In higher weights, the vanishing of cobrackets on relations comes into play.
Write $E_{R}^{\leq k}\subset{{\mathbb{E}}}^{\leq k}({F_{S}}^{\\#})$ for the
vector space of weight $\leq k$ letter linking invariants of $F_{S}$ which
lift to $G$ in the sense of Theorem 4.7. If $E_{R}^{\leq k}$ is known, then we
may compute the cobracket compatible vector subspace
$V_{k+1}\subset{{\mathbb{E}}}^{\leq k+1}({F_{S}}^{\\#})$ as the preimage of
the vector space $E_{R}^{\leq k}\otimes E_{R}^{\leq k}$ under the cobracket
(implicitly identifying ${{\mathbb{E}}}^{\leq k}\subset{{\mathbb{E}}}^{\leq
k+1}$). The next set of invariants, $E_{R}^{\leq k+1}$, is the vector subspace
of $V_{k+1}$ which vanishes on all relations, as calculated by the fast
evaluation algorithm above. Note that $V_{k+1}$ contains $V_{k}$ (as well as
all of ${{\mathbb{E}}}^{0}$), so $E_{R}^{\leq k}\subset E_{R}^{\leq k+1}$.

Implementations of this depend on choices for how these vector spaces are
defined and computed. The second author created one SageMath implementation
[Ozb23], using BCH expansions and his Formula (Corollary 4.3) to check
vanishing on relations. Given a word $r$ in the set of generators $S$, let
$BCH(r)$ be the BCH expansion of $r$, extending the BCH product:

$\mathrm{BCH}(a_{1}a_{2}\dots a_{k})\ =\ \log\bigl{(}e^{a_{1}}e^{a_{2}}\cdots
e^{a_{k}}\bigr{)}\in\mathbb{L}S.$

Write $\mathrm{BCH}_{k}(r)$ for the truncation of $\mathrm{BCH}(r)$ to include
only weight $\leq k$ terms, computing in the free $k$-step nilpotent Lie
algebra, $\mathbb{L}^{\leq k}S$. The code works in $(\mathbb{L}S)^{*}$ using
an previous implementation of Hall basis for $\mathbb{L}S$. If $b$ is a Hall
basis element, then $\bigl{\langle}b^{*},\mathrm{BCH}(r)\bigr{\rangle}$ is the
coefficient of $b$ in the BCH expansion of $r$. Thus, written in terms of a
Hall basis, $\mathrm{BCH}_{k}(r)$ is the matrix of coefficients of the BCH
expansion through weight $k$.

Using SageMath functions for Lie algebras and vector spaces, the code works as
follows.

  1. (1)

Make the vector space $(\mathbb{L}^{\leq 0}S)^{*}$.

  2. (2)

Construct the linear transformation $\mathrm{BCH}_{0}:(\mathbb{L}^{\leq
0}S)^{*}\to\mathbb{Q}^{n}$ evaluating on relations $\\{r_{1},\dots,r_{n}\\}$.

  3. (3)

Get the kernel of $\mathrm{BCH}_{0}$. This is $E_{R}^{0}$, the weight zero
letter linking invariants.

  4. (4)

Loop over the following steps.

    1. (a)

Expand the previous vector space from $(\mathbb{L}^{\leq k-1}S)^{*}$ to
$(\mathbb{L}^{\leq k}S)^{*}$ by including the next weight of (dual) Hall basis
elements.

    2. (b)

Make vector space $(\mathbb{L}^{\leq k-1}S)^{*}\otimes(\mathbb{L}^{\leq
k-1}S)^{*}$

    3. (c)

Make the cobracket linear transformation

$]-[:(\mathbb{L}^{\leq k}S)^{*}\longrightarrow(\mathbb{L}^{\leq
k-1}S)^{*}\otimes(\mathbb{L}^{\leq k-1}S)^{*}$

as the transpose of the $(k+1)$-nilpotent bracket transformation

$[-,-]:(\mathbb{L}^{\leq k-1}S)\otimes(\mathbb{L}^{\leq
k-1}S)\longrightarrow\mathbb{L}^{\leq k}S$

    4. (d)

Calculate the preimage of the vector subspace $E_{R}^{k-1}\otimes
E_{R}^{k-1}\subset(\mathbb{L}^{\leq k-1}S)^{*}\otimes(\mathbb{L}^{\leq
k-1}S)^{*}$. This is $V_{k}$.

    5. (e)

Make the linear transformation $\mathrm{BCH}_{k}:(\mathbb{L}^{\leq
k}S)^{*}\to\mathbb{Q}^{n}$

    6. (f)

Calculate the kernel of $\mathrm{BCH}_{k}$ restricted to $V_{k}$. This is
$E_{R}^{k}$.

There are many possible optimizations to make. We are interested in an
implementation which is more fully Lie coalgebraic, rather than relying on
existing implementations of Hall bases and working through duality. Such an
approach could lead to evaluation through the “one-pass” algorithm outlined at
the beginning of this section, rather than more intensive Baker-Campbell-
Hausdorff series.

###  4.4. The Fundamental Theorem

######  Theorem 4.10 (Universality statement of Theorem 1.2).

The Hopf pairing of $H^{0}_{\mathcal{E}}({A^{*}(X)})$ with the Malcev Lie
algebra of $\pi_{1}(X)$ is a a universal Lie pairing.

The Lifting Criterion immediately gives a way to deduce this theorem for
general groups from the free case. We start with the following refinement of
the calculation in Example 2.10.

######  Lemma 4.11.

Let $X$ be a wedge of circles indexed by a set $S$ and let $i_{s}\colon
S^{1}\hookrightarrow X$ denote the inclusion map of the circles with label
$s\in S$. Under the canonical isomorphism
$H^{0}_{\mathcal{E}}({X})\cong{{\mathbb{E}}}(\mathbb{Q}^{S})$ we have

$\int_{S^{1}}i^{*}_{s}(T)=\begin{cases}f(s)&\text{ if
}T=f\in\mathbb{Q}^{S}\text{ has weight $0$}\\\ 0&\text{
otherwise}.\end{cases}$

for every homogeneous expression $T\in{{\mathbb{E}}}(\mathbb{Q}^{S})$.

###### Proof.

A commutative cochain model for a wedge of circles is the square-zero algebra
$1\oplus\mathbb{Q}^{S}$, whose Harrison complex is
${{\mathbb{E}}}(\mathbb{Q}^{S})$ concentrated in degree $0$. The circle
inclusion $i_{s}$ is modeled at the cochain level by the algebra homomorphism

$A^{*}(X)\simeq 1\oplus\mathbb{Q}^{S}\longrightarrow 1\oplus\mathbb{Q}\simeq
A^{*}(S^{1})$

sending a function $f\in\mathbb{Q}^{S}$ to its evaluation $f(s)$. This
pullback induces a weight preserving map on Harrison complexes
$i_{s}^{*}\colon{{\mathbb{E}}}(\mathbb{Q}^{S})\to{{\mathbb{E}}}(\mathbb{Q})$.
But since the target is one dimensional and concentrated in weight zero, it
follows that all positive weight homogeneous elements are sent to $0$. ∎

###### Proof of universality (Theorem 4.10).

First, if $X$ is a wedge of circles indexed by a set $S$, then on the one hand
$\pi_{1}(X)\cong F_{S}$ the free group on basis $S$ and on the other hand its
Harrison cohomology is isomorphic to ${{\mathbb{E}}}(\mathbb{Q}^{S})$.

To see that the Hopf pairing is universal, let $\langle-,-\rangle_{C}\colon
C\times\mathrm{Mal}({F_{S}})\to\mathbb{Q}$ be any Lie pairing. Then the map
$\varphi_{0}\colon C\to\mathbb{Q}^{S}$ given by

$\varphi_{0}(T)\colon s\longmapsto\langle T,\log(s)\rangle_{C}$

extends uniquely to a Lie coalgebra homomorphism $\varphi\colon
C\to{{\mathbb{E}}}(\mathbb{Q}^{S})$ by cofreeness of the target. We show

$\langle\varphi(T),\ell\rangle_{H}=\langle T,\ell\rangle_{C}$

all $T\in C$ and $\ell\in\mathrm{Mal}({F_{S}})$. First, consider
$\ell=\log(s)$ for a generator $s\in S$. The previous lemma shows that the
Hopf pairing vanishes on homogeneous elements of positive weight in
${{\mathbb{E}}}(\mathbb{Q}^{S})$, so we have

$\langle\varphi(T),\log(s)\rangle_{H}=\langle\varphi_{0}(T)+(\text{higher
weight
terms}),\;\log(s)\rangle_{H}=\langle\varphi_{0}(T),\log(s)\rangle_{H}=\langle
T,\log(s)\rangle_{C}$

by definition of $\varphi_{0}$.

Now, suppose by induction that the equality holds for elements
$\ell_{1},\ell_{2}\in\mathrm{Mal}({F_{S}})$. Then

$\langle\varphi(T),[\ell_{1},\ell_{2}]\rangle_{H}=\langle]\varphi(T)[,\ell_{1}\otimes\ell_{2}\rangle_{H}=\langle(\varphi\otimes\varphi)]T[,\ell_{1}\otimes\ell_{2}\rangle_{H}=\langle]T[,\ell_{1}\otimes\ell_{2}\rangle_{C}=\langle
T,[\ell_{1},\ell_{2}]\rangle_{C}$

since both pairings are Lie pairings. This shows that equality holds on the
Lie algebra generated by $\log(s)$ for $s\in S$. Lastly, suppose $T$ has
weight $n$. The Baker–Campbell–Haudorff relation shows that every
$\ell\in\mathrm{Mal}({F_{S}})$ coincides with an element $\ell^{\prime}$ from
the latter Lie subalgebra up to $(n+1)$-fold nested brackets, so the two
elements have equal pairings with $T$. This completes the proof in the case
that $X$ is a wedge of circles.

