}(X)^{\\#})$, which are essentially Massey products of
$H^{1}(X)$-classes, which ultimately dictate the structure of the zeroth
Harrison cohomology. Relations in a group give rise to products in cohomology,
thus “more” products mean “smaller” fundamental group. Wedges of circles are
an extreme case in which all products vanish. They are also universal: the
fact that wedges of circles can induce surjections onto the fundamental group
of any space is dual to the observation that all Hopf invariants inject into
those of wedges of circles through the Harrison spectral sequence.

At the other extreme, products of circles and their abelian fundamental group
have injective $d_{1}$-differential, killing all but weight zero elements. In
this case the only Hopf invariants come directly from cohomology:

$H_{\mathcal{E}}^{0}\biggl{(}\prod_{s\in S}S^{1}\biggr{)}\cong\mathbb{Q}^{S}$

with zero cobracket.

See Section 5.4 for an intermediate example. Explicitly working through the
Harrison spectral sequence in this case with its $d_{2}$ differential is a
useful exercise.

In the simply connected setting, we show in [SW13] that the corresponding
pairing is perfect between $\pi_{n}(X)\otimes\mathbb{Q}$ and
$H_{\mathcal{E}}^{n-1}(X)$, making Quillen’s well-known isomorphisms of
rational homotopy theory effective for calculations and for use of geometry.
In the present work we focus on the $n=1$ setting, developing Hopf invariants
for fundamental groups and their Malcev Lie algebras with a wide range of
applications.

##  3\. Central Example: Handlebodies and linking of letters for free groups

We connect the topological framework of the previous section with the purely
combinatorial approach to understanding commutators in groups, introduced in
Section 1.2 and developed in [MS22]. We start with some elementary
recollections. If $M$ is a manifold and and $W$ is a codimension $k$
submanifold, there is a Thom $k$-form which is supported in a neighborhood of
$W$ and “counts intersections with $W$” multiplied by a scalar we call mass.
See Appendix B.1.

The simplest example is the most relevant presently, namely $M=S^{1}$ and $W$
is a finite collection of points. A Thom form for $W$ is a sum of “bump forms”
$\omega=\Sigma f_{i}\;d\theta$, with each $f_{i}$ supported in a neighborhood
of a point. If the total integral of such a Thom form is zero, then there is
an anti-derivative zero-form which we call $d^{{\text{-}}1}\omega$, determined
uniquely by the requirement that it vanishes at the basepoint $1\in S^{1}$.
This anti-derivative is a “mostly constant” function whose values are the
partial sums of the masses.

(a) A Thom form $\alpha$ for two points on the circle, with masses $\pm 1$.

(b) The antiderivative of $\alpha$, denoted $d^{{\text{-}}1}\alpha$ is a Thom
form for the interval.

Figure 4. Basic Thom forms on the circle.

Now consider a general smooth manifold $M$ with $\\{W_{i}\\}$ a a collection
of codimension one submanifolds. If $f:S^{1}\to M$ is a smooth loop transverse
to $W_{i}$ (a generic condition) then the pullback of a Thom form for $W_{i}$
is a Thom form for the intersection points $f^{{\text{-}}1}(W_{i})\subset
S^{1}$, whose integral counts intersections between the image of $f$ and
$W_{i}$ with signs. For example, if we take the Borromean rings as presented
in Figure 1(a) and Thom forms for the Seifert surfaces in red and blue, the
pullback will be Thom forms supported in the red and blue points on the left
edge of the square in in Figure 1(b). If we consider the red surface alone,
the pullback will look like the Thom form pictured in Figure 4(a).

###  3.1. First handlebody example

In Example 2.10 we saw that for wedges of spheres the zeroth Harrison
cohomology is a cofree Lie coalgebra. We now calculate the simplest associated
Hopf invariant (simplest aside from evaluation of cohomology).

Let $B_{m}$ denote a three-dimensional handlebody obtained by adding $m$ one-
handles to a three-ball. Then $B_{m}$ is a $3$-manifold with boundary and
homotopy equivalent to a wedge $\bigvee_{m}S^{1}$. We consider the case $m=2$.
Label the two handles by $a$ and $b$, and by abuse we use the same letters for
the generators of $\pi_{1}$ represented by loops along their core disks. Let
$D_{a}$ be a belt disk of the handle $a$ as in Figure 5, and let $A$ be a Thom
form for $D_{a}$ of mass one. The form $A$ counts the (signed) number of times
a closed loop winds around the handle $a$, since if $f\colon S^{1}\to B_{2}$
meets $D_{a}$ transversely,

$\langle A,f\rangle=\int_{S^{1}}f^{*}A=\sum_{\operatorname{Im}(f)\cap
D_{a}}\pm 1.$

Similarly define $D_{b}$ and $B$ for the handle $b$.

Figure 5. A commutator in a handlebody

Consider the loop $f\colon S^{1}\to B_{2}$ with
$[f]=aba^{{\text{-}}1}b^{{\text{-}}1}\in\pi_{1}(B_{2})$ shown in Figure 5. We
calculate the Hopf invariant associated to $\lfloor A\rfloor B$ on $f$. With
appropriate choices of parametrization and Thom forms, $f^{*}A=\alpha$, where
$\alpha$ is shown in Figure 4(a), and we set $\beta=f^{*}B$, which is the time
translate of $\alpha$ by $\frac{1}{5}$. Thus $f^{*}\lfloor A\rfloor
B=\lfloor\alpha\rfloor\beta$. Then recalling $d^{{\text{-}}1}\alpha$ from
Figure 4(b), the Harrison $(-1)$-cochain $\lfloor
d^{{\text{-}}1}\alpha\rfloor{\beta}$ will have total differential

(3) $d_{\mathcal{E}}\Bigl{(}\lfloor
d^{{\text{-}}1}\alpha\rfloor{\beta}\Bigr{)}=\lfloor
d(d^{{\text{-}}1}\alpha)\rfloor{\beta}+\lfloor
d^{{\text{-}}1}\alpha\rfloor{d\beta}-\
(d^{{\text{-}}1}\alpha)\wedge\beta=\lfloor\alpha\rfloor{\beta}\ -\
(d^{{\text{-}}1}\alpha)\wedge\beta.$

This implies that $\lfloor\alpha\rfloor\beta$ is homologous to
$(d^{{\text{-}}1}\alpha)\wedge\beta$ in the Harrison complex of the circle –
the latter is a form of mass one supported near $t=\frac{2}{5}$. We conclude
that $\bigl{\langle}\lfloor A\rfloor
B,\,f\bigr{\rangle}=\int_{S^{1}}d^{{\text{-}}1}\alpha\wedge\beta$, which is
equal to $1$.

