#  Differentials of Sinha’s spectral sequence for long knots
in codimension one

Syunji Moriya
Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University,
3-3-138 Sugimoto, Sumiyoshi-ku Osaka 558-8585 Japan<EMAIL_ADDRESS>
###### Abstract.

We compute the differentials of a few elements of Sinha’s spectral sequence
for cohomology of the space of long knots modulo immersions in codimension one
over a field of characteristic $2$ or $3$. We show that $d_{2}$ of an element
is non-zero in characteristic $2$, which has already been proved by Salvatore
essentially, and $d_{3}$ of another element is non-zero in characteristic $3$.
While even convergence of the sequence is unclear in condimension one, these
results have some application to non-formality of operads. The result in
characteristic $3$ implies planar non-formality of the standard map
$C_{*}(E_{1})\to C_{*}(E_{2})$ in characteristic $3$, where $C_{*}(E_{k})$
denotes the chain little $k$-disks operad. We also reprove the result of
Salvatore which states $C_{*}(E_{2})$ is not formal as a planar operad in
characteristic $2$. For computation, we use a duality between configuration
spaces and fat diagonals.

This work is partially supported by JSPS KAKENHI Grant Number JP17K14192.

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and terminology
  2. 2 Punctured knot model and configuration space model
  3. 3 Fat diagonal model
  4. 4 Spectral sequences
  5. 5 Condensed maps
  6. 6 Computation of differential in characteristic 2
  7. 7 Computation of differential in characteristic 3
    1. 7.1 First bounding chain
    2. 7.2 Second bounding chain
    3. 7.3 Computation of the differential as an element of the first page
    4. 7.4 Non-triviality of the differential
  8. 8 Absolute non-formality in characteristic 2

##  1\. Introduction

A long knot is a smooth embedding $\mathbb{R}\to\mathbb{R}^{d}$ which
coincides with a fixed linear embedding outside of a compact set. Sinha
introduced a spectral sequence for cohomology of the space of long knots based
on Goodwillie-Weiss calculus[4, 26], see [20, 21]. Sinha’s spectral sequence
is closely related to Vassiliev’s spectral sequence and also related to the
little disks operads. Lambrechts-Turchin-Volić [10] proved the rational
collapse of Vassiliev’s sequence for $d\geq 4$ using these relations. For
other coefficients, Boavida de Brito-Horel [1] proved that infinitely many
higher differentials of Sinha’s sequence vanish in $p$-adic integers.
In this paper, we consider Sinha’s sequence for $d=2$, the case of codimension
one. While even convergence is unclear, it is relatively easy to find a new
computational example since the rational collapse is not known in this case.
We compute higher differentials of two elements as follows.

######  Theorem 1.1 (Theorems 6.8, 7.23).

Let $\mathbb{E}_{r}^{p,q}$ be Sinha’s spectral sequence for cohomology of the
space of long knots modulo immersions in codimension one. More precisely,
$\mathbb{E}^{p,q}_{r}$ is the Bousfield-Kan type cohomology spectral sequence
induced by the cosimplicial space $\mathcal{K}^{\bullet}_{2}$ defined in [21],
which consists of compactified configuration spaces of ordered points in
$\mathbb{R}^{2}$ (see Definitions 2.4, 4.5).

  1. (1)

([19]) In characteristic $2$, there exists an element in
$\mathbb{E}_{2}^{-4,2}$ which is not annihilated by $d_{2}$.

  2. (2)

In characteristic $3$, there exists an element in $\mathbb{E}_{3}^{-5,3}$
which is not annihilated by $d_{3}$.

Part 1 (in homological context) has already been proved implicity by Salvatore
[19]. Turchin [24] introduced a notion of divided product which produces
elements of Sinha’s (or Vassiliev’s) sequence in terms of graphs. The whole of
the products forms an interesting subcomplex of a page of Sinha’s sequence.
The elements in parts 1 and 2 of Theorem 1.1 correspond to the products
$\langle Z_{1},Z_{1}\rangle$ and $\langle Z_{1},Z_{2}\rangle$ in the notation
of [24], respectively while we use a version reversing the order of vertexes.
It might be interesting to compute $d_{p}$ of $\langle Z_{1},Z_{p-1}\rangle$
for other primes $p$.
Theorem 1.1 can be related to formality. Let $f:\mathcal{O}\to\mathcal{P}$ and
$f^{\prime}:\mathcal{O}^{\prime}\to\mathcal{P}^{\prime}$ be maps of chain
operads. A quasi-isomorphism from $f$ to $f^{\prime}$ is a commutative square

$\textstyle{\mathcal{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathcal{O^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{\mathcal{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{P^{\prime}}}$

of maps of operads where the horizontal maps induce quasi-isomorphisms of
complexes at each arity. A map $f:\mathcal{O}\to\mathcal{P}$ is said to be
formal if it is connected with the induced map $H_{*}(f)$ on homology by a
zigzag of quasi-isomorphisms. A chain operad $\mathcal{O}$ is formal if the
map from the initial object to $\mathcal{O}$ is formal. Formality of the
little disks operads proved in [23, 8, 12] was essentially used in the proof
of rational collapse of Vassiliev’s sequence. Since Sinha’s sequence comes
from a map of operads, Theorem 1.1 has the following corollary about relative
planar non-formality (see [16]).

######  Corollary 1.2.

Let $E_{k}$ be the little $k$-disks operad and $E_{1}\to E_{2}$ the map
induced by the inclusion $\mathbb{R}\to\mathbb{R}^{2}$ to the first
coordinate. The induced map $C_{*}(E_{1})\to C_{*}(E_{2})$ between singular
chains is not formal as a map of planar (or non-symmetric) operads in
characteristic $3$ ∎

Using the part 1 of Theorem 1.1, we also reprove the (absolute) planar non-
formality in characteristic $2$ first proved in [19].

######  Corollary 1.3 ([19], see section 8).

The chain little $2$-disks operad $C_{*}(E_{2})$ is not formal as a planar
operad in characteristic $2$.

As related works, Turchin-Willwacher [25] proved non-formality of the map
$C_{*}(E_{k})\to C_{*}(E_{k+1})$ over reals for $k\geq 1$. In [11], Livernet
proved non-formality of the Swiss-cheese operads. In [17], non-formality of
the framed little disks operads of odd dimension $\geq 5$ was proved.
Our method of computation is different from [19] which makes use of
obstruction theory and a combinatorial model of the operad. We use a duality
between configuration spaces and fat diagonals, which is similar to the one
given in the former paper [18], to transfer necessary structure to fat
diagonals and resolve the structure by $\check{\text{C}}$ech complex. We use
chains in this resolution for diagram chasing. Perhaps, the main feature of
this method is that the chains are labeled by graphs and defined by relatively
simple maps. So we can use graphical and geometric intuition to some extent.
While the computation in this paper is restricted to condimension one, we
prepare most of basic notions in a form applicable to any codimension.

###  1.1. Notation and terminology

  1. (1)

$\mathcal{CG}$ denotes the category of unpointed compactly generated spaces
and continuous maps and $\mathcal{CG}_{*}$ the corresponding pointed category.
All spaces in this paper belong to $\mathcal{CG}$ or $\mathcal{CG}_{*}$.

  2. (2)

For an unpointed (resp. pointed) topological space $X$, $C_{*}(X)$ (resp.
$\bar{C}_{*}(X)$) denotes the normalized singular chain complex (resp. the
reduced normalized singular chain complex) with coefficients in a fixed field
$\mathsf{k}$. For chains $a\in\bar{C}_{*}(X)$, $b\in\bar{C}_{*}(Y)$, $a\wedge
b\in\bar{C}_{*}(X\wedge Y)$ denotes the Eilenberg-Zilber shuffle product. For
the transposition map $T:X\wedge Y\to Y\wedge X$, we have the following
identity at chain level:

$T_{*}(a\wedge b)=(-1)^{|a||b|}b\wedge a,$

where $|a|$ denotes the degree of $a$. Hereafter, for a pointed map $f:X\to
Y$, the subscript $*$ for the pushforward on homology is omitted if no
confusion occurs. So $f_{*}(a)$ is denoted by $f(a)$.

  3. (3)

Let $X$ be an unpointed space. We denote $X$ with disjoint basepoint by
$X_{+}$ and the one-point compactification of $X$ by $X^{*}$. We set
$S^{k}=(\mathbb{R}^{k})^{*}$ and $[0,\infty]=[0,\infty)^{*}$. We denote the
interval $[0,1]$ by $I$. We fix a fundamental cycle
$w_{S^{2}}\in\bar{C}_{2}(S^{2})$, chains
$w_{\infty}\in\bar{C}_{1}([0,\infty])$ and $w_{I}\in C_{1}(I)$ such that
$dw_{\infty}=\\{0\\}$ and $dw_{I}=\\{1\\}-\\{0\\}$, where $\\{0\\},\\{1\\}$
are cycles represented by $0,1\in[0,1]\subset[0,\infty]$. We set

$w_{k}=(w_{S^{2}})^{\wedge 2}\wedge(w_{\infty})^{\wedge k},\qquad\
w_{kl}=w_{k}\wedge(w_{I})^{\wedge l}\qquad\text{ for }k\geq 0,\ l>0.$

  4. (4)

$|-|$ denotes the standard Euclidean norm. We define elements
$u,v\in\mathbb{R}^{d}$ by

$u=(1,0,\dots,0),\text{ \ and \ }v=(0,1,0,\dots,0).$

  5. (5)

We denote by $=_{1}$, $<_{1}$,$\dots$ etc., the relations between the first
coordinates of elements of $\mathbb{R}^{k}$. For example, for two elements
$x=(x_{1},\dots,x_{k}),y=(y_{1},\dots,y_{k})\in\mathbb{R}^{k}$ and a number
$t\in\mathbb{R}$, $x<_{1}y$ and $x=_{1}y$ mean $x_{1}<y_{1}$ and
$x_{1}=y_{1}$, respectively, and $x<_{1}t$ means $x_{1}<t$.

##  2\. Punctured knot model and configuration space model

For some reason, we mainly deal with a version of the punctured knot model
defined below instead of the cosimplicial model.

######  Definition 2.1.

A partition $P$ of  $[n+1]=\\{0,1,\dots,n+1\\}$ is a set of subsets of $[n+1]$
satisfying the following conditions.

  1. (1)

$\cup_{\alpha\in P}\alpha=[n+1]$.

  2. (2)

Each element of $P$ is non-empty.

  3. (3)

If $\alpha,\beta\in P$, either of $\alpha=\beta$ or
$\alpha\cap\beta=\emptyset$ holds.

  4. (4)

For each element $\alpha\in P$ and pair of numbers $i,j\in\alpha$ with $i<j$,
any number $k$ with $i<k<j$ belongs to $\alpha$.

  5. (5)

$\\#P\geq 2$, in other words the set consisting of the single element $[n+1]$
is not a partition.

We call an element of $P$ a piece of $P$. We regard a partition as a totally
ordered set via the order induced by $[n+1]$. A partition $Q$ is said to be a
subdivision of $P$ if $Q\not=P$ and each piece of $Q$ is contained in some
piece of $P$. $\mathrm{P}_{n}$ denotes the poset of partitions of $[n+1]$. Its
objects are the partitions. A non-identity morphism $P\to Q$ exists if and
only if $Q$ is a subdivision of $P$. By abuse of notation, we identify the
partition $\\{\\{0\\},\\{1\\},\dots,\\{n+1\\}\\}$ consisting of singletons
with $[n+1]$.

######  Example 2.2.

The following sets are examples of partitions of $[5]$:

$P=\\{\\{0\\},\\{12\\},\\{345\\}\\},\qquad
Q=\\{\\{0\\},\\{12\\},\\{3\\},\\{45\\}\\}.$

We omit commas in pieces, so the piece $\\{12\\}$ denotes $\\{1,2\\}$. We see
that $\\#P=3$, $\\#Q=4$, and $Q$ is a subdivision of $P$.

######  Definition 2.3.

  1. (1)

Throughout the paper, we fix positive numbers $\rho,\epsilon$ and $c_{0},\dots
c_{n+1}$ satisfying

$c_{0}+\cdots+c_{n+1}=1,\quad\rho<1,\quad 100\epsilon/\rho<c_{0},\quad
100\bigl{(}\epsilon/\rho+\sum_{j<i}c_{j}\bigr{)}<c_{i}\quad(1\leq i\leq n+1).$

(The last two inequalities are not used until section 5.)

  2. (2)

We define a functor $\mathcal{PK}:\mathrm{P}_{n}\to\mathcal{CG}$ as follows.
Set $b_{i}=c_{0}+\cdots+c_{i-1}$ for $1\leq i\leq n$. For a partition
$P=\\{\alpha_{0}<\cdots<\alpha_{p+1}\\}$, $S_{P}\subset\\{1,\dots,n+1\\}$
denotes the set of minimum elements in each of
$\alpha_{1},\dots,\alpha_{p+1}$. $\mathcal{PK}(P)$ is the space of embeddings
$f:[0,1]-\\{b_{i}\mid i\in S_{P}\\}\to D^{d}$ such that

    1. (a)

$f(0)=-u$ and $f(1)=u$

    2. (b)

In each connected component of the domain, $f(t)=x+ctu$ for some constant
elements $x\in D^{d}$ and $c>0$.

For a subdivision $Q$ of $P$, the map $\mathcal{PK}(P)\to\mathcal{PK}(Q)$ is
the obvious restriction.

Let $\Delta_{n}$ be the category whose objects are $[k]$ ($0\leq k\leq n$) and
whose morphisms are the weakly order preserving maps. We define a functor
$\mathcal{F}:\mathrm{P}_{n}\to\Delta_{n}$ by $P\mapsto[\\#S_{P}-1]$.

######  Definition 2.4.

Let $\mathcal{K}_{d}$ denotes the $d$-dimensional Kontsevich operad defined in
[21]. Its $p$-th term $\mathcal{K}_{d}(p)$ is a version of Fulton-Macpherson
compactification of the ordered configuration space $Conf_{p}(\mathbb{R}^{d})$
of $p$ points. The operad $\mathcal{K}_{d}$ is equipped with a map
$\mathcal{A}\to\mathcal{K}_{d}$ from the associative operad as in [21]. This
map induces a cosimplicial space $\mathcal{K}_{d}^{\bullet}$, which we call
Sinha’s cosimplicial space, via the framework of McClure-Smith. We use the
same notation for its restriction to $\Delta_{n}$ if no confusion occurs.

######  Lemma 2.5.

There is a zigzag of natural homotopy equivalences

$\mathcal{PK}\simeq\mathcal{F}^{*}\mathcal{K}_{d}^{\bullet}.$

###### Proof.

Let $\mathrm{Emb}_{c}([0,1],D^{d})$ be the space of embeddings $[0,1]\to
D^{d}$ to the $d$-dimensional unit closed disk with fixed endpoints in
$\partial D^{d}$ and fixed tangent vectors on them. When we identify
$\mathrm{P}_{n}$ and the poset of non-empty subsets of $\\{1,\dots,n+1\\}$ by
$P\mapsto S_{P}$, after some change of parametrization of $[0,1]$,
$\mathcal{PK}$ is naturally homotopy equivalent to the homotopy fiber of the
map from the punctured knot model of $\mathrm{Emb}_{c}([0,1],D^{d})$ to the
cubical model of $\mathrm{Imm}_{c}([0,1],D^{d})\simeq\Omega S^{d-1}$ given by
$P\mapsto(S^{d-1})^{\\#P-2}$. With this observation, the claim is obvious by
the arguments in [20, 21]. ∎

######  Definition 2.6.

  1. (1)

For $P\in\mathrm{P}_{n}$ and $\alpha,\beta\in P$ with $\alpha<\beta$, we set

$\begin{split}c_{\alpha}:=&\sum_{i\in\alpha}c_{i},\\\
c_{\leq\alpha}:=&\sum_{\gamma\in P,\gamma<\alpha}c_{\gamma}+c_{\alpha}/2,\\\
c_{\geq\alpha}:=&\sum_{\gamma\in P,\gamma>\alpha}c_{\gamma}+c_{\alpha}/2,\\\
c_{\alpha\beta}:=&\sum_{\gamma\in
P,\alpha<\gamma<\beta}c_{\gamma}+(c_{\alpha}+c_{\beta})/2.\end{split}$

In fact, this definition does not depend on the pieces of $P$ other than
$\alpha,\beta$. We abbreviate $c_{\leq\\{i\\}}$ (resp. $c_{\geq\\{i\\}}$) as
$c_{\leq i}$ (resp. $c_{\geq i}$).

  2. (2)

Let $Q$ be a subdivision of $P$ and write
$P=\\{\alpha_{0}<\cdots<\alpha_{p+1}\\}$ and
$Q=\\{\beta_{0}<\cdots<\beta_{q+1}\\}$. We define an affine monomorphism
$e_{P,Q}:\mathbb{R}^{dp}\to\mathbb{R}^{dq}$ as follows. Let $(x_{i})_{1\leq
i\leq p}\in\mathbb{R}^{dp}$ be an element. For convenience, we set

$x_{0}=(-1+\rho c_{\alpha_{0}}/2,0,\dots,0),\quad x_{p+1}=(1-\rho
c_{\alpha_{p+1}}/2,0,\dots,0).$

For $0\leq i\leq p+1$, suppose that $\alpha_{i}$ includes exactly $k$-pieces
of $Q$, say $\beta_{l},\dots,\beta_{l+k-1}$. We create the line segment which
is centered at $x_{i}$, parallel to $u$, and of length $\rho c_{\alpha_{i}}$,
and divide this segment into $k$ little segments of length $\rho
c_{\beta_{l}},\dots,\rho c_{\beta_{l+k-1}}$ arranged from left to right. Let
$y_{l+j-1}$ be the center of the $j$-th little segment. We set
$e_{P,Q}((x_{i})_{i})=(y_{m})_{1\leq m\leq q}$. For $Q=[n+1]$ we write
$e_{P,Q}=e_{P}$. It is clear that $e_{Q,R}\circ e_{P,Q}=e_{P,R}$ for a
subdivision $R$ of $Q$.

  3. (3)

We define a functor $\mathcal{E}:\mathrm{P}_{n}\to\mathcal{CG}$ as follows.

    1. (a)

For $P\in\mathrm{P}_{n}$ with $\\#P-2=p$, $\mathcal{E}(P)$ is the subset of
$\mathbb{R}^{dp}$ consisting of elements $(x_{\alpha})_{\alpha\in P}$
satisfying the following inequalities:

$\begin{split}|x_{\alpha}|&\leq 1-\rho c_{\alpha}/2,\ \text{and}\\\ -1+\rho
c_{\leq\alpha}&\leq_{1}x_{\alpha}\leq_{1}\ 1-\rho c_{\geq\alpha},\
\text{and}\\\ |x_{\alpha}&-x_{\beta}|\geq\rho c_{\alpha\beta}.\end{split}$

    2. (b)

If $Q$ is a subdivision of $P$, the corresponding map
$\mathcal{E}(P)\to\mathcal{E}(Q)$ is given by the map $e_{P,Q}$ defined above.

  4. (4)

We can define an inclusion $\mathcal{E}(P)\to\mathcal{PK}(P)$ by creating the
line segment centered at $x_{\alpha}$ of length $\rho c_{\alpha}$ for
$(x_{\alpha})_{\alpha\in P}$. These inclusions form a natural transformation
$\mathcal{E}\to\mathcal{PK}$.

######  Lemma 2.7.

For each $P\in\mathrm{P}_{n}$, the inclusion
$\mathcal{E}(P)\to\mathcal{PK}(P)$ is a homology isomorphism.

###### Proof.

Write $P=\\{\alpha_{0}<\dots\alpha_{p+1}\\}$. Let $\mathit{Conf}_{p}(D^{d})$
be the ordered configuration space of $p$ points in $D^{d}$. Since the map
$\mathcal{PK}(P)\to\mathit{Conf}_{p}(D^{d})$ which takes a collection of line
segments $(l_{\alpha_{i}})$ to the configuration of centers of
$l_{\alpha_{i}}$ ($i\not=0,p+1$), is a homotopy equivalence, we only have to
prove the composition
$\mathcal{E}(P)\to\mathcal{PK}(P)\to\mathit{Conf}_{p}(D^{d})$ is a homology
isomorphism. This is codimension $0$ embedding and fits into the following
commutative diagram.