Now, suppose $X$ is a two-complex with fundamental group $G$ and one-skeleton
$X^{1}$ a wedge of circles indexed by a set $S$. The inclusion $i\colon
X^{1}\hookrightarrow X$ induces a presentation $i_{*}\colon
F_{S}\twoheadrightarrow G$ with kernel normally generated by the attaching
maps of $2$-cells $r\in R$.

If $\langle-,-\rangle_{C}\colon C\times\mathrm{Mal}({G})\to\mathbb{Q}$ is a
Lie pairing, then it induces a Lie pairing with $F_{S}$ by precomposition with
$\mathrm{Mal}({i_{*}})\colon\mathrm{Mal}({F_{S}})\to\mathrm{Mal}({G})$. Thus,
by the already proved case of a wedge of circles there exists a unique Lie
coalgebra homomorphism $\varphi\colon C\to{{\mathbb{E}}}(\mathbb{Q}^{S})$
pulling back the Hopf pairing to $\langle-,-\rangle_{C}$.

But, since every relation $r\in R$ is killed by the projection $i_{*}$, the
image of $C$ pairs trivially with all relations. That is, the image
$\varphi(C)$ is contained in the subcoalgebra $E_{R}$ from Definition 4.6.
Since the lifting criterion in Theorem 4.7 shows that the pullback
$i^{*}\colon H^{0}_{\mathcal{E}}({A^{*}(X)})\to{{\mathbb{E}}}(\mathbb{Q}^{S})$
is an isomorphism onto $E_{R}$, and since $i^{*}$ is a coalgebra homomorphism
that preserves the Hopf pairing, we get a map

$C\xrightarrow{\varphi}E_{R}\xrightarrow{(i^{*})^{-1}}H^{0}_{\mathcal{E}}({A^{*}(X)})$

pulling back the Hopf pairing to $\langle-,-\rangle_{C}$. Uniqueness of this
map follows from the fact that after composition with $i^{*}$ we must get back
the uniquely determined map $\varphi$, but $i^{*}$ is injective. ∎

In the course of proof we obtain an independent calculation that the Malcev
Lie algebra of a free group is a free Lie algebra. Moreover, as a corollary of
this, Theorem 3.2, and Quillen’s theorem comparing lower central series and
Malcev Lie algebras, we get a topological proof of the main theorem of [MS22],
that letter-linking invariants obstruct the lower central series of free
groups. We can now prove our main combinatorial application.

###### Proof of Theorem 1.10.

The theorem is phrased in terms of braiding functions, so let us first
translate the statement into one regarding Hopf invariants. From Section 3.2
we know that the combinatorially defined braiding functions agree with Hopf
invariants of some realization form. Moreover, braiding functions clearly span
all Hopf invariants for the free group, as seen e.g. by the fact that the
Harrison spectral sequence for the handlebody collapses to
${{\mathbb{E}}}(F_{S}^{\\#})$. Finally, by the lifting criterion in Theorem
4.7, a Hopf invariant of $F_{S}$ descends to $G$ if and only if it belongs to
the Lie dual of $G$. So it suffices to prove that the image of $w\in F_{S}$ in
$G$ has a power that is a product of $k$-fold nested commutators if and only
if all weight $\leq k-1$ elements in the Lie dual of $G$ vanish on $w$.

Let $\gamma_{k}G$ denote the lower central of $G$, that is $\gamma_{1}G=G$ and
$\gamma_{k+1}=[\gamma_{k}G,G]$, and analogously for $\mathrm{Mal}({G})$.
Quillen proved [Qui68, Equation (1.5)] that the natural map $\log\colon
G\to\mathrm{Mal}({G})$ induces an isomorphism

$\bigoplus_{k\geq
1}\nicefrac{{\gamma_{k}G}}{{\gamma_{k+1}G}}\underset{\mathbb{Z}}{\otimes}\mathbb{Q}\to\bigoplus_{k\geq
1}\nicefrac{{\gamma_{k}\mathrm{Mal}({G})}}{{\gamma_{k+1}\mathrm{Mal}({G})}}.$

It follows that some power of $x\in G$ lies in $\gamma_{k+1}G$ if and only if
its logarithm lies in $\gamma_{k+1}\mathrm{Mal}({G})$.

It remains to show that if $E$ is a Lie dual of $G$ then
$\gamma_{k+1}\mathrm{Mal}({G})$ is exactly the kernel of the evaluation on the
weight $\leq k$ elements of $E$. The fact that such evaluations are trivial
follows immediately from the bracket-cobracket duality: if $T\in E$ has weight
$\leq k$ then its ${k+1}$-fold iterated cobracket is zero, which by duality
implies vanishing on $k+1$-fold nested commutators.

For the converse, suppose $\log(w)$ is not in $\gamma_{k+1}\mathrm{Mal}({G})$.
Then it represents a nontrivial element in the finite dimensional Lie algebra
$L=\mathrm{Mal}({G})/\gamma_{k+1}\mathrm{Mal}({G})$. Its linear dual $L^{\\#}$
is a Lie coalgebra whose elements all vanish after $k+1$ iterations of the
cobracket, with an obvious pairing with $\mathrm{Mal}({G})$. Furthermore, by
linear duality it contains a functional $t$ that does not vanish on $\log(w)$.
By the universal property of the Lie dual, there exists a coalgebra map
$\varphi\colon L^{\\#}\to E$ that pulls back the Lie pairing. But then

$0\neq\langle t,\log(w)\rangle_{L^{\\#}}=\langle\varphi(t),\log(w)\rangle_{E}$

exhibiting $T:=\varphi(t)$ as an element of weight $\leq k$ on which $\log(w)$
does not vanish. ∎

##  5\.  Examples

###  5.1. Orientable surfaces and our favorite little two-complex

The torus provides a counterpoint to the free group setting. The standard
model for the torus is a differential graded commutative algebra on generators
$A$ and $B$ in degree one, with non-trivial product $P$. The Harrison complex
for this algebra has homology only given by $A$ and $B$ themselves in weight
zero, and of course these span the dual of the group. All other cochains in
degree zero have a basis whose boundaries are linearly independent. That
higher Hopf invariants are trivial in the abelian setting makes sense as they
arise from derived abelianization.

Surfaces provide an opportunity to see both geometry and algebra. We use Thom
classes of curves, as developed in Appendix B.1, but in this case without a
need to thicken a two-complex to obtain a manifold. We picture the genus two
case below, with “dual” curves $A$, $B$, $C$ and $D$ and choosing associated
Thom forms of the same name as generators of first cohomology. While the
products of pairs of generators are equal in cohomology, we have $A\cdot
B-C\cdot D=dS$ at the cochain level.

In weight zero, Harrison cocycles are $A,B,C$ and $D$. In weight one,
$\left\lfloor A\right\rfloor{C}$ and similar involving pairs of cochains with
zero product at the cochain level are all cocycles. Following the analysis of
Section 3.1, the Hopf invariant associated to $\left\lfloor A\right\rfloor{C}$
would be the count of when a loop crosses curve $C$ in between crossings with
opposite signs of $A$.

Another Harrison cocycle is $\omega=\left\lfloor
A\right\rfloor{B}-\left\lfloor C\right\rfloor{D}-S$. The corresponding
counting invariant would be the sum of the count (with signs) of occurrences
where a curve crosses $B$ in between crossings with $A$, occurrences of
$D$-crossings in between $C$’s, and crossings with $S$. A simple homotopy of a
curve “from left to right” across the octagon would result in this count to
have contribution from different parts of the sum.

If we restrict loops to the one-skeleton, and simple homotopies across the
2-cell, we obtain the standard generators and relations presentation of this
group. We see that because $[a,b][c,d]=1$ the number of $b$’s between
$a-a^{{\text{-}}1}$ pairs and that of $d$’s between $c-c^{{\text{-}}1}$ pairs
can both change individually, but their difference will be invariant. This
difference, represented by the braiding symbol
$\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B-\lfloor\lfloor\mathopen{}C\rfloor\rfloor D$, is the letter-linking
representative of the Hopf invariant associated to $\omega$. From this
algebraic perspective, there is no evidence of a “correction curve” $S$, which
topologically is disjoint from the one-skeleton. In general, model the
cochains of the genus-two surface by the commutative differential graded
algebra generated by $\\{A,B,C,D\\}$ in degree one with all products zero
except $AB=CD$. We define the standard involution on this algebra to be the
one switching $A$ with $C$ and $B$ with $D$. To calculate the homology of
$\mathcal{E}(A)$ we find a suitable basis for ${{\mathbb{E}}}(A)$, namely
graphs whose symbols do not contain the following as sub-symbols:

$\big{\lfloor}\lfloor A\rfloor{B}\big{\rfloor}\,{A},\;\;\big{\lfloor}\lfloor
B\rfloor{A}\big{\rfloor}\,{B},\;\;\big{\lfloor}\lfloor
A\rfloor{B}\big{\rfloor}\,{C},\;\;\big{\lfloor}\lfloor
B\rfloor{A}\big{\rfloor}\,{D},$

as well as subgraphs obtained from these by applying the standard involution.
We call such graphs admissible. That admissible graphs span
${{\mathbb{E}}}{A}$ is an immediate application of the Arnold identity. In
such graphs, any edge connecting an $A$ to $B$ is isolated, as are those
connecting a $C$ to a $D$. Moreover, by eliminating graphs such as
$\big{\lfloor}\lfloor A\rfloor{B}\big{\rfloor}\,{C}$, this isolation remains
when we apply the standard involution to an $\left\lfloor A\right\rfloor{B}$
edge. Calculation in such an admissible basis is immediate, with cocycles
given by linear combinations of $2^{k}$ terms obtained from each admissible
graph with $k$ edges of the form $\left\lfloor A\right\rfloor{B}$ or
$\left\lfloor C\right\rfloor{D}$ by applying the standard involution to those
edges.

