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Interested in pursuing your degree through the Advanced Inquiry Program? Join us for an informational forum on November 13!
Woodland Park Zoo (WPZ) is one of eight institutions across the country that offers the AIP Master’s.
The first AIP cohort at WPZ started in 2011 (many of whom are graduating this December!) and students have already reported positive changes in their personal and professional lives. We asked Carole Parks, AIP graduate student and an instructor with the Community Colleges of Spokane, to illustrate the impact that this program has had on her teaching, her community, and her life.
WPZ: Why did you apply to the Advanced Inquiry Program (AIP)?
Carole: I've always dreamed of studying zoology. When I was growing up in San Diego, we would frequent the zoo there and we would never miss Omaha's Wild Kingdom TV program. Wildlife has always fascinated me and doing something on behalf of wildlife, such as addressing climate change, makes me feel empowered. When the AIP opportunity came up, it was a natural fit.
WPZ: What impact has the program had on you personally and professionally?
Carole: This program has enriched my life deeply. On a personal basis, it has been a source of joy and hope in a difficult time in my life. It's very motivating to work on something that I feel so passionate about... and it's downright fun when we have face to face days at the zoo or in the field. I participated in our Northwest Wildlife Conservation course this summer, where we engaged in four days of learning about and participating in wildlife conservation in Washington’s Columbia Plateau.
We met farmers working to incorporate sustainable practices in their agriculture as well as land managers from the Nature Conservancy and Washington Department of Fish & Wildlife biologists working on pygmy rabbit recovery and golden eagle conservation. We answered our own questions about the area through inquiry investigations of the area’s wildlife and habitat.
We also met Nancy Warner, coordinator with the Initiative for Rural Innovation and Stewardship, who was very inspiring. She helped us challenge ourselves to involve our communities in our local projects.
On a professional level, I teach adult education, much of it developmental education or GED. It has been exciting to be able to implement many of the things I have learned, both content and process, in my classroom. There's nothing better than seeing the "aha" on my students' faces when they understand a science concept that they may have previously struggled with. They also catch my passion. For instance, I taught a lesson about the expense and environmental impact of buying bottled water. They said "Hey! How come this is not on the news? Everyone needs to know this!" Another professional perk is that I participate in citizen science wildlife monitoring, which I hope leads to some paid fieldwork someday, maybe during the summers! I don't think I would want to give up the influential position of being a teacher!
WPZ: What impact has this program had on your community?
Carole: In the beginning of the program, I really thought hard about this question. What kind of impact did I want to have on my community? After all, knowledge without action is pretty worthless. So I decided to focus on a local, urban park: Drumheller Springs, in my hometown of Spokane, Washington. It started out as a website for the Biology in the Age of Technology class, but it has developed into much more than that! I am now doing active research in the park and hope to publish a paper in the next year! We have some heritage plants with Salish names and traditional medicinal uses, thanks to collaboration with my Spokane Tribal friend, Barry Moses, and we hope to get some audio clips with Salish narrative on the website. The native historical aspect of the park is a highlight and is great for visitor information. Now my website is the first that comes up when you put Drumheller Springs, Spokane into a search engine! It's all very exciting and I can't wait to see how this website develops in the future.
I am very proud of the education I am receiving at Woodland Park Zoo and Miami University of Ohio. I already have a MS in education, so the addition of this degree will lead me to even more exciting work. I would encourage anyone who even has a slight interest to check out this program, it's one of the most amazing things that has ever happened to me!
Want to know more?
- Please join us for our informational forum about the Advanced Inquiry Program!
Wednesday, November 13, 6:00 - 7:30 p.m.
Woodland Park Zoo Education Center.
This informational forum includes snacks and a live animal presentation!
To RSVP, please call 206.548.2581 or email AIP@zoo.org
- Check out the Advanced Inquiry Program page on the zoo’s website for more information: Applications for the Advanced Inquiry Program are accepted until February 28, 2014 for summer enrollment. For more information or to apply, see Project Dragonfly’s website. | 125,059 |
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TITLE: Show that a line l intersects all four sides of a quadrilateral given that none of the four points ABCD lies on l.
QUESTION [0 upvotes]: Let ABCD be a quadrilateral with a line l intersecting the segment AB. None of the four points ABCD lies on l.
I have to show that l either intersects two sides of ABCD or all four sides. I get how to show that it intersects two sides(using the Plane separation Postulate), but how do I do it for all four? (And is that even possible???)
REPLY [1 votes]: Suppose e.g. that $l$ intersects sides $AB,BC,CD.$ Then $A,B$ on opposite sides of $l,$ also $B,C$ on opposite sides, finally $C,D$ on opposite sides. This implies that $A,D$ are on opposite sides, so that side $AD$ intersects line $l.$
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\begin{document}
\large
\begin{center}{\bf\Large Milnor-Orr invariants from the Kontsevich invariant}.
\end{center}
\vskip 1.5pc
\begin{center}{Takefumi Nosaka\footnote{
E-mail address: {\tt nosaka@math.titech.ac.jp}
}}\end{center}
\vskip 1pc\begin{abstract}\baselineskip=12pt \noindent
As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich.
We show that the Orr invariant of degree $ k $ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $.
Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariants.
\end{abstract}
\begin{center}
\normalsize
\baselineskip=17pt
{\bf Keywords} \\
\ \ \ Knot, Milnor invariant, nilpotent group, Magnus expansion \ \ \
\end{center}
\large
\baselineskip=16pt
\section{Introduction}
In \cite{Mil1,Mil2}, Milnor defines his $\bar{\mu}$-invariants of links, which extract numerical information from the lower central series of the link groups and the link longitudes. The $\bar{\mu}$-invariants have been studied from topological viewpoints (see \cite{IO,CDM,MY} and references therein, and \cite{KN} for a powerful computation).
Furthermore, as a homotopical approach to $\bar{\mu}$-invariants, Orr \cite{Orr} introduced an invariant of ``based links". This Orr invariant provides obstruction of slicing links in a nilpotent sense \cite{Orr,IO}. However, since the invariant is defined as a homotopy 3-class of a homotopy group, there are few examples of the computation, and it is not clear whether the invariant is properly a generalization of the $\bar{\mu}$-invariants or not
Meanwhile, in quantum topology, a standard way to study nilpotent information is to carefully observe the tree parts of the Kontsevich invariants or LMO functor; see, e.g., \cite{HM,GL,Mas2}. For example, the first non-vanishing term of $\bar{\mu}$-invariants is equal to that of a tree reduction of the Kontsevich invariant \cite{HM}, with a relation to the Chen integral. Concerning the mapping class group, the Johnson-Morita homomorphism and Goldmann Lie algebra can be also nilpotently studied from the LMO functor (see \cite{Mas,Mas2}). Furthermore, such observations of tree parts sometimes approach the fundamental homology 3-class of 3-manifolds, together with relations to Massey products \cite{GL}.
In this paper, inspired by the works of Massuyeau \cite{Mas,Mas2}, we show that the (Milnor-)Orr invariant can be recovered from the Kontsevich invariant. We should note that these invariants are appropriately graded, and that, given $k\in \mathbb{N}$, the Orr invariant of degree $k$ is defined for any based link $L$ whose $\bar{\mu}$-invariants of degree $\leq k$ are zero. The theorem is as follows:
\begin{thm}
[Corollary \ref{thm2244}]\label{thm22442} Given a based link $L$ whose $\bar{\mu}$-invariants of degree $\leq k$ vanish, the Orr invariant of $ L $ is equivalent to the tree-shaped reduction of the Kontsevich invariant of degree $< 2k $.
\end{thm}
\noindent
This theorem is a generalization of the above result \cite{HM} (see Remark \ref{ll}) and gives a topological interpretation of the tree reduction of degree $< 2k $. Moreover, it is natural for us to ask what finite type invariants recover the Orr invariant; as a solution, we suggest a computation of the Orr invariant from HOMFLYPT polynomials (see \S \ref{SHigher}), where this computation is based on \cite{MY}.
Furthermore, we will show the equivalence between the Orr invariant of degree $k$ and Milnor $\bar{\mu}$-invariants of degree $< 2k$; see Theorem \ref{thm24}. Accordingly, while the Orr invariant is a homotopy 3-class, the 3-class turns out to be described by the link longitudes, as is implicitly pointed out in \cite{Mas2,K}; see the figure below as a summary. Our result is analogous to the result \cite{Mas} concerning the mapping class group, which claims an equivalency between ``the Morita homomorphism" of degree $k$ and ``the total Johnson homomorphisms" of degree $< 2k$. Thus, it can be hoped that and the tree parts of the LMO functor can be described in terms of homology 3-classes or algebraic topology.
\begin{center}
\fbox{
\begin{tabular}{c}
\!\!\!\! Milnor $\mu$-invariant \!\!\!\! \\
of degree $<2k$
\end{tabular}} \raisebox{0.95ex}[0pt][0pt]{$\stackrel{\textrm{Theorem \ref{thm24}}}{ =\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=}$}
\fbox{
\begin{tabular}{c}
\!\!\!\! Orr invariant \!\!\!\!\\
of degree $k$
\end{tabular}}\raisebox{0.95ex}[0pt][0pt]{$\stackrel{\textrm{Corollary \ref{thm2244}}}{ \ = \!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=\!\!=}$}
\fbox{
\begin{tabular}{c}
\!\! \!\!Tree-reduced Kontsevich \!\!\!\! \\
invariant of degree $<2k$
\end{tabular}}
\end{center}
\vskip -1.97pc
\hspace{46.83ex}
$ \leftarrow \!\! \!- \!\! \!- \!\! \!-\!\! \!-\!\!\! - \!\!\! -\!\!\! - \!\!\! - $
\vskip -0.97pc
\hspace{47.85ex}
{\small Recover via}
\vskip -0.47pc
\hspace{47.845ex}
{\small HOMFLYPT \cite{MY}}
This paper is organized as follows. Section 2 reviews Milnor-Orr invariants and states Theorem \ref{thm24}, and Section 3 describes the relation to the Kontsevich invariant. Section 4 gives the proofs. Appendix \ref{SHigher} explains the computation from HOMFLYPT polynomials.
\section{Theorem on Milnor-Orr invariant}
\label{Semb}
The first part of this section reviews the Milnor-Orr invariant, and the second part states Theorem \ref{thm24}.
\subsection{Review of Milnor invariant and Orr invariant}
\label{Semb5}
We will start by reviewing string links. Let $I$ be the interval $[0,1]$, and fix $q \in \N $. A {\it ($q$-component) string link} is a smooth embedding of $q$ disjoint oriented arcs $A_1, \dots, A_q$ in the 3-cube $I^3$, which satisfy the boundary condition $A_i = \{ p_i, q_i\}$, where $p_i =( j /2q , 0,0)$ and $q_i =( j /2q , 0,1) \in I^3 $. We define $SL(q)$ to be the set of string links of $q$-components. Given two string links $T $ and $T'$, we can define another string link, $T \cdot T' \subset [0,1]^2 \times [0,2] \cong I^3$, by connecting $q_i$ in $T$ and $ p_i'$ in $T'$. If $T =\{A_i \}$ is a string link, then an oriented link $L = \{L_1,\dots, L_q\}$ can be defined to be $L_i = A_i \cup a_i$, where $a_i$ is a semi-circle in $S^3 \setminus L $ connecting $p_i$ and $q_i$. We call the link {\it the closure of $T$} and denote it by $\overline{T}$; see Figure \ref{STU2}.
\begin{figure}[b]
\begin{center}
\begin{picture}(50,74)
\put(-156,37){\pc{stringlink}{0.202}}
\put(26,37){\pc{stringlink.closure}{0.204}}
\end{picture}
\end{center}
\vskip -1.17pc
\caption{\label{STU2} A string link and the closure as a link in $S^3$.}
\end{figure}
Next, we will describe the groups used throughout this paper. Let $F$ be the free group with generator $x_1, \dots, x_q$.