$\textstyle{H_{*}(\mathcal{E}(P))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}(\mathit{Conf}_{p}(D^{d}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}(D^{dp},\Delta^{\prime}_{\mathrm{fat}}\cup\partial^{\prime}D^{dp})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}(D^{dp},\Delta_{\mathrm{fat}}\cup\partial
D^{dp})}$

Here, $D^{dp}=(D^{d})^{p}$, and $\partial^{\prime}D^{dp}$ is the subspace of
elements which do not satisfy at least one of the first two inequalities in
(3),(a) of Definition 2.6 for some $\alpha\in P$ ( so this is a collar of
$\partial D^{dp}$). $\Delta^{\prime}_{\mathrm{fat}}$ is the subspace of
elements which do not satisfy the third inequality for some pair
$\alpha,\beta\in P$. $\Delta_{\mathrm{fat}}$ is the fat diagonal of
$(D^{d})^{p}$. The vertical arrows are Poincaré duality isomorophisms, and the
bottom horizontal arrow is induced by the identity. We consider the
$\check{\mathrm{C}}$ech spectral sequences of pairs
$(\Delta^{\prime}_{\mathrm{fat}},\Delta^{\prime}_{\mathrm{fat}}\cap\partial^{\prime}D^{dp})$
and $(\Delta_{\mathrm{fat}},\Delta_{\mathrm{fat}}\cap\partial D^{dp})$ with
respect to the coverings $\\{\Delta^{\prime}_{\alpha,\beta}\\}_{\alpha,\beta}$
and $\\{\Delta_{\alpha\beta}\\}_{\alpha,\beta}$, where
$\Delta^{\prime}_{\alpha\beta}$ is the subspace of elements which do not
satisfy the third inequalities in (3)(a) of Definition 2.6 for $\alpha,\beta$
and $\Delta_{\alpha\beta}$ is the subspace of elements whose $\alpha$\- and
$\beta$\- components are the same. The inclusion
$\Delta_{\alpha\beta}\to\Delta^{\prime}_{\alpha\beta}$ is clearly a homotopy
equivalence. The inclusion $\Delta_{\alpha\beta}\cap\partial
D^{dp}\to\Delta^{\prime}_{\alpha\beta}\cap\partial^{\prime}D^{dp}$ is also a
homotopy equivalence since its homotopy inverse is given by the orthogonal
projection to $\Delta_{\alpha\beta}$ followed by the projection to
$\partial(D^{dp})$ from the light source $0$. So, the inclusion induces an
isomorphism between the relative homology of the pairs. We also see that the
inclusion induces an isomorphism between the relative homology of
intersections similarly, so by spectral sequence, we see that the bottom arrow
of the square is an isomorphism. ∎

##  3\. Fat diagonal model

In this section, we define a functor
$\mathcal{T}:\mathrm{P}_{n}^{op}\to\mathcal{CG}_{*}$ consisting of Thom spaces
(or suspensions) of fat diagonals. We prove that this functor is stably
equivalent to the Spanier-Whitehead dual of the punctured knot model
$\mathcal{PK}$.

######  Definition 3.1.

Let $P\in\mathrm{P}_{n}$ be a partition having the minimum piece $\gamma_{m}$
and the maximum piece $\gamma_{M}$.

  1. (1)

We define a positive number $\epsilon_{P}$ by

$\epsilon_{P}=\frac{\epsilon}{8^{n-p}},\quad\text{where}\quad p=\\#P-2.$

Here, $\epsilon$ is the number fixed in Definition 2.3.

  2. (2)

For $P\in\mathrm{P}_{n}$ let $\nu_{P}$ be the $\epsilon_{P}$-neighborhood of
$e_{P}(\mathbb{R}^{dp})$ i.e.,

$\nu_{P}=\\{y\in\mathbb{R}^{dn}\mid|y-e_{P}(x)|<\epsilon_{P}\text{ for some
}x\in\mathbb{R}^{dp}\\}.$

For $\alpha,\beta\in P-\\{\gamma_{m},\gamma_{M}\\}$, we define a subspace
$D_{\alpha\beta}$ of $\mathbb{R}^{dp}$ by

$D_{\alpha\beta}=\\{(x_{\gamma})_{\gamma\in
P-\\{\gamma_{m},\gamma_{M}\\}}\mid|x_{\alpha}-x_{\beta}|\leq
d_{\alpha\beta}(P)\\},$

where

$d_{\alpha\beta}(P)=\rho c_{\alpha\beta}-\epsilon_{P}.$

We denote by $E_{\alpha}$ the subspace of the elements $(x_{\gamma})_{\gamma}$
satisfying the following condition:

$\begin{split}|x_{\alpha}|&\geq 1-\rho c_{\alpha}/2+\epsilon_{P},\
\text{or}\\\ x_{\alpha}&\leq_{1}\,-1+\rho c_{\leq\alpha}-\epsilon_{P},\
\text{or}\\\ x_{\alpha}&\geq_{1}\,1-\rho
c_{\geq\beta}+\epsilon_{P}.\end{split}$

Set

$E_{P}=\bigcup_{\gamma_{m}<\alpha<\gamma_{M}}E_{\alpha},\qquad
D_{P}=E_{P}\cup\bigcup_{\gamma_{m}<\alpha<\beta<\gamma_{M}}D_{\alpha\beta}.$

We define a space $\mathcal{T}(P)\in\mathcal{CG}_{*}$ by

$\mathcal{T}(P)=Th(\nu_{P})/Th(\nu_{P}|_{D_{P}})=\mathbb{R}^{dn}/\\{(\mathbb{R}^{dn}-\nu_{P})\cup\pi_{P}^{-1}(D_{P})\\},$

where $\pi_{P}:\nu_{P}\to\mathbb{R}^{pd}$ is the standard projection. If $Q$
is a subdivision of $P$ we define a map $\mathcal{T}(Q)\to\mathcal{T}(P)$ as
the natural collapsing map and this defines a functor
$\mathcal{T}:(\mathrm{P}_{n})^{op}\to\mathcal{CG}_{*}$ (see Lemmas 3.2 and 3.3
for well-definedness).

  3. (3)

Set

$\mathcal{T}_{\emptyset_{P}}=Th(\nu_{P})/Th(\nu_{P}|_{E_{P}})=\mathbb{R}^{dn}/\\{(\mathbb{R}^{dn}-\nu_{P})\cup\pi_{P}^{-1}(E_{P})\\}.$

For two pieces $\alpha,\beta$ of $P$ with
$\gamma_{m}<\alpha<\beta<\gamma_{M}$, let $\mathcal{T}_{\alpha\beta}$ denotes
the subspace of $\mathcal{T}_{\emptyset_{P}}$ consisting of the basepoint and
the elements represented by $x\in\nu_{P}$ satisfying $\pi_{P}(x)\in
D_{\alpha\beta}$.

######  Lemma 3.2.

Let $Q$ be a subdivision of a partition $P\in\mathrm{P}_{n}$. We have

$(\mathbb{R}^{dn}-\nu_{Q})\cup(\nu_{Q}|_{E_{Q}})\subset(\mathbb{R}^{dn}-\nu_{P})\cup(\nu_{P}|_{E_{P}}).$

In particular, the identity on $\mathbb{R}^{dn}$ induces a collapsing map

$\delta^{\prime}_{P,Q}:\mathcal{T}_{\emptyset_{Q}}\to\mathcal{T}_{\emptyset_{P}}.$

###### Proof.

Let $P^{\prime}\subset P$ (resp. $Q^{\prime}\subset Q$) denote the subset of
pieces which are neither the minimum nor maximum. Let $y\in\mathbb{R}^{dn}$.
Write

$\begin{split}\pi^{\prime}_{Q}(y)&=(x_{\gamma})_{\gamma\in Q^{\prime}},\\\
\pi^{\prime}_{P}(y)&=(\bar{x}_{\gamma^{\prime}})_{\gamma^{\prime}\in
P^{\prime}},\\\ e_{P,Q}(\pi^{\prime}_{P}(y))&=(x^{\prime}_{\gamma})_{\gamma\in
Q^{\prime}}.\end{split}$

Here, $\pi^{\prime}_{Q}:\mathbb{R}^{dn}\to
e_{Q}(\mathbb{R}^{dq})=\mathbb{R}^{dq}$ is the orthogonal projection, where
$q=\\#Q^{\prime}$, and $\pi^{\prime}_{P}$ is the similar map. Suppose
$y\in\nu_{P}$. Since the image of $e_{P}$ is contained in the image of $e_{Q}$
and the map $\pi^{\prime}_{Q}$ sends $y$ to its closest point in the image of
$e_{Q}$, we have

$|y-e_{Q}(x_{\gamma})|\leq|y-e_{Q}(x^{\prime}_{\gamma})|=|y-e_{P}(\pi^{\prime}_{P}(y))|<\epsilon_{P}\
(<\epsilon_{Q}).$ (1)

So we see $y\in\nu_{Q}$. This means
$\mathbb{R}^{dn}-\nu_{Q}\subset\mathbb{R}^{dn}-\nu_{P}$. We shall show
$\nu_{Q}|_{E_{Q}}\subset(\mathbb{R}^{dn}-\nu_{P})\cup\nu_{P}|_{E_{P}}$. Let
$\alpha\in Q^{\prime}$ be a piece and $\alpha^{\prime}$ be the piece of $P$
including $\alpha$. Suppose $|x_{\alpha}|\geq 1-\rho
c_{\alpha}/2+\epsilon_{Q}$. If $y\not\in\nu_{P}$, we have nothing to prove, so
suppose further $y\in\nu_{P}$. By the inequality (1), we have

$|e_{Q}(x^{\prime}_{\gamma})-e_{Q}(x_{\gamma})|\leq|e_{Q}(x^{\prime}_{\gamma})-y|+|y-e_{Q}(x_{\gamma})|\leq
2\epsilon_{P}.$

As $e_{Q}$ is a composition of diagonal map with parallel transport, we have

$|(x^{\prime}_{\gamma})_{\gamma}-(x_{\gamma})_{\gamma}|\leq
2\epsilon_{P}\quad(\text{so}\ |x^{\prime}_{\gamma}-x_{\gamma}|\leq
2\epsilon_{P}\ \text{for each}\ \gamma\in Q^{\prime}).$ (2)

So we have $|x^{\prime}_{\gamma}-x_{\gamma}|\leq 2\epsilon_{P}$ for each
$\gamma\in Q^{\prime}$. Let $\beta\subset\alpha^{\prime}$ be the set of
elements smaller than the minimum of $\alpha$. We easily see

$\bar{x}_{\alpha^{\prime}}-x^{\prime}_{\alpha}=_{1}\frac{\rho}{2}(c_{\alpha^{\prime}}-2c_{\beta}-c_{\alpha}).$
(3)

Putting these (in)equalities together, we see

$\begin{split}|\bar{x}_{\alpha^{\prime}}|=|\bar{x}_{\alpha^{\prime}}|&\geq|x_{\alpha}|-|x_{\alpha}-x^{\prime}_{\alpha}|-|x^{\prime}_{\alpha}-\bar{x}_{\alpha^{\prime}}|\\\
&\geq 1-\rho
c_{\alpha}/2+\epsilon_{Q}-2\epsilon_{P}-\rho(c_{\alpha^{\prime}}-c_{\alpha})/2\\\
&\geq 1-\rho c_{\alpha^{\prime}}/2+\epsilon_{P}\end{split}$

since we see

$\epsilon_{Q}-2\epsilon_{P}=\frac{1-2\cdot
8^{p-q}}{8^{n-q}}\geq\frac{1-2/8}{8^{n-q}}>\epsilon_{P}.$

We have shown that the first inequality in Definition 3.1(2) for $Q$ and the
condition $y\in\nu_{P}$ imply the corresponding inequality for $P$. Suppose
$x_{\alpha}\leq_{1}-1+\rho c_{\leq\alpha}-\epsilon_{Q}$ and $y\in\nu_{P}$. Let
$\alpha^{\prime}$ be as above. We see

$\begin{split}\bar{x}_{\alpha^{\prime}}&=\bar{x}_{\alpha^{\prime}}-x^{\prime}_{\alpha}+x^{\prime}_{\alpha}-x_{\alpha}+x_{\alpha}\\\
&\leq_{1}\frac{\rho}{2}(c_{\alpha^{\prime}}-2c_{\beta}-c_{\alpha})+2\epsilon_{P}+(-1+\rho
c_{\leq\alpha}-\epsilon_{Q}\\\ &=-1+\rho
c_{\leq\alpha^{\prime}}-(\epsilon_{Q}-2\epsilon_{P})\quad(\because\
c_{\leq\alpha}+(c_{\alpha^{\prime}}-2c_{\beta}-c_{\alpha})/2=c_{\leq\alpha^{\prime}})\\\
&<-1+\rho c_{\leq\alpha^{\prime}}-\epsilon_{P}.\end{split}$

This is the second inequality in Definition 3.1(2) for $P$. Similarly, we see
$\bar{x}_{\alpha^{\prime}}\geq_{1}1-\rho c_{\geq\alpha^{\prime}}+\epsilon_{P}$
if $x_{\alpha}\geq_{1}1-\rho c_{\geq\alpha}+\epsilon_{P}$ and $y\in\nu_{P}$.
Thus, we have shown the claimed inclusion. ∎

######  Lemma 3.3.

Let $Q$ be a subdivision of $P$. Let $\alpha,\beta\in Q$ be two pieces which
are neither minimum nor maximum and satify $\alpha<\beta$. Let
$\alpha^{\prime},\beta^{\prime}\in P$ be pieces which include $\alpha$ and
$\beta$ respectively. Let $\delta^{\prime}_{P,Q}:T_{\emptyset_{Q}}\to
T_{\emptyset_{P}}$ denote the map given in Lemma 3.2. We have

$\delta^{\prime}_{P,Q}(\mathcal{T}_{\alpha\beta})\ \subset\
\left\\{\begin{array}[]{cc}\\{*\\}&(\text{if $\alpha^{\prime}=\beta^{\prime}$
or $\alpha^{\prime}$ is the minimum of $P$ or $\beta^{\prime}$ is the maximum
of $P$}),\\\ \mathcal{T}_{\alpha^{\prime},\beta^{\prime}}&(\text{ otherwise
}).\end{array}\right.$

###### Proof.

Let $y\in\mathbb{R}^{dn}$ be an element. We use the same notations
$P^{\prime},Q^{\prime},\pi^{\prime}_{P},\pi^{\prime}_{Q},x_{\gamma},x^{\prime}_{\gamma}$,
and $\bar{x}_{\gamma^{\prime}}$ as in the proof of Lemma 3.2.

We shall show the claim in the case $\alpha^{\prime}=\beta^{\prime}$. We
assume $\delta^{\prime}_{P,Q}(y)\not=*$. So $y\in\nu_{P}$. By the inequality
(2) in the proof of Lemma 3.2 and the definition of the map $e_{P,Q}$, we have
the following inequality.

$\begin{split}|x_{\alpha}-x_{\beta}|&\geq|x^{\prime}_{\alpha}-x^{\prime}_{\beta}|-|x_{\alpha}-x^{\prime}_{\alpha}|-|x_{\beta}-x^{\prime}_{\beta}|\\\
&\geq\rho c_{\alpha\beta}-4\epsilon_{P}>d_{\alpha\beta}(Q).\end{split}$

This inequality implies $\pi_{Q}(y)\not\in D_{\alpha\beta}$.
We shall show the claim in the case that $\alpha^{\prime}$ is the minimum. It
is enough to show the case of $\beta=\beta^{\prime}$ since general
subdivisions factor through the one of this case. In this case, by definition
of $e_{P,Q}$, $x^{\prime}_{\alpha}=(-1+\rho c_{\leq\alpha})u$. Suppose
$\delta^{\prime}_{P,Q}(y)\not=*$ and $\pi_{Q}(y)\in D_{\alpha\beta}$. We have

$\begin{split}x_{\beta}&=x^{\prime}_{\alpha}+(x_{\beta}-x_{\alpha})-(x^{\prime}_{\alpha}-x_{\alpha})\\\
&\leq_{1}-1+\rho
c_{\leq\alpha}+|x_{\beta}-x_{\alpha}|+|x^{\prime}_{\alpha}-x_{\alpha}|\\\
&\leq-1+\rho c_{\leq\alpha}+\rho c_{\alpha\beta}-\epsilon_{Q}+2\epsilon_{P}\\\
&<-1+\rho c_{\leq\beta}-\epsilon_{P}.\end{split}$

So we have the claim. The case that $\beta^{\prime}$ is the minimum is
completely similar.
We shall show the remaining part of the claim. It is enough to the case that
either of the minimum or maximum of $\alpha^{\prime}$ belongs to $\alpha$.
Clearly, we have $e_{P,Q}(\bar{x}_{\gamma^{\prime}})=(x^{\prime}_{\gamma})$.
As we see in the case of $\alpha^{\prime}=\beta^{\prime}$, we have

$\begin{split}|x^{\prime}_{\alpha}-x^{\prime}_{\beta}|\leq&|x^{\prime}_{\alpha}-x_{\alpha}|+|x_{\alpha}-x_{\beta}|+|x_{\beta}-x^{\prime}_{\beta}|\\\
\leq&4\epsilon_{P}+d_{\alpha\beta}(Q)=\rho
c_{\alpha\beta}-\epsilon_{Q}+4\epsilon_{P}.\end{split}$

By the equality (3) in the proof of Lemma 3.2, we have

$|\bar{x}_{\alpha^{\prime}}-x^{\prime}_{\alpha}|=\rho(c_{\alpha^{\prime}}-c_{\alpha})/2.$

By similar equality for $\beta$, we see

$\begin{split}|\bar{x}_{\alpha^{\prime}}-\bar{x}_{\beta^{\prime}}|\leq&|\bar{x}_{\alpha^{\prime}}-x^{\prime}_{\alpha}|+|x^{\prime}_{\alpha}-x^{\prime}_{\beta}|+|\bar{x}_{\beta^{\prime}}-x^{\prime}_{\beta}|\\\
\leq&\rho(c_{\alpha^{\prime}}-c_{\alpha})/2+(\rho
c_{\alpha\beta}-\epsilon_{Q}+4\epsilon_{P})+\rho(c_{\beta^{\prime}}-c_{\beta})/2\\\
<&d_{\alpha^{\prime},\beta^{\prime}}(P).\end{split}$

Thus, we have shown the claim. ∎

######  Definition 3.4.

  1. (1)

A spectrum $X$ is a sequence of pointed spaces $X_{0},X_{1},\dots$ with
structure map $S^{1}\wedge X_{k}\to X_{k+1}$ for each $k\geq 0$. A morphism
(or map) $f:X\to Y$ of spectra is a sequence of pointed maps $f_{0}:X_{0}\to
Y_{0},f_{1}:X_{1}\to Y_{1},\dots$ compatible with the structure maps. Let
$\mathcal{SP}$ denote the category of spectra and their maps. For a spectrum
$X$, $\pi_{k}(X)$ denotes the colimit of the sequence
$\pi_{k}(X_{0})\to\pi_{k+1}(X_{1})\to\cdots$ defined by the structure maps. A
map $f:X\to Y$ is called a stable homotopy equivalence if it induces an
isomorphism $\pi_{k}(X)\to\pi_{k}(Y)$ for any integer $k$.

  2. (2)

We define a functor $\mathcal{T}^{S}:\mathrm{P}_{n}^{op}\to\mathcal{SP}$ as
follows. Set $\mathcal{T}^{S}(P)_{k}=S^{k-dn}\wedge\mathcal{T}(P)$ if $k\geq
dn$, and $\mathcal{T}^{S}(P)_{k}=*$ otherwise. These spaces form a spectra
with the obvious structure map. The map corresponding to a map $P\to Q$ is
also obviously induced from that of $\mathcal{T}$.

  3. (3)

For a positive number $\delta$, we define a spectrum $\mathbb{S}_{\delta}$ as
follows. We set
$\mathbb{S}_{\delta,k}=\\{y\in\mathbb{R}^{k}\\}/\\{y\mid|y|\geq\delta\\}$. The
structure map $S^{1}\wedge\mathbb{S}_{\delta,k}\to\mathbb{S}_{\delta,k+1}$ is
the obvious collapsing map.

  4. (4)

We define a functor $\mathcal{E}^{\dagger}:\mathrm{P}_{n}^{op}\to\mathcal{SP}$
as follows. Set
$\mathcal{E}^{\dagger}(P)=\mathrm{Map}(\mathcal{E}(P)_{+},\mathbb{S}_{\delta})$
where $\delta=\epsilon_{P}/4$. For a map $P\to Q$, the corresponding map is
the restriction of the source followed by the pushforward by the collapsing
map $\mathbb{S}_{\epsilon_{Q}/4}\to\mathbb{S}_{\epsilon_{P}/4}$.

  5. (5)

We define a functor $\mathcal{E}^{\vee}:\mathrm{P}_{n}^{op}\to\mathcal{SP}$ as
follows. Set
$\mathcal{E}^{\vee}(P)=\mathrm{Map}(\mathcal{E}(P)_{+},\mathbb{S})$ For a map
$P\to Q$, the corresponding map is the restriction of the source.