This calculation speaks to the strength of the Eil-graph approach to Lie
coalgebras, as we think it would be difficult to find a suitable basis through
bar elements (long graphs) modulo shuffles. It also speaks to the clarity that
the graphical approach gives rather than just working with symbols, as for
example
$\big{\lfloor}\big{\lfloor}\mathopen{}\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B-\lfloor\lfloor\mathopen{}C\rfloor\rfloor D\rfloor\rfloor B$ is a well-
defined letter linking invariant but
$\big{\lfloor}\big{\lfloor}\mathopen{}\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B-\lfloor\lfloor\mathopen{}C\rfloor\rfloor D\rfloor\rfloor A$ is not.

Higher genus surfaces behave similarly. Another similar example is for the
group $G=\langle a,b,c|aba^{{\text{-}}1}c=id\rangle$, the fundamental group of
a two-complex $X$ homeomorphic to a cylinder with a pair of points on its
boundary identified. It is homotopy equivalent to a wedge of two circles, so
$G$ is isomorphic to a free group on two generators, say $a$ and $b$. Let $A$,
$B$ and $C$ be indicator Thom cochains associated to curves and $P$ a
compatible Thom cochain of a point as pictured below.

In weight zero $A$ and $B-C$ generate the cohomology of $X$. In weight one
$\left\lfloor A\right\rfloor\,{(B-C)}-C$ is a Harrison cocycle. It pulls back
to a cocycle of the same name on the one-skeleton, to which we associate the
letter linking invariant $\lfloor\lfloor\mathopen{}B-C\rfloor\rfloor A-C$. In
group theory, the count of $a$’s between collections of cancelling $b$’s and
$c$’s can change under the relation, increasing by one, but as it does so does
the number of $c$’s. One could also choose $\left\lfloor
A\right\rfloor\,{(B-C)}-B$ as a representative, differing by $B-C$ which is a
cocyle of lower weight.

In higher weights the story is conceptually simple but the combinatorics can
be complicated. Consider the cofree Lie coalgebra generated by $A,(B-C)$. The
Hopf invariants for $G$ must be isomorphic to this algebra. Accordingly, we
claim that any corresponding cochain can be completed to a cocycle. This is
because the Harrison differential introduces products, realized by Thom forms
of points, but these can be cobounded by cochains in for which one replaces
such points by parallel copies of $B$ or $C$. In some cases this replacement
results in further products and thus correction terms needed in lower weights,
so the correspondence between Hopf invariants for $G$ and those for the free
group on two generators is not tautological.

###  5.2. Milnor invariants for Borromean rings

Application to Milnor invariants has been a prime motivation, as first
discussed in Section 1.3. We return to these in light of our general theory.

Consider the Borromean rings, with two presentations given below.

Figure 6. Two presentations of the Borromean rings

We consider the component $L_{0}$ as defining an element in the fundamental
group of the complement of the other components,
$X=\mathbb{R}^{3}-\\{L_{1}\cup L_{2}\\}$. For brevity of discussion we ignore
basepoints and orientations. We again use Thom forms (Appendix B.1).

In the first presentation we assume $L_{1}$ and $L_{2}$ are planar circles,
and let $S_{1}$ be the planar disk which bounds $L_{1}$ and similarly for
$S_{2}$. Because these are disjoint, the associated Thom forms $\tau_{1}$ and
$\tau_{2}$, which form a basis for first cohomology of $X$, define a Harrison
cocycle $\left\lfloor\tau_{1}\right\rfloor{\tau_{2}}$. With appropriate
parametrizations, pull-back of this to $L_{0}$ is a Harrison cocycle on the
circle $\left\lfloor\alpha\right\rfloor{\beta}$ from Section 3.1, pulled back
from a commutator on a handlebody. Thus the Hopf invariant will be one, counts
the intersections of $L_{0}$ with $S_{2}$ which occur between its
intersections with $S_{1}$. On $L_{0}$ this evaluates as $\pm 1$.

Following Cochran [Coc90] we make an alternate choice $S_{1}^{\prime}$ for the
first Seifert surface by removing small disks at the intersection points of
$L_{0}$ with $S_{1}$ and replacing them by a cylinder $S^{1}\times[0,1]$ which
is a unit normal sphere to $L_{0}$ on one side, as pictured. There are two
things which change when using $S_{1}^{\prime}$ in place of $S_{1}$. First,
$L_{0}$ does not intersect $S_{1}^{\prime}$. Secondly,
$\left\lfloor\tau_{1}^{\prime}\right\rfloor\,{\tau_{2}}$ is not a Harrison
cocycle on $X$, since $S_{1}^{\prime}\cap S_{2}$ is a small circle around an
intersection point $p$ of $L_{0}$ with $S_{2}$.

To complete it we can choose any surface $S_{12}$ which cobounds this circle –
a small disk around $p$, or its complement in $S_{2}$, or for example a
cobordism between this circle and $L_{1}$ – and then with appropriate
modifications its Thom form $\tau_{12}$ will complete a Harrison cocycle
$\left\lfloor\tau_{1}^{\prime}\right\rfloor\,{\tau_{2}}-\tau_{12}$. Whatever
choice for $\tau_{12}$ will yield the same value of $\pm 1$ when evaluated on
$L_{0}$, as this will be the linking number of $L_{0}$ with the circle
$S_{1}\cap S_{2}$. Indeed, Cochran’s perspective is that this form of the
Milnor invariant is realized as a “derivative” linking invariant. We can also
modify $S_{2}$ to eliminate its intersections $L_{0}$, but by “nesting” the
cylinder added for that purpose with the existing one, the intersection of
$S_{1}^{\prime}$ and $S_{2}^{\prime}$ will be the same as that of
$S_{1}^{\prime}$ and $S_{2}$, so the analysis from there will coincide.

Next, we work with a standard diagrammatic presentation, given on the right of
Figure 6. Following Seifert’s original algorithm suppose each component to be
planar except on some arbitrarily small intervals near some crossings, with
the $L_{0}$ plane lower than that of $L_{1}$ which is in turn lower than that
of $L_{2}$. Thus where $L_{1}$ over-crosses of $L_{2}$ it must “rise up,”
which can be assumed to happen in an arbitrarily small neighborhood of the
crossing.

We choose “mostly flat” Seifert surfaces $S_{1}$ and $S_{2}$ which are
disjoint except near these crossings, where they intersect because $S_{1}$
goes from being under $S_{2}$ to over it. The intersections are small curves
with one endpoint on $L_{1}$ and the other on $L_{2}$, which in projection are
arbitrarily close to the crossing points. Thus the associated Thom forms
$\tau_{1}$ and $\tau_{2}$ have wedge product which is a Thom form for the
union of these two curves.

To complete a Harrison cocycle we find a cobounding surface for this
intersection. Once again using the projection as faithfully as possible, we
can take the closed loop $L^{\prime}$ consisting of these two curves along
with a portion $L_{1}$ between these crossings, and a portion of $L_{2}$
between them, say the shorter portions for definiteness. We cobound this
closed loop with a disk $S_{12}$ which agrees with $S_{2}$ except near the
crossings and near the $L_{1}$ part of the boundary where it must “fall
downward”. With these new definitions, our Harrison cocycle has the same
formula as above.

By construction, since $L_{0}$ is generally below all of this, the
intersection of $L_{0}$ with these surfaces would occur only at crossings of
$L_{0}$ over $L_{1}$ or $L_{2}$. But $L_{0}$ only crosses under $L_{2}$. Thus
the pullback of $\left\lfloor\tau_{1}\right\rfloor{\tau_{2}}$ to the circle
parametrizing $L_{0}$ will be zero, since the pullback of $\tau_{2}$ is zero.
The $\tau_{12}$ term evaluates as an intersection count of $L_{0}$ with
$S_{12}$, or equivalently a linking number with the “derived” link component
$L^{\prime}$. There is one such intersection point namely near the
$L_{0}$-over-$L_{1}$ crossing which is on the boundary of $S_{12}$.

Thus, in the setting of links, one can work closely from a diagram, or look to
minimize some types of intersections, or to make use of some geometric
structure, giving different approaches to the same set of invariants. As
discussed in Section 1.4.3, part I, we plan to develop this further in a
sequel.

###  5.3. Beyond letter linking

Letter linking is by definition an obstruction theory, as the invariants
require vanishing to be defined. But Proposition 2.11 shows that all Harrison
cocycles on the circle can be weight reduced. Following this through produces
surprisingly rich combinatorics, and also motivated the first author to take
weight reduction seriously on the classical bar complex [Gad23].

We recall the notation of Section 3.1 and calculate the Hopf pairing of the
cocycle $\left\lfloor A\right\rfloor{B}$ and the word $ab$ in the free group
as the fundamental group of the handlebody $B_{2}$. For definiteness,
parametrize so that the curve defining $ab$ passes through the $D_{a}$ at
$t=\frac{1}{3}$ and $D_{b}$ at $t=\frac{2}{3}$. Then $\left\lfloor
A\right\rfloor{B}$ pulls back to
$\left\lfloor\omega_{\frac{1}{3}}\right\rfloor\,{\omega_{\frac{2}{3}}}$.