For a group $G$, we define $G_1 $ to be $G$ and $G_m $ to be the commutator $ [G_{m-1} , G]$ by induction. If $G$ is the free group $G$, then the projection $p_{m-1}: F/F_m \ra F/F_{m-1}$ implies the central extension,
\begin{equation}\label{kihon2} 0 \lra F_{m-1} /F_m \lra F/F_m \xrightarrow{\ \ p_{m-1}\ \ }F/F_{m-1} \lra 0 \ \ \ \ \ (\mathrm{central \ extension})
\end{equation}
The abelian kernel $F_{m-1} /F_m $ is known to be free with a finite basis.
Now let us explain the $m$-th leading terms of the Milnor invariant, according to \cite{Mil1,IO,KN}. We will suppose that the reader has elementary knowledge of knot theory, as can be found in \cite[\S 1 and \S 12]{CDM}. Given a $q$-component link $L $ in the 3-sphere $S^3$ and $ \ell \leq q $, we can uniquely define the (preferred) longitude $ \mathfrak{l}_{\ell} \in \pi_1 (S^3 \setminus L)$ of the $\ell$-th component. In addition, let $f_2 : \pi_1 (S^3 \setminus L) \ra F/F_2 =\Z^q $ be the abelianization $\mathrm{Ab}$. Furthermore, for $k \in \mathbb{N}$, we assume
\begin{itemize}
\item
\noindent
\textbf{Assumption $\mathfrak{A}_{k}$}. There are homomorphisms $f_s : \pi_1 (S^3 \setminus L) \ra F/F_s $ for $s$ with $s \leq k$, which satisfy the commutative diagram,
$$
{\normalsize
\xymatrix{
\pi_1 (S^3 \setminus L) \ar[d]_{f_2} \ar[dr]_{f_3} \ar[drrr]_{f_4}^{} \ar[drrrrrr]^{f_k}_{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \cdots \cdots }& & \\
F/F_2 & F/F_3 \ar[l]^{p_2 }& &F/F_4 \ar[ll]^{p_3} &\cdots \cdots \ar[l] & & F/F_{k} \ar[ll]^{p_{k-1} }.
}}$$
\end{itemize}
\noindent
Here, we should note that if there is another extension $f_k' $ instead of $f_{k}$, then $f_k$ equals $f_k'$ up to conjugacy, by centrality. Further,
we should note following proposition.
\begin{prop}
[\cite{Mil2}]\label{ea211} Suppose Assumption $\mathfrak{A}_{k}$. Then, $f_k$ admits a lift $ f_{k+1} : \pi_1 (S^3 \setminus L) \ra F/F_{k+1} $ if and only if all the Milnor invariants of length $< k $ vanish, i.e., $ f_k ( \mathfrak{l}_{\ell} ) =0 $.
\end{prop}
Thus, the map $f_k$ sends the preferred longitude $ \mathfrak{l}_{\ell} $ to the center $ F_{k-1} /F_k $. Then, the $q$-tuple,
$$ \bigl( f_k( \mathfrak{l}_1 ), \dots, f_k( \mathfrak{l}_{q} ) \bigr) \in (F_{k-1} /F_k)^q, $$
is called {\it the first non-vanishing Milnor $\mu $-invariant} or {\it the Milnor $\mu $-invariant} of length $k-1 $. Proposition \ref{ea211} implies that $\mu $-invariant is known to be a complete obstruction for lifting $f_{m}$.
The paper \cite{KN} gives an algorithm to describe $f_m$ explicitly, and a method of computing the Milnor invariants.
We further review the Orr invariant \cite{Orr}, where $L$ satisfies Assumption $\mathfrak{A}_{k+1}$, that is, all the $\mu $-invariants of length $\leq k$ are zero; $f_{k } (\mathfrak{l}_{\ell})=0$. Fix a homomorphism $\tau : F \ra \piL $ that sends each generator $x_{\ell}$ to some meridian $ \mathfrak{m}_{\ell}$ of the $\ell$-th component for $\ell \leq \# L$. This $\tau$ is called {\it a basing}, and the pair $(L,\tau)$ is referred as to {\it a based link}. As examples that we will refer to later, given a string link $T$, the closure $\overline{T}$ has a canonical basing, where $\tau$ is obtained from choosing the loop circling $\{( j/2q , 0, 1)\} $. Furthermore, for a group homomorphism $f: G \ra H$, we will write $f_*: K( G,1 ) \ra K( H,1 )$ for the induced map between Eilenberg-MacLane spaces. We define the space $K_k$ to be the mapping cone,
$$K_k := \mathrm{Cone} \bigl( (f_k \circ \tau )_*: K( F,1) \ra K( F/F_k ,1 )\bigr). $$
Then, from the assumption $f_k(\mathfrak{l}_{\ell})=0$, $f_k$ gives rise to a continuous map $\rho_L: S^3 \ra K_k . $ It is reasonable to consider the homotopy 3-class,
\begin{equation}\label{hom3} \theta_k (L, \tau) := [ \rho_L ] \in \pi_3 ( K_k ) ,
\end{equation}
which we call {\it the Orr invariant}. The following is a list of known results on the invariant and $\pi_3 ( K_k ).$
\begin{thm}[\cite{Orr,IO}]\label{thdr}
\begin{enumerate}[(I)]
\item Let $N_h \in \N $ be the rank of $H_2 (F/F_h;\Z ) =F_h/F_{h+1 }$. The following are isomorphisms on $\pi_3(K_k)$ and on $H_3(K_k)$:
$$\pi_3(K_k) \cong \bigoplus_{h=k}^{2k-1} \Z^{q N_{h} - N_{h+1} } , \ \ \ \ \ \ \
H_3(K_k;\Z )\cong H_3(K(F/F_k,1 );\Z) \cong \bigoplus_{h=k}^{2k-2} \Z^{q N_{h} - N_{h+1} } .$$
Furthermore, the Hurewicz homomorphism $\mathfrak{H}: \pi_3(K_k)\ra H_3(K_k;\Z)$ is equal to the projection according to the direct sums on the right-hand sides.
\item The lowest summand of $\theta_k (L, \tau) $ is equivalent to the Milnor invariant of length $k$; see \cite{Orr} or \cite[\S 10]{IO} for details.
\item For any element $ \kappa \in \pi_3(K_k ) $, there exist a link $L$ and a homomorphism $g_k : \pi_1 (S^3 \setminus L) \ra F/F_k$ satisfying $ \theta_k (L, \tau) = \kappa. $
\item The Orr invariant has additivity with respect to ``band connected sums"; see \cite[\S 3]{Orr}. As a special case, for two string links $T_1$ and $T_2$ such that the closures $\overline{T}_1$ and $\overline{T}_2$ satisfy Assumption $\mathfrak{A}_{k+1}$, we have $\theta_k ( \overline{T_1 \cdot T_2}, \tau_1 ) = \theta_k ( \overline{T}_1, \tau_1) + \theta_k ( \overline{T}_2, \tau_2) . $
\item Let $\iota_k: K_k \ra K_{k+1}$ be the continuous map arising from the projection $F/F_k \ra F/F_{k+1}$. The Orr invariant has functoriality. To be precise, if $L$ satisfies $\mathfrak{A}_{k+1}$, then the equality $(\iota_{k})_* \bigl( \theta_k (L, \tau)\bigr) = \theta_{k+1}(L, \tau) $ holds in $\mathrm{Im} (\iota_{k})_* \cap \pi_3(K_{k+1}) $.
\end{enumerate}
\end{thm}
Next, we mention the homological reduction of the Orr invariant $ \theta_k (L)$ via the Hurewicz map $\mathfrak{H}: \pi_3(K_k)\ra H_3(K_k;\Z).$ Note that the inclusion $K(F/F_k,1) \ra K_k $ induces the isomorphism,
$$P^{\rm gr} : H_3(K (F/F_k,1), \sqcup^q K(\Z,1) ;\Z) \cong H_3(K_k;\Z), $$
from the relative homology. To summarize, the value $\mathfrak{H} ( \theta_k (L)$) is the reduction of $ \theta_k (L)$ without the top summand $\Z^{q N_{2k-1} - N_{2k} } $. Moreover, by definition, this $\mathfrak{H} ( \theta_k (L)$) can be regarded as the pushforward of the fundamental 3-class $ [S^3 \setminus L , \partial ( S^3 \setminus L)] \in H_3(S^3 \setminus L , \partial ( S^3 \setminus L) ;\Z )\cong \Z $:
\begin{equation}\label{hom553} P^{\rm gr} \circ (f_k)_* [S^3 \setminus L , \partial ( S^3 \setminus L)] = \mathfrak{H} ( \theta_k (L,\tau)) \in H_3(K_k;\Z ).
\end{equation}
The author \cite{Nos} showed that the cohomology $H^3(K (F/F_k,1)) $ are generated by some Massey products; thus, the reduction \eqref{hom553} are characterized by some Massey products of $S^3 \setminus L $.
\subsection{Results: Orr invariant of higher invariants}\label{revi3}
In order to state the theorem, let us briefly review the Milnor $\mu$-invariant for string links (see \cite{HM,K,Le}). For a string link $T \in SL(q)$, let $y_j \in \pi_1 ( [0,1]^3 \setminus T) $ be an element arising from the loop circling $\{( j/2q , 0, 1)\} $; Let $G_m$ be the $m$-th nilpotent quotient of $\pi_1 ( [0,1]^3 \setminus T) $. Then, as is shown \cite{Mil2,Le}, the homomorphism $ F \ra \pi_1 ( [0,1]^3 \setminus T) $ which sends $ x_i $ to $y_i$ descends to an isomorphism between the $m$-th nilpotent quotients:
\begin{equation}\label{dd} \phi_*: F/F_m \cong \pi_1 ( [0,1]^3 \setminus T)/G_m, \textrm{ \ for any } m \in \N.\end{equation}
Here, we should note that, if $T$ is a pure braid, this $\phi_*$ is the identity map. Furthermore, a framing of the $\ell $-th component of $T$ defines a parallel curve which determines an element, $\lambda_\ell \in \pi_1 ([0,1]^3 \setminus T ) $. This $\lambda_\ell $ is referred as to {\it the $\ell$-th longitude of $T$}. We call the reduction $\phi_*^{-1}(\lambda_\ell) \in F /F_m $ modulo $ F_m $ {\it the $\mu$-invariant of $T$} (of degree $\leq m$). Later, we will omit writing $\phi_*^{-1} $ for simplicity. We should notice, from the definitions, that the closure $\overline{T}$ of a string link $T$ satisfies $\mathfrak{A }_{k+1} $ if and only if $ \lambda_j $ lies in $F_k $, and $\lambda_j = f_{k+1} ( \mathfrak{l}_j ) \in F_k / F_{k +1}$ modulo $ F_{k +1}$.
Thus, for such a string link $T$, as in the paper \cite{Le} of Levine, it is reasonable to consider the invariant, $\lambda_j $ modulo $ F_{2k}$. Notice that $F_k/F_{2k}$ is abelian, since $[F_k,F_k] \subset F_{2k}$. Furthermore, as can be seen from \cite[Proposition 4]{Le}, we can verify that the equality,
\begin{equation}\label{le} [x_1, \lambda_1 ][x_2, \lambda_2] \cdots [x_q, \lambda_q] =1 \in F,
\end{equation}
always holds. Thus, for $m \leq 2k$, the sum,
$$ \sum_{j=1}^q(x_j\otimes \lambda_j ) \in \Z^q \otimes_{\Z } F_k/F_{m} \textrm{ modulo } F_{m},$$
is contained in the kernel of the commuting operator,
$$ [ \bullet , \bullet ]_{k,m} : F/F_2 \otimes F_k/F_{m} \lra F_{k+1} /F_{m+1}; \ \ x\otimes y \longmapsto xy x^{-1} y^{-1}. $$
This operation will be used in many times. Now let us show the equivalence of Milnor and Orr invariants:
\begin{thm}
[{See \S \ref{Sher33} for the proof}]\label{thm24}
(I) There is a $\Q$-vector isomorphism,
$$ \Phi \circ \eta^{-1}: \Q \otimes \Ker ([ \bullet , \bullet ]_{ k, 2k -1 } ) \stackrel{\sim }{\lra } H_3(F/F_k;\Q ), $$
such that the following holds for any string link $T$ satisfying $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$:
$$\Phi \circ \eta^{-1} \bigl( (x_1 \otimes \lambda_1)+ \cdots + (x_q \otimes \lambda_q ) \bigr) = \mathfrak{H} \circ \theta_k (\overline{T} , \tau). $$
(II) Furthermore, concerning the homotopy group $\pi_3 (K_k )$, there is a bijection
$$ \overline{\Phi} : \Q \otimes \Ker ([ \bullet , \bullet ]_{ k, 2k } ) \stackrel{\sim }{\lra } \pi_3(K_k) \otimes \Q, $$ as an extention of $\Phi \circ \eta^{-1}$,
such that a similar equality $ \overline{\Phi} \bigl( (x_1 \otimes\lambda_1)+ \cdots + (x_q \otimes \lambda_q ) \bigr) = \theta_k(\overline{T} , \tau)$ holds for any string link $T$ satisfying $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$.