A similar calculation shows that in general, if $\langle A,f\rangle=0$ then we
can pair $a$’s with $a^{{\text{-}}1}$’s in the word $[f]\in\pi_{1}(B_{2})$,
and $\bigl{\langle}\lfloor A\rfloor B,\,f\bigr{\rangle}$ is equal to the total
number of $b$’s which occur between $a$-$a^{{\text{-}}1}$ pairs, counted with
appropriate signs. This count can be seen as the linking number of the zero-
manifolds $f^{{\text{-}}1}D_{a}$ and $f^{{\text{-}}1}D_{b}$ inside $S^{1}$.
Indeed, for any two submanifolds manifolds $W,V\subset S^{n}$ whose dimensions
add to $n-1$, Whitehead’s integral formula for the Hopf invariant [Whi47],
namely $\int_{S^{n}}(d^{{\text{-}}1}\tau_{W})\wedge\tau_{V}$, computes the
linking number of $W$ and $V$, where $\tau_{W}$ and $\tau_{V}$ are respective
Thom forms. When $n=1$ the manifolds $W$ and $V$ are discrete, and when pulled
back from a map representing a word the corresponding invariants are letter
linking invariants, as we now show.

###  3.2. Equating topology and combinatorics of letter linking

In Section 1.2 we used simple combinatorics to define letter braiding
functions on free groups. In Section 2.2 we defined Hopf invariants
topologically. We now show that these coincide when considering fundamental
groups of handlebodies and restricting to “good” symbols.

Let $F_{S}=\langle s_{1},s_{2},\cdots,s_{m}\rangle$ be a free group. We
present it as the fundamental group of a handlebody $B_{m}$, with handles
corresponding to the $s_{i}$, whose first cohomology is
${F_{S}}^{\\#}=\text{Hom}_{Grp}(F_{S},\mathbb{Q})$. Then
${F_{S}}^{\\#}=\text{Hom}_{Grp}(F_{S},\mathbb{Q})$ is the vector space spanned
by the indicator homomorphisms $S_{i}$, respectively sending $s_{i}$ to $1$
and all other generators to $0$.

The braiding symbols defining letter braiding functions in Section 1.2 map to
our presentation of the Harrison homology of the handlebody, basically by
changing $\lfloor\lfloor\mathopen{}\;\rfloor\rfloor{}$’s to
$\lfloor\;\rfloor\,{\\!}$’s. Since the handlebody is formal and has cup
trivial products, there is a canonical isomorphism
$H^{0}_{\mathcal{E}}(B_{m})\cong\mathbb{E}(H^{1}(B_{m}))$. Therefore, the Hopf
invariants of $F_{S}$ are represented by rooted trees decorated by
homomorphisms $h_{i}\colon F_{S}\to\mathbb{Q}$, just as braiding symbols, and
we can let $r$ be the map of vector spaces which sends a linear combination of
braiding symbols to the class it represents in $H^{0}_{\mathcal{E}}(B_{m})$.

While letter braiding invariants of words and Hopf invariants of (free) groups
are globally defined, they only agree with Hopf invariants in the following
special setting.

######  Definition 3.1.

A letter braiding product
$\lfloor\lfloor\mathopen{}f\rfloor\rfloor{{}_{\\!{}_{w}}}g$ is said to be a
linking product when $\int_{[n]}f=0$.

A word $w$ is an eigenword for a braiding symbol $\sigma_{\vec{h}}$ if all of
the constituent braiding products in $w^{*}\sigma_{\vec{h}}$ are linking
products. The restriction of a letter braiding function to the subset of
eigenwords is called a letter linking function.

The term “letter linking” arises as discussed at the end of Section 3.1. For
example, we consider the indicator homomorphism $A$ on a word where the total
count of $a$’s is zero. Pairing off the $a$’s, the values of
$\lfloor\lfloor\mathopen{}A\rfloor\rfloor{{}_{\\!{}_{w}}}$ record how many
$a$–$a^{-1}$ pairs enclose a given letter of $w$. If the total count of $a$’s
is not zero then we do not have such “closed strings” connecting $a$’s and
$a^{-1}$’s, so the calculation involves “open strings”, hence the term letter
braiding.

To relate letter linking to Hopf invariants consider a word of length $\ell$,
and let $f_{w}:S^{1}\to B_{m}$ denote the map which represents $w$ in
$\pi_{1}(B_{m})$, for definiteness later say by traveling through the core of
each handle at constant speed in time $\frac{1}{\ell}$.

######  Theorem 3.2.

If $w$ is an eigenword for $\sigma_{\vec{h}}$ then

$\sigma_{\vec{h}}(w)=\int_{[n]}w^{*}\sigma_{\vec{h}}=\int_{S^{1}}f_{w}^{*}r(\sigma_{\vec{h}}).$

The first equality is by definition of letter braiding functions. We gave a
first example of this in Section 3.1.

The rest of this section is devoted to proving this equality. We relate the
two evaluations on $w$ by carefully choosing cochain representatives for
$H^{0}_{\mathcal{E}}(B_{m})$, generalzing the constructions in Section 3.1. To
do so is elementary but requires some language to specify the Harrison
cochains which make equality apparent.

First, we define a family of Thom forms of belt disks. Fix a non-negative one-
form $\varphi$ with support on the interior of $[0,1]$ with total integral
one. If $J\subset[0,1]$ is an interval, let $\varphi_{J}$ denote the one-form
which on $J$ is the pull-back of $\varphi$ by the order-preserving linear
homeomorphism of $J$ with $[0,1]$ and outside of $J$ is zero. On a handle
$D^{2}\times[0,1]$ let $\Phi_{J}$ denote the pullback of $\varphi_{J}$ along
the projection to the interval.

To consider all possibilities for the placement of such forms on a handle, let
$\kappa_{n}$ be the set of $n$-tuples of subintervals of $[0,1]$. Explicitly,
$\kappa_{n}$ is in natural bijection with $(\Delta^{2})^{\times n}$. (These
sets also have a natural operadic structure, which contains the little one-
disks as a sub-operad, but we will not use this fact.)

Braiding symbols are generated by homomorphisms under the braiding product.
Consider linear combinations of such symbols and declare the braiding product
to be bilinear. Then all letter braiding symbols are spanned by those
$\sigma_{\vec{h}}$ where each $h_{i}$ is an indicator homomorphism $S_{j}$ for
some $j$. Let elements in this spanning set be called indicator symbols.

We now construct a cochain-level version of the map $r$ above which encodes
the correspondance between braiding symbols and the Harrison cohomology of
handlebodies.

######  Definition 3.3.

Let $\sigma_{\vec{h}}$ be a weight $(n-1)$ indicator symbol and
$\vec{J}=(J_{1},\cdots,J_{n})\in\kappa_{n}$. Define the realization
$\rho(\sigma_{\vec{h}},\vec{J})\in\mathcal{E}(\Omega_{dR}(B_{m}))$ to be the
Harrison cochain represented by the same rooted tree (changing
$\lfloor\lfloor\mathopen{}\;\rfloor\rfloor{}$’s to
$\lfloor\;\rfloor\,{\\!}$’s), whose $i$-th vertex is decorated by the form
$\Phi_{J_{i}}$ supported on the handle indexed by $s_{j}$ if the $i$-th vertex
of $\sigma_{\vec{h}}$ has indicator function $h_{i}=S_{j}$.

Extend the realization $\rho$ linearly to all symbols to obtain a map
$\rho:{{\mathbb{E}}}^{n-1}(F_{S}^{\\#})\times\kappa_{n}\to\mathcal{E}(\Omega_{dR}(B_{m}))$.