Neither entry of this Harrison cochain cobounds. But we can still weight
reduce by taking advantage of the fact that by definition any weight one
cochain with repeated entries is zero. Let

$I(t)=\int_{0}^{t}\omega_{\frac{1}{3}}-\omega_{\frac{2}{3}},$

so that $dI=\omega_{\frac{1}{3}}-\omega_{\frac{2}{3}}$. Then

$d_{\mathcal{E}}\left(\left\lfloor
I\,\right\rfloor\,{\omega_{\frac{2}{3}}}\right)=\left\lfloor\omega_{\frac{1}{3}}\right\rfloor\,{\omega_{\frac{2}{3}}}-\left\lfloor\omega_{\frac{2}{3}}\right\rfloor\,{\omega_{\frac{2}{3}}}-I\cdot{\omega_{\frac{2}{3}}}.$

As the term
$\left\lfloor\omega_{\frac{2}{3}}\right\rfloor\,{\omega_{\frac{2}{3}}}$ is
zero, this exhibits a weight reduction of our orignial pullback to the weight
zero cochain $I\cdot{\omega_{\frac{2}{3}}}$. This then can be evaluated to
$\frac{1}{2}$, as predicted by Aydin’s Formula, Corollary 4.3.

###  5.4. Commutators and Massey products

Our theory extends the time-honored connection between vanishing of Massey
products and homomorphisms out from the fundamental group. Consider
$G=\bigl{\langle}a,b,c\ \bigl{|}\;\bigl{[}[a,b],c\bigr{]}=id\bigr{\rangle}$.
We again realize $G$ as the fundamental group of a two-complex with one cell,
and employ Thom cochains for dual curves as pictured.

As discussed in Appendix B.3, the cochains $A$, $B$ and $C$ map to
corresponding cocycles on the $1$-skeleton, and they are themselves cocycle
lifts. Moreover $\left\lfloor A\right\rfloor{B}$ lifts to $\left\lfloor
A\right\rfloor{B}-S$, $\left\lfloor A\right\rfloor{C}$ lifts to $\left\lfloor
A\right\rfloor{C}$ with no correction term needed and similarly for
$\left\lfloor B\right\rfloor{C}$. The Harrison cochain $\big{\lfloor}\lfloor
A\rfloor{B}\big{\rfloor}\,{C}$ however does not complete to a cocycle. It does
partially complete to $\big{\lfloor}\lfloor
A\rfloor{B}\big{\rfloor}\,{C}-\left\lfloor S\right\rfloor{C}$, whose total
differential is in weight zero, but that differential is a Thom cochain of the
intersection point of $S$ and $C$, which does not cobound an available cochain
in weight zero of the Harrison complex.

The value of this total differential on the fundamental class of two-complex
is one, as there is one intersection point, which coincides with the value of
the letter linking invariant
$\big{\lfloor}\big{\lfloor}\mathopen{}\lfloor\lfloor\mathopen{}A\rfloor\rfloor
B\rfloor\rfloor C$ associated to $\big{\lfloor}\lfloor
A\rfloor{B}\big{\rfloor}\,{C}$ on the defining relation
$r=\bigl{[}[a,b],c\bigr{]}$. Such an equality in general is proven in Lemma
4.9. We can see in this example how the process of completing curves to define
Thom forms and intersecting them parallels the cobounding and intersecting of
letter linking. For example, the $A$ curves each are “parallel to” intervals
between $a-a^{{\text{-}}1}$ pairs on the boundary, and their intersections
with $B$ curves can be traced back to occurrences of $b$’s between those
pairs. In other words, if we restrict the extensions of these curves from the
boundary to the two-disk to lie in an epsilon neighborhood of the boundary,
their combinatorics matches that of letter linking invariants. We originally
pursued proofs of the lifting Theorem 4.7 which codified this.

This weight zero differential is also immediately recognizable as a Massey
product, namely $\langle A,B,C\rangle=SC$, in the cohomology of this two-
complex. Indeed, as we discuss in Remark 2.14 and Section A.1, Massey products
define differentials into the weight zero column in the Harrison spectral
sequence. Massey products then tautologically obstruct lifting from the
Harrison complex of the one-skeleton of a CW-complex. But instead of
formulation in terms of Massey products, Lie coalgebraic formalism is the
appropriate language for our structural theorems, in particular the clean
formulation of Theorem 4.7.

###  5.5. Pure braid groups

We apply machinery of the Harrison complex to the space intimately tied to its
definition. Namely, the space ${\rm Conf}_{k}(\mathbb{R}^{2})$ of
configurations of $k$ distinct labelled points in the plane is a manifold
whose fundamental group is the pure braid group and whose $(k-1)$st cohomology
group is isomorphic to ${\mathcal{E}\\!{\mathit{i}l\/}(n)}$ (up to “signs”).
This Harrison complex consists of Eil-graphs decorated by Eil-graphs, though
for sanity we use the more traditional presentation of cohomology as generated
by $\alpha_{ij}=\alpha_{ji}\in H^{1}{\rm Conf}_{k}(\mathbb{R}^{2})$ modulo the
Arnold identity.

At the cochain level $a_{ij}$ be the Thom form associated to the submanifold
defined by having point $i$ over point $j$, so $x_{i}-x_{j}$ is a positive
scalar multiple of $\begin{pmatrix}0\\\ 1\end{pmatrix}$. We (co)orient this
submanifold so that the “positive” normal direction is when point $i$ is
slightly to the left. Then for example the wedge product $a_{ij}\wedge a_{jk}$
will be a Thom form for the submanifold where point $i$ is above $j$ which is
above $k$.

Consider the codimension one submanifold with boundary $Ar_{ijk}$ defined by
having point $i$ over point $j$ and point $k$ with $x$-coordinate which is
greater than or equal to the shared $x$-coordinate of points $i$ and $j$. The
boundary is when the $x$-coordinate of $k$ is equal to that $x$-coordinate.
There are naturally three components, depending on whether point $k$ is over
$i$, in between, or below $j$. Thus $Ar_{ijk}$ cobounds the Arnold identity:

$dAr_{ijk}=a_{ki}\wedge a_{ij}+a_{ik}\wedge a_{kj}+a_{ij}\wedge a_{jk}.$

Immediately we can construct a Harrison cocycle

$\left\lfloor a_{ki}\right\rfloor{a_{ij}}+\left\lfloor
a_{ik}\right\rfloor{a_{kj}}+\left\lfloor
a_{ij}\right\rfloor{a_{jk}}-Ar_{ijk}.$

By bracket-cobracket compatibility, it pairs non-trivially on the commutator
$[b_{ki},b_{ij}]=[b_{ik},b_{kj}]=[b_{ij},b_{jk}].$ (This equality of
commutators is the defining relation of a Drinfeld-Kohno Lie algebra .)

The corresponding Hopf invariant will have four terms, which we interpret in
terms of standard braid diagrams. The $Ar_{ijk}$ term counts $i$-over-$j$
crossings, but only those for which strand $k$ is to the right. Weight
reduction on $\left\lfloor a_{ki}\right\rfloor{a_{ij}}$ will yield a count for
example of $a_{ki}$ between canceling pairs $a_{ij}^{\pm 1}$. Geometrically,
in this setting strands $i$ and $j$ are not linked, but strand $k$’s being
counted are “coming between them” and so possibly linking.

###  5.6. An infinite example

Consider the group generated by $y$ and $x_{i},i\in\mathbb{N}$ with
$[x_{2i-1},x_{2i}]=y$ for all $i$. Such a group is the fundamental group of a
countable union of genus one surfaces with one boundary component, with all
such boundary components identified with each other. We consider closed dual
curves as in Section 5.1 (with the genus two surface being the $n=2$ version
of this infinite complex), so that $X_{2i-1}$ intersects $X_{2i}$ in a point
$p_{i}$ and otherwise these curves are disjoint.

Possibly infinite unions of such curves are define proper submanifolds of the
thickening of this complex and thus Thom cochains. These realize the first
cohomology group, which is an infinite product rather than sum.

Harrison cocycles $\lfloor X_{i}\rfloor\,{X_{j}}$, for $i,j\neq 2k,2k-1$ for
any $k$, are a dual basis to the corresponding $[x_{i},x_{j}]$. But $y$ is
detected by $\lfloor\bigcup X_{2i-1}\rfloor\,{\bigcup X_{2i}}$, which also
pairs with any $[x_{2i-1},x_{2j}]$. This Harrison cocycle is not a finite sum
of more basic such. Key here is that the first cohomology is already a direct
product so infinite sums can be “embedded” in the standard Harrison complex.

###  5.7. Beyond residual nilpotence using the product totalization

First recall from Definition 2.8 that the Harrison complex is defined as the
standard total complex of a bicomplex $\mathcal{E}(A,\ \mu_{A},\ d_{A})$.
There is an alternative totalization of a complex ${\rm Tot}_{\Pi}$ defined
using direct product in each total degree, from which we get a variant of our
theory.

######  Definition 5.1.

Define the completed Harrison complex as the direct product totalization

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

We will not fully develop a theory based on the completed Harrison complex
here, but do give an example to show that it detects phenomena beyond the
residually nilpotent setting, which has been a universal limitation for group
theory through computational algebraic topology.

Consider the group generated by $x_{i}$, $i\in{\mathbb{N}}$ with relations
$x_{2i-1}=[x_{2i},x_{2i+1}]$. Thus any $x_{2i-1}$ is in the intersection of
all commutator subgroups. We realize this as the fundamental group of the
quotient of the union $\bigsqcup_{\mathbb{N}}\Sigma_{i}$, where each
$\Sigma_{i}$ is a genus one surfaces with one boundary component, by
identifying the boundary of $\Sigma_{i+1}$ with the longitude of $\Sigma_{i}$,
as in the theory of gropes. Explicitly, we build this two-complex as a union
of pentagons as pictured on the left below. By abuse we use $x_{i}$ to
decorate edges, which under identifications are loops, though for the
fundamental group paths from the basepoint must be given.

We use Thom cochains for a fattened four-manifold homotopy equivalent to this
two-complex as in Appendix B.1. Though we normally prefer to use capitalized
versions of generator names for dual cochains, for clarity let $Y_{2i+1}$ be
the dual curve to $x_{2i+1}$ in $\Sigma_{i}$, a segment between midpoints of
the $x_{2i+1}$ edges in the pentagon preimage of $\Sigma_{i}$, and similarly
for $Y_{2i}$, as pictured on the right above. Let $Z_{2i-1}$ be the segment
between the midpoint of $x_{2i-1}$ and the intersection point of $Y_{2i}$ and
$Y_{2i+1}$.