\end{thm}
We conjecture that the bijection in (II) is an isomorphism. However, from Theorem \ref{thdr} (III), we have the realizability result as a generalization of \cite[Proposition 5]{Le}:
\begin{cor}\label{cor24}
Let $(\alpha_1,\dots, \alpha_q ) \in F_k/F_{2k} $ satisfy $[x_1, \alpha_1 ]\cdots [x_q, \alpha_q] =1 \in F/F_{2k}$. There exists a string link with $\bar{\mu}$-invariants $(\lambda_1, \dots, \lambda_q)\in (F_k)^q $ such that $ \lambda_j \equiv \alpha_j $ modulo $F_{2k}$.
\end{cor}
\section{As a tree reduction of the Kontsevich invariant}
\label{2thm}
As described in the Introduction, we will relate the $\bar{\mu}$-invariants with the Kontsevich invariant.
\
{\bf Notation}. Throughout this paper, the expression $O(n)$ will be used to denote terms of degree greater than or equal to $n$.
\subsection{A brief review of the Kontsevich invariants of (string) links}
\label{review}
Let us start by briefly reviewing the definition of the $\Q$-vector space $\A(\uparrow^q)$, where {\it a chord diagram (of $q$-components)} is a union $ (\sqcup^q_{j=1} [0,1]) \cup \Gamma$ such that $\Gamma$ is a uni-trivalent graph, whose univalent vertices lie in the interior of $ (\sqcup^q_{j=1} [0,1])$, and each component of $\Gamma$ is required to have a univalent vertex. It is customary to refer to the components of $\Gamma$ as dashed. Then, $\A(\uparrow^q)$ is defined by the $\Q$-vector space generated by all chord diagrams subject to the STU, AS, and IHX relations. The three relations are described in Figure \ref{STU}. The space $\A( \uparrow^q )$ is graded by the degree, where the degree of a diagram is half the number of vertices of $\Gamma $. We denote the subspace of $\A(\uparrow^q)$ of degree $n$ by $\A_n(\uparrow^q ) $. By abuse of notation, we will denote the graded completion of $\A(\uparrow^q)$ by $\A(\uparrow^q)$ as well. We define $\A^t(\uparrow^q)$ as the subspace of $\A(\uparrow^q)$ generated by the chord diagram such that all trivalent diagrams containing a non-simply connected dashed component are considered relations. Furthermore, the stucking connection of $(\sqcup^q_{j=1} [0,1])$ and $(\sqcup^q_{j=1} [0,1])$ gives rise to a ring structure of $\A(\uparrow^q)$. Moreover, there is a cocommutative multiplication $\Delta : \A(\uparrow^q) \ra \A(\uparrow^q)^{\otimes 2 } $ (see, e.g., \cite[Chapter 4]{CDM}, for the details), and $\A(\uparrow^q )$ is made into a Hopf algebra.
\begin{figure}[tpb]
\begin{center}
\begin{picture}(50,74)
\put(-174,38){\large \ \ \ \ $=$}
\put(-94,39){\large $- $}
\put(11,38){\large \ \ \ \ $= \ \!\! -$}
\put(128,38){\large \ \ \ \ \ \ $= \ \ \ \ \ \ \ \ \ \ \ \ \ \ -$}
\put(-16,37){\pc{AS2}{0.2532}}
\put(-206,37){\pc{IHX3}{0.354}}
\put(101,37){\pc{STU3}{0.354}}
\end{picture}
\end{center}
\vskip -1.7pc
\caption{\label{STU} The IHX, AS, and STU relations among chord diagrams. }
\end{figure}
There are several formulations of the Kontsevich invariant.
In this paper, we emply
the Kontsevich invariant defined for ``$q$-tangles", as in \cite{HM}.
Since we can chose an injection from the set of string links into the set of $q$-tangles which is invariant with respect to the composite of ($q$-)tangles,
we can regard the Kontsevich invariant of $q$-tangles that of string links (Throughout this paper, we fix such an injection).
In this paper, we only consider some of the properties of the Kontsevich invariant of string links $T$.
Thus, while we refer the reader to, e.g., \cite[\S\S 8--10]{CDM}, \cite[\S 6]{Mas2}, and references therein for the definition of the Kontsevich invariant for $q$-tangles, here are the properties that we will use later.
\begin{enumerate}[(I)]
\item The invariant, $ Z(T)$, is defined as an element of $ \A (\uparrow^q)$.
\item $Z(T) $ is multiplicative, i.e., $Z(T_1) Z(T_2) = Z (T_1 \cdot T_2)$ holds for two string links $T_1 , T_2$.
\item Every $Z(T)$ is group-like in $ \A (\uparrow^q)$, i.e., $\Delta(Z(T) ) = Z(T) \otimes Z(T)$.
\item (Doubling Formula) For $i \leq q$ and $T \in \A (\uparrow^q)$, one has that
\[ \Delta_i (Z(T))=Z ( D_i(T)) \in \mathcal{A}(\uparrow^{q+1})\]
where $D_i (T) $ denotes the double of $T $ along the $i$-th component, $C_i $, and $\Delta_i $ of a diagram is the sum over all lifts of vertices on $C_i$, to vertices on the two components over $C_i$.
\end{enumerate}
Recalling the tree subspace $\A^t(\uparrow^q)$, we take the projection $p^t: \A (q) \ra \A^t(\uparrow^q)$. We denote by $Z^t_{<m}(T)$ the composite $p^t \circ Z (T)$ subject to $O(m)$. In other words, this $Z^t_{<m}(T)$ is a tree reduction of the Kontsevich invariant $ Z_{<m}(T)$.
Furthermore, we need to use the notion of primitive subspaces. Here, $m \in \mathcal{A}(\uparrow^q)$ is called {\it primitive} if $\Delta(m) = 1 \otimes m + m \otimes 1 $. We denote by $\mathcal{P}^t(\uparrow^q)$ the subspace of primitive elements of $\A^t (\uparrow^q)$, and by $\mathcal{P}^t_{h}(\uparrow^q )$ the subspace of degree $h$. As is known, $\mathcal{P}^t(q )$ is the graded subspace of $\A^t (\uparrow^q)$ generated by chord diagrams such that the the dashed graph $\Gamma $ is simply connected.
Furthermore, the rank of $\mathcal{P}^t_{h}(\uparrow^q )$ is known to be $q N_h-N_{h+1}$; see \eqref{bbcc} and Theorem \ref{thma1}.
\subsection{Results}
\label{review5}
Before stating the theorem, we should mention the following easily proven lemma.
\begin{lem}\label{lem122}
(1) Fix $k \in \Z$. Every elements $a , b $ in $\oplus_{h=k}^{2k-1} \A_h (\uparrow^q) $ satisfy $(1+a) \cdot (1+b )\equiv 1+a + b+ O (2k) \in \A (\uparrow^q)$.
\noindent
(2) \ Let $a \in \A (\uparrow^q) $ satisfy $ a = 1+ O(k )$, and $\Delta (a) = a \otimes a$. If we decompose $a =1 +b +c $ such that $b \in \A_{< 2k } (\uparrow^q) $ and $c\in O(2k)$, then $b$ is primitive.
\end{lem}
As is known \cite{HM}, if a string link $T$ satisfies $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$, $Z^t(T)=1+O(k -1 )$; thus, we can see that $ Z^t(T)_{< 2k } -1 \in \mathcal{P}^t_{< 2k }(q ) $ from property (III). The theorem is as follows:
\begin{thm}\label{thm224}
There is a linear isomorphism $R : \oplus_{j= k}^{2k-1} \mathcal{P}_{j}^t ( q ) \ra \Ker([\bullet,\bullet ]_{k,2k } \otimes \Q ) $ such that the following holds for any string link $T$ satisfying $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$:
\begin{equation}\label{bbc2}R ( Z^t_{< 2k } (T) -1 ) = x_1 \otimes \lambda_1 + \cdots + x_q \otimes \lambda_q .
\end{equation} \end{thm}
Since Theorem \ref{thm24} implies that the left-hand side is equivalent to the Orr invariant, we have the following equivalence:
\begin{cor}\label{thm2244}
For any string link $T$ satisfying $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$, the Orr invariant $ \theta_k( L,\tau)$ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k $.
\end{cor}
\begin{exa}\label{kk}
As an example, we give a computation of the Boromean rings $6_2^3 $.
Let $PB(q+1) $ denote the pure braid group on $q+1$. Let $\sigma_i$ be the geometric braid formed by crossing the $i$-th string over the $(i+1)$-th one.
Consider the string link $T$ presented by $ \sigma_1^{-1} \sigma_2 \sigma_1^{-1} \sigma_2 \sigma_1^{-1} \sigma_2 $. Then $\overline{T} = 6_2^3 $.
We can easily verify $\mathfrak{A }_{3} $ and the expressions of the longitudes as
$$ \lambda_{i}= x_{i+2} x_i x_{i+1} x_i^{-1} x_{i+2}^{-1} x_i x_{i+1}^{-1} x_i^{-1}= [x_{i+2}, x_i x_{i+1} x_i^{-1}]. $$
where $ i \in \Z/3$. Thus, the $\mu$-invariant forms $ \sum_{i=1}^3 x_i \otimes [x_{i+2}, x_i x_{i+1} x_i^{-1}]$ modulo $F_4$. This value can be computed as something quantitative by using Magnus expansion $\mathcal{M}_4$ (see \S \ref{Sp34} for the definition of $\mathcal{M}_m$).
\end{exa}
\section{Proof of Theorem \ref{thm224}}
\label{Sp34}
As preparation, let us review the notion of group-like expansions and look at Example \ref{lie22}. Let $ \mathcal{I}_m \subset \Q \langle X_1, \dots, X_q \rangle $ be the both-sided ideal generated by polynomials of degree $\geq m. $ Consider the augmentation $\varepsilon : \Q \langle X_1, \dots, X_q \rangle\ra \Q $ with $\varepsilon (X_i)=1$, and a coproduct defined by $\Delta(X_i) =X_i \otimes 1 + X_i \otimes 1$. Then, the involution $S : \Q \langle X_1, \dots, X_q \rangle \ra \Q \langle X_1, \dots, X_q \rangle $ which sends $X_i$ to $-X_i$ makes it into a Hopf algebra. A {\it Magnus expansion} (modulo $O(m)$) is a group homomorphism $\theta : F \ra (\Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_m )^{ \times }$, that satisfies $\theta (y )=1+ [ y] + O(2)$ for any $y \in Y$. Furthermore, a {\it group-like expansion} is a Magnus expansion $\theta$ satisfying $\Delta (\theta(y))= \theta(y) \otimes \theta(y)$ and $ \varepsilon (\theta(y)) =1$ for any $y \in Y$. For example, the homomorphism $\mathcal{M}_m : F \ra \Q \langle X_1, \dots, X_q \rangle/ \mathcal{I}_m $ which sends $ x_i $ to $1+X_i$ is not a group-like expansion, but a Magnus expansion.
\begin{rem}\label{ska}
We should mention some of the properties of these expansions (see \cite[Theorem 1.3]{Ka})
(1) Given another Magnus expansion $\theta'$, there is a ring automorphism $S_{\theta'}$ on $\Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_m $ such that $ \theta_{\rm str} = S_{\theta'} \circ \theta'$.