In summary, we are realizing braiding symbols at the cochain level by linear
combinations of Thom forms for belt disks which are supported on sets of
intervals parametrized by $\kappa_{n}$. Below, we utilize a particular subset
of such realizations.

Recall that the vertices of a rooted tree are partially ordered, with $w<v$ if
$v$ is on the path from $w$ to the root (and the root a maximal element). Call
a pair $(\sigma_{\vec{h}},\vec{J})$ order compatible if whenever $w<v$ in
$\sigma$ the interval corresponding to $v$ is entirely to the left of the one
corresponding to $w$. We claim that an order-compatible realization
$\rho(\sigma_{\vec{h}},\vec{J})\in\mathcal{E}(\Omega_{dR}(B_{m}))$ pulls back
to the circle along the map $f_{w}\colon S^{1}\to B_{m}$ so that this pullback
is homologous to a single form that matches the d-function
$w^{*}\sigma_{\vec{h}}$ combinatorially.

To make this precise, we say a form on the interval represents a d-function
$f\colon[\ell]\to\mathbb{Q}$ if it is supported on the interiors the sub-
intervals $(\frac{i-1}{\ell},\frac{i}{\ell})$ and its mass on the $i$-th sub-
interval is $f(i)$. Clearly, the integral over $S^{1}$ of a form which
represents a d-function is equal to the discrete integral over $[n]$ of that
d-function. Thus Theorem 3.2 is an immediate corollary of the following.

######  Theorem 3.4.

Let $w$ is an eigenword for $\sigma_{\vec{h}}$, and let $\vec{J}\in\kappa_{n}$
be order compatible with $\sigma_{\vec{h}}$. Then the pullback of the
realization
$f^{*}_{w}\rho(\sigma_{\vec{h}},\vec{J})\in\mathcal{E}(\Omega_{dR}(S^{1}))$
weight reduces to a form which represents $w^{*}\sigma_{\vec{h}}$.

###### Proof.

We will prove the following stronger statement by induction on weight. If
$J\subseteq(0,1)$ is a sub-interval, let
$J(i)\subset(\frac{i-1}{\ell},\frac{i}{\ell})$ denote its image under the
increasing linear homeomorphism $(0,1)\to(\frac{i-1}{\ell},\frac{i}{\ell})$ if
$w$ has $\epsilon_{i}=1$, and the decreasing linear homeomorphism when
$\epsilon_{i}=-1$. Then the pullback $f^{*}_{w}\rho(\sigma_{\vec{h}},\vec{J})$
can be weight reduced to a form whose restriction to the interval
$(\frac{i-1}{\ell},\frac{i}{\ell})$ has total mass $w^{*}\sigma_{\vec{h}}(i)$
and is supported on the interval $J_{o}(i)$, where $J_{o}\in\vec{J}$ is the
interval corresponding to the root vertex.

In weight zero, no weight reduction is necessary. If $h$ is a homomorphism
decorating the unique vertex in $\sigma$ then the realization
$\rho(h,J_{1})\in\Omega^{1}(B_{m})$ is the form restricting to each handle
$s_{j}$ as some bump form with mass $h(s_{j})$. Thus its pullback along the
loop $f_{w}$ is indeed has a sum of bump forms with respective mass $h(w_{i})$
supported in the sub-interval $J_{1}(i)$. Note that if a letter in $w$ is the
inverse of a generator $s_{j}^{-1}$ then the loop $f_{w}$ travels through the
handle $s_{j}$ backwards, mapping through the order reversing homeomorphism
$(\frac{i-1}{\ell},\frac{i}{\ell})\to(0,1)$.

Proceeding by induction, consider a braiding product
$\sigma=\lfloor\lfloor\mathopen{}\alpha\rfloor\rfloor\beta$, suppressing
decorating homomorphisms from the notation. Note that the eigenword assumption
means in particular that $w$ is an eigenword for both $\alpha$ and $\beta$.

Let $J_{k}\in\vec{J}$ denote the sub-interval corresponding to the root of
$\alpha$. Then by the inductive hypothesis the pullback
$f^{*}_{w}\rho(\alpha,\vec{J})$ can be weight reduced to a form $\alpha_{0}$
on $S^{1}$ with support in $J_{k}(1)\cup\ldots\cup J_{k}(\ell)$ and total mass
$\int_{[\ell]}w^{*}\alpha$. The eigenword assumption implies that this mass is
zero. Therefore $\alpha_{0}$ has an anti-derivative
$d^{{\text{-}}1}\alpha_{0}$ as a form (as in Figure 4(b)), which is constant
outside of the intervals $J_{k}(i)$ and whose constant values given by the
partial sums of the masses of $\alpha_{0}$. Similarly,
$f^{*}_{w}\rho(\beta,\vec{J})$ can be weight reduced to a form $\beta_{0}$
supported on the intervals $J_{o}(i)$ for $1\leq i\leq\ell$.

It follows that the entire pullback
$f^{*}_{w}\rho(\lfloor\lfloor\mathopen{}\alpha\rfloor\rfloor\beta)$ can be
weight reduced to the form $d^{{{\text{-}}1}}\alpha_{0}\wedge\beta_{0}$. This
product is immediately seen to have support contained in that of $\beta_{0}$,
as claimed. Furthermore, the intervals $J_{k}(i)$ are disjoint from
$J_{o}(i)$, so the form $d^{{{\text{-}}1}}\alpha_{0}$ is locally constant on
the support of $\beta_{0}$.

Finally, we calculate the mass of the weight reduced form on
$(\frac{i-1}{n},\frac{i}{n})$. Since $\vec{J}$ was chosen to be order
compatible with $\lfloor\lfloor\mathopen{}\alpha\rfloor\rfloor\beta$, the
interval $J_{k}$ is entirely to the left of $J_{o}$. After applying the
homeomorphism $(0,1)\to(\frac{i-1}{n},\frac{i}{n})$, we see that $J_{k}(i)$ is
still to the left of $J_{o}(i)$. In this case, the (constant) value of
$d^{{{\text{-}}1}}\alpha_{0}$ on $J_{o}(i)$ is the partial sum of
$w^{*}(\alpha)$ up to and including the $i$-th value. However, if
$\epsilon_{i}=-1$ then the orientation of $J_{o}(i)$ is reversed so that
$J_{k}(i)$ is on the right of $J_{o}(i)$, which implies that the value of
$d^{{{\text{-}}1}}\alpha_{0}$ on $J_{o}(i)$ is the partial sum not including
the $i$-th value. This is exactly the form of the d-function
$\lfloor\lfloor\mathopen{}w^{*}\alpha\rfloor\rfloor{}_{w}$ multiplying the
d-function $w^{*}\beta$, as claimed. ∎

###  3.3. Configuration braiding of trees and words

Rooted trees give an alternate way to view the evaluation of letter braiding
functions $\sigma_{\vec{h}}(w)$ from Section 1.2 akin to the configuration
pairing of [SW11]. Let $w=s_{1}^{\epsilon_{1}}\dots s_{n}^{\epsilon_{n}}$ be a
word in $F_{S}$ and let $T$ be a rooted tree with vertices labeled by
homomorphisms $(F_{S})^{\\#}$. Braiding symbols $\sigma_{\vec{h}}$ are
equivalent to (sums of) such rooted trees by formally converting
$\lfloor\lfloor\mathopen{}\,\rfloor\rfloor$ to $\lfloor\,\rfloor\,$.