With appropriate choices of Thom forms, the infinite sum

$Y_{1}+\left\lfloor
Y_{2}\\!\right\rfloor\,{Y_{3}+Z_{3}}+\Big{\lfloor}\big{\lfloor}Y_{2}\big{\rfloor}\,{Y_{4}}\Big{\rfloor}\,{(Y_{5}+Z_{5})}+\cdots$

will be a cocycle in the product Harrison complex which will evaluate non-
trivially on $x_{1}$. Algebraically, the corresponding letter linking count is
“sensitive to infinitely many possible subwords”, but is well defined since
only finitely many of them could occur in a word.

This infinitary construction echos other recent developments in non-simply
connected homotopy theory. As discussed in Section 1.4.3, Rivera–Zeinalian
[RZ21] and Rivera–Medina [MMR23] show that the cobar construction on singular
chains of a space faithfully encodes the fundamental group, but in a way that
does not lend itself to calculation because of lack of quasi-isomorphism
invariance and convergent spectral sequences. The completed Harrison complex
might serve as a bridge to theories such as these, with some ability for
calculations by hand while being able to see beyond traditional nilpotence
restrictions.

##  Appendix A Bar and Harrison complexes

###  A.1. Basics and connection with Sullivan models

Bar constructions are standard in homological algebra of modules, but are less
familiar in the context of algebras, for functors such as indecomposibles. We
share this perspective, which is key to understanding the Harrison complex and
its relationship to the well-known Sullivan models of rational homotopy theory
(from which we have borrowed our title).

######  Definition A.1.

Let $(A,\ \mu_{A},\ d_{A})$, be an associative differential graded algebra,
with augmentation ideal $\bar{A}$.

The classical Bar complex denoted $\mathcal{B}(A)$ is the second-quadrant
bicomplex for which the $-n$th column is $(\bar{A})^{\otimes n+1}$, with
elements denoted $a_{0}|a_{1}|\cdots|a_{n}$. The vertical differential, on
$(\bar{A})^{\otimes n+1}$, is $d_{A}$, extended by the Leibniz rule (with
Koszul signs reflecting a desuspension of every tensor factor of $\bar{A}$).
The horizontal differential is the alternating sum of $\mu_{A}$ on consecutive
factors, sending

$a_{0}|a_{1}|\cdots|a_{n}\;\;\longmapsto\;\;a_{0}a_{1}|a_{2}|\cdots|a_{n}\;-\;a_{0}|a_{1}a_{2}|a_{3}\cdots|a_{n}\;\pm\cdots\pm\;a_{0}|a_{1}|\cdots|a_{n-1}a_{n}.$

The spectral sequence of a bicomplex applies to give the Bar spectral
sequence, a standard organizational tool. Taking homology with respect to the
vertical differential first, the spectral sequence has $E_{1}$ given by
$\mathcal{B}(H_{*}(A))$. The first differential reflects product structure,
for example leaving the indecomposibles of $H_{*}(A)$ in the zero column at
the $E_{2}$-page. Higher differentials reflect Massey products, and indeed
this can be taken as a definition of Massey products. For example, if $ab=dx$
and $bc=dy$ then

$\alpha=a|b|c\;\mp\;a|y\;\pm\;x|c$

will have total Bar differential $d(\alpha)=\mp ay\pm xc$, which represents
the Massey product $\langle a,b,c\rangle$. This will give rise to a non-zero
$d_{2}$-differential in the Bar spectral sequence if and only if the Massey
product is non-zero modulo standard indeterminacy.

The classical Bar complex arises in topology in a few ways. If $A$ is a group
ring, it computes group homology with trivial coefficients. If $A$ is the
cochain algebra of a simply connected space, it models cochains on the loop
space. There are many perspectives for this latter fact, with two relevant for
the present theory. One is that if $a_{i}$ are Thom forms as in Appendix B.1
or geometric cochains, then $\tau_{W_{0}}|\cdots|\tau_{W_{n}}$ essentially
counts intersections of a loop (within a family of loops defining a smooth
chain on the loopspace) with the $W_{i}$, in order. If $W_{i}$ has boundary,
this count can change through a homotopy of a loop or more generally a
homology of a cycle of such loops. The count can also change if the $W_{i}$
intersect, as we saw in discussion of the Whitehead link in Section 1.3. These
two possible ways this count of ordered intersections can change correspond to
the vertical and horizontal differentials in the Bar complex, respectively.

Another perspective is that of defining invariants on homotopy groups through
cohomology. We let cohomology evaluate on $f:S^{n}\to X$ by pulling back and
integrating cohomology classes. If a cohomology class is a non-trivial
product, its pull-back will be zero in the cohomology of the sphere. Thus
evaluation of cohomology on homotopy factors through the indecomposibles of
cohomology. Hopf invariants can be viewed as a derived version of this, in
which case a key fact is that the Bar complex computes derived
indecomposibles. Recall that by the Leibniz role, the differential on an
associative differential graded algebra passes to its indecomposibles.

######  Theorem A.2.

Let $F$ be an associative differential graded algebra which is quasi-
isomorphic to $A$ and whose underlying algebra is a free (tensor) algebra.
Then the complex of indecomposibles of $F$ is quasi-isomorphic to
$\mathcal{B}(A)$.

###### Proof.

We first apply the basic fact that the Bar complex is quasi-isomorphism
invariant to replace $\mathcal{B}(A)$ by $\mathcal{B}(F)$. Then we show that
$\mathcal{B}(F)$ is quasi-isomorphic to the indecomposibles of $F$.

To do so, identify $F$ as a the free associative algebra on
$x_{1},x_{2},\cdots$, in which case $F$ has a monomial basis which determines
one on $\mathcal{B}(F)$. Define a chain homotopy $P$ on $\mathcal{B}(F)$ by
sending a monomial basis element $m_{0}|m_{1}|\cdots|m_{n}$ to either $0$ if
$m_{0}$ is a generator $x_{i}$ for some $i$ or otherwise, when $m_{0}=\prod
x_{i_{k}}$, to $x_{i_{1}}|m_{0}^{\prime}|m_{1}|\cdots m_{n}$, where
$m_{0}^{\prime}=\prod_{k\geq 2}x_{i_{k}}$. Immediate calculation shows that
$P$ is a homotopy between the identity and the zero map except on the
indecomposibles of $F$, through the inclusion of $A$ as the zeroth column,
modulo the first (product) differential. ∎

For example, if $x\cdot y=0$ in some associative algebra (with zero
differential) then $x|y$ will be a cycle in $\mathcal{B}(A)$. In some quasi-
isomorphic $F\overset{\simeq}{\to}A$ sending $\hat{x}\mapsto x$ and
$\hat{y}\mapsto y$ we could have $\hat{x}\cdot\hat{y}=dz$ for some $z$. Then
$z$ would be the indecomposible of $F$ corresponding to the original cycle
$x|y$. Indeed in $\mathcal{B}(F)$ we have the lifted cycle
$\hat{x}|\hat{y}\;\mp\;z$, with the original trivial product and its
corresponding derived indecomposible both accounted for.

If $A$ is commutative, then there are some canonical cocycles in
$\mathcal{B}(A)$, starting with $a|b\;\mp\;b|a$. Reflective of this, this bar
complex does not represent derived indecomposibles when restricted to the
commutative setting (replacing the free associative algebra $F$ in the
previous theorem by a symmetric algebra). Harrison’s original construction
[Har62] is to quotient $\mathcal{B}(A)$ by shuffle relations to obtain a
complex which does compute the commutative derived indecomposables.

In [SW11, Definition 2.14], we use directed, acyclic, connected graphs with
formal span denoted $\mathcal{G}\\!{\mathit{r}\/}(n)$ to build
$\mathcal{E}\\!{\mathit{i}l\/}(n)$, a model for the Lie cooperad given by the
quotient of $\mathcal{G}\\!{\mathit{r}\/}(n)$ by relations similar to those of
Definition 2.1:

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######  Proposition A.3.

Rooted trees modulo root change and Arnold is isomorphic to directed, acyclic,
connected graphs modulo arrow reversal and Arnold – that is,
$\mathcal{E}\\!{\mathit{i}l\/}\mathcal{T}\\!{\mathit{r}\/}_{*}\cong\mathcal{E}\\!{\mathit{i}l\/}$
– through the map
$\mathcal{T}\\!{\mathit{r}\/}_{*}(n)\to\mathcal{G}\\!{\mathit{r}\/}(n)$
converting a rooted tree to a directed graph by orienting all edges towards
the root.

###### Proof.

The map converting a rooted tree to a directed acyclic graph with all edges
oriented towards the root is well-defined modulo the tree relations, since
root change and root Arnold for trees are sent to graph arrow reversal and
graph Arnold (after reversing one arrow in each term) relations respectively.
This map is a surjection because given a graph representative we can pick an
arbitrary vertex to be the “root” and, modulo arrow reversal, orient all edges
appropriately to obtain a directed graph in the image of the map from rooted
trees. Injectivity holds true because an arbitrary Arnold relation for graphs
is the image of a root Arnold relation for rooted trees after orienting edges
appropriately by repeated arrow reversals. ∎

The most popular approach to rational homotopy theory, by far, is given by
Sullivan models which are free commutative differential graded algebras. Our
present work advertises the usefulness of bar constructions. These are related
by a version of Theorem A.2.