  6. (6)

We define a map
$\widetilde{\Phi}=\widetilde{\Phi}_{P,k}:\mathbb{R}^{k}\to\mathcal{E}^{\dagger}(P)_{k}$
by

$\mathbb{R}^{k}\ni
y\longmapsto\\{(x_{\gamma})\mapsto(y-(0,e_{P}(x_{\gamma}))\\}\in\mathcal{E}^{\dagger}(P)/$

$\mathcal{T}^{S}_{k}$ is naturally identified with Thom space associated to
the tubular neighborhood $\mathbb{R}^{k-dn}\times\nu_{P}$ of the embedding
$0\times e_{P}:\mathbb{R}^{dp}\to\mathbb{R}^{k}$ (with some extra collapsed
points). $\Phi_{P,k}$ factors through $\mathcal{T}^{S}(P)_{k}$ as in Theorem
3.5, and these maps form a natural transformation
$\Phi:\mathcal{T}^{S}\to\mathcal{E}^{\dagger}$. We see that this is well-
defined below.

  7. (7)

A natural transformation $p_{*}:\mathcal{E}^{\vee}\to\mathcal{E}^{\dagger}$ is
defined by the pushforward by the collapsing map
$p:\mathbb{S}\to\mathbb{S}^{\delta}$.

The following equivalence is a variation of the one given in [18]. If it is
projected to the stable homotopy category, it is a special case of Atiyah
duality which is also a special case of $S$-duality. We need point-set level
compatibility so we have been taking care about parameters.

######  Theorem 3.5.

Under the notations of Definition 3.4, the map $\Phi$ is well-defined, and the
two maps $\Phi$ and $p_{*}$ are stable homotopy equivalences.

###### Proof.

We shall show the map $\widetilde{\Phi}$ factors through
$\mathcal{T}^{S}(P)_{k}$. For notational simplicity, we consider the case of
$k=dn$. The other cases will follow completely similarly. It is clear that
$\widetilde{\Phi}(\mathbb{R}^{dn}-\nu_{P})=\\{*\\}$. Let $y\in\mathbb{R}^{k}$
be an element with $\widetilde{\Phi}(y)\not=*$. There exists an element
$(x_{\gamma})\in\mathcal{E}(P)$ such that
$|y-e_{P}(x_{\gamma})|<\epsilon_{P}/4$ holds. So we have
$|y-e_{P}(\pi_{P}y)|<\epsilon_{P}/4$ and

$|\pi_{P}y-(x_{\gamma})|\leq|e_{P}(\pi_{P}y)-e_{P}(x_{\gamma})|\leq|e_{P}(\pi_{P}y)-y|+|y-e_{P}(x_{\gamma})|<\epsilon_{P}/2$

If we write $\pi_{P}(y)=(\bar{x}_{\gamma})$, it follows that
$|\bar{x}_{\alpha}-x_{\alpha}|<\epsilon_{P}/2$ for each non-minimum and non-
maximum $\alpha\in P$. We see

$\begin{split}|\bar{x}_{\alpha}|&\leq|x_{\alpha}|+|\bar{x}_{\alpha}-x_{\alpha}|\\\
&\leq 1-\rho c_{\alpha}/2+\epsilon_{P}/2<1-\rho c_{\alpha}/2+\epsilon_{P},\\\
\bar{x}_{\alpha}&=x_{\alpha}+(\bar{x}_{\alpha}-x_{\alpha})\\\ &\geq_{1}-1+\rho
c_{\leq\alpha}-|\bar{x}_{\alpha}-x_{\alpha}|\\\ &\geq-1+\rho
c_{\leq\alpha}-\epsilon_{P}/2>-1+\rho c_{\leq\alpha}-\epsilon_{P},\\\
\bar{x}_{\alpha}&\leq x_{\alpha}+(\bar{x}_{\alpha}-x_{\alpha})\\\
&\leq_{1}1-\rho c_{\geq\alpha}+|\bar{x}_{\alpha}-x_{\alpha}|\\\ &\leq 1-\rho
c_{\geq\alpha}+\epsilon_{P}/2<1-\rho c_{\geq\alpha}+\epsilon_{P}.\\\
\end{split}$

These inequalities imply $\Phi(\pi_{P}^{-1}(E_{\alpha}))=*$ in the notations
of Definition 3.1. We also see

$\begin{split}|\bar{x}_{\alpha}-\bar{x}_{\beta}|&\geq|x_{\alpha}-x_{\beta}|-|x_{\alpha}-\bar{x}_{\alpha}|-|x_{\beta}-\bar{x}_{\beta}|\\\
&>\rho c_{\alpha\beta}-\epsilon_{P}=d_{\alpha\beta}(P).\end{split}$

This implies $\widetilde{\Phi}(\pi_{P}^{-1}(D_{\alpha\beta}))=*$. Thus,
$\widetilde{\Phi}$ factors through $\mathcal{T}^{S}$. Now the claim of the
theorem follows from the classical Atiyah duality (see [2] for example).

∎

######  Remark 3.6.

In [14], Malin proved homotopy invariance of homogeneous layers of stable
embedding calculus tower using a duality similar to Theorem 3.5.

######  Definition 3.7.

  1. (1)

Let $\mathcal{CH}_{\mathsf{k}}$ be the category of chain complexes and chain
maps over $\mathsf{k}$.

  2. (2)

For a chain complex $C_{*}$, $C_{*}[k]$ is the chain complex given by
$C_{l}[k]=C_{k+l}$ with the same differential as $C_{*}$ (without extra sign).

  3. (3)

Functors $C^{*}(\mathcal{PK}),\
\bar{C}_{*}(\mathcal{T}):\mathrm{P}_{n}^{op}\to\mathcal{CH}_{\mathsf{k}}$ is
given by taking cochains and reduced chains of $\mathcal{PK}$ and
$\mathcal{T}$ in the objectwise manner respectively.
$\bar{C}_{*}(\mathcal{T})[dn]:\mathrm{P}_{n}^{op}\to\mathcal{CH}_{\mathsf{k}}$
is given by taking the shift of $\bar{C}_{*}(\mathcal{T})$ in the objectwise
manner.

######  Proposition 3.8.

$C^{*}(\mathcal{PK})$ and $\bar{C}_{*}(\mathcal{T})[dn]$ are connected by a
zigzag of natural quasi-isomorphisms

###### Proof.

In [18], a chain functor $C_{*}$ for symmetric spectra is defined. The same
definition works for our category of spectra as it is, so we adopt this
functor in this proof. It preserves stable equivalence between semistabe
spectra. By Lemma 5.3 of [18], Lemma 2.7, and Theorem 3.5. we have the
following chain of natural quasi-isomorphisms

$C^{*}(\mathcal{PK})\simeq C^{*}(\mathcal{E})\simeq
C_{*}(\mathcal{E}^{\vee})\simeq C_{*}(\mathcal{T}^{S})\simeq
C_{*}(\mathcal{T})[dn],$

where the last morphism is the canonical one in view of definition of the
chain functor. ∎

##  4\. Spectral sequences

######  Definition 4.1.

  1. (1)

For a partition $P\in\mathrm{P}_{n}$, a graph $G$ on $P$ consists of the set
of vertexes $V(G)=P$, a finite set of edges $E(G)$ and a map $\phi_{G}:E(G)\to
P_{1,2}(P)$ called the incidence map, where $P_{1,2}(P)=\\{S\subset
P\mid\\#S=1\text{ or }2\\}$. So, the vertexes of $G$ are the pieces of $P$. We
say that an element of $\phi_{G}(e)$ is incident with an edge $e$. An edge $e$
is called a loop if $\\#\phi_{G}(e)=1$. two edges $e,e^{\prime}$ are called
double edges if $\phi_{G}(e)=\phi_{G}(e^{\prime})$ and $\\#\phi_{G}(e)=2$.
(Other edges may have the same incident vertexes as double edges.) Let
$\tilde{\mathsf{G}}(P)$ denote the set of all graphs on $P$. Let $\gamma_{m}$
and $\gamma_{M}$ be the minimum and maximum pieces of $P$, respectively.
$\mathsf{G}(P)\subset\tilde{\mathsf{G}}(P)$ denotes the subset of graphs with
the edge set $E(G)\subset\\{(\alpha,\beta)\mid\alpha,\beta\in P,\
\gamma_{m}<\alpha<\beta<\gamma_{M}\\}$ and the natural incidence map
$(\alpha,\beta)\mapsto\\{\alpha,\beta\\}$. The set $E(G)$ of a graph
$G\in\mathsf{G}(P)$ is regarded as a totally ordered set by the
lexicographical order. Let $\emptyset_{P}\in\mathsf{G}(P)$ denote the graph
with empty edge set. We sometimes denote a graph in $\mathsf{G}(P)$ by a
formal product of edges (see Example 4.2 below). Let $e$ be the $i$-th edge of
$G\in\mathsf{G}(P)$. $\partial_{e}G$ (or $\partial_{i}G$) denotes the subgraph
of $G$ with $E(\partial_{e}G)=E(G)-\\{e\\}$. Similarly, if $e^{\prime}$ is the
$j$-th edge, $\partial_{ee^{\prime}}G$ (or $\partial_{ij}G$) denotes the
subgraph made by removing $e$ and $e^{\prime}$. For two vertexes
$\alpha,\beta$ of a graph $G$, we write $\alpha\sim_{G}\beta$ when $\alpha$
and $\beta$ belong to the same connected component of $G$. By abusing
notations, for $i\in[n+1]$, we write $i\sim_{G}\beta$ if there is a (unique)
piece $\alpha$ satisfying $i\in\alpha$ and $\alpha\sim_{G}\beta$. Similarly,
if $i$ belongs to a piece $\alpha$ which belongs to a connected component $S$
of $G$, we write $i\in S$ instead of $i\in\alpha\in S$.

  2. (2)

For a map $P\to Q$ of partitions, $\delta_{P,Q}:Q\to P$ denotes the map of
sets sending $\alpha\in Q$ to the piece of $P$ containing $\alpha$. This map
induces a map $\delta_{P,Q}:\tilde{\mathsf{G}}(Q)\to\tilde{\mathsf{G}}(P)$.
The graph $\delta_{P,Q}(G)$ has the same edge set as $G$ and the incidence map
given by the composition

$E(G)\stackrel{{\scriptstyle\phi_{G}}}{{\to}}P_{1,2}(P)\stackrel{{\scriptstyle(\delta_{P,Q})_{*}}}{{\to}}P_{1,2}(Q).$

For a graph $G\in\mathsf{G}(Q)$, if $\delta_{P,Q}(G)$ has no loop nor double
edges and the minimum and maximum pieces of $P$ are not incident with any edge
of $\delta_{P,Q}(G)$, we always identify $\delta_{P,Q}(G)$ with a unique graph
in $\mathsf{G}(P)$ which has the same image of the incidence map. If $P$ is
made by unifying the $i+1$-th and $i+2$-th pieces of $Q$, we denote $P$ and
$\delta_{P,Q}$ by $\delta_{i}Q$ and $\delta_{i}$, respectively. Similarly, if
$P$ is made by unifying each of the $i+1$-th and $i+2$-th pieces, and $j+1$-th
and $j+2$-th pieces, $P$ and $\delta_{P,Q}$ are denoted by $\delta_{ij}Q$ and
$\delta_{ij}$. $\delta_{ijk}$ is similarly understood.

######  Example 4.2.

Let $G_{1},G_{2}\in\mathsf{G}([5])$ be the graphs with two edges given by

$G_{1}=(1,4)(2,3),\quad G_{2}=(1,3)(2,4).$

These graphs are drawn in Figure 1. The straight line is not a part of data,
and the chords denote the edges. The vertexes are intersections of the line
and chords. They are labeled by $\\{1\\},\dots,\\{4\\}$ in the order from left
to right. The minimum and maximum vertexes $\\{0\\},\\{5\\}$ are usualy
omitted. We have $E(\partial_{1}G_{1})=\\{(2,3)\\}$ and
$E(\partial_{2}G_{1})=\\{(1,4)\\}$. $G_{1}$ has the four connected components
$\\{\\{0\\}\\}$, $\\{\\{1\\},\\{4\\}\\}$, $\\{\\{2\\},\\{3\\}\\}$, and
$\\{\\{5\\}\\}$. By definition, we have

$\delta_{1}[5]=\\{\\{0\\},\\{12\\},\\{3\\},\\{4\\},\\{5\\}\\},$

and $\delta_{1}G_{1}=\delta_{1}G_{2}$ since
$E(\delta_{1}G_{1})=E(\delta_{1}G_{2})=\\{(\\{12\\},\\{3\\}),(\\{12\\},\\{4\\})\\}$.
Similarly, we have $\delta_{3}G_{1}=\delta_{3}G_{2}$. The graph
$\delta_{1}G_{1}$ has the three connected components $\\{\\{0\\}\\}$,
$\\{\\{12\\},\\{3\\},\\{4\\}\\}$, and $\\{\\{5\\}\\}$.

######  Definition 4.3.

For a graph $G\in\mathsf{G}(P)$, set
$\mathcal{T}_{G}=\bigcap_{(\alpha,\beta)\in
E(G)}\mathcal{T}_{\alpha\beta}\subset T_{\emptyset_{P}}$.
We define a triple complex $\mathbb{T}_{\bullet*\star}$ as follows. As a
module, we set

$\mathbb{T}_{pqs}=\bigoplus_{P,G}\bar{C}_{q}(\mathcal{T}_{G})$

where $P$ runs through partitions of $P_{n}$ such that $\\#P=p+2$ and $G$ runs
through graphs of $\mathsf{G}(P)$ such that $\\#E(G)=s$. $\mathbb{T}$ has
three differentials $\delta,d,\partial$ of degree $(1,0,0)$, $(0,1,0)$,
$(0,0,1)$. $d$ denotes the differential on singular chains, $\partial$ is the
Cech differential given by

$\partial=\sum_{1\leq k\leq s}(-1)^{k-1}\partial_{k},$

where
$\partial_{k}:\bar{C}_{q}(\mathcal{T}_{G})\to\bar{C}_{q}(\mathcal{T}_{\partial_{k}G})$
is the pushforward by the inclusion. To define $\delta$, we shall define a map
$\delta_{i}$ of degree $(1,0,0)$. Put $P=\\{\alpha_{0}<\cdots\alpha_{p+1}\\}$.
If $\delta_{i}G$ has a loop or double edge, or at least one of the connected
components of $\delta_{i}G$ which include the minimum or maximum is not
discrete, we set $\delta_{i}=0$ on $\bar{C}_{*}(\mathcal{T}_{G})$. Suppose
otherwise i.e., $\delta_{i}G$ belongs to $\mathsf{G}(P_{i})$. The set of edges
of $G$ whose smaller incident vertex is $\alpha_{i}$ or $\alpha_{i+1}$ and the
set of edges of $\delta_{i}(G)$ whose smaller incident vertex is
$\alpha_{i}\cup\alpha_{i+1}$ are one-to-one correspondence via the incidence
map. This correspondence induces the permutation $\sigma_{G,i}$ of the
lexicographical order of the edges. We set
$\delta_{i}=sgn(\sigma_{G,i})(\delta^{\prime}_{i})_{*}$. Here,
$\delta^{\prime}_{i}:\mathcal{T}_{G}\to\mathcal{T}_{\delta_{i}G}$ is the
collapsing map for subdivision $P$ of $\delta_{i}P$ given in Lemma 3.3. (By
definition of $\mathcal{T}_{G}$ and the same lemma, the image of
$\delta^{\prime}_{i}$ is contained in $\mathcal{T}_{\delta_{i}G}$.) We set
$\delta=\sum_{i=0}^{p}(-1)^{i}\delta_{i}$.

######  Lemma 4.4.

The triple complex $\mathbb{T}$ is well defined. i.e., the three differentials
are commutative without signs.

###### Proof.

Only non-trivial commutativity is that of $\partial$ with $\delta$. Let $e$ be
the $k$-th edge among those whose smaller incident vertex is $\alpha_{i}$.
Suppose exactly $m$ edges pass through $e$ by $\sigma_{G,i}$. On $G$, the sign
in removing $e$ is $(-1)^{k}$ and On $\delta_{i}G$, the sign in removing the
corresponding edge is $(-1)^{k+m}$. On the other hand, we have
$sgn(\sigma_{G,i})=sgn(\sigma_{\partial_{e}G,i})$. So we have the
commutativity. ∎

######  Definition 4.5.

  1. (1)

Let $\operatorname{Tot}(\mathbb{T})$ be a total complex given by

$\operatorname{Tot}_{k}(\mathbb{T})=\bigoplus_{p+q+s=k}\mathbb{T}_{pqs}$

with the differential $\tilde{D}=\delta+(-1)^{p}d+(-1)^{p+q}\partial$ on
$\mathbb{T}_{pqs}$. Let $\\{F_{l}\\}_{l}$ be a filtration of
$\operatorname{Tot}(\mathbb{T})$ given by $F_{l}=\oplus_{p\leq
l}\mathbb{T}_{pqs}$. This filtration induces a spectral sequence
$\check{\mathbb{E}}_{r}$. We give the grading on $\check{\mathbb{E}}_{r}$ such
that $\check{\mathbb{E}}_{r}^{-p,q}$ is a subquotient of the submodule
$\oplus_{a,b}\mathbb{T}_{pab}$ where $(a,b)$ runs through pairs satisfying
$a+b=dn-q$. We also consider a truncated version
$tr_{m}(\operatorname{Tot}(\mathbb{T}))$ which consists of the chains in
$\mathbb{T}_{pqs}$ with $s\geq m$. The differential is given by the same
formula as $\tilde{D}$ but the $\check{\mathrm{C}}$ech differential on the
graph with exactly $m$ edges are understood as zero. Considering the
filtration on $p$, we form a spectral sequence
$\\{tr_{m}\check{\mathbb{E}}_{r}\\}_{r}$. There is an obvious map
$\operatorname{Tot}(\mathbb{T})\to tr_{m}\operatorname{Tot}(\mathbb{T})$ given
by the identity on $\mathbb{T}_{pqs}$ with $s\geq m$ and zero on the other
summands. This map commutes with differentials and filtrations so induces the
map of spectral sequences $\check{\mathbb{E}}_{r}\to
tr_{m}\check{\mathbb{E}}_{r}$.

  2. (2)

Let $D(C^{*}(\mathcal{K}^{\leq n}_{d}))$ be the unnormalized complex of
singular cochain of Sinha’s cosimplicial space restricted to $\Delta_{\leq n}$
(see [21]). So, $D_{p}(C^{*}(\mathcal{K}_{d}^{\leq
n}))=C^{*}(\mathcal{K}_{d}^{p})$ for $p\leq n$, and it is zero for $p>n$, and
the differential is the signed sum of face maps. We give this double complex
the filtration by cosimplicial degree $p$. The resulting spectral sequence is
denoted by $\bar{\mathbb{E}}_{r}$. This is a version of Sinha’s spectral
sequence restricted to the part of cosimplicial degree $\leq n$. The (full)
Sinha’s spectral sequence $\mathbb{E}_{r}$ is defined as the Bousfield-Kan
type cohomological spectral sequence associated the cosimplicial space
$\mathcal{K}^{\bullet}_{d}$ (see Definition 2.4).

First part of the following lemma follows from Lemma 2.5 and Proposition 3.8.
Second part is seen by observing the spectral sequence associated to
$(\mathbb{T}_{p\,*\,\star},d,\partial)$.

######  Lemma 4.6.

  1. (1)

The spectral sequence $\\{\bar{\mathbb{E}}_{r}\\}_{r}$ is isomorphic to
$\\{\check{\mathbb{E}}_{r}\\}_{r}$ after $E_{1}$-page.

  2. (2)

The natural map $\check{\mathbb{E}}_{1}^{pq}\to
tr_{m}\check{\mathbb{E}}_{1}^{pq}$ induces an isomorphism between the summands
labeled by the graphs with a larger number of edges than $m$ and a
monomorphism between the summands of graphs with exactly $m$ edges. ∎

##  5\. Condensed maps

In this section, we define a class of maps used to define chains in the triple
complexes $\mathbb{T}$ and prove their properties.

######  Definition 5.1.

  1. (1)

Let $X$ be an unpointed topological space and $G\in\mathsf{G}(P)$ a graph. A
map $f=(f_{1},\dots,f_{n}):X\to\mathbb{R}^{dn}$ is $G$-condensed if the
following conditions hold.

    1. (a)

$f$ is proper.

    2. (b)

For each connected component $S$ of $G$, there is a vertex $\alpha\in S$,
called a base of $S$ (for $(G,f)$), which satisfies the following conditions.

      1. (i)

For each $\beta\in S$ and each $i\in\beta$, $f_{i}(x)$ belongs to the convex
hull of $\\{f_{j}(x)\mid j\in\alpha\\}$,

      2. (ii)

there is no edge in $S$ both of whose incident vertexes are smaller than
$\alpha$.

(A discrete vertex is regarded as a base.)

  2. (2)

An edge $e$ of a graph $G$ is called a bridge if
$\\#\pi_{0}(\partial_{e}G)>\\#\pi_{0}(G)$.