######  Definition 3.5.

A configuration of $T$ is a weakly decreasing function
$c:\mathrm{Vertices}(T)\to[n]$ with respect to the partial ordering on
$\mathrm{Vertices}(T)$ induced by the root. That is, the root is a maximum
elements and $c(b)\geq c(a)$ if the unique monotone path from the root to $a$
passes through $b$.

A configuration is proper with respect to $w$ if $c$ is strictly decreasing at
$i$ whenever $\epsilon_{i}=1$. That is, the inverse image $c^{-1}(i)$ must not
contain comparable vertices ($a$,$b$ with either $a<b$ or $b<a$).

The associated pairing is $\langle
T,w\rangle_{c}=\prod\ell_{v}\bigl{(}w_{c(v)}\bigr{)}$ where
$\ell_{v}\in(F_{S})^{\\#}$ is the label of vertex $v$ and
$w_{i}=s_{i}^{\epsilon_{i}}$.

The restriction to proper configurations is necessary to account for the
effect of $\epsilon_{i}$ when computing
$\lfloor\lfloor\mathopen{}f\rfloor\rfloor{{}_{\\!{}_{w}}}$ in Definition 1.5.

It is instructive to compare $\lfloor\lfloor\mathopen{}A\rfloor\rfloor A$
evaluated on $aa$ versus $a^{{\text{-}}1}a^{{\text{-}}1}$. On $aa$ there is
only one proper configuration $\lfloor\lfloor\mathopen{}1\rfloor\rfloor 2$,
leading to a value of $1$. On $a^{{\text{-}}1}a^{{\text{-}}1}$ there are three
proper configurations $\lfloor\lfloor\mathopen{}1\rfloor\rfloor 1$,
$\lfloor\lfloor\mathopen{}1\rfloor\rfloor 2$, and
$\lfloor\lfloor\mathopen{}2\rfloor\rfloor 2$, leading to a value of $3$.

######  Proposition 3.6.

Letter braiding functions count proper configurations of $\sigma_{\vec{h}}$ in
$w$.

$\displaystyle\sigma_{\vec{h}}(w)=\sum_{\begin{subarray}{c}\mathrm{proper}\\\
\mathrm{configuration}\end{subarray}}\bigl{\langle}\sigma_{\vec{h}},\,w\bigr{\rangle}_{c}$

######  Example 3.7.

Consider evaluating $\lfloor\lfloor\mathopen{}A\rfloor\rfloor B$ on the word
$aba^{{\text{-}}1}b^{{\text{-}}1}$. Only three configurations yield nonzero
pairings.

  * •

$\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B\mapsto\lfloor\lfloor\mathopen{}1\rfloor\rfloor 2$, contributing
$A(a)\,B(b)=1$

  * •

$\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B\mapsto\lfloor\lfloor\mathopen{}1\rfloor\rfloor 4$, contributing
$A(a)\,B(b^{{\text{-}}1})=-1$

  * •

$\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B\mapsto\lfloor\lfloor\mathopen{}3\rfloor\rfloor 4$, contributing
$A(a^{{\text{-}}1})\,B(b^{{\text{-}}1})=1$

The sum of these is $1$, recovering
$\bigl{(}\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B\bigr{)}(aba^{{\text{-}}1}b^{{\text{-}}1})$.

##  4\. Structural theorems

We establish the main formal theorems governing Hopf invariants, starting with
the fact that the Hopf pairing establishes a Lie coalgebraic duality, then
giving a lifting theorem which relates Hopf invariants of two complexes and
their one skeleton, and finally we establish universality using the lifting
theorem.

###  4.1. Lie pairing

We use the standard approach to Lie (co)algebras through (co)associative
algebras and Hopf algebras. Fundamental group invariants in (co)associative
setting were developed by Chen.

We first recall the standard presentation of the Malcev Lie algebra. For a
group $G$ let $\widehat{\mathbb{Q}[G]}$ be the completion of its group ring at
the ideal generated by elements of the form $(\gamma-1)$ for $\gamma\in G$.
For example, the power series

$\log(\gamma)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}(\gamma-1)^{n}$

converges in $\widehat{\mathbb{Q}[G]}$. This ring is naturally a Hopf algebra,
with product $(\gamma,\gamma^{\prime})\mapsto\gamma\gamma^{\prime}$ and
coproduct $\Delta(\gamma)=\gamma\otimes\gamma$ for all group elements
$\gamma,\gamma^{\prime}\in G$. Recall that primitives in a Hopf algebra form a
Lie algebra.

######  Definition 4.1.

The Malcev Lie algebra $\mathrm{Mal}({G})$ of the group $G$ is the Lie algebra
of primitive elements in $\widehat{\mathbb{Q}[G]}$.

Quillen showed in [Qui69] that this is a complete filtered Lie algebra,
topologically generated by elements of the form $\log(\gamma)$ for $\gamma\in
G$. Moreover, he showed that the logarithm function preserves filtrations when
$G$ and $\mathrm{Mal}({G})$ are filtered by their respective lower central
series, inducing an isomorphism of associated graded vector spaces after
tensoring with $\mathbb{Q}$. (For a thorough treatment see Appendix A in
[Qui69], and Section 2 of [Mas12].) Recall the notion of a Lie pairing from
Definition 1.1.

######  Theorem 4.2 (Theorem 1.2, Lie pairing).

Hopf invariants extend uniquely to a Lie pairing, with

$\bigl{\langle}T,\log(w)\bigr{\rangle}_{H}=\int_{S^{1}}w^{*}(T),$

for every $T\in H^{0}_{\mathcal{E}}(X)$ and group element $w\in\pi_{1}(X)$.

Uniqueness of such a pairing will follow from the stated bracket-cobracket
duality, which forces the pairing to be continuous with respect to the
filtration by commutator depth, while $\mathrm{Mal}\big{(}{\pi_{1}(X)}\big{)}$
is topologically generated by the $\log(\gamma)$. We call the resulting
pairing the Hopf pairing.

Recall that the Baker-Campbell-Hausdorff series is a formal power series in a
completed Lie algebra satisfying

$\exp(\ell)\exp(\ell^{\prime})=\exp\bigl{(}\mathrm{BCH}(\ell,\ell^{\prime})\bigr{)}.$

This series governs values of Hopf invariants.

######  Corollary 4.3 (Aydin’s Formula).

Hopf invariants of products in $\pi_{1}(X)$ are given by the Baker-Campbell-
Hausdorff formula:

$\displaystyle\int_{S^{1}}(ab)^{*}(T)$ $\displaystyle=\bigl{\langle}T,\
\mathrm{BCH}\bigl{(}\log(a),\log(b)\bigr{)}\bigr{\rangle}_{H}$
$\displaystyle=\Bigl{\langle}T,\
\log(a)+\log(b)+\frac{1}{2}\bigl{[}\log(a),\log(b)\bigr{]}+\frac{1}{12}(\cdots)+(\cdots)\Bigr{\rangle}_{H}.$

###### Proof.

We apply the defining property of the BCH series to
$\bigl{\langle}T,\log(ab)\bigr{\rangle}_{H}=\int_{S^{1}}(ab)^{*}(T)$ taking
$a=\exp(\log(a))$ and $b=\exp(\log(b))$. ∎

######  Remark 4.4.

The second author originally found a proof of Corollary 4.3 with finite
generation hypotheses using the theory of $C_{\infty}$-algebras. Briefly, a
$C_{\infty}$-algebra is a graded vector space $C$ with a coderivation on
${{\mathbb{E}}}(C)$ which squares to zero, defining a Harrison complex.
Morphisms are just maps on Harrison complexes.