######  Theorem A.4.

Let $A$ and $P$ be quasi-isomorphic commutative differential graded algebras,
where the undelying algebra of $P$ is a free (symmteric) algebra. Then the
subcomplex of indecomposibles of $P$ is quasi-isomorphic to $\mathcal{E}(A)$

For Sullivan models, one assumes through minimality that the indecomposibes of
$P$ have no induced differential. Thus one can obtain a proof through Sullivan
models that the Harrison homology of $X$ is linearly dual to its rational
homotopy groups in the simply connected setting; though the use of Harrison
homology is closer to Quillen’s original approach.

The proof of Theorem A.4 parallels that of Theorem A.2. The quasi-isomorphism
invariance of the Harrison complex was proven in Proposition 2.10 of [BFMT20],
without any connectivity hypotheses. A quasi-isomorphism between
$\mathcal{E}(P)$ and its indecomposibles is then given via the chain homotopy
argument in [SW13, Lemma 2.15]. This argument is more involved than that for
Theorem A.2. While the basic idea of “pulling out a generator” is the same, it
requires an inductive filtration approach. This gives the first more explicit
proof of this standard fact, to our knowledge.

###  A.2. Explicit treatment of signs

Recall the coproduct defined on $\mathcal{T}\\!{\mathit{r}\/}_{*}$ by excising
subsymbols. The induced coproduct in ${{\mathbb{E}}}(W)$ can be written
similarly, applying our convention of placing elements of $W$ as labels of
respective vertices in tree symbols for ${{\mathbb{E}}}W$. For example if
$a,b,c,d,e\in W$, the coalgebra ${{\mathbb{E}}}(W)$ has a coproduct

$\Delta\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e=c\otimes\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e\;+\;\lfloor a\rfloor\lfloor
b\rfloor\,d\otimes\lfloor
c\rfloor\,e\;+\;a\otimes\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor
b\rfloor\,d\big{\rfloor}\,e\;+\;b\otimes\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor
a\rfloor\,d\big{\rfloor}\,e.$

(Recall Example 2.3 for a picture of this tree.) Writing elements as labels
for ${{\mathbb{E}}}W$ automatically encodes the label-shifting of the
coproduct in $\mathcal{T}\\!{\mathit{r}\/}_{*}$, greatly simplifying
logistics. This is the rooted tree analog of the convention in [SW11,
Definition 3.3] of $W$-labeled directed graphs.

For calculations in the graded setting, it is necessary incorporate Koszul
signs. However, our convention that tree symbols labeled by $W$ elements
correspond to left-to-right ordering in the corresponding tensor product
$W^{\otimes n}$ (as remarked after Proposition 2.7), means that Koszul signs
may be computed following the same rules as for tensor algebras. This stands
in great contrast to the Koszul sign challenges of [SW11].

In particular if $W$ is graded, then the corolla commutativity relations
should carry Koszul signs. For example $\lfloor
a\rfloor\,b=(-1)^{|a|\,|b|}\,b\,\lfloor a\rfloor\,$ and also $\lfloor
a\rfloor\lfloor b\rfloor\,c=(-1)^{|a|\,|b|}\,\lfloor b\rfloor\lfloor
a\rfloor\,c$. Note that we carefully chose the form of root arrow reversing
and root Arnold written via symbols in Section 2.1 so as to not require Koszul
signs. This is the benefit of allowing the root vertex to commute inside
corolla products in our notation.

The Koszul signs for graded corolla commutativity induce similar signs on
graded coproduct and cobracket maps, similarly computed the same way as for
tensor algebras. For example, the graded version of the coproduct shown above
has Koszul signs as follows.

$\displaystyle\Delta\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor
a\rfloor\lfloor b\rfloor\,d\big{\rfloor}\,e$ $\displaystyle=$ $\displaystyle
c$ $\displaystyle\otimes\big{\lfloor}\lfloor a\rfloor\lfloor
b\rfloor\,d\big{\rfloor}\,e$ $\displaystyle+$
$\displaystyle(-1)^{|c|\bigl{(}|a|+|b|+|d|\bigr{)}}$ $\displaystyle\ \lfloor
a\rfloor\lfloor b\rfloor\,d$ $\displaystyle\otimes\lfloor c\rfloor\,e$
$\displaystyle+$ $\displaystyle(-1)^{|c||a|}$ $\displaystyle\ a$
$\displaystyle\otimes\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor
b\rfloor\,d\big{\rfloor}\,e$ $\displaystyle+$
$\displaystyle(-1)^{\bigl{(}|a|+|c|\bigr{)}|b|}$ $\displaystyle\ b$
$\displaystyle\otimes\big{\lfloor}c\big{\rfloor}\big{\lfloor}\lfloor
a\rfloor\,d\big{\rfloor}\,e.$

The signs above are computed similar to those of the graded unshuffle
coproduct of a graded tensor algebra.

The Harrison complex of a graded algebra also gains Koszul signs. In
particular the “contract and multiply” differential $d_{\mu}$ involves a sign
due to a degree $1$ operator, $d_{\mu}$, moving across a tensor (the same as
in the bar complex), plus additional Koszul signs due to required
rearrangements of a tree symbol. For example the tree symbol shape used above,
appearing in the Harrison complex, would have $d_{\mu}$ as follows.

$\displaystyle
d_{\mu}\Bigl{(}\big{\lfloor}s^{{{\text{-}}1}}c\big{\rfloor}\big{\lfloor}\lfloor
s^{{{\text{-}}1}}a\rfloor\lfloor
s^{{{\text{-}}1}}b\rfloor\,s^{{{\text{-}}1}}d\big{\rfloor}\,s^{{{\text{-}}1}}e\Bigr{)}=$
$\displaystyle\pm(-1)^{|c|}\,\big{\lfloor}\lfloor
s^{{{\text{-}}1}}a\rfloor\lfloor
s^{{{\text{-}}1}}b\rfloor\,s^{{{\text{-}}1}}d\big{\rfloor}\,s^{{{\text{-}}1}}(ce)$
$\displaystyle\pm(-1)^{|a|}\,\big{\lfloor}s^{{{\text{-}}1}}c\big{\rfloor}\big{\lfloor}\lfloor
s^{{{\text{-}}1}}b\rfloor\,s^{{{\text{-}}1}}(ad)\big{\rfloor}\,s^{{{\text{-}}1}}e$
$\displaystyle\pm(-1)^{|b|}\,\big{\lfloor}s^{{{\text{-}}1}}c\big{\rfloor}\big{\lfloor}\lfloor
s^{{{\text{-}}1}}a\rfloor\,s^{{{\text{-}}1}}(bd)\big{\rfloor}\,s^{{{\text{-}}1}}e$
$\displaystyle\pm(-1)^{|d|}\,\lfloor s^{{{\text{-}}1}}c\rfloor\lfloor
s^{{{\text{-}}1}}a\rfloor\,\lfloor
s^{{{\text{-}}1}}b\rfloor\,s^{{{\text{-}}1}}(de)$

where the $\pm$ are Koszul signs from applying graded commutativity of
corollas, reordering the symbol so that the “contract and multiply” terms are
adjacent; as well as signs from $d_{\mu}$ moving across terms in the symbol to
reach the location where it is applied. The Koszul sign for the first term
above is
$(-1)^{\bigl{(}|c|-1\bigr{)}\bigl{(}(|a|-1)+(|b|-1)+(|d|-1)\bigr{)}+\bigl{(}(|a|-1)+(|b|-1)+(|d|-1)\bigr{)}}$
from applying corolla commutativity to move $s^{{{\text{-}}1}}c$ next to
$s^{{{\text{-}}1}}e$ and also from $d_{\mu}$ crossing terms. The Koszul sign
of the last term is $(-1)^{(|c|-1)+(|a|-1)+(|b|-1)}$ from $d_{\mu}$ crossing
the $s^{{{\text{-}}1}}c$, $s^{{{\text{-}}1}}a$, and $s^{{{\text{-}}1}}b$
terms. To get all Koszul signs correct and complete calculations with greatest
confidence, we have found it generally worthwhile to explicitly include the
desuspensions $s^{{\text{-}}1}$ in the Harrison complex.

##  Appendix B Graphical (partial) models for groups through Thom Forms

We outline a theory which makes rigorous the calculations of Section 5, for
example giving graphical calculations of Massey products. We plan to give a
general theory of graphical models for groups in future work. We first review
elementary constructions of de Rham cochains associated to submanifolds in
Section B.1. These allow for geometric access to cochains. We summarize basic
definitions and properties of Thom forms from [BT82] and prove that one can
find a Thom form for a manifold with boundary which is compatible with a
chosen form on the boundary.

We apply these submanifold tools to (non-smooth) two-complexes by constructing
four-manifold thickenings $M_{G}$ which deformation retract onto their sub-
two-complexes $X_{G}$ in Section B.2. This construction is a standard,
immediate application of handlebody theory, though we prefer an approach
through blow-ups which is new.

In Section B.3, we build on the immediate observation that points and curves
in the two-disks of $X_{G}$ immediately thicken to two- and three-dimensional
submanifolds of $M_{G}$ and thus support Thom forms. We also associate Thom
forms to belt disks in the one-handles(as we did in Section 3.2). Some of the
boundary and product behavior of these forms is determined by the boundary and
intersection behavior of curves. In some cases, this data is enough to give a
complete geometric approach to real-valued cochains associated to a
fundamental group presented by generators and relations. This ultimately
affords calculations needed as input for our theory to be performed
pictorially.

###  B.1. Thom forms

######  Definition B.1.

Let $W\subseteq M$ be a co-orientable submanifold, whose boundary is contained
in that of $M$. A Thom form $\tau_{W}$ is a closed form which is supported on
a tubular neighborhood of $W$, so that the integral of $\tau_{W}$ over each
transverse slice of the tubular neighborhood is a fixed number called the mass
of the Thom form.

The degree of a Thom form is thus the codimension of the submanifold.

The following is a summary of results from Chapter 6 of [BT82].

######  Theorem B.2.

  1. (1)

Thom forms exist for any submanifold $W\subseteq M$, whose boundary is
contained in that of $M$. They represent a unique de Rham cohomology class
$[W]$.