  3. (3)

Let $f:X\to\mathbb{R}^{dn}$ be a map, and $e=(\alpha,\beta),\ e^{\prime}$ two
bridges of a graph $G$. For $1\leq i\leq n$ and $s\in[0,\infty)$, we set

$A_{e}^{i}(s)=\left\\{\begin{array}[]{cl}s&\text{if
$i\sim_{\partial_{e}G}\alpha$}\\\ -s&\text{if $i\sim_{\partial_{e}G}\beta$}\\\
0&\text{otherwise}\end{array}\right.$

and define $A_{e}:[0,\infty)\to\mathbb{R}^{dn}$

$A_{e}(s)=(A_{e}^{i}(s)v)_{1\leq i\leq n},$

where $v=(0,1,0,\dots,0)$ as in subsection 1.1. The contracting homotopy
$F:X\times[0,\infty)\to\mathbb{R}^{dn}$ of $f$ in removing $e$ from $G$ (in
short, $e$-contraction of $f$ for $G$) is defined by

$F(x,s)=f(x)+A_{e}(s).$

The $(e,e^{\prime})$-contraction
$F^{\prime}:X\times[0,\infty)^{2}\to\mathbb{R}^{dn}$ of $f$ for $G$ is defined
by

$F^{\prime}(x,s_{1},s_{2})=f(x)+A_{e}(s_{1})+A_{e^{\prime}}(s_{2}).$

  4. (4)

For two maps $f,g:X\to\mathbb{R}^{dn}$, the straight homotopy $h:X\times
I\to\mathbb{R}^{dn}$ from $f$ to $g$ is defined by $h(x,t)=(1-t)f(x)+tg(x)$.

  5. (5)

Let $G\in\mathsf{G}([n+1])$ be a graph with exactly $4$ connected components.
Define a map $f_{G}=(f_{1},\dots f_{n}):\mathbb{R}^{2d}\to\mathbb{R}^{dn}$ by

$f_{i}(x,y)=\left\\{\begin{array}[]{cc}x&(i\sim_{G}1)\\\
y&(\text{otherwise}).\end{array}\right.$

Clearly, $f_{G}$ is $G$-condensed, and a vertex is a base if it satisfies the
condition (ii) on a base. So, the minimum and the second to minimum vertexes
of each connected component are bases.

  6. (6)

For a graph $G\in\mathsf{G}(P)$, we define subsets $U_{G}$ and $U^{1}_{G}$ of
$\mathbb{R}^{dn}$ by

$\begin{split}U_{G}=&(\mathbb{R}^{dn}-\nu_{P})\cup\pi_{P}^{-1}(\cap_{(\alpha,\beta)\in
E(G)}D_{\alpha\beta}),\\\
U_{G}^{1}=&(\mathbb{R}^{dn}-p_{1}^{-1}p_{1}(\nu_{P}))\cup
p_{1}^{-1}p_{1}(\cap_{(\alpha,\beta)\in
E(G)}\pi_{P}^{-1}D_{\alpha\beta}).\end{split}$

Here, $p_{1}:\mathbb{R}^{dn}\to\mathbb{R}^{n}$ is the product of the
projection to the first coordinate $\mathbb{R}^{d}\to\mathbb{R}$.

######  Example 5.2.

Set $n=4$ and let $G_{1}=(1,4)(2,3),G_{2}=(1,3)(2,4)\in\mathsf{G}([5])$.
$f_{G_{1}}$ (resp. $f_{G_{2}}$) is given by $(x,y)\mapsto(x,y,y,x)$ (resp.
$(x,y)\mapsto(x,y,x,y)$). Put $e=(1,4)$ and $e^{\prime}=(2,3)$. The
$e$-contraction of $f_{G_{1}}$ for $G_{1}$ is given by
$(x,y,s)\mapsto(x+sv,y,y,x-sv)$ and the $(e,e^{\prime})$-contraction of
$f_{G_{1}}$ for $G_{1}$ is given by
$(x,y,s_{1},s_{2})\mapsto(x+s_{1}v,y+s_{2}v,y-s_{2}v,x-s_{1}v)$. $f_{G_{1}}$
is also $\delta_{1}G_{1}$-condensed. $\\{1,2\\}$ is a base of a connected
component of $\delta_{1}G_{1}$. The other vertexes (except for $\\{0\\}$ and
$\\{5\\}$) are not bases. The $e$-contraction of $f_{G_{1}}$ for
$\delta_{1}G_{1}$ is given by $(x,y,s)\mapsto(x+sv,y+sv,y+sv,x-sv)$ and is
different from that for $G_{1}$ (since connected components are different).
The straight homotopy $h$ from $f_{G_{1}}$ to $f_{G_{2}}$ is given by
$(x,y,t)\mapsto(x,y,(1-t)y+tx,(1-t)x+ty)$. This is also
$\delta_{1}G_{1}$-condensed. its base is $\\{1,2\\}$.

######  Lemma 5.3.

Let $f:X\to\mathbb{R}^{dn}$ be a $G$-condensed map. We have $f(X)\subset
U_{G}\cap U^{1}_{G}$. In particular, $f$ induces a map
$f:X^{*}\to\mathcal{T}_{G}$.

###### Proof.

Let $S$ be a connected component of $G$, $\alpha=\\{j,j+1,\dots,k\\}$ and
$\beta=\\{l,\dots,m\\}$ two vertexes of $S$. For $x\in X$, suppose
$f(x)\in\nu_{P}$. Write $\pi_{P}(f(x))=(y_{\gamma})$. First suppose further
that $\alpha$ is a base of $S$. By definitions of $e_{P}$ and $\nu_{P}$, for
$i\in\beta$, we have

$|f_{i}(x)-y_{\beta}|\leq\rho\frac{c_{\beta}-\min\\{c_{l},c_{m}\\}}{2}+\epsilon_{P}.$

Since $f_{i}(x)$ belongs to the convex hull of $\\{f_{r}(x)\\}_{r\in\alpha}$,
we have

$|f_{i}(x)-y_{\alpha}|\leq\rho\frac{c_{\alpha}-\min\\{c_{j},c_{k}\\}}{2}+\epsilon_{P}.$

So, we have
$|y_{\alpha}-y_{\beta}|<\rho(c_{\alpha}+c_{\beta})/2-\epsilon_{P}\leq
d_{\alpha\beta}(P)$, which implies $\pi_{P}(f(x))\in D_{\alpha\beta}$. If
neither of $\alpha$ nor $\beta$ is a base, we take a base $\gamma$ of $S$. By
definition of the base, we may assume $\alpha<\gamma<\beta$ or
$\gamma<\alpha<\beta$. By the above argument, we see

$\begin{split}|y_{\alpha}-y_{\beta}|&<\frac{\rho}{2}(c_{\alpha}+c_{\beta}+2c_{\gamma}-\min\\{c_{l},c_{m}\\}-\min\\{c_{j},c_{k}\\})+4\epsilon_{P}\\\
&<d_{\alpha\beta}(P).\end{split}$

The second inequality follows from the condition on $c_{r}$ imposed in
Definition 2.3 (1). We have shown $f(X)\subset U_{G}$. The proof of the
inclusion to $U^{1}_{G}$ is completely similar. Since $f$ is proper, $f$
induces the map from the one-point compactification. ∎

######  Lemma 5.4.

Let $Q$ be a partition, $G\in\mathsf{G}(Q)$ a graph, and
$f:X\to\mathbb{R}^{dn}$ a $G$-condensed map.

  1. (1)

Let $e$ be a bridge of $G$ such that each non-discrete connected component of
$\partial_{e}G$ contains a base of some connected component of $G$. The
$e$-contraction $F$ of $f$ is $\partial_{e}G$-condensed.

  2. (2)

Let $e,e^{\prime}$ be two bridges of $G$ such that each non-discrete connected
component of $\partial_{ee^{\prime}}G$ contains a base of some connected
component of $G$. The $(e,e^{\prime})$-contraction $F$ of $f$ is
$\partial_{ee^{\prime}}G$-condensed.

  3. (3)

Set $P=\delta_{i}Q$. Suppose that $\delta_{i}G\in\mathsf{G}(P)$ and both of
the $i+1$-th and $i+2$-th pieces of $Q$ are bases for $f$. Then, $f$ is
$\delta_{i}G$-condensed.

###### Proof.

For part 1, we only have to prove that $F$ is proper. The other conditions are
clear by definition. Let $\beta$ and $\gamma$ be the vertexes incident with
$e$, $\alpha$ a base of the connected component of $G$ including $e$. Say
$\alpha\not\sim_{\partial_{e}G}\beta$ and $\gamma<\beta$. Suppose $F(x)\in
D^{dn}_{R}=(D^{d}_{R})^{n}$, the product of disks of radius $R$. For
$i\in\beta$ and $j\in\alpha$, we have

$2R\geq|F_{i}(x)-F_{j}(x)|=|f_{i}(x)-sv-(f_{j}(x)+sv)|\geq||f_{i}(x)-f_{j}(x)|-2s|.$

This implies $s\leq R+|f_{i}(x)-f_{j}(x)|/2$. Since the convex hulls of
$\\{f_{j}(x)+sv\\}_{j\in\alpha}$ and $\\{f_{j}(x)\\}_{j\in\alpha}$ are
congruent and the diameter of former one is $2R$ or less. So, we have
$|f_{i}(x)-f_{j}(x)|\leq 2R$. By these inequalities, we have $s\leq 2R$, which
implies $|f_{k}(x)|\leq 3R$ for a number $k$ belonging to a piece included in
$S$. Thus, we have $F^{-1}(D^{dn}_{R})\subset f^{-1}(D^{dn}_{3R})$ and
conclude that $F$ is proper. Other conditions of $\partial_{e}G$-condensed map
is clear from the assumption. Parts 2 and 3 are similar. ∎

The proof of the following lemma is clear and omitted.

######  Lemma 5.5.

  1. Let $G\in\mathsf{G}(P)$ be a graph.

  2. (1)

Let $f,g:X\to\mathbb{R}^{dn}$ be two $G$-condensed maps which have a common
base $\alpha$ in each connected component $S$ satisfying $f_{i}=g_{i}$ for
each $i\in\alpha$. Then, the straight homotopy $h$ from $f$ to $g$ is
$G$-condensed, so it induces a map $h:X^{*}\wedge(I_{+})\to\mathbb{R}^{dn}$.

  3. (2)

Let $f:X\to\mathbb{R}^{dn}$ be a $G$-condensed map and $e$ an edge of $G$.
Suppose that $\delta_{i}G\in\delta_{i}P$, the $i+1$-th and $i+2$-th pieces of
$P$ are bases for $f$, each non-discrete connected component of
$\partial_{e}G$ contains a base of some connected component of $G$, and $e$ is
a bridge of $\delta_{i}G$. Let $F$ (resp. $F^{\prime}$) be the $e$-contraction
of $f$ for $G$ (resp. $\delta_{i}G$.) The straight homotopy from $F$ to
$F^{\prime}$ is $\delta_{k}\partial_{e}G$-condensed.

  4. (3)

Let $f:X\to\mathbb{R}^{dn}$ be a $G$-condensed map and $e,e^{\prime}$ be two
edges of $G$. Suppose that $\delta_{i}G\in\delta_{i}P$, the $i+1$-th and
$i+2$-th pieces of $P$ are bases for $f$, each non-discrete connected
component of $\partial_{ee^{\prime}}G$ contains a base of some connected
component of $G$, and $e$, $e^{\prime}$ are bridges of $\delta_{i}G$. Let $F$
(resp. $F^{\prime}$) be the $(e,e^{\prime})$-contraction of $f$ for $G$ (resp.
$\delta_{i}G$.) The straight homotopy from $F$ to $F^{\prime}$ is
$\delta_{i}\partial_{ee^{\prime}}G$-condensed.

∎

Notation and terminology :  As written in Lemmas 5.3, 5.4, and 5.5, we use the
same symbol for the induced map between the pointed spaces as the original
condensed map. We also use the terms ‘contraction’ and ‘straight homotopy’ for
the induced map.

The following lemma is clear.

######  Lemma 5.6.

Let $f=(f_{1},\dots,f_{n}):X\to\mathbb{R}^{dn}$ be a proper map, $x\in X$, and
$P\in\mathrm{P}_{n}$.Put $f_{0}(x)=(-1+\rho c_{0}/2)u$ and $f_{n+1}(x)=(1-\rho
c_{n+1}/2)u$. For the induced map $f:X^{*}\to\mathcal{T}_{\emptyset_{P}}$, if
$f(x)\not=*$, the following inequalities hold for any $\alpha\in P$, numbers
$1\leq k\leq n$, and $i,j\in\alpha$ with $i<j$:

$\begin{split}-1+\rho c_{\leq k}-2\epsilon_{P}&<_{1}f_{k}(x)<_{1}1-\rho
c_{\geq k}+2\epsilon_{P},\\\ \rho
c_{ij}-2\epsilon_{P}&<_{1}f_{j}(x)-f_{i}(x)<_{1}\rho
c_{ij}+2\epsilon_{P}.\end{split}$

In particular if $f_{j}(x)\leq_{1}f_{i}(x)$ for $i,j\in\alpha$ with $i<j$,
$f(x)=*$.

######  Lemma 5.7.

Let $P\to Q\in\mathrm{P}_{n}$ be a subdivision, $G\in\mathsf{G}(Q)$ a graph
with an edge $e=(\alpha,\beta)$.

  1. (1)

Let $f:X\to\mathbb{R}^{dn}$ be a proper map whose image is contained in
$U^{1}_{G}$. Suppose that either of the following two conditions holds.

    1. (a)

$\alpha\cup\beta$ is included in a piece of $P$.

    2. (b)

At least one of $\alpha$, $\beta$ is included in either of the minimum or
maximum of $P$.

Then, the induced map $f:X^{*}\to\mathcal{T}_{\emptyset_{P}}$ is $*$ (the
constant map to the base point). In particular, if $f$ is the $e$-contraction
of some $G$-condensed map for $G$, or more generally, $f_{l}=_{1}g_{l}$
($1\leq l\leq n$) for some $G$-condensed map $g$, the induce map to
$\mathcal{T}_{\emptyset_{P}}$ is $*$.

  2. (2)

Let $f:X\to\mathbb{R}^{dn}$ be a $G$-condensed map. Let $\gamma$ be a piece of
$Q$ with $\beta>\alpha<\gamma$, Suppose that $\alpha$ is a base for $f$,
$\gamma\sim_{G}\alpha$, and $\beta\cup\gamma$ is included in a piece of $P$.
The $e$-contraction of $f$ followed by the collapsing map
$\delta_{P,Q}|_{\mathcal{T}_{\partial_{e}G}}:\mathcal{T}_{\partial_{e}G}\to\mathcal{T}_{\emptyset_{P}}$
is the constant map to $*$.

###### Proof.

We shall show part 1 for the condition (a). For simplicity, we assume
$\alpha<\beta$ and $\alpha$ is a base of $f$. Let $i$ (resp. $j$) be the
minimum of $\beta$ (resp. the maximum of $\alpha$. Suppose $f(x)\in\nu_{P}$
for some $x\in X$. By definitions of $e_{P}$ and $\nu_{P}$, we have
$f_{i}(x)-f_{j}(x)>_{1}\rho(c_{i}+c_{j})/2-2\epsilon_{P}$. Since $f(x)\in
U^{1}_{G}\cap\nu_{P}$, by an argment similar to the proof of Lemma 3.2, we
have

$f_{i}(x)-f_{j}(x)<_{1}\rho(c_{i}+c_{j})/2-\epsilon_{Q}+2\epsilon_{P}<\rho(c_{i}+c_{j})/2-2\epsilon_{P}.$

These inequalities contradict and we have proved part 1. The proofs of other
parts are similar. ∎

######  Definition 5.8.

Let $F=(F_{1},\dots,F_{n}):X\to\mathbb{R}^{dn}$ be a map. For $\epsilon=+$ or
$-$, we define a map $F^{i\epsilon}:X\times[0,\infty)\to\mathbb{R}^{dn}$
called the $(i,\epsilon)$-contraction of $f$ as follows.

$F^{i\epsilon}_{k}(x)=\left\\{\begin{array}[]{cl}F_{i}(x)+\epsilon
su&(k=i),\\\ F_{i+1}(x)-\epsilon su&(k=i+1),\\\
F_{k}(x)&(\text{otherwise}).\end{array}\right.$

On the right hand side, $\epsilon=\pm$ are regaded as $\pm 1$ respectively.

The $(i,\epsilon)$-contraction is not necessarily $G$-condensed it induces a
map to the Thom space.

######  Lemma 5.9.

Let $f=(f_{1},\dots,f_{n}):X\to\mathbb{R}^{dn}$ be a $G$-condensed map for a
graph $G\in\mathsf{G}(P)$, $e,e^{\prime}$ bridges of $G$ and $i$ a number with
$1\leq i\leq n-1$. Say $i$ belongs to the $k+1$-th piece of $P$. Suppose that
$i\sim_{G}i+1$ and $\delta_{k}\partial_{e}G$ (resp.
$\delta_{k}\partial_{ee^{\prime}}G$) belongs to $\mathsf{G}(\delta_{k}P)$. If
any base of the component including $i$ includes $i^{\prime}=i$ or $i+1$,
suppose further that for any $x\in X$ there exist $l,m\not=i,i+1$ such that

$|f_{i^{\prime},1}(x)-f_{l,1}(x)|\leq|f_{m,1}(x)-f_{l,1}(x)|,$

where the subscript $1$ denotes the first coordinate. Let $F$ be the
$e$-contraction (resp. $(e,e^{\prime})$-contraction) of $f$ for $G$. The
$(i,\epsilon)$-contraction $F^{i\epsilon}$ of $F$ is proper and its image is
contained in $U_{\delta_{k}\partial_{e}G}\cap U^{1}_{\delta_{k}\partial_{e}G}$
(resp. $U_{\delta_{k}\partial_{ee^{\prime}}G}\cap
U^{1}_{\delta_{k}\partial_{ee^{\prime}}G}$). In particular, $F^{i\epsilon}$
induces a map
$F^{i\epsilon}:X^{*}\wedge[0,\infty]\to\mathcal{T}_{\delta_{k}\partial_{e}G}$
(resp. $F^{i\epsilon}:X^{*}\wedge[0,\infty]^{\wedge
2}\to\mathcal{T}_{\delta_{k}\partial_{ee^{\prime}}G}$).

###### Proof.

Put $F=(F_{1},\dots F_{n})$ and
$F^{i\epsilon}=F^{\prime}=(F^{\prime}_{1},\dots,F^{\prime}_{n})$. Similarly to
the proof of Lemma 5.3, we can see that $F^{\prime}$ is proper. (The extra
assumption for the case that any base includes one of $i,i+1$ is used here).
We shall show $F^{\prime}(X\times[0,\infty)^{2})\subset U_{\delta_{k}G}$ in
the case that $F$ is the $e$-contraction and there is a base $\beta$ which
does not include $i,i+1$. Suppose $F^{\prime}(\tilde{x})\in\nu_{P}$ for
$\tilde{x}=(x,s_{1},s_{2})\in X\times[0,\infty)^{2}$. Put
$\pi_{P}(F^{\prime}(\tilde{x}))=(y_{\gamma})_{\gamma}$. Let
$\alpha\in\delta_{k}P$ be the piece including $i,i+1$ By definition, we have

$|y_{\alpha}-\frac{1}{2}(F^{\prime}_{i}(x)+F^{\prime}_{i+1}(x))|<\frac{\rho
c_{\alpha}}{2}-50\epsilon.$

Clearly,
$\frac{1}{2}(F^{\prime}_{i}(x)+F^{\prime}_{i+1}(x))=\frac{1}{2}(F_{i}(x)+F_{i+1}(x))$
and the first coordinate of $\frac{1}{2}(F_{i}(x)+F_{i+1}(x))$ is in the image
of projection of the convex hull of $\\{F_{j}(x)\\}_{j\in\beta}$ to the first
coordinate, so

$|y_{\beta 1}-\frac{1}{2}(F_{i1}(x)+F_{i+1,1}(x))|<\frac{\rho
c_{\beta}}{2}-50\epsilon.$

By these (in)equalities, we see $|y_{\alpha 1}-y_{\beta
1}|<\rho(c_{\alpha}+c_{\beta})/2-100\epsilon$. Since $\alpha\simeq_{G}\beta$,
there exists $i_{1}\in\alpha$ such that $F_{i_{1}}(x)$ is in the convex hull
of $\\{F_{i}(x)\\}_{l\in\beta}$. We also have
$F^{\prime}_{i_{2},2}(x)=F_{i_{2},2}(x)$ for any $i_{2}$ by definition, where
the extra subscript means the $n-1$-tuple of the second to $n$-th coordinate.
So we see $|y_{\alpha 2}-y_{\beta 2}|<4\epsilon_{P}$. Thus, we have shown
$|y_{\alpha}-y_{\beta}|<d_{\alpha\beta}(P)$. ∎

Part 1 of the following lemma is the reason why we need both signs in
$(i,\pm)$-contractions.