(2) We have $ \theta (F_m)=0$, and $\theta $ induces an injection $\theta : F/F_m \ra \Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_m . $ In fact, since the above $\mathcal{M}_m $ is injective, the injectivity inherits every $\theta$, by (1).
Furthermore, since the restriction on $F_j/F_{j+1} $ of $\mathcal{M}_{j+1} $ is additive by definition, that of $\theta $ is also an additive map.
(3) There is a Lie algebra isomorphism from $\mathcal{L}/ \mathcal{L}_{\geq m} $ to the subspace,
\begin{equation}\label{lie}
\mathcal{P} (F/F_m ) := \{ a \in \Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_m \ | \ \Delta(a)= a \otimes 1 +1 \otimes a \ \}
\end{equation}
with the Lie bracket $[a,b]=ab - ba.$ The restricted image $\mathcal{M}_m (F_m /F_{m+1}) $ is contained in this $\mathcal{P} (F/F_m ) $.
\end{rem}
Next, the isomorphism \eqref{qq} below is related to another example of $\theta $ arising from the Kontsevich invariant. For an index pair $(i, j) \in \{1,\dots , q+1 \}^2 $, let $t_{i,j } \in \A_{1}^t (\uparrow^{q+1}) $ be the Jacobi diagram with only one edge connecting the $i$-th strand to the $j$-th one. Namely,
\begin{figure}[h]
\begin{center}
\begin{picture}(50,18)
\put(-63,-8){\Large $t_{ij}=$}
\put(-16,-7){\pc{tij}{0.144}}
\put(-15,-30){\large \ $ 1 \ \cdots i $ \ \ \ \ \ $j \cdots q+1 $}
\end{picture}
\end{center}
\end{figure}
\
\noindent
We denote $\A^{\rightleftharpoons} (\uparrow^{q},* ) $ by the subalgebra of $\A (\uparrow^{q+1}) $ generated by $t_{1, q+1}, t_{2, q+1} \dots, t_{q, q+1}$. Let $FI$ denote the framing independence relation in $\A^{\rightleftharpoons} (\uparrow^{q},* ) $, where any Jacobi diagram with an isolated chord on the same interval is equal to 0. Then, as shown \cite[(6.2)]{Mas2}, we can verify that the map,
\begin{equation}\label{qq}
\Q \langle X_1, \dots, X_q \rangle/ \mathcal{I}_m \lra \A^{\rightleftharpoons} (\uparrow^{q} ,* )/ (\A_{\geq m }^{\rightleftharpoons} (\uparrow^{q} ,* ) + FI),
\end{equation}
which sends $X_i$ to $t_{i, q+1}$ is a Hopf algebra isomorphism.
\begin{exa}
[{\cite[Proposition 6.2]{Mas2}}]\label{lie22}
We use notation on the pure braid group $PB(q+1) $ in Example \ref{kk}.
Let $\sigma_{j, q+1}\in PB(q +1 ) $ be $ \sigma_q \sigma_{q-1} \cdots \sigma_{j+1} \sigma_j^2 \sigma_{j+1}^{-1 } \cdots \sigma_j^{-1}$; see Figure \ref{tw}.
As is well known, there is a semi-direct product decomposition $ PB(q+1) \cong F(q) \ltimes PB(q)$, where $ F(q) $ is the free group generated by $\sigma_{1,q+1}, \dots, \sigma_{q,q+1}$, and $ PB(q) $ is embedded into $PB(q+1 )$ via $\beta \mapsto \beta \times \uparrow. $
Thus, any element $g$ of the free group $F (q) $ can be regarded as a pure-braid $ PB(q+1) \subset SL (q+1)$. Therefore, we can define $Z^t(g) \in \A^t (\uparrow^{q+1})$. As is shown \cite{Mas2}, $Z^t(g)$ lies in the subalgebra $\A^{\rightleftharpoons} (\uparrow^{q} ,* )$, and the composite,
\begin{equation}\label{qqqq}\theta^Z: F \stackrel{Z }{\lra }
\A^{\rightleftharpoons} (\uparrow^{q} ,* )/ (\A_{\geq m }^{\rightleftharpoons} (\uparrow^{q+1}) + FI) \stackrel{\eqref{qq}^{-1}}{\lra } \Q \langle X_1, \dots, X_q \rangle/ \mathcal{I}_m,
\end{equation}
turns out to be a group-like expansion (Moreover, it was shown to be a ``special expansion").
\end{exa}
Before turning back to Theorem \ref{thm224}, we should mention \eqref{qqqq2} from \cite[Corollary 12.2]{HM}. For a pure braid $\sigma \in PB(q)$, the $\ell$-th longitude, $\lambda_{\ell} \in F (q)$, of $\sigma$ is equal to $ ( \sigma \times 1 )^{-1} \beta_{\ell} D_{\ell } (\sigma) \beta_{\ell} ^{-1}$ in $ PB(q+1)$. Here, $\beta_{\ell}$ is a braid of the form $ \sigma_1 \sigma_2 \cdots \sigma_{\ell-1} \in B_n .$ Thus, the $\theta^Z(\lambda_{\ell} )$ can be computed as
\begin{equation}\label{qqqq2} \theta^Z (\lambda_{\ell})= Z^t( \sigma \times 1 )^{-1} Z^t(\beta_{\ell}) \Delta_{\ell } (Z^t (\sigma)) Z^t(\beta_{\ell})^{-1} \in \A^{\rightleftharpoons} (\uparrow^{q} ,* ). \end{equation}
\begin{figure}[tpb]
\begin{center}
\begin{picture}(110,74)
\put(16,37){\pc{delta}{0.200}}
\put(18,3){$1$}
\put(75,3){$j $}
\put(118,3){$q+1$}
\end{picture}
\end{center}
\vskip -1.7pc
\caption{\label{tw} The pure braid $\sigma_{j,q+1}$ }
\end{figure}
\begin{proof}[Proof of Theorem \ref{thm224}]
Inspired by \eqref{qqqq2}, for $\ell \leq q $, we set up a homomorphism,
$$ \Upsilon_\ell^{(j)} : \mathcal{P}^{t}_j (\uparrow^{q} ) \lra \A^{\rightleftharpoons}_{\leq j} (\uparrow^{q} ,* ) /FI \cong \Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_{j+1} ; \ a \longmapsto ( (a+1) \times 1 )^{-1} Z^t(\beta_{\ell}) \Delta_{\ell } (a +1) Z^t(\beta_{\ell})^{-1}. $$
Furthermore, we consider the linear homomorphism,
$$ \Upsilon^{(j)} : \mathcal{P}^{t}_j (\uparrow^{q} ) \lra F/F_2 \otimes_{\Z } \Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_{j+1} ; \ \
a \longmapsto \sum_{\ell : \ 1 \leq \ell \leq q }x_i \otimes \Upsilon_\ell^{(j)} (a). $$
We will show the injectivity of $\Upsilon^{(j)} $ and that the image is equal to
\begin{equation}\label{qqqq5} \{ \sum_{1 \leq \ell \leq q} x_{\ell} \otimes S_{\theta^Z}(a_{\ell}) \in F/F_2 \otimes (\mathcal{I}_{j} / \mathcal{I}_{j+1})\ | \ \Delta(a_i) = a_i \otimes a_i, \ \sum_{1 \leq \ell \leq q} a_{\ell}X_{\ell} - X_{\ell} a_{\ell}=0 \ \}. \end{equation}
From Remark \ref{ska} (3), this subspace \eqref{qqqq5} can be identified with the kernel of $[\bullet, \bullet]_{j,j+1 } : F/F_2 \otimes \mathcal{L}_{j}/ \mathcal{L}_{j+1} \ra \mathcal{L}_{j+1}/ \mathcal{L}_{j+2 }$. Thus, the rank of \eqref{qqqq5} is $q N_j - N_{j+1}. $ As is known (see \cite{Le,HM}), if some $ (\alpha_1,\dots, \alpha_q ) \in F_j/F_{j+1} $ satisfies $[x_1, \alpha_1 ]\cdots [x_q, \alpha_q] =1 \in F/F_{j+1}$, then there exists a string link $T_{\alpha}$ with $ \bar{\mu}$-invariants $( \lambda_1, \dots, \lambda_q)\in (F_j)^q$ such that $ \lambda_j \equiv \alpha_j $ mod $F_{j+1}$. Therefore, by \eqref{qqqq2}, the image of $\Upsilon^{(j)} $ is generated by $\Upsilon^{(j)} (Z^t_{\leq j} ( T_{\alpha})-1 )= \sum_{i=1}^{q } x_\ell \otimes \theta^Z (\lambda_\ell ) $, where $\alpha $ runs over a basis of $\Ker[\bullet, \bullet]_{j,j +1 }. $ Since the rank of $ \mathcal{P}^{t}_j ( \uparrow^q )$ is $q N_j - N_{j+1}$, $\Upsilon^{(j)} $ must be injective, and the image is \eqref{qqqq5}, as required.
To complete the proof, we will construct the isomorphism $R$ and show the equality \eqref{bbc2}.
Consider
$$\mathrm{Id}_{\Z^q} \otimes \theta^Z :
\Z^q \otimes_{\Z} F_k /F_{2k } \lra F/F_2 \otimes_{\Z } \Q \langle X_1, \dots, X_q \rangle / \mathcal{I}_{2k},$$
which is an injective homomorphism, by Remark \ref{ska} (2). The image of $\Ker[\bullet, \bullet]_{j,j+1} $ is contained in \eqref{qqqq5}. We define the isomorphism $R : \Ker([\bullet,\bullet ]_{k,2k-1 } \otimes \Q ) \ra \oplus_{j=k}^{2k-1} \mathcal{P}^t_{j } ( q ) $ as a linear extension of $ \mathrm{Id}_{\Z^q} \otimes (\mathrm{id}_{\Z^q} \otimes \theta^Z ) \circ\oplus_{j=k}^{2k-1}( \Upsilon^{(j)} )^{-1} )$.
We now prove the equality \eqref{bbc2} where the string link $T$ is a pure braid. Noting that $Z_{<2k } (T)-1$ is primitive by Lemma \ref{lem122}, we compute $\Upsilon^{(\ell )}( Z^t_{< 2k } (T)-1) $ as
\[ \sum_{\ell=1}^{q} \bigl( x_\ell \otimes \sum_{j=k}^{2k-1} (\Upsilon^{(j)}_{\ell})( Z^t_{< 2k } (T)-1)\bigr) = \sum_{\ell=1}^{q} x_\ell \otimes ( \theta^Z(\lambda_\ell ) )= \sum_{\ell=1}^{q} \theta^Z( x_\ell) \otimes ( \theta^Z(\lambda_\ell ) )
\]
which is immediately deduced from from \eqref{qqqq2}.
By the injectivity of $\theta^Z$, this equality implies the desired \eqref{bbc2}.
Finally, we will prove \eqref{bbc2} for any string link $T$ satisfying $\mathfrak{A }_{k+1} $ of the closure $\overline{T}$.
As implicitly shown in the proofs of \cite[Propositions 10.6 and Theorem 6.1]{HM},
there is a pure braid $\sigma$ such that the Milnor invariants of $T$ and $\sigma$ in $F_k/F_{2k}$ are equal. Thus, the invariant of $ T \sigma^{-1}$ is zero. Thus, \cite[Theorem 6.1]{HM} immediately implies $ Z^t( T \sigma^{-1}) = 1 +O(2k)$,
which leading to $ Z^t( T ) =Z^t( \sigma)$ modulo $ O(2k)$ by Lemma \ref{lem122} (1).
Hence, $ R (Z^t( T ) -1 ) =R ( Z^t( \sigma)-1) $ is equal to $\sum_{i=\ell}^q x_i \otimes \lambda_\ell $ by the above paragraph, as desired. It completes the proof.