  2. (2)

The value of a Thom form on a smooth cycle which meets $W$ transversally, or
on the fundamental class of some closed manifold mapping to $M$ which is
transverse to $W$, is the count with signs of the preimages of $W$ weighted by
the form’s mass.

  3. (3)

The pullback of a Thom form by a map which is transverse to $W$ is a Thom form
for the preimage of $W$.

  4. (4)

The wedge product of Thom forms for submanifolds which intersect transversally
is a Thom form of their intersection, whose mass is their product.

For example the Thom form of any linear
$\mathbb{C}P^{n-i}\subset\mathbb{C}P^{n}$ represents a generator of
$H^{2i}_{dR}(\mathbb{C}P^{n}).$ This class is non-zero because it evaluates
non-trivially on the fundamental class of a $\mathbb{C}P^{i}$ subspace which
intersects $\mathbb{C}P^{n-i}$ in a point. By considering complex linear
subspaces which intersect transversally we see that
$\tau_{\mathbb{C}P^{n-i}}\wedge\tau_{\mathbb{C}P^{n-j}}=\tau_{\mathbb{C}P^{n-(i+j)}}$.
Thus the cohomology ring structure of projective spaces is reflecting basic
intersection properties of linear subspaces.

For many applications it is necessary to extend the notion to Thom forms to
submanifolds with boundary.

######  Definition B.3.

Let $W\subset M$ be a submanifold of a manifold with boundary, and possibly
with corners modeled by $[0,\infty)^{n}$ which are contained in the boundary
of $M$. Let $\partial^{\circ}W$ denote the subset of $\partial W$ in the
interior of $M$.

A Thom form for $W$ is a form $\tau_{W}$ supported in a tubular neighborhood
$\nu_{W}$ of $W$, whose exterior derivative is a Thom form for
$\partial^{\circ}W$, supported in $\nu_{\partial^{\circ}W}$. Outside of the
support of $d\tau_{W}$, the integral of $\tau_{W}$ over any transverse slice
of the tubular neighborhood of $W$ is the mass of the form.

######  Theorem B.4.

Let $W$ be a submanifold with boundary whose collar neighborhood extends from
some $[0,\varepsilon)\times\partial^{\circ}W\hookrightarrow W$ to
$(-\varepsilon,\varepsilon)\times\partial^{\circ}W\hookrightarrow M$. Let
$\tau_{\partial^{\circ}W}$ be a Thom form for $\partial^{\circ}W$. There
exists a Thom form $\tau_{W}$ for $W$ so that
$d\tau_{W}=\tau_{\partial^{\circ}W}$.

###### Proof.

We assume a mass of one, with the general case following by a scalar multiple.
And assume first there exists a Thom form $\hat{\tau}_{W}$ with
$d\hat{\tau}_{W}$ some Thom form for $\partial^{\circ}W$, likely different
from the given $\tau_{\partial^{\circ}W}$. Then because $\partial^{\circ}W$ is
a submanifold whose boundary is in $\partial M$, $d\hat{\tau}_{W}$ and
$\tau_{\partial^{\circ}W}$ differ by some $d\nu$ by Theorem B.2 (1). Set
$\tau_{W}=\hat{\tau}_{W}-\nu$ to have the desired exterior derivative.

To find such a $\hat{\tau}_{W}$ make use of the extension of the collar
neighborhood of $\partial^{\circ}W$. We construct a Thom form
$\tilde{\tau}_{W}$ for a tubular neighborhood $\nu_{e}$ of this extended
submanifold as in Chapter 6 of [BT82]. Let $t$ be the collar coordinate and
$\sigma(t)$ be a smooth monotone function with $\sigma(-\varepsilon)=0$ and
$\sigma(\varepsilon)=1$. Let $f:\nu_{e}\to\mathbb{R}$ denote the function with
value $1$ on outside the tubular neighborhood of the collar and on that
neighborhood is the projection onto the $(-\varepsilon,\varepsilon)$
coordinate followed by $\sigma$. We let $\hat{\tau}_{W}$ equal
$f\wedge\tilde{\tau}_{W}$. By construction, its exterior derivative
$d(f\wedge\tilde{\tau}_{W})=df\wedge\tilde{\tau}_{W}$ is a Thom form for
$\partial^{\circ}W$ and which has fiber integrals equal to one outside a
neighborhood of $\partial^{\circ}W$, as desired. ∎

Thus we can use submanifolds to define de Rham cochains, and use these to get
concrete models for de Rham cochain algebras. We may in fact use submanifolds
whose boundaries match up and thus support cancelling Thom forms by Theorem
B.4. The general idea is to first construct cohomology ring generators. Then
one can use part (4) of Theorem B.2 to calculate products. Afterwards one
typically needs to apply Theorem B.4 again to realize relations among
products.

An important technical point is that since non-trivial submanifolds can never
intersect themselves transversally, part (4) of Theorem B.2 cannot be applied
immediately to understand squaring (in even degrees). The same can be said for
products involving a manifold and its boundary or two manifolds which meet at
their boundaries. For calculations, the submanifolds in question need to be
“moved off,” but that in turn will produce more intersections to analyze.

###  B.2. Thickening two-complexes

The methods in this paper require a cochain model for a space with a given
fundamental group, which we informally call a cochain model for the group.
Flavors of models include manifolds of interest, the simplicial model for the
classifying space of a group, and two-dimensional CW-complexes. We have found
the last to be a rich source for examples, intuition and technical use, in
particular with our Lifting Criterion Theorem 4.7 and the universal property
of Theorem 1.2. We construct manifold models for two-complexes presently, and
then show how to associate de Rham cochains to curves on two-complexes in the
next section.

######  Definition B.5.

Let $G=\langle S\;|\;R\rangle$ be a presentation of a group via generators $S$
and relations $R$.

Define $X_{G}$ to be the $2$-complex whose $1$-skeleton is a wedge of circles
for each generator in $S$, with $2$-disks attached according to each relation
in $R$, uniformly parametrized.

We will mimic this construction in surgery theory using handle attachments to
build $M_{G}$, a smooth four-manifold with boundary, such that $M_{G}\simeq
X_{G}$. This is standard, though we use a version of surgery theory defined
via blowup and attaching handles onto resulting manifolds with corners since
that makes it simpler to identify submanifolds.

######  Proposition B.6.

Let $W$ be an embedded submanifold of $\partial M$ with codimension $d$ in the
boundary. Then there is a manifold with corners $M_{\widehat{W}}$, which we
call the blow-up of $M$ at $W$, equipped with a smooth map
$\beta:M_{\widehat{W}}\to M$, with the following properties.

  * •

$M_{\widehat{W}}$ is homeomorphic to $M$, with interior diffeomorphic to that
of $M$.

  * •

There is a closed stratum $\widehat{W}\subset M_{\widehat{W}}$ diffeomorphic
to the unit disk bundle of the normal bundle of $W$ in $\partial M$. Under
this identification, the restriction of $\beta$ to $\widehat{W}$ coincides
with the bundle map projection to $W$. Outside of $\widehat{W}$, $\beta$ is a
diffeomorphism.

  * •

The interior end of a collar neighborhood of $\widehat{W}$ can be identified
with the “southern hemisphere” of the normal bundle of $W$ in $M$.

Now consider the blowup when $S$ is a framed, embedded sphere in $\partial M$.
The stratum $\widehat{S}$ has a collar neighborhood in $M_{\widehat{S}}$ which
is diffeomorphic to $\hat{S}\times(-1,0]$. Observe as well that the boundary
of $D^{k+1}\times D^{d}$ as a manifold with corners has a component
$S^{k}\times D^{d}$ which has a collar neighborhood diffeomorphic to
$S^{k}\times D^{d}\times[0,1)\cong\hat{S}\times[0,1)$.

######  Definition B.7.

If $S$ is a framed, embedded sphere in $\partial M$, the handle attachment
along $S$ is the identification of $M_{\widehat{S}}$ and $D^{k+1}\times D^{d}$
by $S^{k}\times D^{d}\sim\hat{S}$, with smooth structure defined through

$\widehat{S}\times(-1,0]\ \cup\ S^{k}\times D^{d}\times[0,1)/_{\sim}\ \cong\
\widehat{S}\times(-1,0).$

See the figure below illustrating a two-handle attachment defined by blow-up.

There is a similar construction for handle attachment to the interior, where
the blow-up is a manifold with boundary and the identifications have
tautological smooth structure once collar neighborhoods are chosen.

We apply these ideas to construct a manifold $M_{G}$ realizing a group
presentation $G=\langle S\;|\;R\rangle$.

######  Definition B.8.

Let $M_{0}$ be a four dimensional $0$-handle – that is $M_{0}\cong D^{4}$.

For each generator $\ell\in S$, choose an embedding of a $0$-sphere
$S^{0}=\\{\pm\\}$ in the boundary of $M_{0}$. Blow-up at each of these points
to obtain $({M_{0}})_{\widehat{S}}$, a manifold with corners, homeomorphic to
$D^{4}$.

The interior of $({M_{0}})_{\widehat{S}}$ is identical to that of $D^{4}$, but
its boundary now has a “large” component we call $\partial_{0}$ diffeomorphic
to $S^{3}\setminus(S\times\\{\pm\\})$ as well as “small” boundary components
contributed by each blowup. For each generator $\ell\in S$, write
$\partial^{\pm}_{\ell}$ for the boundary components of
$({M_{0}})_{\widehat{S}}$ contributed by the blowup of the corresponding
$S^{0}$ points embedded in $M_{0}$. Note that the $\partial^{\pm}_{\ell}$ are
diffeomorphic to open three-disks, each abutting the “large” boundary
component $\partial_{0}$ at an $S^{2}$. These two-spheres constitute the
codimension-two corners of $({M_{0}})_{\widehat{S}}$.