######  Lemma 5.10.

Let $f=(f_{1},\dots,f_{n}):X\to\mathbb{R}^{dn}$ be a $G$-condensed map for a
graph $G\in\mathsf{G}(P)$, and $i,j$ two numbers such that $1\leq i,j\leq
n-1$. Say $i$ and $j$ belongs to the $k$-th and $l$-th pieces of $P$,
respectively. Suppose $i\sim_{\delta_{kl}G}j$. Let $e$ be a bridge of $G$ such
that $\delta_{kl}\partial_{e}G\in\mathsf{G}(\delta_{kl}P)$, and $F$ the
$e$-contraction of $f$ for $G$.

  1. (1)

Suppose that either of the following conditions holds.

    1. (a)

$f_{i}=f_{j}$ and $f_{i+1}=f_{j+1}$, or

    2. (b)

$f_{i}=f_{j+1}$ and $f_{i+1}=f_{j}$

We put $\epsilon^{\prime}=-\epsilon$ (resp. $\epsilon$) in the case (a) (resp.
(b)). The straight homotopy $h$ from $F^{i\epsilon}$ to
$F^{j\epsilon^{\prime}}$ is proper and its image is contained in
$U_{\delta_{kl}\partial_{e}G}\cap U^{1}_{\delta_{kl}\partial_{e}G}$. So, $h$
induces a map $h:X^{*}\wedge[0,\infty]^{\wedge
2}\wedge(I_{+})\to\mathcal{T}_{\delta_{kl}\partial_{e}G}$.

  2. (2)

Suppose that $i,i+1,j,j+1$ is contained in a single connected component of $G$
which has a base which does not contain any of the four numbers. For any pair
$\epsilon,\epsilon^{\prime}=\pm$, the straight homotopy $h$ from
$F^{i\epsilon}$ to $F^{j\epsilon^{\prime}}$ is proper and its image is
contained in $U_{\delta_{kl}\partial_{e}G}\cap
U^{1}_{\delta_{kl}\partial_{e}G}$ . So, $h$ induces a map
$h:X^{*}\wedge[0,\infty]^{\wedge
2}\wedge(I_{+})\to\mathcal{T}_{\delta_{kl}\partial_{e}G}$.

A similar claim holds for $(e,e^{\prime})$-contraction.

###### Proof.

The proof is similar to that of Lemma 5.9 so we omit details. The choice of
$\epsilon^{\prime}$ in part 1 ensures $h$ is proper. ∎

##  6\. Computation of differential in characteristic 2

In this section, we set $n=4$ and $d=2$ and assume that the base field
$\mathsf{k}$ is of characteristic $2$. As is well-known, cohomology
$H^{*}(\mathit{Conf}_{n}(\mathbb{R}^{2}))$ of the ordered configuration space
of $n$ points is generated by elements $g_{ij}$ ($1\leq i<j\leq n$) which
satisfy $(g_{ij})^{2}=0$ and the 3-term relation. This cohomology appears in
the $E_{1}$-page of Sinha’s spectral sequence. The element
$g_{14}g_{23}+g_{13}g_{24}+g_{12}g_{34}$ is a cycle for $d_{1}$ in
characteristic $2$. We consider $d_{2}$ of the corresponding element in
$\check{\mathbb{E}}$. More precisely, we consider its projection to the
truncated spectral sequence $tr_{1}\check{\mathbb{E}}$. Our computation is
based on the well-known description of the differential of spectral sequence
associated to a double complex in terms of zigzag of horizontal and vertical
differentials.
We define three graphs in $\mathsf{G}([5])$ as follows:

$G_{1}=(1,4)(2,3),\qquad G_{2}=(1,3)(2,4),\qquad G_{3}=(1,2)(3,4).$

See Figure 1 and Example 4.2 for explanation of the figure. Throughout this
section, $G_{i}$ denotes one of these graphs (not those in section 7).

$G_{1}$$G_{2}$$G_{3}$$x$$y$$y$$x$$x$$y$$x$$y$$x$$x$$y$$y$ Figure 1. graphs in
sections 3, 5 and 6 and corresponding maps $f_{G}$

######  Definition 6.1.

For $G=G_{1},G_{2},$ and $G_{3}$, put $f=f_{G}$ and $E(G)=\\{e_{1}<e_{2}\\}$
(see Definition 5.1). For $j=1,2$, let $f_{j}$ be the $e_{j}$-contraction of
$f$ for $G$. Set

$c(G)=f(w_{0})+f_{1}(w_{1})+f_{2}(w_{1})\quad\in\bar{C}_{4}(\mathcal{T}_{G})\oplus\bar{C}_{5}(\mathcal{T}_{\partial_{1}G})\oplus\bar{C}_{5}(\mathcal{T}_{\partial_{2}G})$

Here, by our convention, $f_{j}(w_{1})$ denotes the pushforward of $w_{1}$ by
$f_{j}:S^{4}\wedge[0,\infty]\to\mathcal{T}_{\partial_{i}G}$ and $f(w_{0})$ is
a similar abbreviation.

See Figure 2 where the dotted chords denote the removed edges and the signs
denote those of the term $sv$ added to the corresponding component.

$+$$-$$+$$-$$+$$+$ Figure 2. $c(G_{1})$

######  Example 6.2.

Let $G=G_{1}$. See Example.5.2 for the concrete formulas of $f$ and $f_{1}$.
$f_{2}$ is given by $(x,y,s)\mapsto(x,y+sv,y-sv,x)$.

Set $D=d+\partial$.

######  Lemma 6.3.

For $G=G_{1},G_{2},$ and $G_{3}$, $c(G)$ is a cycle in
$(tr_{1}\mathbb{T},D)=(tr_{1}\check{\mathbb{E}}_{0},d_{0})$.

###### Proof.

Under the notations of Definition 6.1, $f_{j}$ is $\partial_{j}G$-condensed by
Lemma 5.4. So by Lemma 5.3, each pushfoward in the definition of $c(G)$ is
well-defined. Clearly, we have $df(w_{0})=0$ and
$\partial_{j}f(w_{0})=df_{j}(w_{1})$. These equality imply the claim. ∎

######  Definition 6.4.

For $(G,H,i)=(G_{1},G_{2},3),(G_{1},G_{2},1)$ and $(G_{2},G_{3},2)$, we shall
define a bounding chain of $\delta_{i}(c(G)+c(H))$. Set $f=f_{G}$ and
$f^{\prime}=f_{H}$. If the $i$-th components of $f$ and $f^{\prime}$ are
identical, $\psi$ denotes the straight homotopy from $f$ to $f^{\prime}$.
Otherwise, $\psi$ is the straight homotopy from $f$ to $f^{\prime}\circ T$,
where $T:\mathbb{R}^{4}\to\mathbb{R}^{4}$ is the transposition $T(x,y)=(y,x)$.
Put $E(G)=\\{e_{1}<e_{2}\\}$ and $E(H)=\\{e^{\prime}_{1}<e^{\prime}_{2}\\}$.
We regard $E(\delta_{i}G)=E(G)$ and $E(\delta_{i}H)=E(H)$ in the way
compatible with the incidence maps. Let

  1. (1)

$\lambda_{j}$ be the straight homotopy from $e_{j}$-contraction of $f$ for $G$
to $e_{j}$-contraction of $f$ for $\delta_{i}G$,

  2. (2)

$\lambda^{\prime}_{j}$ the straight homotopy from $e^{\prime}_{j}$-contraction
of $f^{\prime}$ for $H$ to $e^{\prime}_{j}$-contraction of $f^{\prime}$ for
$\delta_{i}H$,

  3. (3)

$\psi$ the straight homotopy from $f$ to $f^{\prime}$, and

  4. (4)

$\psi_{j}$ be the $e_{j}$-contraction of $\psi$ for $\delta_{i}G$.

Set

$\begin{split}c(G,H,i)=\psi(w_{01})+\sum_{j=1,2}&(\lambda_{j}+\lambda^{\prime}_{j}+\psi_{j})(w_{11})\\\
&\in\bar{C}_{*}(\mathcal{T}_{\delta_{i}G})\oplus\bar{C}_{*}(\mathcal{T}_{\delta_{i}\partial_{1}G})\oplus\bar{C}_{*}(\mathcal{T}_{\delta_{i}\partial_{2}G})\end{split}$

Here, we compose $\psi$ with the transposition $[0,\infty]\wedge(I_{+})$
implicitly since definition of $\psi_{j}$ put $[0,\infty]$ at the rightmost
component.

######  Example 6.5.

For $(G,H,i)=(G_{1},G_{2},3)$, we have

$\begin{split}\lambda_{1}(x,y,s,t)&=(x+sv,y-stv,y-stv,x-sv)\\\
\lambda_{2}(x,y,s,t)&=(x-stv,y+sv,y-sv,x-stv),\\\
\lambda^{\prime}_{1}(x,y,s,t)&=(x+sv,y-stv,x-sv,y-stv),\\\
\lambda^{\prime}_{2}(x,y,s,t)&=(x-stv,y+sv,x-stv,y-sv),\\\
\psi(x,y,t)&=((1-t)x+ty,tx+(1-t)y,y,x),\\\
\psi_{1}(x,y,t,s)&=((1-t)x+ty+sv,tx+(1-t)y-sv,y-sv,x-sv),\\\
\psi_{2}(x,y,t,s)&=((1-t)x+ty-sv,tx+(1-t)y+sv,y-sv,x-sv),\end{split}$

See Figure 3 where the rectangles denote the vertexes unified by $\delta_{3}$.

$+$$-$$+$$-$$-$$-$$\simeq$$e_{1}$-cont. of $f$ for $G_{1}$$e_{1}$-cont. of $f$
for
$\delta_{3}G_{1}$$\underline{x}$$\underline{y}$$y$$x$$y$$x$$y$$x$$\simeq$$\psi:$$\lambda_{1}:$
Figure 3. $\psi$ and $\lambda_{1}$ for $(G_{1},G_{2},3)$: The underlines
denote the components moved by the straight homotopy.

######  Lemma 6.6.

For $(G,H,i)=(G_{1},G_{2},3),(G_{1},G_{2},1)$ and $(G_{2},G_{3},2)$, the chain
$c(G,H,i)$ is well-defined and satisfies $D(c(G,H,i))=\delta_{i}(c(G)+c(H))$.
If we set $C=c(G_{1},G_{2},3)+c(G_{1},G_{2},1)+c(G_{2},G_{3},2)$, we have

$D(C)=\delta(c(G_{1})+c(G_{2})+c(G_{3})).$

(In particular, $c(G_{1})+c(G_{2})+c(G_{3})$ represents a cycle in
$(tr_{1}\check{\mathbb{E}}^{-2,2}_{1},d_{1})$.)

###### Proof.

Clearly, we have $\delta_{i}G=\delta_{i}H$. This equation, Lemmas 5.4 and 5.5
imply that $\psi$ is $\delta_{i}G$-condensed. $\psi_{j}$ and $\lambda_{j}$ are
$\delta_{i}\partial_{j}G$-condensed and $\lambda^{\prime}_{j}$ is
$\delta_{i}\partial_{j}H$-condensed similarly. By Lemma 5.3, each pushforward
in the definition of $c(G,H,i)$ is well-defined. Concatenation of $\psi_{j}$,
$\lambda_{j}$, and $\lambda^{\prime}_{j^{\prime}}$ gives homotopy between the
$e_{j}$-contraction of $f$ for $G$ and $e^{\prime}_{j^{\prime}}$-contraction
of $f^{\prime}$ or $f^{\prime}\circ T$ for $H$, where $j^{\prime}=j$ if $\psi$
does not alternate edges, and $j^{\prime}=3-j$ otherwise. For example, if
$(G,H,i,j)=(G_{1},G_{2},1,1)$, the concatenation gives a homotopy from
$e_{1}$-contraction of $f$ to $e^{\prime}_{2}$-contraction of $f^{\prime}$.
The concatenation induces a homotopy between maps to the Thom space
$\mathcal{T}_{\delta_{i}\partial_{j}G}$ since the maps are
$\delta_{i}\partial_{j}G$-condensed. Since $\lambda_{j}|_{s=0}$ and
$\lambda^{\prime}_{j}|_{s=0}$ are constant for the variable $t\in I$, the
pushforwards by these maps are zero in the normalized chain complex. The rest
of boundaries cancel with each other. Thus, we have the first equation. The
chains $\delta_{j}(c(G_{k}))$ which are not covered by $c(G,H,i)$ are zero by
Lemma 5.7 (1). For example, $\delta_{2}c(G_{1})=0$ by the part (a) of it. The
equation of $D(C)$ follows from the first equation and this observation. ∎

While we give a rigorous proof below, we shall explain the reason why
$\delta^{\prime}_{3}\circ\psi_{j}$ for $(G_{1},G_{2},1)$ collapses to the
basepoint and $\delta^{\prime}_{1}\circ\psi_{j}$ for $(G_{1},G_{2},3)$ does
not intuitively ($\delta^{\prime}_{i}$ is the collapsing map in Definition
4.3). Let $\psi^{1}_{j}$ (resp.$\psi^{2}_{j}$) be $\psi_{j}$ for
$(G_{1},G_{2},1)$ (resp. $(G_{1},G_{2},3)$). The homotopy $\psi^{1}_{j}$
(resp. $\psi^{2}_{j}$) alternates $x$ and $y$ in the third and fourth
components (resp. the first and second components). See Figure 4. If the image
of a point $(x,y,s,t)$ by these maps is not the basepoint, the image must be
contained in $\nu_{P}$ for $P=\\{\\{0\\},\\{12\\},\\{34\\},\\{5\\}\\}$. So the
difference between the first coordinates of the first and second components is
approximately $\rho(c_{1}+c_{2})/2\pm\epsilon_{P}$ and that between the third
and fourth components is approximately $\rho(c_{3}+c_{4})/2\pm\epsilon_{P}$.
In the figure, $x$ and $y$ must be in the interior of the solid circles on the
right and left, respectively. (The radius of these circles represents the
width of tubular neighborhood $\nu_{P}$ and we ignore addition of $\pm sv$
since the second coordinate is not important.) The components moved by the
homotopy must be in the interior of the dotted circles. It is seen that this
is not the case for $\delta^{\prime}_{3}\circ\psi^{1}_{j}$ while the moving
points pass through the dotted circle for
$\delta^{\prime}_{1}\circ\psi^{2}_{j}$ as $c_{1}+c_{2}<<c_{3}+c_{4}$.

$\rho(c_{1}+c_{2})/2$$\rho(c_{3}+c_{4})/2$$y$$x$$\rho(c_{1}+c_{2})/2$$\rho(c_{3}+c_{4})/2$$\delta_{1}^{\prime}\circ\psi^{2}_{j}$$\delta^{\prime}_{3}\circ\psi^{1}_{j}$$y$$x$
Figure 4. $\delta^{\prime}_{3}\circ\psi^{1}_{j}$ collapses to $*$ while
$\delta^{\prime}_{1}\circ\psi^{2}_{j}$ does not.

######  Lemma 6.7.

Let $C$ be the chain defined in Lemma 6.6, and $\psi$, $\psi_{j}$,
$\lambda_{j}$, and $\lambda^{\prime}_{j}$ the maps given in Definition 6.4 for
$(G,H,i)=(G_{1},G_{2},3)$. Put $G_{0}=\delta_{13}\partial_{1}G_{1}$. $G_{0}$
is the graph with only one edge $(\\{12\\},\\{34\\})$.

  1. (1)

Each pushforward which appears as the terms of $c(G,H,i)$ are annihilated by
$\delta_{k}$ unless $(G,H,i,k)=(G_{1},G_{2},3,1)$. The terms of
$c(G_{1},G_{2},3)$ except for $\psi_{j}(w_{11})$ is annihilated by
$\delta_{1}$.

  2. (2)

$\delta C=\delta_{1}(\psi_{1}(w_{11})+\psi_{2}(w_{21}))$ is a fundamental
cycle of $\mathcal{T}_{G_{0}}$.

###### Proof.

For part 1, we consider the case of $(G,H,i)=(G_{1},G_{2},1)$. We have
$\delta_{1}\psi(w_{01})=0$ as $\delta_{1}\partial_{1}G$ has a loop. The graph
of $\delta^{\prime}_{1}\circ\lambda_{2}$ is $\delta_{1}\partial_{2}G$. While
this does not have a loop nor double edges, by Lemma 5.7 (1), we have
$\delta^{\prime}_{1}\circ\lambda_{2}=*$. We also have
$\delta^{\prime}_{2}\circ\psi_{j}=*$ by (2) of the same lemma. The other terms
vanish similarly.
We shall show part 2. By definition, $\mathcal{T}_{G_{0}}$ is a Thom space
associated to the disk bundle $\nu_{P}|_{N}$, where $N$ is a open tubular
neighborhood of the diagonal $\Delta$ in the product of disks $(D^{2})^{2}$
(with different radius). Precisely speaking, in the notation of Definition
3.1, $(D^{2})^{2}=\mathbb{R}^{4}-E_{P}$, and
$N=D_{\alpha\beta}\cap(\mathbb{R}^{4}-E_{P})$, where
$\alpha=\\{12\\},\beta=\\{34\\}$ and $P=\\{\\{0\\},\alpha,\beta,\\{5\\}\\}$.
So, $\mathcal{T}_{G_{0}}$ is homeomorphic to $S^{4}\wedge
S^{2}\wedge(IntD^{2}_{+})$, where $S^{4}$ and $S^{2}\times\\{0\\}\subset
S^{2}\wedge(IntD^{2}_{+})$ correspond to a fiber and $\Delta$, respectively.
We consider a retract to $S^{4}\wedge(S^{2}\times\\{0\\})\cong
Th(\nu_{P}|_{\Delta})$ and show that the composition of $\psi_{j}$ with this
retract is of degree one. Write $\nu=\nu_{P}|_{N}$. The orthogonal projection
$r:N\to\Delta$ induces the bundle map $\tilde{r}:\nu\to\nu|_{\Delta}$ which is
given by the parallel transport taking the center to the center on each fiber.
This map induces a map $\tilde{r}:\mathcal{T}_{G_{0}}\to Th(\nu|_{\Delta})$.
We consider $\nu$ and $\nu|_{\Delta}$ as subspaces of $\mathbb{R}^{8}$. Put

$F_{j}:=\tilde{r}\circ\psi_{j}|_{(\psi_{j})^{-1}(\nu)}:(\psi^{1}_{i})^{-1}(\nu)\to\nu|_{\Delta}\qquad\text{for}\qquad
j=1,2.$

We shall show $F_{j}$ is a monomorphism and $Im(F_{1})\cup
Im(F_{2})=\nu|_{\Delta}$ and $Im(F_{1})\cap
Im(F_{2})=Im(F_{1})|_{s=0}=Im(F_{2})|_{s=0}$, where $s\in[0,\infty)$ is the
variable in the $e_{j}$-contraction. These claims imply the lemma. By
definition, $\nu$ is the tubular neighborhood of the map

$e_{P}:(a,b)\mapsto\left(a-\frac{\rho}{2}c_{2}u,\ a+\frac{\rho}{2}c_{1}u,\
b-\frac{\rho}{2}c_{4}u,\ b+\frac{\rho}{2}c_{3}u\right).$

The projection $\pi_{P}$ of $\nu$ sends $(c,d,e,f)\in\mathbb{R}^{8}$ to the
point $(a,b)$ which minimize the distance $|(c,d,e,f)-e_{P}(a,b)|$. By
elementary calculation, the point is given by

$(a,b)=\left(\frac{c+d}{2}+\frac{\rho}{4}(c_{2}-c_{1})u,\
\frac{e+f}{2}+\frac{\rho}{4}(c_{4}-c_{3})u\right).$

Similarly, we see that $r:N\to\Delta$ is given by $r(a,b)=(a+b)/2$. We shall
verify $F_{1}$ is a monomorphism. Since
$\psi_{1}(x,y,s,t)=((1-t)x+ty+sv,tx+(1-t)y-sv,x-sv,y-sv)$, we have

$r\circ\pi\circ\psi_{1}(x,y,s,t)=\frac{x+y}{2}-\frac{sv}{2}+\frac{\rho}{8}(c_{2}+c_{4}-c_{1}-c_{3})u.$

We denote the right hand side by $w$. For simplicity, we move the fiber of
$\nu$ over $\pi_{P}(\psi_{1}(x,y,s,t))$ by the parallel transport which sends
its center to $0$. By this move, $\psi_{1}(x,y,s,t)$ is sent to

$\psi_{1}(x,y,s,t)-e_{P}(\pi_{P}(\psi(x,y,s,t))=(p_{1},-p_{1},q,-q)$

where

$p_{1}=(t-1/2)(y-x)+sv+\frac{\rho}{4}(c_{1}+c_{2})u,\
q=\frac{-1}{2}(y-x)+\frac{\rho}{4}(c_{3}+c_{4})u.$

Similarly, If $F_{2}(x,y,s,t)$ is in the fiber over a point sent to $w$ by
$r$, the point is given by $(p_{2},-p_{2},q,-q)$ where

$p_{2}=(t-1/2)(y-x)-sv+\frac{\rho}{4}(c_{1}+c_{2})u$

The fiber over $\pi_{P}$ is a disk of radius $\epsilon_{P}$. It is enough to
show there exists a unique $(x,y,s,t)$ such that $q^{\prime}=q$ and
($p^{\prime}=p_{1}$ or $p^{\prime}=p_{2}$) for given point
$(p^{\prime},q^{\prime},w)$ with $|(p^{\prime},q^{\prime})|\leq\epsilon_{P}$,
and for the point $(x,y,s,t)$, $p^{\prime}=p_{1}=p_{2}$ holds if and only if
$s=0$. Suppose that $(p^{\prime},q^{\prime})=(p_{j},q)$. By the above formulas
for $p_{j}$ and $q$, for both of $j=1,2$ we have

$t=\frac{{p^{\prime}}^{1}-{q^{\prime}}^{1}+\rho(c_{3}+c_{4}-c_{1}-c_{2})/4}{-2{q^{\prime}}^{1}+\rho(c_{3}+c_{4})/2},$

where ${p^{\prime}}^{1}$ and ${q^{\prime}}^{1}$ denote the first components o
$p^{\prime}$ and $q^{\prime}$, respectively. The assumptions on $c_{j}$ and
$|(p^{\prime},q^{\prime})|$ ensure $0<t<1$. Similar computation shows that
$s>0$ holds for exactly one of $p_{1}$, $p_{2}$ or otherwise, $s=0$ holds for
both of $p_{1}$ and $p_{2}$. Thus we have proved the claim. ∎

######  Theorem 6.8.

In dimension $d=2$ and over a field of characteristic $2$, there exists an
element $g\in\mathbb{E}^{-2,2}_{2}$ satisfying $d_{2}(g)\not=0$.