\end{proof}
\section{Proof of Theorem \ref{thm24}}
\label{Sher33}
\subsection{Review of the infinitesimal Morita-Milnor homomorphism, and tree reduction}
\label{Sher}
As a preliminary to prove Theorem \ref{thm24}, we review some of the results in \cite{Mas,IO,K}.
We will start by briefly reviewing the Lie algebra homology of $F/F_k$. Let $H$ be the $\Q$-vector space of rank $q$ with basis $X_1, \dots, X_q$, i.e., $H= \mathrm{Span}_{\Q} \langle X_1, \dots, X_q \rangle. $ Let $\mathfrak{L}$ be the free Lie algebra generated by $H$. This $\mathfrak{L}_{\geq k}$ is the subspace generated by the commutator of length $\geq k$. We have the quotient Lie algebra $\mathfrak{L} / \mathfrak{L}_{\geq k} $. Then, {\it the Koszul complex} of $\mathfrak{L} / \mathfrak{L}_{\geq k} $ is the exterior tensor algebra $\Lambda^{*} (\mathfrak{L} / \mathfrak{L}_{\geq k} ) $ with the boundary map $\partial_n: \Lambda^{n} (\mathfrak{L} / \mathfrak{L}_{\geq k}) \ra \Lambda^{n-1} (\mathfrak{L} / \mathfrak{L}_{\geq k}) $ given by
$$ \partial_n(h_1 \wedge \cdots \wedge h_n ) = \sum_{i <j} (-1)^{i+j} [h_i, h_j]\wedge h_1
\wedge \cdots \wedge \check{h}_{i} \wedge \cdots \wedge \check{h}_{j} \wedge \cdots \wedge h_n. $$
Later, we will use the known isomorphism $H_n( \Lambda^{*} (\mathfrak{L} / \mathfrak{L}_{ \geq k} ) ) \cong H_n(F/F_{k}; \Q )$, which is called Pickel's isomorphism; see, e.g., \cite{IO, Mas,SW}.
Next, let us review Jacobi diagrams. A {\it Jacobi diagram} is a uni-trivalent graph whose univalent vertices are labeled by one of $\{ 1,2,\dots, q\}$, where each trivalent vertex is oriented. Consider the graded $\Q$-vector space generated by Jacobi diagrams, where the degree of such a diagram is half the number of vertices. Let $\mathcal{J}(q)$ be the quotient space subject to the AS and IHX relations, and let $\mathcal{J}^t (q) \subset \mathcal{J}(q)$ be the subspace generated by Jacobi diagrams which are simply connected. As a diagrammatic analogy of the Poincar\'{e}-Birkhoff-Witt theorem, we can construct a graded vector isomorphism,
\begin{equation}\label{bbcc}\chi_q : \mathcal{P}^t_n (\uparrow^q) \cong \mathcal{J}^t_n (q). \end{equation}
See, e.g., \cite[\S 5.7]{CDM} for details.
\begin{rem}\label{ll}
Theorem \ref{thm224} implies a generalization of the main theorem 6.1 of \cite{HM}.
In fact, recalling from Theorem \ref{thdr} (II) the $k$-th summand of $\theta_k (L,\tau)$ is the Milnor invariant of length $k$,
the equivalence between the Milnor invariant and $\chi_q(Z^t_{k}(T)-1 )$ coincides with the theorem 6.1 of \cite{HM} exactly.
\end{rem}
Furthermore, we consider the subspace $ \mathcal{J}_n (q, 0)$ of $ \mathcal{J}_n (q+1) $ generated by tree diagrams in which the label $q+1$ occurs exactly once. Given such a diagram $J \in \mathcal{J}_n (q, 0) $, all of its univalent vertices apart from one labeled by $r$ defines $\mathrm{comm}(J)$ in $F_n /F_{n+1}$ in a canonical way. For example,
\begin{figure}[h]
\begin{center}
\begin{picture}(50,74)
\put(-38,63){\large \ \ \ \ \ $x_1 \ \ \ \!\! x_2 \ \ \!\! x_3 \ x_4 $}
\put(8,19){\large \ \ \ $q+1 $}
\put(-63,38){\large comm $\Bigl(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Bigr) = [ x_1,[[x_2,x_3], x_4 ]] \in F_4 . $}
\put(-16,37){\pc{tree}{0.2114}}
\end{picture}
\end{center}
\end{figure}
\vskip -2.7pc
\noindent
Then, we can easily see that the following correspondence is an isomorphism,
$$ \mathrm{comm}: \mathcal{J}_n ( q, 0) \lra F_n /F_{n+1}\otimes \Q . $$
Next, we will review the infinitesimal Morita-Milnor homomorphism, $M_k$, defined in \cite[\S 5.1]{K} (cf. the infinitesimal Morita map in \cite{Mas2}). For this, a string link $T \in SL (q)$ is {\it of degree $k$} if the closure $\overline{T}$ satisfies Assumption $\mathfrak{A}_{k+1} $. Let $SL (q)_k $ be the subset consisting of string links of degree $k$. Then, we have a filtration,
$$SL (q)=SL (q)_0 \supset SL (q)_1 \supset \cdots \supset SL (q)_k \supset \cdots. $$
Further, we recall the $\bar{\mu}$-invariants $\lambda_{\ell} \in F_k/F_{2k -1 }$ and expand them as $\lambda_{\ell} = \lambda_{\ell}^{(k)} + \cdots + \lambda_{\ell}^{(2k -2 )}$ with $\lambda_{\ell}^{(j)} \in F_j/F_{j+1} $ according to $F_k/F_{2k-1} \cong \oplus_{j=k}^{2k-2} F_j/F_{j+1} $. Recall that, if $ L \in SL (q)_k $ is of degree $k$, the $\bar{\mu}$-invariants $ \lambda_\ell $ are contained in $F_k $. Then, identifying $F_{j}/F_{j+1} \otimes \Q$ with $ \mathfrak{L}_j/\mathfrak{L}_{j+1}$ as a $\Q$-vector space, let us consider a 2-form,
\begin{equation}\label{bbc9}
\sigma_L := \sum_{j=k}^{ 2k-2 } \sum_{\ell =1}^{q} X_\ell \wedge \lambda_\ell^{(j)} \in \Lambda^2 (\mathfrak{L}/\mathfrak{L}_{\geq 2k -1 } ) .
\end{equation}
As is shown in \S 5.1 of \cite{K}, there exists $t_L \in \Lambda^3 (\mathfrak{L}/\mathfrak{L}_{\geq 2k -1} )$ satisfying $\partial_3(t_L) =\sigma_L $.
Since $ \lambda_\ell \in F_k$ by Assumption $\mathfrak{A}_{k+1}$, the 2-form $ \sigma_L $ reduced in $\Lambda^2 (\mathfrak{L}/\mathfrak{L}_{ \geq k} ) $ is zero. Hence, $t_L$ is a 3-cycle. It has been shown \cite[Lemma 5.1.2]{K} that the homology 3-class $[\{ t_L\}] \in H_3 (\mathfrak{L}/\mathfrak{L}_{ \geq k} )$ is independent of the choice of $t_L$. To summarize, we have a map,
$$M_{k} : SL(q)_k \lra H_3 (\mathfrak{L}/\mathfrak{L}_{ \geq k} ); \ \ L \longmapsto [\{ t_L\}] .$$
Kodani \cite[\S 5]{K} showed that $M_{k}$ is a monoid homomorphism and its kernel is $SL(q)_{2k-1}$.
Now let us review the tree description of the third homology \cite{Mas}. Fix a tree diagram $J \in \mathcal{J}^t_j (q)$. For each trivalent vertex $r$ of $J$, $J$ is the union of the three tree diagrams rooted as $r$. We denote the three tree diagrams by $J_r^{(1)}$, $J_r^{(2)}$, and $J_r^{(3)}$, by clockwise rotation. Here, the numbering 1,2,3 is according to the cyclic ordering of $r$. Furthermore, if we label a univalent $r$ by $q+1$, each $J_r^{(j)}$ can be regarded as an element in $\mathcal{J}^t(q,0)$. Then, the {\it fission map} $\phi: \mathcal{J}^t_j (q) \ra \Lambda^{3} (\mathfrak{L} / \mathfrak{L}_{ \geq k} ) $ is defined by
$$ \phi(J)= \sum_{r: \mathrm{trivalent \ vertex \ of \ }J } \mathrm{comm}(J_r^{(3)} )\wedge \mathrm{comm}(J_r^{(2)} )\wedge \mathrm{comm}( J_r^{(1)}). $$
\begin{thm}
[\cite{Mas}]\label{thma1} If $ k \leq j \leq 2k - 2 $, this $\phi(J)$ is a 3-cycle. Moreover, the fission map gives rise to a linear isomorphism,
$$ \Phi : \bigoplus_{j=k }^{2k-2} \mathcal{J}^t_j ( q ) \stackrel{\sim}{\lra} H_3( \mathfrak{L} / \mathfrak{L}_{ \geq k} ) . $$
\end{thm}
Next, let us review the isomorphism \eqref{bbc} below. For a Jacobi diagram $J^{(j)} \in \mathcal{J}_j^t (q) $, we define $ \eta_{j} ( J^{(j)})$ to be the sum,
\begin{equation}\label{bbc6} \sum_{v: \mathrm{univalent \ vertex \ of \ }J^{(j)}} [ x_{\mathrm{col}(v)}] \otimes \mathrm{comm}(J_v^{(j)}) \in (F/F_2) \otimes_{\Q} (\mathcal{L}/\mathcal{L}_{> j} ), \end{equation}
where ${\mathrm{col}(v)} \in \{ 1, \cdots, q\} $ is the label on $v$, and $J_v^{(j)} \in \mathcal{J}_j^t (q,0) $ is the labelled tree obtained by replacing the label on $v $ with $q+1$. Taking the bracket $ [ -, -]:F/F_2 \otimes \mathcal{L}_{ j} \ra \mathcal{L}_{ j+1} $, which $(x,y)$ sends $xyx^{-1} y^{-1}$, we can easily verify that the sum \eqref{bbc6} lies in the kernel. Then, this $\eta_j$ defines a linear homomorphism,
\begin{equation}\label{bbc}\eta_{j} : \mathcal{J}_j^t (q) \lra \Ker([ -, -]: F/F_2 \otimes \mathcal{L}_{ j}/\mathcal{L}_{ j+1} \ra\mathcal{L}_{ j+1}/ \mathcal{L}_{ j+2} ), \end{equation}
which is known to be an isomorphism; see \cite{Le2}. We denote by $\eta $ the sum of the isomorphisms $\oplus_{j=k}^{2k -2 } \eta_{j} $ for short.
\subsection{Proof of Theorem \ref{thm224}}
\label{Sp2}
Then, the following lemma relates to the 3-class $[t_L]$ from tree diagrams:
\begin{lem}\label{oo2}
For $T \in SL(q)_{k}$, the composite $ \Phi \circ \eta^{-1} (\sum_{\ell=1}^q x_{\ell} \otimes \lambda_{\ell}) $ coincides with $M_k(T)=[t_L] $.
\end{lem}
\begin{proof}
Let $J^{(j)} \in \mathcal{J}^t_j(q)$ be $(\eta)^{-1}_j ( \sum_{i=1}^qx_i^{(j)}\otimes \lambda_i^{(j)})$. As is shown in \cite[Lemma 1.2]{Mas}, the composite $\partial_3(\phi ( J^{(j)})) $ is formed as
$$ \partial_3(\phi ( J^{(j)})) = \sum_{v: \mathrm{univalent \ vertex \ of \ }J^{(j)}} X_{\mathrm{col}(v)} \wedge \mathrm{comm}(J^{(j)}) \in \wedge^2 (\mathcal{L}/\mathcal{L}_{ \geq 2k} ). $$
Compared with \eqref{bbc6}, this $ \partial_3(\phi ( J^{(j)})) $ is equal to $\sigma_L \in \wedge^2 (\mathcal{L}/\mathcal{L}_{ \geq 2k} ) $ by definition. Thus, letting $t_L$ be $ \sum_{j=k}^{2k-2} \phi (J^{(j)}) $, we get a 3-chain $t_L$ satisfying $ \partial_3(t_L) =\sigma_L $, leading to the desired coincidence.