In [BFMT20] the authors show that the simplicial cochains on a simplicial set
have a $C_{\infty}$ structure extending cup product which is quasi-isomorphic
to the PL de Rham cochains on that simplicial set. In the case of the one-
simplex this is dual to the Lawrence-Sullivan interval.

The “universal product” on fundamental groups is given by a collapse map
$S^{1}\to S^{1}\vee S^{1}$. The second author made explicit calculations in
this model to see that weight reduction in the Harrison complex leads to
Baker-Campbell-Hausdorff product. He used this to achieve our first proof of
the Lifting Criterion below, in the finitely generated setting.

###### Proof of Theorem 4.2.

Let $A^{*}(X)$ be the de Rham forms (smooth or piecewise linear) for $X$ and
consider a loop $\gamma\colon S^{1}\to X$. We consider the interplay between
the coassociative and coLie bar constructions

$\mathcal{H}^{0}\bigl{(}\mathcal{B}(A^{*}(X)\bigr{)}\twoheadrightarrow\mathcal{H}^{0}\bigl{(}\mathcal{E}(A^{*}(X)\bigr{)}.$

Chen defined an invariant of loops using the iterated integral. If $\gamma$ is
a loop, $\omega_{i}\in A^{1}(X)$ are a $1$-forms and
$\gamma^{*}(\omega_{i})=f_{i}(\theta)d\theta$ then

$\bigl{\langle}(\omega_{0}|\ldots|\omega_{n}),\gamma\bigr{\rangle}_{C}=\int_{t_{0}<t_{2}<\ldots<t_{n}}f_{0}(t_{0})\ldots
f_{n}(t_{n})\;dt_{0}\ldots dt_{n}.$

Chen’s invariant gives a linear map
$\langle-,\gamma\rangle_{C}\colon\mathcal{H}^{0}(\mathcal{B}(A^{*}(X)))\longrightarrow\mathbb{Q}$
which extends to a bilinear pairing with the group ring $\mathbb{Q}[\pi_{1}]$.
Letting $\bar{\Delta}$ denote the reduced coproduct of
$\mathcal{H}^{0}(\mathcal{B}(A^{*}(X)))$, Chen proves in [Che71, Eq. (1.6.1)]
that

$\bigl{\langle}\bar{\Delta}(T),(\gamma-1)\otimes(\gamma^{\prime}-1)\bigr{\rangle}_{C}=\bigl{\langle}T,(\gamma-1)(\gamma^{\prime}-1)\bigr{\rangle}_{C}$

from which it follows that weight $p$ Bar cocycles vanish on the $p$-th power
of the augmentation ideal in $\mathbb{Q}[\pi_{1}]$. Thus the pairing extends
to the completion $\widehat{\mathbb{Q}[\pi_{1}]}$. Define a pairing on
$\mathcal{H}^{0}(\mathcal{E}(A^{*}(X))$ by restriction to primitive elements,
so if $\ell\in\widehat{\mathbb{Q}[\pi_{1}]}$ is primitive, set

$\langle\overline{T},x\rangle_{\hat{H}}=\langle T,x\rangle_{C}$

where $T$ is any lift of
$\overline{T}\in\mathcal{H}^{0}\bigl{(}\mathcal{E}(A^{*}(X)\bigr{)}$ to the
coassociative Bar complex, which exists by Barr’s splitting [Bar68]. To see
that this is independent of the choice of lift $T$, recall that under Chen’s
pairing the shuffle product on the Bar complex is dual to the group ring
coproduct [Che71, Eq. (1.5.1)] from which it follows that nontrivial shuffles
must vanish on primitive elements. Since the Harrison complex is the quotient
of the coassociative bar complex by nontrivial shuffles, the pairing on
primitives descends to Harrison homology.

Bracket-cobracket duality of this pairing follows quickly from Chen’s product-
coproduct duality,

$\bigl{\langle}\,]\overline{T}[,\ell\otimes\ell^{\prime}\bigr{\rangle}_{\hat{H}}=\bigl{\langle}\,]T[,\ell\otimes\ell^{\prime}\bigr{\rangle}_{C}=\bigl{\langle}\Delta(T),\ell\otimes\ell^{\prime}-\ell^{\prime}\otimes\ell\bigr{\rangle}_{C}=\bigl{\langle}T,\ell\cdot\ell^{\prime}-\ell^{\prime}\cdot\ell\bigr{\rangle}_{C}=\bigl{\langle}\overline{T},[\ell,\ell^{\prime}]\bigr{\rangle}_{\hat{H}}.$

Vanishing of weight $p$ cocycles on $p$-fold nested commutators follows
immediately from this identity.

It remains to show that the pairing $\langle\,,\rangle_{\hat{H}}$ agrees with
the Hopf pairing, by weight reduction on the Harrison complex of the circle.
By work of Gugenheim [Gug77], Chen’s iterated integrals extend to natural
isomorphisms between the top two rows of the commutative diagram

$\textstyle{\mathcal{H}^{0}(\mathcal{B}\bigl{(}C^{*}(X))\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{*}}$$\textstyle{\mathcal{H}^{0}(\mathcal{B}\bigl{(}C^{*}(S^{1})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Q}[e]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\log}$$\textstyle{\mathbb{Q}e}$$\textstyle{\mathcal{H}^{0}\bigl{(}\mathcal{B}(A^{*}(X))\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\cong}$
$\textstyle{\mathcal{H}^{0}(\mathcal{B}\bigl{(}A^{*}(S^{1})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Q}[d\theta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{H}^{0}\bigl{(}\mathcal{E}(A^{*}(X))\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{*}}$$\textstyle{\mathcal{H}^{0}\bigl{(}{{\mathbb{E}}}(A^{*}(S^{1})\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Q}d\theta.}$

$\textstyle{T\vphantom{\mathcal{H}^{0}\bigl{(}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{T}}$

Chen’s invariant is the dashed arrow (this holds in all higher dimensional as
well). Here, the surjection between the bottom two rows is Harrison’s quotient
by shuffles.