###### Proof.

By degree reason, we may consider $\bar{\mathbb{E}}_{r}$ instead of
$\mathbb{E}_{r}$. By Lemma 4.6, we see the class $g^{\prime}$ represented by
$\sum_{1\leq i\leq 3}c(G_{i})$ is lifted to a class
$g\in\bar{\mathbb{E}}^{-2,2}_{1}$. By Lemma 6.6, $d_{1}g^{\prime}=0$. Again by
Lemma 4.6, $d_{1}g=0$. By Lemma 6.6 and 6.7, we have $d_{2}g^{\prime}\not=0$,
and we have $d_{2}g\not=0$. ∎

##  7\. Computation of differential in characteristic 3

Throughout this section we set $n=5$ and $d=2$ and assume that $\mathsf{k}$ is
a field of characteristic $3$. We compute $d_{3}$ of the element in
$tr_{1}\check{\mathbb{E}}$ corresponding to
$-g_{13}g_{23}g_{45}+g_{14}g_{24}g_{35}+g_{14}g_{25}g_{34}+g_{15}g_{24}g_{34}\in
H^{*}(Conf_{5}(\mathbb{R}^{2}))$. (In the integral coefficient, this element
does not represents a cycle for $d_{1}$.) The computation is similar to the
one in section 6. We define four graphs in $\mathsf{G}([6])$ as follows:

$G_{1}=(1,3)(2,3)(4,5),\quad G_{2}=(1,4)(2,4)(3,5),\quad
G_{3}=(1,4)(2,5)(3,4),\quad G_{4}=(1,5)(2,4)(3,4),$

see Figure 5. Throughout this section, $G_{i}$ ($1\leq i\leq 4$) denotes one
of these graphs.

$G_{1}$$G_{2}$$G_{3}$$G_{4}$ Figure 5. graphs in section 7

######  Definition 7.1.

Let $G$ be one of $G_{i}$. Put $E(G)=\\{e_{1}<e_{2}<e_{3}\\}$ and
$f=f_{G}:\mathbb{R}^{4}\to\mathbb{R}^{10}$, see Definition 5.1. Let $f_{j}$ be
the $e_{j}$-contraction of $f$, $f_{jk}$ the $(e_{j},e_{k})$-contraction of
$f$. We define a chain $c(G)$ by

$\begin{split}c(G)=f(w_{0})+\sum_{j=1}^{3}(-1)^{j}f_{j}(w_{1})&+\sum_{1\leq
j<k\leq 3}(-1)^{j+k+1}f_{jk}(w_{2}).\\\
&\in\quad\bar{C}_{*}(\mathcal{T}_{G})\oplus\bigoplus_{j}\bar{C}_{*}(\mathcal{T}_{\partial_{j}G})\oplus\bigoplus_{j<k}\bar{C}_{*}(\mathcal{T}_{\partial_{jk}G})\end{split}$

Set $D=d+(-1)^{*}\partial$, where $*$ is the singular degree. The following
lemma is obvious.

######  Lemma 7.2.

$c(G)$ is a cycle in $(tr_{1}\check{\mathbb{E}}_{0},d_{0})$. ∎

###  7.1. First bounding chain

######  Definition 7.3.

Let $(G,H,i)$ be one of
$(G_{1},G_{2},3),(G_{2},G_{3},2),(G_{2},G_{3},4),(G_{3},G_{4},1),$ and
$(G_{4},G_{3},4)$.

Set $f=f_{G}$, $f^{\prime}=f_{H}$, and $E(G)=\\{e_{1}<e_{2}<e_{3}\\}$. We
identify edge sets through the bijection $E(G)\cong E(\delta_{i}G)\cong
E(\delta_{i}H)\cong E(H)$ commuting with the incidence maps. (By this
convention, $e_{j}$ does not necessarily represent the $j$-th edge of $E(H)$
or $E(\delta_{i}H)$.) If the $i$-th components of $f$ and $f^{\prime}$ are
identical, $\psi$ denotes the straight homotopy from $f$ to $f^{\prime}$.
Otherwise, $\psi$ is the straight homotopy from $f$ to $f^{\prime}\circ T$,
where $T:\mathbb{R}^{4}\to\mathbb{R}^{4}$ is the transposition $T(x,y)=(y,x)$.
Let

  1. (1)

$\psi_{j}$ (resp. $\psi_{jk}$) be the $e_{j}$-contraction (resp.
$(e_{j},e_{k})$-contraction) of $\psi$ for $\delta_{i}G$,

  2. (2)

$\lambda_{j}$ (resp. $\lambda_{jk}$) the straight homotopy from the
$e_{j}$-contraction (resp. $(e_{j},e_{k})$-contraction) of $f$ for $G$ to the
$e_{j}$-contraction (resp. $(e_{j},e_{k})$-contraction) of $f$ for
$\delta_{i}G$, and

  3. (3)

$\lambda^{\prime}_{j}$ (resp. $\lambda^{\prime}_{jk}$) the straight homotopy
from the $e_{j}$-contraction (resp. $(e_{j},e_{k})$-contraction) of
$f^{\prime}$ for $H$ to $e_{j}$-contraction (resp.
$(e_{j},e_{k})$-contraction) of $f^{\prime}$ for $\delta_{i}H$.

Then, we set

$c(G,H,i):=\psi(w_{01})+\sum_{1\leq j\leq
3}(-1)^{j+1}(\psi_{j}+\lambda_{j}-\lambda^{\prime}_{j})(w_{11})+\sum_{1\leq
j<k\leq 3}(-1)^{j+k+1}(\psi_{jk}+\lambda_{jk}-\lambda^{\prime}_{jk})(w_{21}),$

Here, we compose $\psi_{j}$ with the transposition of
$[0,\infty]\wedge(I_{+})$ implicitly since definition of $\psi_{j}$ put
$[0,\infty]$ at the rightmost component. $\psi_{jk}$ is also composed with the
transposition $(I_{+})\wedge[0,\infty]^{\wedge 2}\cong[0,\infty]^{\wedge
2}\wedge(I_{+}),\ (t,s_{1},s_{2})\mapsto(s_{1},s_{2},t)$. Throughout this
section, definitions of chains are understood to include similar implicit
transposition.

######  Lemma 7.4.

We have

$\begin{split}Dc(G_{1},G_{2},3)&=-\delta_{3}c(G_{1})+\delta_{3}c(G_{2}),\\\
Dc(G_{2},G_{3},2)&=-\delta_{2}c(G_{2})-\delta_{2}c(G_{3}),\\\
Dc(G_{2},G_{3},4)&=-\delta_{4}c(G_{2})+\delta_{4}c(G_{3}),\\\
Dc(G_{3},G_{4},1)&=-\delta_{1}c(G_{3})-\delta_{1}c(G_{4}),\\\
Dc(G_{4},G_{3},4)&=-\delta_{4}c(G_{4})+\delta_{4}c(G_{3}).\end{split}$

###### Proof.

The proof is similar to that of Lemma 6.6 with some care about signs. Let
$(G,H,i),f,$ and $f^{\prime}$ be as in Definition 7.3. $\delta_{i}:E(G)\to
E(\delta_{i}(G))$ preserves the order of edges so the restriction of each map
to $0\in[0,\infty)$ cancels with $\check{\text{C}}$ech differential of another
map. Let $f_{j}$ and $f^{\prime}_{j}$(resp. $f_{jk}$, and $f^{\prime}_{jk}$)
denotes the $e_{j}$-(resp. $(e_{j},e_{k})$-) contractions of $f$ and
$f^{\prime}$ for $G$ and $H$, respectively. The concatenation of $\psi_{j}$,
$\lambda_{j}$, and $\lambda^{\prime}_{j}$ defines a homotopy from $f_{j}$ to
$f^{\prime}_{j}$ possibly composed with a transposition. We have a similar
homotopy for $\psi_{jk},\lambda_{jk}$, and $\lambda^{\prime}_{jk}$. For
$(G,H,i)=(G_{1},G_{2},3)$, the bijection $E(G)\cong E(H)$ in Definition 7.3
preserves the order. Straightforward computation shows the first equation. The
proof of the third and fifth equations is similar. For the second equation,
consider the case of $(G,H,i)=(G_{2},G_{3},2)$. The bijection alternate second
and third edges, so the homotopy gives the terms of $f^{\prime}_{j}$ and
$f^{\prime}_{jk}$ the signs opposite to $c(H)$ except for the terms of
$f^{\prime}$ and $f^{\prime}_{1}$. (For $f^{\prime}_{23}$, the homotopy
alternates the components of $[0,\infty)^{2}$, which produces a sign.) This
exception is covered by the extra sign on $\delta_{2}$ in permutating edges
since we have $sgn(\sigma_{H,2})=sgn(\sigma_{\partial_{1}H,2})=-1$ and
$sgn(\sigma_{H^{\prime},2})=1$ for the other subgraphs $H^{\prime}$ of $H$
(see Definition 4.3). The proof of the fourth equation is similar. ∎

######  Definition 7.5.

Let $(G,i)$ be one of $(G_{1},1),(G_{2},1)$ and $(G_{4},2)$. Set $f=f_{G}$ and
$E(G)=\\{e_{1}<e_{2}<e_{3}\\}$ Let $f_{j}$ be $e_{j}$-contraction of $f$, and
$f_{jk}$ $(e_{j},e_{k})$-contraction of $f$. Let $f_{j}^{\epsilon}$
(resp.$f_{jk}^{\epsilon}$) be the $(i,\epsilon)$-contraction of $f_{j}$ (resp.
$f_{jk}$) for $\epsilon=+$ or $-$, see Definition 5.8. We set

$\begin{split}c(G,i)=\sum_{1\leq j\leq
3}&(-1)^{j+1}f^{\pm}_{j}(w_{2})+\sum_{1\leq j<k\leq
3}(-1)^{j+k+1}f^{\pm}_{jk}(w_{3})\\\
&\in\bigoplus_{j}\bar{C}_{*}(\mathcal{T}_{\delta_{i}\partial_{j}G})\oplus\bigoplus_{j<k}\bar{C}_{*}(\mathcal{T}_{\delta_{i}\partial_{jk}G})\end{split}$

Here, $f^{\pm}_{j}(w_{2})$ denotes $(f^{+}_{j}(w_{2})+f^{-}_{j}(w_{2}))/2$ and
$f^{\pm}_{jk}$ is similarly understood. If $\delta_{i}(\partial_{j}G)$ has
double edges, the corresponding term $f^{\pm}_{j}(w_{2})$ is regarded as $0$,
and we identify two complexes in the direct sums if their graphs are the same.
For example $c(G_{1},1)$ belongs to
$\bar{C}_{*}(\mathcal{T}_{\delta_{1}\partial_{1}G_{1}})\oplus\bar{C}_{*}(\mathcal{T}_{\delta_{1}\partial_{12}G_{1}})\oplus\bar{C}_{*}(\mathcal{T}_{\delta_{1}\partial_{13}G_{1}})$
since $\delta_{1}\partial_{13}G_{1}=\delta_{1}\partial_{23}G_{1}$.

######  Lemma 7.6.

Let $(G,i)$ be as in Definition 7.5. We have $D(c(G,i))=0$. Set

$C_{1}=c(G_{1},G_{2},3)+c(G_{2},G_{3},2)+c(G_{2},G_{3},4)-c(G_{3},G_{4},1)+c(G_{4},G_{3},4)-c(G_{1},1)+c(G_{2},1)-c(G_{4},2).$

We have $D(C_{1})=-\delta(-c(G_{1})+c(G_{2})+c(G_{3})+c(G_{4}))$.

###### Proof.

$c(G,i)$ is well-defined by Lemma 5.9, For $c(G_{1},1)$, let $s_{1}$ (resp.
$s_{2}$) be the variable for $e_{j}$-contraction (resp.
$(i,\epsilon)$-contraction). Since
$\delta_{i}\partial_{1}G_{1}=\delta_{i}\partial_{2}G_{1}$ and the restrictions
of $f_{1}^{\epsilon}$ and $f_{2}^{\epsilon}$ to $s_{1}=0$ are the same, the
corresponding terms cancel with each other. Put
$f^{\epsilon}_{j}=({f^{\epsilon}_{j}}^{1},\dots,{f^{\epsilon}_{j}}^{5})$. We
have
${f^{\epsilon}}^{1}_{j}|_{s_{2}=0}=_{1}{f^{\epsilon}_{j}}^{2}|_{s_{2}=0}$. By
Lemma 5.6, the induced map is $*$. Thus we see the first equation. The
equation for $D(C_{1})$ follows from Lemma 7.4 with an argument similar to the
proof of Lemma 6.6. ∎

###  7.2. Second bounding chain

######  Definition 7.7.

Let $(G,H,i,j)$ be one of those which appears in Lemmas 7.9,7.10, 7.11, and
7.12 below. For these 4-tuples, $G,H$ satisfy the condition that
$\delta_{i}G=\delta_{i}H$ and $\delta_{ij}G$ has double edge. Let
$\psi_{k},\psi_{kl},\lambda_{k},\lambda_{kl},\lambda^{\prime}_{k},$ and
$\lambda^{\prime}_{kl}$ be the maps given in Definition 7.3 for $(G,H,i)$. The
superscript $\epsilon$ on these maps represents $(j,\epsilon)$-contraction of
the map for $\epsilon=+$ or $-$.

  1. (1)

For $(G,H,i,j)\not=(G_{3},G_{4},1,2)$, we set

$\begin{split}c(G,H,i,j)=\sum_{1\leq k\leq
3}&(-1)^{k}(\psi_{k}^{\pm}+\lambda_{k}^{\pm}-{\lambda^{\prime}_{k}}^{\pm})(w_{21})+\sum_{1\leq
k<l\leq
3}(-1)^{k+l+1}(\psi_{kl}^{\pm}+\lambda_{kl}^{\pm}-{\lambda^{\prime}}^{\pm}_{kl})(w_{31})\\\
&\in\bigoplus_{k}\bar{C}_{*}(\mathcal{T}_{\delta_{ij}\partial_{k}G})\oplus\bigoplus_{k<l}\bar{C}_{*}(\mathcal{T}_{\delta_{ij}\partial_{kl}G})\end{split}$

  2. (2)

For $(G,H,i,j)=(G_{3},G_{4},1,2)$, we set

$c(G,H,i,j)=\sum_{1\leq k\leq
3}(-1)^{k}sgn(\sigma_{\partial_{k}G,j})(\psi_{k}^{\pm}+\lambda_{k}^{\pm}-{\lambda^{\prime}_{k}}^{\pm})(w_{21})+\sum_{1\leq
k<l\leq
3}(-1)^{j+k+1}(\psi_{jk}^{\pm}+\lambda_{jk}^{\pm}-{\lambda^{\prime}}^{\pm}_{jk})(w_{31})$

Here, we use notations similar to those in Definition 7.5. For example,
$\psi_{k}^{\pm}$ is the average of $\psi_{k}^{+}$ and $\psi_{k}^{-}$, and the
term of a graph with a loop is regarded as zero. See Definition 4.3 for
$sgn(\sigma_{\partial_{k}G,j})$

$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$$\simeq$$\simeq$$+$$\biggl{(}$$\biggr{)}$$-$$\stackrel{{\scriptstyle
D}}{{\longmapsto}}$$c(G_{4},G_{3},4,2)$$\delta_{3}c(G_{4},2)$$\delta_{2}c(G_{4},G_{3},4)$$c^{\prime}(G_{3},2)$
Figure 6. the differential of $c(G_{4},G_{3},4,2)$: We acturally consider
subgraphs of these graphs.

######  Definition 7.8.

Let $(G,i,j)$ be one of those which appears in Lemmas 7.9,7.10, 7.11, and 7.12
below. Put $f=f_{G}$ and $E(G)=\\{e_{1}<e_{2}<e_{3}\\}$. Let $f_{k}$ (resp.
$f_{kl}$) be the $e_{k}$-contraction (resp. $f_{kl}$) of $f$ for $G$. Let
$\mu^{\epsilon}_{k}$ (resp. $\mu^{\epsilon}_{kl}$) be the straight homotopy
$h$ defined in Lemma 5.10 (1) for $F=f_{k}$ (resp. $f_{kl}$) and numbers
$i,j$. We set

$\begin{split}c(G,i,j)=\sum_{1\leq k\leq
3}&(-1)^{k}\mu^{\pm}_{k}(w_{21})+\sum_{1\leq l<k\leq
3}(-1)^{k+l+1}\mu^{\pm}_{lk}(w_{31})\\\
&\in\bigoplus_{k}\bar{C}_{*}(\mathcal{T}_{\delta_{ij}\partial_{k}G})\oplus\bigoplus_{k<l}\bar{C}_{*}(\mathcal{T}_{\delta_{ij}\partial_{kl}G})\end{split}$

Here we use notations similar to Definition 7.5.

The proofs of the following three lemmas are similar to that of Lemma 7.4, and
we omit them.

######  Lemma 7.9.

We have

$\begin{split}Dc(G_{2},G_{3},2,4)&=\delta_{3}c(G_{2},G_{3},2)+c^{\prime}(G_{2},4)+c^{\prime}(G_{3},4),\\\
Dc(G_{2},G_{3},4,2)&=\delta_{2}c(G_{2},G_{3},4)+c^{\prime}(G_{2},2)-c^{\prime}(G_{3},2),\\\
Dc(G_{4},G_{3},4,2)&=\delta_{2}c(G_{4},G_{3},4)+\delta_{3}c(G_{4},2)-c^{\prime}(G_{3},2),\\\
Dc(G_{2},2,4)&=c^{\prime}(G_{2},2)-c^{\prime}(G_{2},4),\\\
Dc(G_{3},2,4)&=c^{\prime}(G_{3},2)-c^{\prime}(G_{3},4).\end{split}$

Here, $c^{\prime}(G_{2},4)$ is the chain defined by the same formula as
$c(G,i)$, where $j$ in the first sum runs through the labels of double edges
of $\delta_{24}G_{2}$. (While this chain is not well-defined in the Thom space
of the subgraphs of $\delta_{2}G_{2}$, it is well-defined in that of
$\delta_{24}G_{2}$.) The other $c^{\prime}(G_{k},i)$’s are defined completely
similarly. If we set

$C_{21}=c(G_{2},G_{3},2,4)-c(G_{2},G_{3},4,2)-c(G_{4},G_{3},4,2)+c(G_{2},2,4)+c(G_{3},2,4),$

We have

$DC_{21}=\delta_{3}(c(G_{2},G_{3},2)-c(G_{4},2))-\delta_{2}(c(G_{2},G_{3},4)+c(G_{4},G_{3},4)).$

∎

See Figure 6, where $\leftrightarrow$ specifies the vertexes whose
corresponding components of the map $\psi$ is added $\pm su$ in defining
$(i,\pm)$-contraction.