\end{proof}
We can prove Theorem \ref{thm24} by showing that the 3-class $[t_L]$ is equal to the Orr invariant of degree $k $. For this, we will have to prove the following lemma.
\begin{lem}\label{a2}
From the identification $H_3(\mathcal{L}/\mathcal{L}_{\geq k}) \cong H_3^{\rm gr}(F/F_k;\Q )$, the homomorphism $ M_k $ is equal to a map which sends the 3-class $t_L$ to $ \mathfrak{H} \circ \theta_k (\overline{T},\tau).$
\end{lem}
Before proving this lemma, we now prove Theorem \ref{thm24}.
\begin{proof}
[Proof of Theorem \ref{thm24}] The proof of (I) immediately follows from Lemmas \ref{oo2} and \ref{a2}.
Now let us show (II). Fix a basis $\{ b_s \} $ of $H_3(F/F_k;\Z ) $, where $s$ ranges over $1 \leq s \leq \mathrm{rk}H_3(F/F_k;\Z ) $. Thanks to Theorem \ref{thdr} (III), we can choose a string link $L_{s} $ such that $ \mathfrak{H} (\theta_k ( \overline{L_s} ,\tau))= b_s$ and the component of $ \theta_k ( \overline{L_s} ,\tau) $ in the kernel $ \Ker \mathfrak{H}$ is zero. Then, from (1), the $\bar{\mu}$-invariant of $L_s $ in $F_k /F_{2k}$ is $ \eta \circ \Phi^{-1} (b_s ) + \sum_{j=1}^q x_j \otimes \lambda_{j,s}^{(2k)} $ for some $ \lambda_{j,s}^{(2k)} \in F_{2k-1}/F_{2k} \otimes \Q $. Recalling the isomorphism $\Ker \mathfrak{H} \otimes \Q \cong \Q^{q N_{2k-1} - N_{2k} } $, we fix the isomorphism $\iota : \Q^{q N_{2k-1} - N_{2k} } \ra \Ker[ \bullet, \bullet ]_{2k-1,2k}$ and define the bijection,
$$ \overline{\Psi}_k : \bigl( H_3(F/F_k;\Z ) \oplus \Ker \mathfrak{H}\bigr) \otimes \Q \lra \Ker[ \bullet, \bullet ]_{k,2k }, $$
by setting
$$\overline{\Psi}_k \bigl(( b_s , a) \otimes r \bigl) = r \bigl(\eta \circ \Phi^{-1 } (b_s )- (x_1 \otimes \lambda_{1,s}^{(2k)} + \cdots + x_q \otimes \lambda_{q,s}^{(2k)}) + \iota( a) \bigl), \ \ \ \ r \in \Q. $$
Let $ \overline{\Phi}$ be $\overline{\Psi}_k ^{-1}$. Then it suffices to show the desired equality $x_1\otimes \lambda_1+ \cdots + x_q\otimes \lambda_q = \overline{\Psi}_k \bigl( \theta_k(\overline{T} , \tau)\bigr) $ for any string link $T$ of degree $k$. Suppose $\mathfrak{H} \circ \theta_k (\overline{T} , \tau) = \sum_s \alpha_s b_s $ for some $\alpha_s \in \Z$. Then, the stucking $ T \sharp (\sharp_s (L_s)^{-\alpha_s} )$ is of degree $k+1.$ Let $\mathfrak{H}'$ be the Hurewicz map $\pi_3(K_{k+1}) \ra H_3(F/F_{k+1}) $. Using the additivity of $\theta_k $ and of the $\bar{\mu}$-invariants, we can compute $\overline{\Psi}_k \bigl( \theta_k(\overline{T} , \tau)\bigl) $ restricted to the kernel of $[\bullet, \bullet]: F/F_ 2 \times F_{2k-1}/F_{2 k} \ra F_{2k}/F_{ 2k +1 } $ as
\[\overline{\Psi}_k \bigl( \theta_k(\overline{T} , \tau)\bigr) = \overline{\Psi}_k \bigl( \theta_k(\overline{T}\sharp (\sharp_s (L_s)^{-\alpha_s} ) , \tau) + \theta_k(\sharp_s (L_s)^{\alpha_s} , \tau)\bigr) \]
\[= \overline{\Psi}_k \bigl( \theta_k(\overline{T}\sharp (\sharp_s (L_s)^{-\alpha_s} ) , \tau)\bigr) + \overline{\Psi}_k \bigl( \theta_k(\sharp_s (L_s)^{\alpha_s} , \tau)\bigr) \]
\[=\overline{\Psi}_{k+1} \circ \mathfrak{H}' \bigl( \theta_{k+1}(\overline{T}\sharp (\sharp_s (L_s)^{-\alpha_s} ) , \tau) \bigr) + \overline{\Psi}_k \bigl( \sum_s \alpha_s (b_s,0) \bigr) \]
\[=\sum_{j=1}^q X_i \otimes \lambda_j( T \sharp (\sharp_s (L_s)^{-\alpha_s} ) + \sum_s \alpha_s (x_1 \otimes \lambda_{1,s}^{(2k)} + \cdots + x_q \otimes \lambda_{q,s}^{(2k)}) = \sum_{j=1}^q x_i \otimes \lambda_j( T ),\]
as desired. Here, the third equality is obtained from (II), and the last is done from additivity of $\lambda $.
\end{proof}
\begin{proof}
[Proof of Lemma \ref{a2}] First, we set up some complexes and chain maps. In what follows, we consider only complexes over $\Q$ and omit writing the coefficients $\Q$. Let $( C_*^{\rm gr} (F/F_k), \partial_* )$ be the non-homogenous group complex of $F/F_k$. Then, on the basis of Suslin and Wodzicki's paper \cite{SW}, Massuyeau (see the proof of \cite[Proposition 4.3]{Mas}) showed the natural existence of a chain map $ \kappa: \bigwedge^*( \mathcal{L} / \mathcal{L}_{\geq k}) \ra C_*^{\rm gr} (F/F_k) $ that induces an isomorphism on the homology. Thus, it is enough for us to show that the 3-cycle $\kappa( t_L)$ is equivalent to the pushforward $(f_k)_* [S^3 \setminus \overline{T},\partial (S^3 \setminus \overline{T})]$.
To do so, we can study the 3-cycle $t_L$ from the viewpoint of the group complex. For $\ell \leq q$, let $K_\ell$ be the abelian subgroup of $F$ generated by the meridian-longitude pair $(\mathfrak{m}_{\ell} , \mathfrak{l}_{\ell} )$. Let us consider the commutative diagram,
$$
{\normalsize
\xymatrix{
0 \ar[r] & \oplus_{\ell=1}^q C_3^{\rm gr} (K_\ell ) \ar[r]^{\iota_3} \ar[d]_{\partial_3} & C_3^{\rm gr } ( F ) \ar[d]_{\partial_3} \ar[r]^{\!\!\!\!\!\!\!\! \!\!\!\! P_3} &C_3^{\rm gr } (F, K_1 \cup \cdots \cup K_q ) \ar[d]_{\partial_3} \ar[r] & 0 & (\mathrm{exact}) \\
0 \ar[r] & \oplus_{\ell=1}^q C_2^{\rm gr} (K_\ell ) \ar[r]^{\iota_2} & C_2^{\rm gr } (F ) \ar[r] ^{\!\!\!\!\!\!\!\! \!\!\!\! P_2} &C_2^{\rm gr } (F, K_1 \cup \cdots \cup K_q ) \ar[r] & 0& (\mathrm{exact}),
}}$$
where the right-hand sides are defined as the cokernel of $\iota_*$. In the subcomplex $C_2^{\rm gr} (K_\ell ) $, the cross product $\mathfrak{m}_{\ell} \times \mathfrak{l}_{\ell}$ is a 2-cycle that generates $ H_2^{\rm gr} (K_\ell ) \cong \Z $. Let $\tau_L$ be a 2-cycle $\sum_{\ell=1}^q \mathfrak{m}_{\ell} \times \mathfrak{l}_{\ell} \in \oplus_{\ell=1}^q C_2^{\rm gr} (K_\ell )$.
Accordingly, since $H_2^{\rm gr}(F)=0$, we can choose a 3-cycle $\eta_L $ in $ C_3^{\rm gr } (F, K_1 \cup \cdots \cup K_q )$ such that $\delta_*(\eta_L )= \tau_L.$
Next, we will examine the diagrams subject to $F_k$ and $F_{2k}$ with regard to their functoriality:
$$
{\normalsize
\xymatrix{
0 \ar[r] & \oplus_{\ell=1}^q C_2^{\rm gr} (K_\ell ) \ar[r] \ar[rdd] \ar[d]_{\partial_3} & C_3^{\rm gr} ( F/F_{2k -1 } ) \ar[d]_{\partial_3} \ar[r]^{P^{\rm gr}} \ar[rdd] &C_3^{\rm gr} (F/F_{2k -1 }, \cup_{\ell }K_\ell ) \ar[d]_{\partial_3} \ar[r] \ar[rdd] & 0 & & \\
0 \ar[r] & \oplus_{\ell=1}^q C_2^{\rm gr} (K_\ell ) \ar[r] \ar[rdd] & C_2^{\rm gr} ( F/F_{2k-1 }) \ar[r] \ar[rdd] &C_2^{\rm gr} (F/F_{2k -1 }, \cup_{\ell } K_\ell) \ar[r] \ar[rdd] & 0& & \\
& 0 \ar[r] & \oplus_{\ell=1}^q C_3^{\rm gr} (K_\ell ) \ar[d]_{\partial_3} \ar[r] & C_3^{\rm gr} ( F/F_k ) \ar[d]_{\partial_3} \ar[r]_{ \!\!\!\!\!\!\!\!\!\!\!\! P^{\rm gr}} &C_3^{\rm gr} (F/F_k, \cup_{\ell }K_\ell ) \ar[d]_{\partial_3} \ar[r] & 0 \\
& 0 \ar[r] & \oplus_{\ell=1}^q C_2^{\rm gr} (K_\ell ) \ar[r] & C_2^{\rm gr} ( F/F_k ) \ar[r] &C_2^{\rm gr} (F/F_k, \cup_{\ell } K_\ell ) \ar[r] & 0.
}}$$
Here, the horizontal arrows are exact, and the slanting ones are the maps induced from the projection $F/F_{2k -1 } \ra F/F_k. $ Since the above quasi-isomorphism $\kappa $ was constructed from the projective resolution of the augmentation $\varepsilon : \Q[G] \ra \Q$, this $\kappa $ replaces the wedge product $\wedge $ by the cross product $\times$. Therefore, recalling the isomorphism $\phi^*$ in \eqref{dd}, the 2-cycle $\phi_*^{-1}(\tau_L) $ modulo $F_{2k}$ is exactly equal to $\sigma_L$ in \eqref{bbc9}. Thus, the 3-cycle $t_L \in C_3^{\rm gr}(F/F_{2k -1 } )$ satisfies $P^{\rm gr} (t_L) = \phi_*^{-1} ( \eta_L) \in H_3^{\rm gr}(F/F_{k} )$.