Letting $I_{\gamma}\colon\mathcal{H}^{0}(\mathcal{B}(A^{*}(X)))\to\mathbb{Q}e$
denote the dashed arrow, observe that it is the corestriction onto the
cogenerator of the cofree coalgebra $\mathbb{Q}[e]$. Since Gugenheim’s maps
between the top two rows are coalgebra homomorphisms, the image of $T$ in
$\mathbb{Q}[e]$ is

$\sum_{n=1}^{\infty}I_{\gamma}^{\otimes
n}\bigl{(}\overline{\Delta}^{n-1}(T)\bigr{)}\cdot
e^{n}=\sum_{n=1}^{\infty}\,\bigl{\langle}T,\,(\gamma-1)^{n}\bigr{\rangle}_{C}\cdot
e^{n}.$

On the other hand, the map denoted by $\log$ is the corestriction of the
inverse to Gugenheim’s isomorphism, which is computed in [BS16, Proposition
1.7] (where both $e$ and $d\theta$ are denoted by $dt$) to be

$e^{n}\longmapsto\frac{(-1)^{n+1}}{n}\cdot d\theta.$

With this, the image of $T$ in the bottom-right corner is seen to be

$\sum_{n=1}^{\infty}\bigl{\langle}T,(\gamma-1)^{n}\bigr{\rangle}_{C}\cdot\frac{(-1)^{n+1}}{n}\cdot
d\theta=\bigl{\langle}T,\log(\gamma)\bigr{\rangle}_{C}\cdot d\theta.$

This shows that

$\int_{S^{1}}\gamma^{*}(\overline{T})=\bigl{\langle}T,\log(\gamma)\bigr{\rangle}_{C}=\bigl{\langle}\overline{T},\log(\gamma)\bigr{\rangle}_{\hat{H}}$

for all $\overline{T}\in\mathcal{H}^{0}\bigl{(}\mathcal{E}(A^{*}(X)\bigr{)}$.
∎

###  4.2. The lifting criterion

We relate the Harrison cohomology of a two-dimensional CW-complex to that of
its one-skeleton. We use this criterion in the next two sections to prove the
fundamental theorem that the Hopf pairing is a universal Lie coalgebraic
extension of homomorphisms from a group to the rational numbers and to extend
the theory of letter linking invariants to general groups.

For any simplicial set $X$ let $A^{*}X$ denote the commutative differential
graded algebra of PL differential forms on $X$. Let $X$ be a two-dimensional
complex with one-skeleton $X^{(1)}\simeq\vee_{S}S^{1}$. Then $X$ is built by
attaching $2$-cells along maps $r\colon\partial D^{2}\to X^{(1)}$ for some set
$r\in R$, and its fundamental group is the free group $F_{S}$ modulo the
images in $\pi_{1}$ of the maps $r$ which by abuse we also call $r$.

As discussed in Example 2.10 the zeroth Harrison cochomology of a wedge of
circles is just the cofree Lie coalgebra on the linear dual ${F_{S}}^{\\#}$.
Our goal is to understand the map induced by inclusion of the one-skeleton
$i^{*}:H^{0}_{\mathcal{E}}(X)\to
H^{0}_{\mathcal{E}}(\vee_{S}S^{1})\cong{{\mathbb{E}}}({F_{S}}^{\\#})$, which
by the following also yields a calculation of the zeroth Harrison cohomology
of $X$.

######  Proposition 4.5.

The map $i^{*}$ is injective.

###### Proof.

As shown in Remark 2.14, there are no differntials in to degree zero for the
Harrison spectral sequence of any connected space. So zeroth Harrison
cohomology includes in the $E_{1}$-page. As $i^{*}$ is injective on first
cohomology, it is also injective in total degree zero on this $E_{1}$-page.
But the spectral sequence collapses for a wedge of circles. So $i^{*}$ is a
composite of injective maps. ∎

######  Definition 4.6.

Let $E_{R}$ denote the maximal sub-Lie coalgebra of
${{\mathbb{E}}}({F_{S}}^{\\#})\cong H^{0}_{\mathcal{E}}(\vee_{S}S^{1})$ whose
Hopf invariants vanish on relations.

######  Theorem 4.7 (Lifting Criterion).

The image of $i^{*}$ is $E_{R}$.

Informally, a Harrison cocycle lifts if and only if it and all of its
cobrackets vanish on relations. The image of $i^{*}$ vanishing on relations is
immediate, since by naturality the Hopf pairing can be computed on $X$, in
which case the lift of $r$ is null-homotopic. We will see examples below for
which the closure under cobracket, whose necessity is also immediate, excludes
some classes which vanish on all relations.

We analyze $i^{*}$ at the level of Harrison complexes, letting
$Z\subseteq\mathcal{E}^{0}({A^{*}X})$ denote the collection of
$d_{\mathcal{E}}$-closed elements. Let
$\mathcal{E}_{R}\subseteq\mathcal{E}^{0}({A^{*}(\vee S^{1})})$ be the maximal
sub-coalgebra whose Hopf invariants all vanish on the attaching maps. Thus
restriction to the $1$-skeleton gives a commutative diagram

(4) $\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.61389pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.77084pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.26988pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\operatorname{res}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-31.30444pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-1.99997pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.26988pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0}({A^{*}X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
61.64629pt\raise-16.45485pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.77362pt\hbox{$\scriptstyle{i^{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 61.64629pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-1.99997pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
8.55043pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
32.55042pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.55042pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0}({A^{*}(\vee\\!S^{1})})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

The Lifting Criterion of Theorem 4.7 follows immediately from the following.

######  Lemma 4.8 (Lifting Lemma).

There exists a section $s\colon\mathcal{E}_{R}\to Z$ which is a Lie coalgebra
homomorphism.

The proof of this claim is the most technical aspect of our work. The key to
our argument is to relate Hopf invariants of attaching maps to the Harrison
complex of the attached disks.

Recall that Harrison cochains of weight zero in $\mathcal{E}^{k}({A^{*}X})$
are simply elements in $A^{k+1}(X)$. In text below we will abusively write
$\alpha$ for both a weight zero element of $\mathcal{E}^{k}({A^{*}X})$ as well
as the underlying form in $A^{k+1}X$.

######  Lemma 4.9 (Circles and Disks).

Consider $b:S^{1}\hookrightarrow D^{2}$, the inclusion map of the boundary.
Every Harrison cocycle $\alpha\in\mathcal{E}^{0}{(A^{*}S^{1})}$ is realized as
a pullback from the disc $\alpha=b^{*}(\bar{\alpha})$ for some cochain
$\bar{\alpha}\in\mathcal{E}^{0}({A^{*}D^{2}})$ such that:

  1. (1)

$d_{\mathcal{E}}\bar{\alpha}$ has weight $0$;

  2. (2)

The $2$-form $d_{\mathcal{E}}\bar{\alpha}$ is supported on the interior of
$D^{2}$;

  3. (3)

$\int_{D^{2}}d_{\mathcal{E}}\bar{\alpha}=\int_{S^{1}}\alpha$.

The integral on the left of (3) is the ordinary integration of a $2$-form on
the disc, while the integral on the right is evaluation defined in Proposition
2.11. An illustration of (3) is discussed in Section 5.4.