######  Lemma 7.10.

We have

$\begin{split}Dc(G_{2},G_{3},4,1)&=\delta_{1}c(G_{2},G_{3},4)+\delta_{3}c(G_{2},1)-c^{\prime}(G_{3},1),\\\
Dc(G_{4},G_{3},4,1)&=\delta_{1}c(G_{4},G_{3},4)+c^{\prime}(G_{4},1)-c^{\prime}(G_{3},1),\\\
Dc(G_{3},G_{4},1,4)&=\delta_{3}c(G_{3},G_{4},1)+c^{\prime}(G_{3},4)+c^{\prime}(G_{4},4),\\\
Dc(G_{4},4,1)&=c^{\prime}(G_{4},4)-c^{\prime}(G_{4},1),\\\
Dc(G_{3},4,1)&=c^{\prime}(G_{3},4)-c^{\prime}(G_{3},1).\end{split}$

Here, $c^{\prime}(G_{k},i)$’s are defined completely similarly as in Lemma
7.9. If we set

$C_{22}=c(G_{2},G_{3},4,1)-c(G_{3},G_{4},1,4)+c(G_{4},G_{3},4,1)+c(G_{3},4,1)+c(G_{4},4,1),$

we have

$DC_{22}=\delta_{1}(c(G_{2},G_{3},4)+c(G_{4},G_{3},4))-\delta_{3}c(G_{3},G_{4},1)+\delta_{3}c(G_{2},1).$

∎

######  Lemma 7.11.

We have

$Dc(G_{1},G_{2},3,1)=\delta_{1}c(G_{1},G_{2},3)+\delta_{2}c(G_{1},1)-\delta_{2}c(G_{2},1).$

∎

######  Lemma 7.12.

$\begin{split}Dc(G_{2},G_{3},2,1)&=\delta_{1}c(G_{2},G_{3},2)+\delta_{1}c(G_{2},1)+c^{\prime}(G_{3},1),\\\
Dc(G_{3},G_{4},1,2)&=\delta_{1}c(G_{3},G_{4},1)+c^{\prime}(G_{3},2)+\delta_{1}c(G_{4},2),\\\
Dc^{\prime}(G_{3},1,2)&=c^{\prime}(G_{3},1)-c^{\prime}(G_{3},2).\end{split}$

Here, $c^{\prime}(G_{3},i)$’s are defined similarly as in Lemma 7.9 except for
the extra sign $sgn(\sigma_{H,1})=-1$ on the term of $f_{1}^{\pm}$, where
$H=\delta_{1}(\partial_{1}G_{3})$. $c^{\prime}(G_{3},1,2)$ is also similar to
$c(G,i,j)$ but has the same extra sign on the term $\mu_{1}^{\pm}$. If we set

$C_{23}=c(G_{2},G_{3},2,1)-c(G_{3},G_{4},1,2)-c^{\prime}(G_{3},1,2),$

we have

$DC_{23}=\delta_{1}(c(G_{2},G_{3},2)-c(G_{3},G_{4},1)+c(G_{2},1)-c(G_{4},2)).$

###### Proof.

In the first equation, the only point which we need to care is the sign of the
third term of the right hand side. The straight homotopy between the maps
corresponding to $G_{2}$ and $G_{3}$ alternates the second and third edges.
This homotopy negates the terms except for the one corresponding to the
removal of the first edge. This exception is covered by the extra sign in
$c^{\prime}(G_{3},1)$. In the second equation, the extra sign of $\delta_{1}$
of $c(G_{3},G_{4},1,2)$ is negative only for the term corresponding to
$\delta_{1}\partial_{1}(G_{3})$. This is covered by the extra sign in the
definition of $c(G_{3},G_{4},1)$. The straight homotopy from $G_{3}$ to
$G_{4}$ alternates the first and second edges. This negates the term except
for the removal of the third edge. This is covered by the extra sign of
$\delta_{1}$ on $c(G_{4},2)$. ∎

######  Lemma 7.13.

Let $C_{1},C_{21},C_{22}$, $C_{23}$ be as in Lemmas 7.6, 7.9, 7.10, and 7.12.
If we set

$C_{2}=C_{21}+C_{22}+C_{23}+c(G_{1},G_{2},3,1),$

we have

$D(C_{2})=-\delta(C_{1}).$

###### Proof.

Let $s$ be the variable for $(i,\epsilon)$-contraction in $\mu^{\epsilon}_{k}$
or $\mu^{\epsilon}_{kl}$ in Definition 7.8. We see that
$\mu^{\epsilon}_{k}|_{s=0}$ and $\mu^{\epsilon}_{kl}|_{s=0}$ are constant for
$t\in I$, and the pushforwards by these maps are zero. This observation and
arguments similar to the proof of Lemmas 6.6, 6.7, together with Lemmas 7.9,
7.12, 7.11, and 7.10 imply the claim. ∎

###  7.3. Computation of the differential as an element of the first page

######  Lemma 7.14.

We have

$-\delta(C_{2})=\delta_{1}(-c(G_{4},G_{3},4,2)+c(G_{2},2,4)+c(G_{3},2,4)+c(G_{2},G_{3},4,1)).$

Any pushforward which appears as a term of the left hand side, other than the
following four terms, is zero:

$\begin{split}\delta_{1}\psi^{-}_{kl}(w_{31})&\text{ of
}\delta_{1}c(G_{4},G_{3},4,2),\ \delta_{1}\mu_{kl}^{+}(w_{31})\text{ of
}\delta_{1}c(G_{2},2,4),\\\ \delta_{1}\mu_{kl}^{-}(w_{31})&\text{ of
}\delta_{1}c(G_{3},2,4),\text{ and }\ \delta_{1}\psi^{+}_{kl}(w_{31})\text{ of
}\delta_{1}c(G_{2},G_{3},4,1).\end{split}$

###### Proof.

We will see that all the maps which appears as terms of $\delta(C_{2})$, other
than the four maps in the claim are $*$. If $\delta_{abc}G_{i}$ has a loop,
the maps corresponding to its subgraphs are $*$. For example, we consider the
term $\delta_{2}\psi^{\epsilon}_{23}(w_{31})$ of
$\delta_{2}c(G_{2},G_{3},2,4)$. $\delta_{134}G_{2}$ has loops but
$\delta_{134}\partial_{23}G_{2}$ does not. The image of $\psi^{\epsilon}_{2}$
is contained in $U^{1}_{\delta_{2}\partial_{2}G_{2}}$ by Lemma 5.9 and
$\psi^{\epsilon}_{2,l}=_{1}\psi^{\epsilon}_{23,l}$ for the $l$-th components
by definition ($1\leq l\leq 5$) so the image of $\psi^{\epsilon}_{23}$ is also
contained in $U^{1}_{\delta_{2}\partial_{2}G_{2}}$. By Lemma 5.7, we have
$\delta_{2}^{\prime}\circ\psi^{\epsilon}_{23}=*$. Some of the other terms are
zero by the latter part of Lemma 5.6. For example, we consider the term
$\delta_{1}\psi^{\epsilon}_{kl}(w_{31})$ of $\delta_{1}c(G_{2},G_{3},2,4)$.
$\psi^{+}_{kl}$ alternates $x$ and $y$ by the homotopy in the first, fourth,
and fifth components and adds $-su$ and $su$ to the fourth and fifth
components respectively, so it may be expressed as follows.

$\\{\underline{x}\ x\ y\\}\\{\overrightarrow{\underline{x}}\
\overleftarrow{\underline{y}}\\}$

Here, the arrows $\rightarrow,\leftarrow$ denote addition of $su,-su$,
respectively, $\underline{x}$ (resp. $\underline{y}$) represents the homotopy
from $x$ to $y$ (resp. $y$ to $x$), and $\\{\ \\}$ specifies a vertex (the
vertex set of the graph corresponding to the map
$\delta_{1}\circ\psi^{+}_{kl}$ is $\\{\\{0\\},\\{123\\},\\{45\\},\\{6\\}\\}$).
We ignore $\pm sv$ as it does not affect the first coordinate. Write
$\psi^{+}_{kl}=(\psi_{1},\dots,\psi_{5})$ for simplicity. If $x<_{1}y$,
$\psi_{1}=\underline{x}>_{1}x=\psi_{2}$, and if $x>_{1}y$,
$\psi_{2}=x>_{1}y=\psi_{3}$ since $\underline{x}$ and $\underline{y}$ always
lie between $x$ and $y$. So we have $\delta_{1}^{\prime}\circ\psi^{+}_{kl}=*$
by Lemma 5.6. $\delta_{1}\circ\psi^{-}_{kl}$ is expressed as
$\\{\underline{x}\ x\ y\\}\\{\overleftarrow{\underline{x}}\
\overrightarrow{\underline{y}}\\}$ and we have
$\delta_{1}^{\prime}\circ\psi^{-}_{kl}=*$ similarly. As another example, we
consider the term $\delta_{1}\mu^{+}_{kl}(w_{31})$ of
$\delta_{1}c(G_{4},4,1)$. $\delta_{1}^{\prime}\circ\mu^{+}_{kl}$ is expressed
as

$\\{\overrightarrow{x}\ \overleftarrow{y}\ y\\}\\{\overrightarrow{y}\
\overleftarrow{x}\\}$

Write $\mu^{+}_{kl}=(\mu_{1},\dots,\mu_{5})$. We see that
$\mu_{4}=\overrightarrow{y}>_{1}\overleftarrow{x}=\mu_{5}$ if $x<_{1}y$, and
$\mu_{1}>\mu_{2}$ otherwise, which implies
$\delta_{1}^{\prime}\circ\mu^{+}_{kl}=*$. The terms which cannot be seen to be
zero by the above two kinds of observations are the following four terms,
where we omit subscripts.

$\begin{split}\delta^{\prime}_{1}\circ\psi^{\epsilon}&\text{ of
}\delta_{1}c(G_{2},G_{3},4,2),\ \delta^{\prime}_{1}\circ\mu^{-}\text{ of
}\delta_{1}c(G_{2},2,4),\ \\\ \delta^{\prime}_{1}\circ\psi^{-}&\text{ of
}\delta_{1}c(G_{2},G_{3},4,1),\text{ and
}\delta_{2}^{\prime}\circ\psi^{-}\text{ of
}\delta_{2}c(G_{3},G_{4},1,2).\end{split}$

We consider the first term. The map $\delta^{\prime}_{1}\circ\psi^{+}$ is
expressed as

$\\{x\ \overrightarrow{\underline{x}}\ \overleftarrow{\underline{y}}\\}\\{x\
y\\}$

Put $P=\\{\\{0\\},\\{123\\},\\{45\\},\\{6\\}\\}$. Suppose
$\psi^{+}(\tilde{x})\in\nu_{P}$ for
$\tilde{x}=(x,y,\dots)\in\mathbb{R}^{4}\times[0,\infty)^{3}\times I$. By
fourth and fifth components, we have $x<_{1}y$. Put
$\psi^{+}=(\psi_{1},\dots,\psi_{5})$ Since the average of
$\psi_{2}(\tilde{x})=\overrightarrow{\underline{x}}$ and
$\psi_{3}(\tilde{x})=\overleftarrow{\underline{y}}$ is equal to $(x+y)/2$, we
see

$\psi_{3}(\tilde{x})-\psi_{1}(\tilde{x})>_{1}\frac{1}{2}(\psi_{5}(\tilde{x})-\psi_{4}(\tilde{x})).$

This implies

$c_{1}+2c_{2}+c_{3}>(c_{4}+c_{5})/2-4\epsilon_{P}$

by the assumption of $\tilde{x}$ and an argument similar to the proof of Lemma
3.3. This is impossible when $c_{i}/c_{i-1}$ is sufficiently large as assumed
in Definition 2.3, which implies $\delta^{\prime}_{1}\circ\psi^{+}=*$. The
other three terms are shown to be zero similarly.∎

######  Definition 7.15.

Put $G_{0}=\delta_{124}\partial_{12}G_{2}$
($=\delta_{124}\partial_{12}G_{3}=\delta_{124}\partial_{12}G_{4}$). $G_{0}$ is
the graph having only one edge $(\\{123\\},\\{45\\})$.

  1. (1)

For $(G,H,i,j)=(G_{2},G_{3},4,1)$ (resp. $(G_{2},G_{3},4,2)$,
$(G_{4},G_{3},4,2)$, $(G_{4},G_{3},4,1)$), let $\tilde{\psi}^{1+}_{kl}$ (resp.
$\tilde{\psi}^{2-}_{kl}$, $\hat{\psi}^{2-}_{kl}$, $\hat{\psi}^{1+}_{kl}$)
denote $\psi^{+}_{kl}$ (resp. $\psi^{-}_{kl}$, $\psi^{-}_{kl}$,
$\psi^{+}_{kl}$) in Definition 7.7. Let $\tilde{\eta}_{kl}$ (resp.
$\hat{\eta}_{kl}$) be the straight homotopy from $\tilde{\psi}^{1+}_{kl}$ to
$\tilde{\psi}^{2-}_{kl}$ (resp. from $\hat{\psi}^{2-}_{kl}$ to
$\hat{\psi}^{1+}_{kl}$). Set

$B_{1}=\sum_{k<l}(-1)^{k+l+1}\tilde{\eta}_{kl}(w_{32}),\qquad
B_{2}=\sum_{k<l}(-1)^{k+l+1}\hat{\eta}_{kl}(w_{32})\
\in\bar{C}_{*}(\mathcal{T}_{G_{0}}).$

  2. (2)

Put $G=G_{3}$, $E(G)=\\{e_{1}<e_{2}<e_{3}\\}$, and $f=f_{G}$. Let
$\lambda_{kl}$ be the straight homotopy from $(e_{k},e_{l})$-contraction of
$f$ for $G$ to $(e_{k},e_{l})$-contraction of $f$ for $\delta_{4}G$. Let
$\tilde{\lambda}_{kl}$ be the straight homotopy from $\lambda^{2+}_{kl}$ and
$\lambda^{4+}_{kl}$. Set

$B_{3}=\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}\tilde{\lambda}_{kl}(w_{32})\
\in\bar{C}_{*}(\mathcal{T}_{G_{0}})$

Put $\tilde{\mu}^{-}_{3kl}=\tilde{\lambda}_{kl}|_{t_{1}=1}$, where $t_{1}\in
I$ is the variable in $\lambda$. Let $\theta_{kl}$ be the straight homotopy
from $\tilde{\mu}^{-}_{3kl}$ to $\hat{\eta}|_{t_{1}=1}$ (see part 1). Set

$B_{4}=\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}\theta_{kl}(w_{32})\
\in\bar{C}_{*}(\mathcal{T}_{G_{0}})$

$B_{1},B_{2}$ are well-defined by Lemma 5.10, and well-definedness of
$B_{3},B_{4}$ is verified similarly to the proof of the lemma.

######  Lemma 7.16.

The cycle $-\delta C_{2}$ is homologous to
$\delta_{1}\left(\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}\mu_{2kl}^{+}(w_{31})\right)$
in $(tr_{1}\check{\mathbb{E}}_{0},d_{0})$. Here, $\mu_{2kl}^{+}$ is the map
$\mu_{kl}^{+}$ in Definition 7.8 for $(G,i,j)=(G_{2},2,4)$.

###### Proof.

By arguments similar to the proof of Lemma 7.14, we have

$\begin{split}D(B_{1})&=\sum_{k<l}(-1)^{k+l}(\tilde{\psi}_{kl}^{1+}(w_{31})+\tilde{\eta}_{kl}|_{t_{1}=1}(w_{31}))\\\
&=-\delta_{1}c(G_{2},G_{3},4,1)+\tilde{K},\\\
D(B_{2})&=\sum_{k<l}(-1)^{k+l}(\hat{\psi}_{kl}^{2-}(w_{31})+\hat{\eta}_{kl}|_{t_{1}=1}(w_{31}))\\\
&=-\delta_{1}c(G_{4},G_{3},4,2)+\hat{K},\\\
D(B_{3})&=\frac{1}{2}\sum_{k<l}(-1)^{k+l}(\tilde{\mu}_{3kl}^{-}-\mu_{3kl}^{-})(w_{31}),\\\
D(B_{4})&=\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}(\hat{\eta}|_{t_{1}=1}-\tilde{\mu}_{3kl}^{-})(w_{31}).\end{split}$

Here, we set

$\tilde{K}=\sum_{k<l}(-1)^{k+l}\tilde{\eta}_{kl}|_{t_{1}=1}(w_{31}),\qquad\hat{K}=\sum_{k<l}(-1)^{k+l}\hat{\eta}_{kl}|_{t_{1}=1}(w_{31}).$

$\mu_{3kl}^{-}$ is the map $\mu_{kl}^{-}$ in Definition 7.8 for
$(G,i,j)=(G_{3},2,4)$. Putting these equations into together, we have

$-\delta
C_{2}+D(B_{1}-B_{2}-B_{3}+B_{4})=\tilde{K}-2\hat{K}+\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}\mu_{2kl}^{+}(w_{31}).$

$\hat{K}$ is homologous to $-\tilde{K}$ since the parameter for the homotopy
between $(i,\pm)$-contractions are reversed. We have
$[\tilde{K}-2\hat{K}]=[-3\hat{K}]=0$. Thus, we have obtained the claim. ∎

We shall compute the remaining term.

######  Lemma 7.17.

The chain
$\delta_{1}\left(\frac{1}{2}\sum_{k<l}(-1)^{k+l+1}\mu_{2kl}^{+}(w_{31})\right)$
in Lemma 7.16 is a fundamental cycle of $\mathcal{T}_{G_{0}}$.

###### Proof.

By definition, $\mathcal{T}_{G_{0}}$ is a Thom space associated to the disk
bundle $\nu_{P}|_{N}$, where $N$ is a open tubular neighborhood of the
diagonal $\Delta$ in the product of disks $(D^{2})^{2}$ (with different
radius). Precisely speaking, in the notation of Definition 3.1,
$(D^{2})^{2}=\mathbb{R}^{4}-E_{P}$, and
$N=D_{\alpha\beta}\cap(\mathbb{R}^{4}-E_{P})$, where
$\alpha=\\{1,2,3\\},\beta=\\{4,5\\}$ and
$P=\\{\\{0\\},\alpha,\beta,\\{6\\}\\}$. Write $\nu=\nu_{P}$. Fiber dimension
of $\nu$ is $6$. The orthogonal projection $r:N\to\Delta$ induces the bundle
map $\tilde{r}:\nu\to\nu|_{\Delta}$ which is given by the parallel transport
taking center to center on each fiber. This map induces a map
$\tilde{r}:\mathcal{T}_{G_{0}}\to Th(\nu|_{\Delta})$. $Th(\nu|_{\Delta})$ is
homeomorphic to $S^{8}$. We consider $\nu$ and $(\nu|_{\Delta})$ as subspaces
of $\mathbb{R}^{10}$. Put

$F_{kl}:=\tilde{r}\circ\mu^{+}_{2kl}|_{(\mu^{+}_{2kl})^{-1}(\nu)}:(\mu^{+}_{2kl})^{-1}(\nu)\to\nu|_{\Delta}\qquad\text{for}\qquad
1\leq k<l\leq 3$

We shall show $F_{kl}$ is a monomorphism and
$\cup_{k<l}Im(F_{kl})=\nu|_{\Delta}$ and $Im(F_{kl})\cap
Im(F_{k^{\prime},l^{\prime}})$ is contained in the union of the image by
$F_{kl}$ of subspace where at least one of the contracting parameters
$s_{1},s_{2}$ in the direction of $v$ is zero if
$(k,l)\not=(k^{\prime},l^{\prime})$. These claims imply the lemma. By
definition, $\nu$ is the tubular neighborhood of the map

$e_{P}:(a,b)\mapsto\left(a-\frac{\rho}{2}(c_{2}+c_{3})u,\
a+\frac{\rho}{2}(c_{1}-c_{3})u,\ a+\frac{\rho}{2}(c_{1}+c_{2})u,\
b-\frac{\rho}{2}c_{5}u,\ b+\frac{\rho}{2}c_{4}u\right).$

The projection $\pi_{P}$ of $\nu_{P}$ sends $(c,d,e,f,g)\in\mathbb{R}^{10}$ to
the point $(a,b)$ which minimize the distance $|(c,d,e,f,g)-e_{P}(a,b)|$. By
elementary calculation, the point is given by

$(a,b)=\left(\frac{1}{3}(c+d+e+\rho(c_{3}-c_{1})u),\
\frac{1}{2}(f+g+\frac{\rho}{2}(c_{4}-c_{3})u)\right).$

Similarly, we see that $r:N\to\Delta$ is given by $r(a,b)=(a+b)/2$. We shall
verify that $F_{kl}$ is a monomorphism. For a while, we omit the subscript
$kl$ and $A_{e_{k},e_{l}}(s_{1},s_{2})$ in Definition 5.1 is also abbreviated
as $A=(A^{1},\dots,A^{5})v$, and $\mu^{+}_{2kl}$ is abbreviated as $\mu$. For
$\tilde{x}=(x,y,s_{1},s_{2},s_{3},t)$, we have