Finally, we give a relation to the link complement $S^3 \setminus \overline{T}$. Let $E$ denote $S^3 \setminus \overline{T}$. Similarly, let us consider the long exact sequence on the cellular homology:
$$ 0 \ra H_3^{\rm cell } (E, \partial E;\Q ) \stackrel{\delta}{\lra} H_2^{\rm cell } ( \partial E;\Q ) \lra H_2^{\rm cell } (E;\Q )\lra H_2^{\rm cell } (E, \partial E;\Q ).$$
Here are some well-known facts from knot theory: The first term is $\Q$ generated by the fundamental 3-class $[E,\partial E]$, and the second is $\Q^{q}$ generated by the cross products $\mathfrak{m}_{\ell} \times \mathfrak{l}_{\ell}$. Further, the sum $\sum_{\ell =1}^q \mathfrak{m}_{\ell} \times \mathfrak{l}_{\ell}$ is zero in $ H_2^{\rm cell } (E ;\Q )$ by relation \eqref{le}. Thus, we have a cellular 3-chain $ \overline{b_L} \in C_3^{\rm cell } (E, \partial E;\Q )$ such that $ (P_3)_*[ \overline{b_L} ] = [E,\partial E]$ in the relative $C_3^{\rm cell } (E, \partial E;\Q )$ and $\partial_3( \overline{b_L} ) = \sum_{\ell =1}^q \mathfrak{m}_{\ell} \times \mathfrak{l}_{\ell}.$ In particular, letting $I: [0,1]^3 \setminus T \ra S^3 \setminus \overline{T}$ be the inclusion, we have $[E,\partial E] = I_* ( \eta_L) $. Notice that $f_k \circ I_* : \pi_1( [0,1]^3 \setminus T ) \ra F/F_k $ is equal to the reduction $F \ra F/F_k. $
Hence, noting $ (f_k)_* \circ I_* = \phi_*^{-1}$, we obtain the computation,
$$ (f_k)_* [E,\partial E] =(f_k)_* \circ I_*(\eta_L) = \phi_*^{-1}(\eta_L)= P^{\rm gr}(t_L) \in H_3^{\rm gr} (F/F_k, \cup_{\ell}K_{\ell}). $$
Since $ K_{\ell} $ modulo $F_k$ is isomorphic to $ \Z $ from Assumption $\mathfrak{A}_{k+1}$, $P^{\rm gr}$ induces $H_3^{\rm gr} (F/F_k ) \cong H_3^{\rm gr} (F/F_k, \cup_{\ell }K_\ell )$. Hence, $ t_L$ is equivalent to the pushforward $(f_k)_* [E,\partial E]$, as desired.
\end{proof}
\appendix
\section{HOMFLYPT polynomials and Orr invariants}
\label{SHigher}
Let $T$ be a string link and $(\overline{T}, \tau)$ be the associated based link. According to Theorem \ref{thm24}, the computation of the Orr invariant of $(\overline{T}, \tau)$ is equivalent to the $\bar{\mu}$-invariant of $ T$ of degree $< 2k$. In general, it is hard to get a presentation of longitudes $\lambda_{\ell}$ as a word of $x_1, \dots, x_q $ (However, if $T$ is a pure braid, we can easily get such a presentation). Furthermore, in quantum topology, it is natural to ask what finite type invariants recover the $\bar{\mu}$-invariants. As a solution, we will show that the result of Meilhan-Yasuhara \cite{MY} give a computation of the $\bar{\mu}$-invariants from HOMFLYPT polynomials, without having to write longitudes (Theorem \ref{ooth2}).
To describe the result, we should recall the HOMFLYPT polynomial and some of its properties. {\it The HOMFLYPT polynomial} $P ( L; t, z) \in \Z [t^{\pm 1} , z^{\pm 1}]$ of an oriented link $L \subset S^3 $ is defined formulas as follows:
\begin{enumerate}[(I)]
\item Concerning the unknot $U$ in $S^3$, the polynomial $ P ( U; t, z)$ is $1$.
\item The skein relation $t \cdot P ( L_+ ; t, z) + t^{-1} \cdot P ( L_- ; t, z) = z \cdot P ( L_0 ; t, z)$ holds, where $L_-$, $ L_+$, and $L_0$ are links formed by crossing and smoothing changes on a local region of a link diagram, as indicated in Figure \ref{t321}.
\end{enumerate}
Furthermore, given a $q $-component link $L$, we can expand the HOMFLYPT polynomial as
$$ P ( L ; t, z) = \sum_{k=1}^N P_{2k-1-q} (L;t ) z^{2k-1-q}, $$
for some $N \in \mathbb{N}$, where $P_{2k-1-q} (L;t ) \in \Z[t^{\pm 1}]$. Consider the logarithm $ \log ( P_0 (L,t))$ as a smooth function in $C^{\infty}(\R)$. We denote by $(\log ( P_0 (L,t))^{(n)} $ the $n$-th derivative of $ \log ( P_0 (L,t))$ evaluated at $t= 1$.
\begin{figure}[tpb]
\begin{center}
\begin{picture}(110,51)
\put(-96,27){\pc{kouten.p}{0.300}}
\put(16,27){\pc{kouten.n}{0.300}}
\put(126,27){\pc{arrowedmove}{0.300}}
\put(-116,43){\Large $L_+$}
\put(-6,43){\Large $L_-$}
\put(106,43){\Large $L_0$}
\end{picture}
\end{center}
\vskip -1.7pc
\caption{\label{t321} The links $L_+$, $ L_-$, and $L_0$.}
\end{figure}
Next, we review the notation used in \cite{MY}. Let $T = \cup_{i=1}^q T_i$ be a $q$-component string link in the 3-cube, and take an index $I= i_1 i_ 2 \cdots i_m \in \{ 1, \dots, q\}^m $. Let $r_i$ be the cardinality $\# \{ k | i_{k}=i \}$. We can define another link $D_I (T)$ of $q$-components in the following manner:
\begin{enumerate}[(1)]
\item Replace each string $T_i$ by $r_i$ zero-framed parallel copies of $T_i$. Here, we write $T_{(i,j)}$ for the $j$-th copy. If $r_i=0$ for some index $i$, we delete $L_i$.
\item We define $D_I(T)$ to be the $m$-component link $\cup_{i,j} T_{(i,j)}$ with the order induced by the lexicographic order of the index $(i,j)$. This ordering defines a bijection $\{ (i,j) | \ 1 \leq i \leq q, \ 1 \leq j \leq r_i\} \ra \{ 1, \dots, m\} $.
\end{enumerate}
In addition, we define a sequence $D_I (T) \in \{ 1, \dots, m\}^m $ without a repeated index as follows. First, we take a sequence of elements of $\{ (i,j) | \ 1 \leq i \leq q, \ 1 \leq j \leq r_i\} \ra \{ 1, \dots, m\} $ by replacing each $i$ in $I$ with $(i,1), \dots, (i,r_i )$ in this order. Next, we replace each term $(i,j)$ of this sequence with $\varphi((i,j)). $
In addition, given a subsequence $H < D(I)$, we define another link $D_I(T)_J$. Let $B_I$ be an oriented $2m$-gon, and denote by $p_j$ $(j=1, \dots, m)$ a set of $m$ nonadjacent edges of $B_I$ according to the boundary orientation. Suppose that $B_I$ is embedded in $S^3$ such that $B_I \cap L = \cup_{j=1}^m p_j$ and such that each $p_j$ is contained in $L_{i_j}$ with opposite orientation. We call such a disk {\it an $I$-fusion disk} of $D_I(L)$. For any subsequence $J$ of $D(I)$, we define the oriented link $L_J$ as the closure of
$$\bigcup_{j \in \{ J \} } (L_j \cap \partial B_I) \setminus \bigl( \bigcup_{j \in \{ J \} } (L_j \cap B_I) \bigr), $$
where $\{ J\}$ is the subset of $\{ 1, \dots, n\}$ of all indices appearing in the sequence $J $.
Consider the homomorphism $\mathcal{M}_m: F \ra \Z[X_1, \dots, X_q]/ {\mathcal{I}_m}$ defined by $ \mathcal{M}_m (x_i) =1+X_i $. For a sequence in $I=i_1 \cdots i_m \in \{ 1,2, \dots, q\}^m$, $\mu_{I}(T)$ is defined by the coefficient of $X_{i_1 } \cdots X_{i_{m-1}}$ in $\mathcal{M}_m(\lambda_{i_m})$ as in \cite{Mil2,Le,Orr,MY}.
\begin{thm}
[\cite{MY}]\label{ooth2} Let $T$ be a $q$-component string link which satisfies $\mathfrak{A}_{k+1} $. Assume $3 \leq m \leq 2k+2 $. Let $I$ be a sequence in $\{ 1,2, \dots, q\}^m$ of length $m$. For any $D_I$-fusion disk for $D_I (T)$, we have
$$ \mu_I (T ) = \frac{(-1)}{m ! 2^m }\sum_{J < D(I) } (-1)^{|J|} \log P_0(\overline{D_I( T )_J }) ^{(m)}. $$
\end{thm}
The original statement \cite[Theorem 1.3]{MY} dealt with only links $S^3$ and takes the same formula modulo some integers. However, as can be seen in their proof, after the authors proved Theorem \ref{ooth2} for string links before they proved the original statement. Thus, we do not need to give a detailed proof of Theorem \ref{ooth2}.
\subsection*{Acknowledgments}
The author expresses his gratitude to Professor Akira Yasuhara for his helpful discussion and for referring him to the paper \cite{Le}. He also thanks Yusuke Kuno, Jean-Baptiste Meilhan, and Kent Orr for giving him valuable comments on this work. | 188,871 |
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TITLE: Finding the volume using Washers
QUESTION [6 upvotes]: Problem:
Find the volume generated when the region bounded by the given curves and line is revolved about
the x-axis.
$$ y = 3x - x^2$$
$$ y = 3x $$
Answer:
Let $V$ be the volume we are trying to find. The first step is to find the points where
$3x - x^2$ and $y = 3x$ intersect.
\begin{align*}
3x - x^2 &= x \\
-x^2 &= 2x \\
x = 0 &\text{ or } x = 2 \\
V &= \int_0^2 \pi \left( (3x - x^2)^2 - x^2 \right) \,\, dx \\
\dfrac{V}{\pi} &= \int_0^2 (3x - x^2)^2 \,\, dx - \int_0^2 x^2 \,\, dx \\
\end{align*}
Now we have two integrals to evaluate.
\begin{align*}
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 (x^2-3x)^2 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 x^4 - 6x^2 + 9 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{x^5}{5} - \dfrac{6x^3}{3} + 9x \Big|_0^2 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - \dfrac{6(8)}{3} + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - 16 + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{42}{5} \\
\end{align*}
For the second integral we have:
\begin{align*}
\int_0^2 x^2 \,\, dx &= \dfrac{x^3}{3} \Big|_0^2 = \dfrac{8}{3} \\
\dfrac{V}{\pi} &= \dfrac{42}{5} - \dfrac{8}{3} = \dfrac{ 3(42) - 5(8)}{15} \\
\dfrac{V}{\pi} &= \dfrac{86 }{15 } \\
V &= \dfrac{86\pi}{15}
\end{align*}
However, the book gets: $ \dfrac{ 56 \pi}{15} $.
Where did I go wrong?
Based upon a comment from John Douma, I realized that I copied the question incorrectly. Here is the revised question with my solution which still has the wrong answer.
Problem:
Find the volume generated when the region bounded by the given curses and line is revolved about
the x-axis.
$$ y = 3x - x^2 $$
$$ y = x $$
Answer:
Let $V$ be the volume we are trying to find. The first step is to find the points where
$3x - x^2$ and $y = 3x$ intersect.
\begin{align*}
3x - x^2 &= x \\
-x^2 &= -2x \\
x = 0 &\text{ or } x = 2 \\
V &= \int_0^2 \pi \left( (3x - x^2)^2 - x^2 \right) \,\, dx \\
\dfrac{V}{\pi} &= \int_0^2 (3x - x^2)^2 \,\, dx - \int_0^2 x^2 \,\, dx \\
\end{align*}
Now we have two integrals to evaluate.
\begin{align*}
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 (x^2-3x)^2 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 x^4 - 6x^2 + 9 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{x^5}{5} - \dfrac{6x^3}{3} + 9x \Big|_0^2 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - \dfrac{6(8)}{3} + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - 16 + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{42}{5} \\
\end{align*}
For the second integral we have:
\begin{align*}
\int_0^2 x^2 \,\, dx &= \dfrac{x^3}{3} \Big|_0^2 = \dfrac{8}{3} \\
\dfrac{V}{\pi} &= \dfrac{42}{5} - \dfrac{8}{3} = \dfrac{ 3(42) - 5(8)}{15} \\
\dfrac{V}{\pi} &= \dfrac{86 }{15 } \\
V &= \dfrac{86\pi}{15}
\end{align*}
However, the book gets: $ \dfrac{ 56 \pi}{15} $.