###### Proof.

The map $A^{*}b:A^{*}D^{2}\to A^{*}S^{1}$ is a surjection, so the induced map
on Harrison complexes
$\mathcal{E}A^{*}b:\mathcal{E}^{*}({A^{*}D^{2}})\to\mathcal{E}^{*}({A^{*}S^{1}})$
is as well. Define $\mathcal{E}^{*}({D^{2},S^{1}})$ to be the kernel of
$\mathcal{E}A^{*}b$ and write $H^{*}_{\mathcal{E}}({D^{2},S^{1}})$ for the
cohomology of the resulting complex. The long exact sequence of relative
Harrison cohomology reads

$\cdots\to H^{*}_{\mathcal{E}}({D^{2}})\to
H^{*}_{\mathcal{E}}({S^{1}})\overset{\delta}{\longrightarrow}H^{*+1}_{\mathcal{E}}({D^{2},S^{1}})\to
H^{*+1}_{\mathcal{E}}({D^{2}})\to\cdots.$

Homotopy invariance implies $H^{*}_{\mathcal{E}}({D^{2}})=0$, so the
connecting homomorphism $\delta$ is an isomorphism.

Let $\alpha\in\mathcal{E}^{0}({S^{1}})$. The connecting homomorphism is
defined by lifting and then applying $d_{\mathcal{E}}$, namely
$\delta\alpha=d_{\mathcal{E}}\hat{\alpha}\in\mathcal{E}^{1}({D^{2},S^{1}})$.
The connecting homomorphism thus respects filtration by weight, since lifting
and $d_{\mathcal{E}}$ do. As $H^{0}_{\mathcal{E}}({S^{1}})$ is supported in
(spanned by) weight zero, so is $H^{1}_{\mathcal{E}}({D^{2},S^{1}})$. We can
thus choose an $\eta\in\mathcal{E}^{0}({D^{2},S^{1}})$ such that
$d_{\mathcal{E}}\hat{\alpha}+d_{\mathcal{E}}\eta=d(\hat{\alpha}+\eta)$ has
weight zero. Let $\bar{\alpha}=\hat{\alpha}+\eta$. Since $\eta|_{S^{1}}=0$
(every summand of $\eta$ contains at least one form that vanishes on $\partial
D^{2}$), the cochain $\bar{\alpha}$ itself restricts to $\alpha$, and as
desired has coboundary in weight zero, proving statement (1).

To prove statement (3), we first apply Proposition 2.11 to obtain an
$\alpha_{0}=\alpha+d_{\mathcal{E}}\xi$ of weight zero. Extending
$\alpha_{0}\in A^{1}(S^{1})$ to the interior of the disc results in a form
$\bar{\alpha}_{0}\in A^{1}(D^{2})$ such that

$\int_{D^{2}}d\bar{\alpha}_{0}=\int_{S^{1}}\alpha_{0}=\int_{S^{1}}\alpha,$

where the first equality is by Stokes’ Theorem. So it suffices to show that
$d_{\mathcal{E}}\bar{\alpha}=d\bar{\alpha}_{0}+d\zeta_{0}$ for some relative
$1$-form $\zeta_{0}\in A^{1}(D^{2},S^{1})$.

Observe that both $d_{\mathcal{E}}\bar{\alpha}$ and $d\bar{\alpha}_{0}$
represent the element $\delta\alpha\in\mathcal{E}^{1}({D^{2},S^{1}})$ and they
therefore differ by some $d_{\mathcal{E}}\zeta$ for
$\zeta\in\mathcal{E}^{0}({D^{2},S^{1}})$, possibly of positive weight. If we
can show that $\zeta=\zeta_{0}+d_{\mathcal{E}}\mu$ where
$\mu\in\mathcal{E}^{-1}({D^{2},S^{1}})$ and $\zeta_{0}$ has weight zero, then

$d_{\mathcal{E}}\bar{\alpha}=d\bar{\alpha}_{0}+d_{\mathcal{E}}(\zeta_{0}+d_{\mathcal{E}}\mu)=d\bar{\alpha}_{0}+d\zeta_{0}$

as desired.

To see we can weight reduce $\zeta$, suppose it has weight $p>0$ and let
$\zeta_{p}$ be its summand of weight $p$. The vertical component of
$d_{\mathcal{E}}$ must vanish on $\zeta_{p}$ since this is the weight $p$ term
of $d_{\mathcal{E}}\zeta$ which is only supported in weight zero. So
$\zeta_{p}$ is a $0$-cocycle in the complex $\mathcal{E}^{*}({D^{2},S^{1}})$
equipped with only the internal (vertical) differential. But both
$\mathcal{E}^{*}({D^{2}})$ and $\mathcal{E}({S^{1}})$ have no vertical
cohomology in degree zero in this weight by the Künneth Theorem, so neither
does $\mathcal{E}({D^{2},S^{1}})$. Thus $\zeta_{p}$ is the image under the
vertical differential of some $\mu_{p}\in\mathcal{E}^{-1}({D^{2},S^{1}})$. It
follows that $\zeta^{\prime}=\zeta-d_{\mathcal{E}}\mu_{p}$ has weight $p-1$
and still cobounds the difference of
$d_{\mathcal{E}}\bar{\alpha}-d\bar{\alpha}_{0}$. Performing this weight
reduction iteratively lets us subtract coboundaries from $\zeta$ until the
result $\zeta_{0}$ has weight zero, completing the proof.

∎

With these facts in place we can proceed to proving the Lifting Lemma.

###### Proof of Lemma 4.8.

Extend the diagram from Equation (4) to the right using maps $\pi$ projecting
onto cogenerators:

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}{A^{*}X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\mathcal{E}i^{*}}$$\textstyle{A^{*}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\textstyle{X}$$\textstyle{\mathcal{E}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{s}}$$\scriptstyle{\bar{s}}$$\scriptstyle{s}$$\textstyle{\mathcal{E}{A^{*}(\vee\\!S^{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A^{*}(\vee\\!S^{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{X^{(1)}\simeq\vee
S^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$

We define the desired section by climbing the weight filtration
$\\{\mathcal{E}_{R}^{\leq n}\\}$ inherited from
$\mathcal{E}^{0}({A^{*}(\vee\\!S^{1})})$. To start, choose splittings
$\mathcal{E}_{R}^{\leq
n}=\mathcal{E}_{R}^{\leq(n-1)}\oplus\mathcal{E}_{R}^{n}$ so that
$\mathcal{E}_{R}=\bigoplus\mathcal{E}_{R}^{n}$. Furthermore, pick an arbitrary
section $\sigma:A^{*}(\vee\\!S^{1})\to A^{*}X$ by extending the domain of
definition of differential forms to the interiors of the $2$-cells in $X$. By
construction, $\sigma$ is a section of the restriction map $i^{*}:A^{*}X\to
A^{*}(\vee\\!S^{1})$.