Each $\partial^{\pm}_{\ell}$ has a neighborhood in $({M_{0}})_{\widehat{S}}$
which is diffeomorphic, as manifolds with corners, to
$D^{3}\times(-\varepsilon,0]$. Thus after choosing an explicit identification
with $D^{3}$, we may perform the following.

######  Definition B.9.

Let $M_{1}$ be the quotient of
$({M_{0}})_{\widehat{S}}\;\coprod\;(\bigcup_{S}D^{3}\times D^{1})$ smoothly
attaching 1-handles at each $\partial^{\pm}_{\ell}$.

The result is a manifold with boundary, as the corners are now “internal” due
to gluing the $D^{3}\times(-\varepsilon,0]$ at $\partial^{\pm}_{0}$ in
$(M_{0})_{\hat{S}}$ to the $D^{3}\times[0,\varepsilon)$ in the attached
handle, resulting in $D^{3}\times(-\varepsilon,\varepsilon)$ in the quotient.
We call each $0\times D^{1}$ the core of the $1$-handle, and the images of
$0\times 0$ and $0\times 1$ in $(M_{0})_{\hat{S}}$ are the outgoing and
incoming handle attachment points respectively. The manifold with boundary
$M_{1}$ retracts onto the wedge product $\bigvee_{S}S^{1}$ through the
homotopy equivalence sending $M_{0}$ to a point and each 1-handle to its core.

Next, for each relation $r\in R$, we claim there exists a framed embedding of
$S^{1}$ into the boundary $\partial M_{1}$ such that its image in
$\pi_{1}(M_{1})\cong\langle S\rangle$ is the corresponding word, and these can
be made disjoint. Disjoint embeddings are generic because $\partial M_{1}$ is
three-dimensional. To have canonical framings we can choose such an embedding
so that its image in any of the one handles is of the form $x\times D^{1}$
where $x\in\partial D^{3}$, so that a choice at either end determines a
“product framing.” Then we can use the fact that $\partial M_{0}\cong S^{3}$
has a trivial tangent bundle to transport framings when the image is in
$\partial_{0}$.

######  Definition B.10.

Perform the blowup of $M_{1}$ along each embedded image of $S^{1}$ labeled by
a relation as above to obtain $(M_{1})_{\hat{R}}$, a manifold with corners
homeomorphic to $M_{1}$.

The boundary of $(M_{1})_{\hat{R}}$ consists of a “large” boundary component
$\partial_{1}$ coming from $\partial M_{1}$ as well as “small” boundary
components $\\{\partial_{r}\\}$, diffeomorphic to $S^{1}\times\mathbb{R}^{3}$,
contributed by each individual blowup. The $\partial_{r}$ and $\partial_{1}$
components connect at codimension 2 corners in $(M_{1})_{\hat{R}}$ (each
diffeomorphic to the boundary of an annulus). Each $\partial_{r}$ has a
“collar” neighborhood in $(M_{1})_{\hat{R}}$ which is diffeomorphic, as
manifolds with corners, to $S^{1}\times D^{3}\times(-\varepsilon,0]$.

######  Definition B.11.

Choosing such a diffeomorphism, we can then smoothly “fill in the relation
$r$”, attaching a $2$-handle $D^{2}\times D^{2}$ by gluing along the
identification $(\partial D^{2})\times D^{2}=S^{1}\times D^{2}$. This glues
the collar neighborhood $\bigl{(}S^{1}\times[0,\varepsilon)\bigr{)}\times
D^{2}$ of $(\partial D^{2})\times D^{2}$ inside of $D^{2}\times D^{2}$ to the
collar neighborhood $S^{1}\times D^{2}\times(-\varepsilon,0]$ of
$\partial_{r}$ inside of $(M_{1})_{\hat{R}}$ along the common circle
$S^{1}\times D^{3}\times\\{0\\}$.

The result is a manifold with boundary but no corners near the attachment.

######  Definition B.12.

Define ${M}_{G}$ to be the manifold resulting from $(M_{1})_{\hat{R}}$ by
attaching a $2$-handles associated to each relation.

Then $M_{G}$ retracts onto $X_{G}$ by having each handle retract onto its core
cell.

###  B.3. de Rham cochains associated to group generators and curves on disks

We next provide a method to define deRham cochains on $M_{G}$, which is used
throughout Section 5. The central players in our model are Poincaré duals to
the $1$-handles of $M_{G}$.

Recall that in the construction of $M_{1}$ we attached $1$-handles
$D^{3}\times D^{1}$ at $\partial^{\pm}_{\ell}$ for each generator $\ell\in S$.
Write $D^{3}_{\ell,t}=D^{3}\times\\{t\\}\subset D^{3}\times D^{1}\subset
M_{1}$ for the belt disk at time $t$ in the $1$-handle corresponding to
$\ell\in S$. In blowing up to obtain $(M_{1})_{\hat{R}}$, each such disk is
blown up at a finite collection of points on its boundary, with one point on
the boundary blown up for each occurrence of $\ell^{\pm 1}$ in a relation of
$R$. Call the resulting manifold $(D^{3}_{\ell,t})_{\hat{R}}$. Write
$D^{2}_{\ell\in r,t}$ for the $2$-disks on the boundary of
$(D^{3}_{\ell,t})_{\hat{R}}$ resulting from blowing up the relation $r\in R$.

The identification resulting in ${M}_{G}$ identifies each $2$-disk
$(D^{2}_{\ell\in r,t})$ with some $\\{y(t)\\}\times D^{2}$ in the $2$-handle
$(D^{2}\times D^{2})_{r}$ corresponding to the relation $r$. Here
$y(t)\in\partial D^{2}$. The scalar multiples of $y_{t}$ with
$C=[1-\varepsilon,1]$ crossed with $D^{2}$ form a tab extending
$(D^{3}_{\ell,t})_{\hat{R}}$ into $(D^{2}\times D^{2})_{r}$.

######  Definition B.13.

The union of $(D^{3}_{\ell,t})_{\hat{R}}$ with all of its tabs is a
submanifold (with corners) of $\overline{M}_{G}$ which we call an indicator
submanifold associated to $\ell$ and will denote by ${\mathcal{I}}_{\ell,t}$.

The variable $t$ lets us choose multiple representatives, as will be needed
for our models . By abuse, if $t$ is not provided it can be assumed to be
$\frac{1}{2}$. Indicator submanifolds are codimension one, and their Thom
forms restrict to Thom forms on $M_{1}$ which generate the first cohomology,
as the indicator homomorphisms, as in Secction 1.2, where multiple belt disks
were also considered. The boundary of ${\mathcal{I}}_{\ell,t}$ internal to
$M_{G}$ consists of a union of $\\{y\\}\times D^{2}$ where $y$ is
$\varepsilon$ away from the boundary of $D^{2}$ for each occurrence of
$\ell^{\pm}$ in any relation.

######  Definition B.14.

For each generator $\ell\in S$, a Thom form for $\mathcal{I}_{\ell}$ is called
an indicator form. A choice of such will be denoted by $\tau_{\ell}$.

Indicator forms are supported one one-handles within $\varepsilon$ of the
boundary of two-handles and are fixed, except for the possibility to perturb
them using the $t$ coordinate in the one-handle. We next define forms with
complementary domains of definition, the ones which are drawn in the examples
of Section 5.

######  Definition B.15.

Assume a collection of indicator submanifolds for each generator has been
fixed, and consider some two-handle $(D^{2}\times D^{2})_{r}$. We define the
representative two-disk to be the disk inside of $D^{2}\times 0$ which
excludes points within $\varepsilon$ of the boundary, and the heart of the
two-handle to be the product of the representative two disk with $D^{2}$

The boundary of the representative two-disk is marked by the points $y$ so
that $y\times D^{2}$ is a boundary component of some indicator submanifold.
Call these indicator points, which are labeled by the corresponding generator.
We say a curve in the representative two-disk is admissible if each endpoint
is either an interior point or an indicator point.

######  Definition B.16.

Let $S$ be a submanifold (points or curves) in representative two-disk. The
thickening of $S$ is just $S\times D^{2}\subset(D^{2}\times D^{2})_{r}$.

We use $\tau_{S}$ to denote a Thom form for the thickening of $S$, which we
call a point form or curve form when $S$ is a point or curve respectively.

Thus the thickenting of the representative two-disk itself is the heart of the
two-handle.

######  Definition B.17.

A partial graphical model is a partially defined sub-algebra of
$\Omega_{dR}^{*}(M_{G})$ specified by a collection of admissible curves in
representative two-disks for all relations. These admissible curves are
assumed to intersect transversally, in their interiors.

The model consists of indicator forms, Thom forms for indicator points, Thom
forms for the admissible curves, and Thom forms for any intersection points.
The coboundary of an indicator form is the sum of chosen Thom forms for its
indicator points. The coboundary of a Thom form for a curve is the sum (with
signs) of the Thom forms for its boundary points. And some products are
defined by intersection of admissible curves in their interiors.

We use partial graphical models throughout Section 5, though we do not
explicitly check that the Thom forms can be chosen consistently. In our
examples it suffices to first take indicator forms, setting their boundary as
the Thom forms of indicator points. Those Thom forms for indicator points are
then used, applying Theorem B.4, to set the boundary behavior of Thom forms of
curves with boundary at indicator points. These curves’ intersection points
are assigned Thom forms which are given as the wedge product of the Thom forms
for curves. And curves with some of those as as boundary points once again
have their boundary behavior set by Theorem B.4.

We use simple models to make calculations in Harrison complexes in Section 5.
In further work we plan to develop partial graphical models fully, addressing
when, as partially defined algebras, they suffice as models of the cochains of
two-complexes. There are also interesting questions such as for which groups
such models are finite, and seeing whether finer geometry can be used to fill
in products of curves which share a boundary point.

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