$\mu(\tilde{x})=(x+A^{1}v,x+(1-t)s_{3}u+A^{2}v,y-(1-t)s_{3}u+A^{3}v,x-ts_{3}u+A^{4}v,y+ts_{3}u+A^{5}v).$

Straightforward computation shows

$r\circ\pi_{P}\circ\mu(\tilde{x})=\frac{1}{12}\left(7x+5y+\rho\bigl{(}2c_{3}-2c_{1}+\frac{3}{2}c_{5}-\frac{3}{2}c_{4}\bigr{)}u+2(A^{1}+A^{2}+A^{3})v+3(A^{4}+A^{5})v\right).$

We denote right hand side of this equation by $w$. For simplicity, we move the
fiber of $\nu$ over $\pi_{P}(\mu(\tilde{x}))$ by the parallel transport which
sends its center to $0$. By this move, $\mu(\tilde{x})$ is sent to to

$\mu(\tilde{x})-e_{P}(\mu(\tilde{x}))=(p,q,-p-q,r,-r)$

where

$\begin{split}p&=\frac{1}{3}(x-y)+\frac{\rho}{6}(2c_{1}+3c_{2}+c_{3})u+\frac{1}{3}(2A^{1}-A^{2}-A^{3})v\\\
q&=\frac{1}{3}(x-y)+\frac{\rho}{6}(c_{3}-c_{1})u+(1-t)s_{3}u+\frac{1}{3}(-A^{1}+2A^{2}-A^{3})v\\\
r&=\frac{1}{2}(x-y)+\frac{\rho}{4}(c_{4}+c_{5})u-ts_{3}u+\frac{1}{2}(A^{4}-A^{5})v\\\
\end{split}$

The fiber over $\pi_{P}$ is a disk of radius $\epsilon_{P}$. It is enough to
show that there exists a point $\tilde{x}$ such that $p^{\prime}=p$,
$q^{\prime}=q$, $r^{\prime}=r$ for a given point
$(p^{\prime},q^{\prime},r^{\prime})$ with
$|(p^{\prime},q^{\prime},r^{\prime})|\leq\epsilon_{P}$, and the combination of
such a point and numbers $k,l$ is unique unless $s_{1}$ or $s_{2}=0$. We fix
$w,p^{\prime},q^{\prime},r^{\prime}$ and suppose
$(p^{\prime},q^{\prime},r^{\prime})=(p,q,r)$. By using the formula for $r$, we
eliminate $x-y$ in the formulas of $p,q$. We have

$\begin{split}p^{\prime}&=\frac{1}{3}(2r^{\prime}+(2ts_{3}+\frac{\rho}{2}(2c_{1}+3c_{2}+c_{3}-c_{4}-c_{5}))u+(2A^{1}-A^{2}-A^{3}-A^{4}+A^{5})v),\\\
q^{\prime}&=\frac{1}{3}(2r^{\prime}+((3-t)s_{3}+\frac{\rho}{2}(c_{3}-c_{1}-c_{4}-c_{5}))u+(-A^{1}+2A^{2}-A^{3}-A^{4}+A^{5})v).\end{split}$

By these formulas, we have

$\begin{split}s_{3}&=\frac{1}{4}(2p^{\prime}_{1}+2q^{\prime}_{1}-4r^{\prime}_{1}+\rho(c_{4}+c_{5}-c_{2}-c)),\\\
t&=\frac{6p^{\prime}_{1}-4r^{\prime}_{1}+\rho(c_{4}+c_{5}-2c_{1}-3c_{2}-3c_{3})}{2p^{\prime}_{1}+4q^{\prime}_{1}-4r^{\prime}_{1}+\rho(c_{4}+c_{5}-c_{2}-c_{3}),}\end{split}$

where the subscript $1$ means the first coordinate. The conditions on $c_{i}$
and $(p^{\prime},q^{\prime},r^{\prime})$ ensure $s_{3}>0$ and $0<t<1$. We
consider the last terms of the above formulas for $p^{\prime},q^{\prime}$. For
$(k,l)=(1,2),(1,3),$ and $(2,3)$, we have

$\begin{split}(2A^{1}-A^{2}&-A^{3}-A^{4}+A^{5},-A^{1}+2A^{2}-A^{3}-A^{4}+A^{5})\\\
&=(4s_{1}-2s_{2},-2s_{1}+4s_{2}),\quad(4s_{1}-2s_{2},-2s_{1}-2s_{2}),\ \text{
and }\ (-2s_{1}-2s_{2},4s_{1}-2s_{2}),\end{split}$

respectively. When $s_{1},s_{2}\geq 0$ vary, this point runs through a domain
$D_{kl}$. It is easy to see that $\cup_{kl}D_{kl}=\mathbb{R}^{2}$, $\partial
D_{kl}$ corresponds to $s_{1}$ or $s_{2}=0$, and $D_{kl}\cap
D_{k^{\prime},l^{\prime}}\subset\partial D_{kl}\cap\partial
D_{k^{\prime},l^{\prime}}$ if $(k,l)\not=(k^{\prime},l^{\prime})$. So, the
coefficients of $v$ can take any value. When we fix
$(p^{\prime},q^{\prime},r^{\prime})$, $x-y$ is fixed but $7x+5y$ can take any
value, so we can set $w$ freely. Thus, we have proved the claim. ∎

###  7.4. Non-triviality of the differential

To prove non-triviality of the differential $d_{3}$, we need to prove that a
generator of $\mathbb{E}^{-2,1}_{2}$ persists to $E_{3}$-page. For this, we
compute $d_{2}:\mathbb{E}_{2}^{-4,2}\to\check{\mathbb{E}}_{2}^{-2,1}$. The
module $\mathbb{E}^{-4,2}_{2}$ is generated by $g_{13}g_{24}$. We actually
consider the spectral sequence $tr_{1}\check{\mathbb{E}}$. A cycle which
represents $g_{13}g_{24}$ is a linear combination of chains in the Thom spaces
of the following three graphs (and their subgraphs).

$G_{5}=(1,3)(2,4),\qquad G_{6}=(1,3)(1,4),\qquad G_{7}=(1,4)(2,4)$

The computation of $d_{2}$ is similar to those done in section 6 and previous
subsections. One difference is that we need to deal with the 3-term relation.
To make the computation easier, we modify the definition of chains.

######  Definition 7.18.

Let $f:X\to\mathbb{R}^{8}$ be $G$-condensed map and $e=(\alpha,\beta)$ be a
bridge of $G$. In this subsection, we call the $e$-contraction given in
Definition 5.1 the $(e,+)$-contraction. The $(e,-)$-contraction of $f$ for $G$
is a version of $e$-contraction whose contracting direction is reversed. We
add $-sv$ (resp. $sv$) to the $i$-th component if
$i\sim_{\partial_{e}G}\alpha$ (resp. $i\sim_{\partial_{e}G}\beta$).

######  Example 7.19.

Put $f=f_{G_{5}},e=(1,3)$. the $(e,-)$-contraction of $f$ (for $G_{5}$) is
given by $(x,y,s)\mapsto(x-sv,y,x+sv,y)$.

######  Definition 7.20.

For $G=G_{5},G_{6},$ and $G_{7}$, set $f=f_{G}$ and $\\{e_{1}<e_{2}\\}=E(G)$.
Let $f^{\epsilon}_{j}$ be the $(e_{j},\epsilon)$-contraction of $f$. Set

$c^{\prime}(G)=f(w_{0})-f^{\pm}_{1}(w_{1})+f^{\pm}_{2}(w_{1}).$

Here, $f^{\pm}_{j}$ denotes the average as before. We consider the chain

$C^{\prime}=-c^{\prime}(G_{5})+c^{\prime}(G_{6})+c^{\prime}(G_{7}).$

Clearly, we have $D(C^{\prime})=0$.
We shall construct a bounding chain of $\delta C^{\prime}$. We define two
chain $c^{\prime}(G_{5},G_{6},1)$ and $c^{\prime}(G_{5},G_{7},3)$ satisfying

$Dc^{\prime}(G_{5},G_{6},1)=\delta_{1}c^{\prime}(G_{6})-\delta_{1}c^{\prime}(G_{5}),\qquad
Dc^{\prime}(G_{5},G_{7},1)=\delta_{1}c^{\prime}(G_{7})-\delta_{1}c^{\prime}(G_{5}),$

respectively. The definition is similar to that of $c(G,H,i)$ with taking
$(e,\epsilon)$-contractions into consideration and we omit details. We shall
construct a bounding chain of $\delta_{2}C^{\prime}$. Put
$f_{1}=f_{G_{5}},f_{2}=f_{G_{6}},$ and $f_{3}=f_{G_{7}}$. Below, we denote
edges of all involved graphs with two edges by the same notation
$e_{1}<e_{2}$. Since $\delta_{k}$ which appears here does not permute edges,
this does not cause confusion. Let $\lambda_{ij}^{\epsilon}$ be the straight
homotopy from $(e_{j},\epsilon)$-contraction of $f_{i}$ for $G_{i}$ to
$(e_{j},\epsilon)$-contraction of $f_{i}$ for $\delta_{2}G_{i}$, $\psi$ the
straight homotopy from $f_{1}$ to $f_{2}$, $\phi$ the straight homotopy from
$f_{1}$ to $f_{3}$, $\psi_{j}^{\epsilon}$ $(e_{j},\epsilon)$-contraction of
$\psi$ for $\delta_{2}G_{6}$, $\phi_{j}^{\epsilon}$
$(e_{j},\epsilon)$-contraction of $\phi$ for $\delta_{2}G_{7}$. Let
$G_{8}\in\mathsf{G}(\delta_{2}[5])$ be the graph with
$E(G_{8})=\\{(1,\\{23\\}),(1,4),(\\{23\\},4)\\}$. We set

$c(G_{5},G_{6},G_{7},2)=f_{1}(w_{0})+(\psi+\phi)(w_{01})+\sum_{j=1,2}(\psi^{\pm}_{j}+\phi^{\pm}_{j}+\lambda^{\pm}_{1j}-\lambda^{\pm}_{2j}-\lambda^{\pm}_{3j})(w_{11})$

where the pushforward by $f_{1}$, (resp. $\psi$, $\phi$,
$\psi^{\epsilon}_{j}$, $\phi^{\pm}_{j}$) belongs to $\bar{C}_{*}(G_{8})$,
(resp. $\bar{C}_{*}(\delta_{2}G_{6})$, $\bar{C}_{*}(\delta_{2}G_{7})$,
$\bar{C}_{*}(\delta_{2}\partial_{j}G_{6})$,
$\bar{C}_{*}(\delta_{2}\partial_{j}G_{7})$). This is well-defined as $f_{1}$
is also $\delta_{2}G_{6}$\- and $\delta_{2}G_{7}$-condensed.

######  Lemma 7.21.

We have $Dc(G_{5},G_{6},G_{7},2)=\delta_{2}C^{\prime}$. If we set
$\Gamma^{\prime}=c(G_{5},G_{6},1)+c(G_{5},G_{7},3-)-c(G_{5},G_{6},G_{7},2)$,
we have $D\Gamma^{\prime}=-\delta C^{\prime}$.

###### Proof.

Since $\delta_{2}\partial_{1}G_{6}=\delta_{2}\partial_{2}G_{7}$, the
pushforwards by $\psi^{\epsilon}_{1}$ and $\phi^{\epsilon}_{2}$ belongs to the
same summand, and we see $\psi^{\pm}_{1}|_{t=0}=\phi^{\mp}_{2}|_{t=0}$, where
$t$ is the variable for straight homotopy. (This twist of sign is the reason
why we introduce $(e,-)$-contraction.) These observations and elementary
computation imply first equation. Second one follows from first one. ∎

######  Proposition 7.22.

Let $\Gamma^{\prime}$ be the chain defined in Lemma 7.21.
$\delta\Gamma^{\prime}$ is null-homologous in the chain complex
$(tr_{1}\mathbb{T},D)=(tr_{1}\check{\mathbb{E}}_{0},d_{0})$. In particular,
$d_{2}(g_{13}g_{24})=0$ in $\mathbb{E}_{2}$.

###### Proof.

By degree reason, we may consider $\bar{\mathbb{E}}_{r}$ instead of
$\mathbb{E}_{r}$. Among the terms of $\delta\Gamma^{\prime}$,
$\delta_{2}(c^{\prime}(G_{5},G_{6},1))$ only contains non-zero terms. Each of
$(e,+)$\- and $(e,-)$-contractions of this term forms a cycle and we can
easily see their classes cancel with each other similarly to the proof of
Lemmas 6.7 and 7.17. Latter part easily follows from former one and Lemma 4.6.
∎

In view of Lemma 7.17 and Proposition 7.22, the proof of the following theorem
is completely similar to Theorem 6.8.

######  Theorem 7.23.

In dimension $d=2$ and over a field of characteristic $3$, the element
$[-c(G_{1})+c(G_{2})+c(G_{3})+c(G_{4})]\in
tr_{1}\check{\mathbb{E}}^{-3,5}_{1}$ lifts to an element
$g\in\mathbb{E}^{-3,5}_{1}$ which persists up to $\mathbb{E}^{-3,5}_{3}$ and
satisfies $d_{3}(g)\not=0$. ∎

##  8\. Absolute non-formality in characteristic 2

In this section, we prove Corollary 1.3. We assume that $\mathsf{k}$ is a
field of characteristic $2$ and $d=2$. Let $\mathcal{A}_{\infty}$ denote the
cellular chain operad of Stasheff’s associahedral operad. Precisely speaking,
$\mathcal{A}_{\infty}$ is generated by a set
$\\{\,\mu_{k}\in\mathcal{A}_{\infty}(k)\,\\}_{k\geq 2}$ ( $|\mu_{k}|=k-2$ )
with partial compositions. The differential is given by the following formula:

$d\mu_{k}=\sum_{l,p,q}\,\mu_{l}\circ_{p+1}\mu_{q}.$

where $l,p,q$ runs through the range $l,q\geq 2,0\leq p\leq l-1,$ and
$l+q=k+1$ .

######  Definition 8.1.

For a $\mathsf{k}$-vector space $U$, $U^{\vee}$ denotes its linear dual. Let
$f:\mathcal{A}_{\infty}\to\mathcal{O}$ be a map of chain operads. Let
$\mu^{\prime}_{l}\in\mathcal{O}(l)$ be the image of
$\mu_{l}\in\mathcal{A}_{\infty}(l)$ by $f$. we define a linear map
$(-\circ_{i}\mu_{l}):\mathcal{O}(m)^{\vee}\to\mathcal{O}(m-l+1)^{\vee}$ for
integers $m\geq l$ and $1\leq i\leq l$ as the following composition

$\textstyle{\mathcal{O}(m)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-\circ_{i}-)^{\vee}\qquad\qquad}$$\textstyle{(\mathcal{O}(l)\otimes\mathcal{O}(m-l+1))^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(m-l+1)^{\vee}}$

where the right arrow is the evaluation of the first factor on
$\mu^{\prime}_{l}$. We also define
$(\mu_{l}\circ_{i}-):\mathcal{O}(m)^{\vee}\to\mathcal{O}(m-l+1)^{\vee}$ for
integers $m\geq l$ and $1\leq i\leq m-l+1$ similarly. We define a chain
complex $(\mathrm{CH}\mathcal{O},\tilde{d})$ called Hochschild complex of
$\mathcal{O}$, as follows. Set
$\mathrm{CH}^{-p,q}\mathcal{O}=(\mathcal{O}(p)_{q})^{\vee}$. The differential
$\tilde{d}$ is given as a map

$\tilde{d}=d+\delta:\underset{q-p=k}{\bigoplus}\mathrm{CH}^{-p,q}\mathcal{O}\longrightarrow\underset{q-p=k+1}{\bigoplus}\mathrm{CH}^{-p,q}\mathcal{O}.$

Here $d$ is the internal (original) differential on $\mathcal{O}(p)^{\vee}$
and $\delta$ is given by the formula $\delta(x)=\sum_{l\geq 2}\mu_{l}*x$ where

$\mu_{l}*x=x\circ_{1}\mu_{l}+x\circ_{l}\mu_{l}+\sum_{i=1}^{p-l+1}\mu_{l}\circ_{i}x$

for $x\in\mathcal{O}(p)^{\vee}$. We define a spectral sequence
$E^{-p,q}_{r}(\mathcal{O})$ by filtering $(\mathrm{CH}\mathcal{O},\tilde{d})$
by the arity $p$.

We call a map $f:\mathcal{O}\to\mathcal{P}$ of chain operads a weak-
equivalence if it induces a quasi-isomorphism
$\mathcal{O}(p)\to\mathcal{P}(p)$ for each $p$. The following lemma is clear.

######  Lemma 8.2.

  1. (1)

Let $\mathcal{A}_{\infty}\to\mathcal{O}$ and
$\mathcal{A}_{\infty}\to\mathcal{O}^{\prime}$ be two maps of operads and
$f:\mathcal{O}\to\mathcal{O}^{\prime}$ a weak equivalence compatible with the
maps from $\mathcal{A}_{\infty}$. Then $f$ induces an isomorphism
$E_{r}(\mathcal{O})\cong E_{r}(\mathcal{O}^{\prime})$ compatible with the
differentials for $r\geq 1$.

  2. (2)

Let $\mathcal{A}_{\infty}\to C_{*}(\mathcal{K}_{2})$ be the composition
$\mathcal{A}_{\infty}\to\mathcal{A}\to C_{*}(\mathcal{K}_{2})$ of the fixed
maps (see Definition 2.4). The spectral sequence
$\\{E_{r}(C_{*}(\mathcal{K}_{2}))\\}$ is isomorphic to $\\{\mathbb{E}_{r}\\}$
(see Definition 4.5). ∎

The following lemma is easily obtained by unwinding the definition of the
spectral sequence $E_{r}(\mathcal{O})$.

######  Lemma 8.3.

Let $\mathcal{A}_{\infty}\to\mathcal{O}$ be a map of operads.

  1. (1)

$d_{1}([x])=[\mu_{2}*x]$ for an element $[x]\in E^{-p,q}_{1}(\mathcal{O})$
represented by $x\in\mathcal{O}(p)^{\vee}_{q}$.

  2. (2)

For an element $[x]\in E^{-p,q}_{2}(\mathcal{O})$ represented by
$x\in\mathcal{O}(p)^{\vee}_{q}$, we can take an element
$y\in\mathcal{O}(p-1)_{q-1}^{\vee}$ with $dy=\mu_{2}*x$. We have
$d_{2}[x]=[\mu_{2}*y+\mu_{3}*x]$. ∎

###### Proof of Corollary 1.3.

Let $\mathcal{A}_{\infty}\to C_{*}(\mathcal{K}_{2})$ be the map given in Lemma
8.2. We use the projective model structure on the category of planar chain
operads (see e.g. [16]). We take a cofibrant replacement
$\mathcal{A}_{\infty}\rightarrowtail\mathcal{O}\stackrel{{\scriptstyle\sim}}{{\twoheadrightarrow}}C_{*}(\mathcal{K})$.
Suppose that $C_{*}(\mathcal{K}_{2})$ is formal. By this assumption, we can
take a weak equivalence of operads $\mathcal{O}\to H_{*}(\mathcal{K})$. By
considering the composition $f:\mathcal{A}_{\infty}\to\mathcal{O}\to
H_{*}(\mathcal{K})$, we obtain an isomorphism of spectral sequences
$E_{r}(\mathcal{O})\cong E_{r}(H_{*}(\mathcal{K}))$. Let $\mu^{\prime}_{l}\in
H_{l-2}(\mathcal{K}_{2}(l))$ be the image of $\mu_{l}$ by $f$. By definition
of $\mathcal{A}_{\infty}$, we see

$\mu^{\prime}_{2}\circ_{1}\mu^{\prime}_{3}+\mu^{\prime}_{2}\circ_{2}\mu^{\prime}_{3}+\mu^{\prime}_{3}\circ_{1}\mu^{\prime}_{2}+\mu^{\prime}_{3}\circ_{2}\mu^{\prime}_{2}+\mu^{\prime}_{3}\circ_{3}\mu^{\prime}_{2}=d\mu^{\prime}_{4}=0.$

This means that $\mu^{\prime}_{3}$ is a cycle for the differential of the
unnormalized complex of the cosimplicial vector space associated to the fixed
map $\mathcal{A}\to H_{*}(\mathcal{K})$. By easy (and well-known) computation,
this differential is a monomorphism on $H_{1}(\mathcal{K}(3))$, so we have
$\mu^{\prime}_{3}=0$. This observation and Lemma 8.3 imply $d_{2}=0$ for
$E_{2}(H_{*}(\mathcal{K}))$. Since $E_{r}(\mathcal{O})$ is isomorphic to
$E_{r}(C_{*}(\mathcal{K}))\cong\mathbb{E}_{r}$, this vanishing of differential
contradicts to Theorem 6.8. ∎

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