We build $s:\mathcal{E}_{R}\to Z$ by constructing a coalgebra homomorphism
$\bar{s}:\mathcal{E}_{R}\to\mathcal{E}{A^{*}X}$ landing in
$d_{\mathcal{E}}$-closed elements. Since $\mathcal{E}{A^{*}X}$ is cofreely
cogenerated by $A^{*}X$, such a map must be the cofree coextension of some
$\hat{s}:\mathcal{E}_{R}\to A^{*}X$. We use the weight filtration on
$\mathcal{E}_{R}$ to define $\hat{s}$. Begin with

$\hat{s}_{-1}:\mathcal{E}_{R}\xrightarrow{\
}{{\mathbb{E}}}{A^{*}(\vee\\!S^{1})}\xrightarrow{\;\pi\;}A^{*}(\vee\\!S^{1})\xrightarrow{\;\sigma\;}A^{*}X,$

and let $\bar{s}_{-1}$ be the cofree extension to a homomorphism into
$\mathcal{E}{A^{*}X}$. The induced homomorphism $\bar{s}_{-1}$ is indeed a
section of the restriction ${{\mathbb{E}}}i^{*}$, however it does not
necessarily send $\mathcal{E}_{R}$ to $d_{\mathcal{E}}$-closed elements. We
perform a modification at each step to get maps $\hat{s}_{n}$ whose cofree
extensions to homomorphisms $\bar{s}_{n}$ map $\mathcal{E}_{R}^{\leq n}$ to
$d_{\mathcal{E}}$-closed elements. We first consider this modification in
detail to get $\hat{s}_{0}$ from $\hat{s}_{-1}$ . Given
$v\in\mathcal{E}_{R}^{0}$, we calculate

$\Big{]}d_{\mathcal{E}}\bar{s}_{-1}(v)\Big{[}\;=(d_{\mathcal{E}}\otimes
I+I\otimes d_{\mathcal{E}})\circ(\bar{s}_{-1}\otimes\bar{s}_{-1})\ ]v[\;=0,$

since in $\mathcal{E}_{R}^{0}$ we have $]v[=0$. Because cobrackets are
injective in positive weight by Proposition 2.7 we deduce that
$d_{\mathcal{E}}\bar{s}_{-1}(v)$ has weight zero. By assumption,
$v\in\mathcal{E}_{R}$ has vanishing Hopf invariants on all $r\in R$ so by the
Circles and Disks Lemma 4.9,

$\int_{D^{2}_{r}}d_{\mathcal{E}}\bar{s}_{-1}(v)=\int_{S^{1}}r^{*}v=0,$

where $D^{2}_{r}$ is the two-disk attached by $r$. By the the relative
cohomology long exact sequence for $X$ and its one-skeleton, any cohomology
class which evaluates to zero on all $D^{2}_{r}$ must cobound. Thus the
$2$-form $d_{\mathcal{E}}\bar{s}_{-1}(v)\in A^{2}(X)$ has a cobounding
$\omega_{v}\in A^{1}(X,\vee S^{1})$ such that
$d\omega_{v}=d_{\mathcal{E}}\bar{s}_{-1}(v)$.

Pick a basis $\\{v_{i}\\}_{i\in I}\subset\mathcal{E}_{R}^{0}$ and define
$\hat{s}_{0}\colon V\to C^{1}X$ by linearly extending

$\hat{s}_{0}(v_{i})=\hat{s}_{-1}(v_{i})-\omega_{v_{i}}$

on $\mathcal{E}_{R}^{0}$ and $\hat{s}_{0}=\hat{s}_{-1}$ elsewhere. Since the
forms $\omega_{v_{i}}$ restrict to zero on $\vee S^{1}$, we have not changed
images after restriction $i^{*}\,\hat{s}_{0}=i^{*}\,\hat{s}_{-1}$, and
$\bar{s}_{0}(v)$ is now $d_{\mathcal{E}}$-closed by construction.

The general case is almost identical. Given $\hat{s}_{n}$ extending to
$\bar{s}_{n}$ with the property that $\bar{s}_{n}$ sends
$\mathcal{E}_{R}^{\leq n}$ to $d_{\mathcal{E}}$-closed elements, we first note
that $v\in\mathcal{E}_{R}^{n+1}$ must have weight zero, again by applying
injectivity of cobracket, now using the fact that tensor factors of $]v[$ have
weight $\leq n$ but $d_{\mathcal{E}}\bar{s}_{n}$ of such terms is zero. That
is,

$\Big{]}d_{\mathcal{E}}\bar{s}_{\leq n}(v)\Big{[}\;=(d_{\mathcal{E}}\otimes
I+I\otimes d_{\mathcal{E}})\circ(\bar{s}_{\leq n}\otimes\bar{s}_{\leq n})\
]v[\;=0$

Thus we can apply the Circles and Disks Lemma as before to find coboundings of
$d_{\mathcal{E}}\bar{s}_{n}(v)$ and use these to define $\hat{s}_{n+1}$ such
that $i^{*}\bar{s}_{n+1}=i^{*}\bar{s}_{n}$ and $\bar{s}_{n+1}$ of
$\mathcal{E}_{R}^{\leq(n+1)}$ is now $d_{\mathcal{E}}$-closed. ∎

###  4.3. Letter linking algorithms

There is a simple, quick algorithm which can be used to evaluate letter
braiding functions, from Section 1.2, on words. Rather than being recursive, a
single pass across the word suffices. To evaluate we store homogeneous symbols
as their corresponding tree where each node contains a d-function $f_{k}$ and
two variables $s_{k}$ and $\Delta_{k}$, initially set to 0. Evaluation is
computed letter-by-letter in the word, depth-first (away from the root) in the
symbol. For each letter $\ell_{i}$ and node $k$, the $s_{k}$ and $\Delta_{k}$
values are updated as follows.

  * •

First $s_{k}$ is incremented by the current $\Delta_{k}$ value, and
$\Delta_{k}$ is reset to 0.

  * •

Next the sum of all parents’ $s$-values is multiplied by the d-word value at
the given letter $f_{k}(i)$.

  * •

If letter $\ell_{i}$ is an inverse then this product is added to $s_{k}$.
Otherwise it is added to $\Delta_{k}$.

After evaluating over all nodes at all letters, the value of the braiding
symbol on the given word is the sum $s+\Delta$ at the root node of the
invariant. During the course of computation, at letter $i$ of the word and
node $k$ of the braiding symbol,
$s_{k}=\lfloor\lfloor\mathopen{}\Phi_{k}\rfloor\rfloor{{}_{\\!{}_{w}}}(i)$
where $\Phi_{k}$ is the subsymbol supported above node $k$. Thus the sequence
of values of $s_{k}$ give the d-word
$\lfloor\lfloor\mathopen{}\Phi_{k}\rfloor\rfloor{{}_{\\!{}_{w}}}$.

It is substantially more involved to generate letter braiding functions which
are well-defined on a group $G=\langle S\,|\,R\rangle$. Applying the Lifting
Criterion from Section 4.2 we know these are invariants on $F_{S}$ whose
iterated cobrackets all vanish on the relations in $R$. While the vanishing of
iterated cobrackets on relations seems like a large set of conditions to
check, it leads to fairly simple algorithms for generating letter linking
invariants up to a fixed weight.

At an abstract level, start in weight zero, where we recall that the indicator
homomorphisms $\\{S_{i}\\}_{i=1}^{m}$ are a basis for
$\text{Hom}(F_{S},\mathbb{Q})$. These evaluate on relations $r\in R$ by
counting (signed) occurrences of letters. Thus vanishing on a given relation
$r$ imposes a linear condition on the coefficients of $S_{*}$. Calculating the
kernel of all such conditions yields
$E_{R}^{0}=\text{Hom}(G,\mathbb{Q})\subset\text{Hom}(F_{S},\mathbb{Q})$, the
weight zero letter linking invariants for $G$.