Where did I go wrong?
Here is an updated answer based upon the comments from DougM.
Answer:
Let $V$ be the volume we are trying to find. The first step is to find the points where
$3x - x^2$ and $y = x$ intersect.
\begin{align*}
3x - x^2 &= x \\
-x^2 &= -2x \\
x = 0 &\text{ or } x = 2 \\
V &= \int_0^2 \pi \left( (3x - x^2)^2 - x^2 \right) \,\, dx \\
\dfrac{V}{\pi} &= \int_0^2 (3x - x^2)^2 \,\, dx - \int_0^2 x^2 \,\, dx \\
\end{align*}
Now we have two integrals to evaluate.
\begin{align*}
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 (x^2-3x)^2 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 x^4 - 6x^3 + 9 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{x^5}{5} - \dfrac{6x^4}{4} + 9x \Big|_0^2 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - \dfrac{6(16)}{4} + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - 24 + 18 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{2}{5} \\
\end{align*}
For the second integral we have:
\begin{align*}
\int_0^2 x^2 \,\, dx &= \dfrac{x^3}{3} \Big|_0^2 = \dfrac{8}{3} \\
\dfrac{V}{\pi} &= \dfrac{2}{5} - \dfrac{8}{3} = \dfrac{ 6 - 24}{15} \\
\dfrac{V}{\pi} &= -\dfrac{18 }{15 } \\
V &= -\dfrac{18\pi}{15}
\end{align*}
This answer is obviously wrong.
The book gets: $ \dfrac{ 56 \pi}{15} $.
Where did I go wrong?
Here is an updated answer based upon the comment from N. F. Taussig. I now have a correct solution.
Answer:
Let $V$ be the volume we are trying to find. The first step is to find the points where
$3x - x^2$ and $y = x$ intersect.
\begin{align*}
3x - x^2 &= x \\
-x^2 &= 2x \\
x = 0 &\text{ or } x = 2 \\
V &= \int_0^2 \pi \left( (3x - x^2)^2 - x^2 \right) \,\, dx \\
\dfrac{V}{\pi} &= \int_0^2 (3x - x^2)^2 \,\, dx - \int_0^2 x^2 \,\, dx \\
\end{align*}
Now we have two integrals to evaluate.
\begin{align*}
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 (x^2-3x)^2 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \int_0^2 x^4 - 6x^3 + 9x^2 \,\, dx \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{x^5}{5} - \dfrac{6x^4}{4} + \dfrac{9x^3}{3} \Big|_0^2 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - \dfrac{6(16)}{4} + \dfrac{9(8)}{3} \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} - 24 + 24 \\
\int_0^2 (3x - x^2)^2 \,\, dx &= \dfrac{32}{5} \\
\end{align*}
For the second integral we have:
\begin{align*}
\int_0^2 x^2 \,\, dx &= \dfrac{x^3}{3} \Big|_0^2 = \dfrac{8}{3} \\
\dfrac{V}{\pi} &= \dfrac{32}{5} - \dfrac{8}{3} = \dfrac{ 96 - 40}{15} \\
\dfrac{V}{\pi} &= \dfrac{56 }{15 } \\
V &= \dfrac{56\pi}{15}
\end{align*}
This answer matches that given in the book.
REPLY [1 votes]: With problems like this, it is usually solved by the "Washer Method." This method deals when we have a gap about the rational axis.
Let's have the red line be $y = 3x - x^2$ and the blue line be $y = x$. Now by looking at the graph we can see that the points of intersection are $0$ and $2$, but we can also find this by some algebra:
$$\text{Let's set the two equations equal to each other: $3x-x^2 = x$}
\\
x^2 = 2x
\\
\text{We see $x=0$ is a solution for both sides. To find the other value, let $x\neq0$}
\\
\frac{x^2}{x} = \frac{2x}{x}
\\
x = 2$$
We see that are points of intersection are $0$ and $2$ by the graph and verified by algebra. Now let's look at the Washer Method. This method states:
$$V = \int_a^b \pi [f(x)^2 - g(x)^2]dA$$
Let's think of $f(x)$ as the top and $g(x)$ the bottom of the area we want to calculate. $dA$ is if we are going to integrate with respect to $x$ or $y$. Since we rotation about the $x$-axis we will be integrating with respect to $x$, and the "top" will be $3x-x^2$ and the bottom will be $x$. Now we will perform the integration:
$$V = \int_0^2 \pi[(3x-x^2)^2 - (x)^2]dx
\\
=\pi \int_0 ^2 [(9x^2 - 6x^3 +x^4) - (x^2)]dx
\\
=\pi \int_0 ^ 2 [x^4 -6x^3+8x^2]dx \\ =\pi \big[ \frac{x^5}{5} - \frac{6x^4}{4} + \frac{8x^3}{3}\big] \bigg|_0^2
\\
= \pi \big[ \frac{32}{5} - \frac{48}{2} - \frac{64}{3}\big]
\\ = \boxed{\frac{56 \pi}{15}}$$ | 34,460 |
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\begin{document}
\title[Hurewicz map]{The Hurewicz map in motivic homotopy theory}
\subjclass[2000]{14F42}
\author{Utsav Choudhury}
\author{Amit Hogadi}
\begin{abstract} For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$,
we prove that the Hurewicz map $\pi_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ is surjective. We also observe that the Hurewicz map for $\P^1_k$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $\phi$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $\phi$ are also strongly $\A^1$-invariant.
\end{abstract}
\maketitle
\section{Introduction}
For a field $k$, let $Sm/k$ denote the category of smooth $k$-varieties with Nisnevich topology. Let $\Delta^{op}Sh(Sm/k)$ denote the category of simplicial sheaves on the category $Sm/k$. This cateogry with its $\A^1$-model structure as defined in \cite{morel-voevodsky} is one of the main objects of study in $\A^1$-homotopy theory. For any pointed simplicial sheaf $\sX$ in $\Delta^{op}Sh(Sm/k)$ one defines the $\A^1$-homotopy group sheaves, $\pi_i^{\A^1}(\sX)$, to be the sheaves of simplicial homotopy groups of a fibrant replacement of $\sX$ in the $\A^1$-model structure. Morel, in his foundational work in \cite[Ch. 6]{morel} has defined, for every integer $i$, $\A^1$-homology groups $H_i^{\A^1}(\sX)$ and canonical Hurewicz morphisms $$\pi_i^{\A^1}(\sX) \to H_i^{\A^1}(\sX)$$
The above maps are analogous to the Hurewicz map that we have in topology. In the topological setup, the Hurewicz morphism for $i=1$ is known to be the abelianisation when the underlying space is connected. We will refer to this result as the Hurewicz theorem. Hurewicz theorem is expected in $\A^1$-homotopy theory (see \cite[6.36]{morel}), but not yet known. However the following theorem by Morel is the closest known result to the Hurewicz theorem.
\begin{theorem}\cite[6.35]{morel}\label{morel-hurewicz}
For a connected simplicial sheaf $\sX$, the Hurewicz morphism $$ \pi_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$$ is a universal map to a strictly $\A^1$-invariant sheaf of abelian groups.
\end{theorem}
Recall that a sheaf of groups $\sG$ is called strongly $\A^1$-invariant if for $i=0,1$ the maps
$$ H^i(U, \sG) \to H^i(U\times \A^1, \sG)$$ are bijective for all $U$ in $Sm/k$. If $\sG$ is abelian, then it is called strictly $\A^1$-invariant if
the above isomorphism holds for all $i\geq 0$. $\pi_1^{\A^1}(\sX)$ is strongly $\A^1$-invariant and $H_1^{\A^1}(\sX)$ is known to be strictly $\A^1$-invariant (see \cite[6.1, 6.23]{morel}).
In topology, the surjectivity of the Hurewicz map is almost a direct consequence of the definitions. This is not the case in $\A^1$-homotopy theory. The main source of difficulty lies in the non-explicit nature of $\A^1$-fibrant replacements; non-explicit from the viewpoint of making explicit calculations.
In this paper we prove this surjectivity by using Giraud's theory of non-abelian cohomology.
\begin{theorem} \label{hurewicz}
Let $k$ be a perfect field and $\sX$ be a pointed simplicial sheaf on $Sm/k$ in the Nisnevich topology. Then the Hurewicz map
$ \pi_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX) $ is surjective.
\end{theorem}
The above theorem will be deduced from the following result, which is of independent interest.
\begin{thm} \label{main theorem} Let $k$ be a perfect field. Let $G$ be a strongly $\A^1$-invariant sheaf of groups on $Sm/k$ and $G\to H$ be an epimorphism. Then $H$ is strongly $\A^1$-invariant iff it is $\A^1$-invariant.
\end{thm}
\begin{remark} If $k$ is perfect field, a theorem of Morel \cite[5.46]{morel} says that any strongly $\A^1$-invariant Nisnevich sheaf of abelian groups on $Sm/k$ is also strictly $\A^1$-invariant. Unfortunately it is not yet known if this statement holds for imperfect fields. This is the sole reason for assuming $k$ to be perfect in Theorems \ref{main theorem} and \ref{hurewicz}. Also note that strongly $\A^1$-invariant is a stronger notion than just $\A^1$-invariant. In particular, there exists $\A^1$-invariant sheaves which are not strongly $\A^1$-invariant (see \cite[Lemma 5.6]{ch}).
\end{remark}
For a morphism of strongly $\A^1$-invariant abelian sheaves over a perfect field, the kernel and image of the morphism are also strongly $\A^1$-invariant. This result is a consequence of a nontrivial theorem of Morel (see \cite[6.24]{morel}) that the category of strongly $\A^1$-invariant sheaves of abelian groups is an abelian category, as it is obtained as a heart of a $t$-structure. The theorem below, can be viewed as a generalization of this result for non-abelian strongly $\A^1$-invariant sheaves. Moreover the proof of this generalization is completely different and is more direct in the sense that it does not appeal to the existence of $t$-structures.
\begin{theorem}
Let $G\xrightarrow{\phi} H$ be a morphism of strongly $\A^1$-invariant sheaves of groups. Then the image and the kernel of $\phi$ are strongly $\A^1$-invariant.
\end{theorem}
\begin{proof}
The image ${\rm Image}(\phi)$ is $\A^1$-invariant, since it is a subsheaf of an $\A^1$-invariant sheaf $H$. Thus by Theorem \ref{main theorem} it is strongly $\A^1$-invariant. The kernel $K$ is strongly $\A^1$-invariant as it fits in the following exact sequence
$$ 1\to K \to G \to {\rm Image(\phi)} \to 1$$
where the other two sheaves are strongly $\A^1$-invariant.
\end{proof}
\noindent {\bf Acknowledgement}: We thank Fabien Morel, Tom Bachmann and O. R{\"o}ndigs for their comments.
\section{Preliminaries on Gerbes and Giraud's non-abelian cohomology}
Let $\sC$ be any small site (e.g. $Sm/k$ with Nisnevich topology) and $\Delta^{op}(\sC)$ be the category of simplicial sheaves on $\sC$. The goal of this section is to recall the main results of Giraud on non-abelian cohomology. Everything in this section is a subset of \cite{giraud}. We work with the following definition of a gerbe.
\begin{definition} A simplicial sheaf $\sX$ on $\sC$ is called a gerbe if it is connected and if for any $U\in \sC$ and any $x\in \sX(U)$, the homotopy sheaves of groups $\pi_i(\sX_{|U}, x) = 0$ for all $i \in \N$.
\end{definition}
Given any simplicial sheaf $\sX$ (not necessarily a gerbe), one gets a category fibered in groupoids over $\sC$ defined by the fundamental groupoid construction: i.e. for every $U\in \sC$, the fiber category $\sX_{|U}$ is the fundamental groupoid of the space $\sX(U)$, a category whose objects are elements of $\sX(U)$ and morphisms are paths up to homotopy.
This category fibered in groupoids is in fact a gerbe in the sense of \cite[3.2]{laumon} if $\sX$ is connected. If $\sX$ was a gerbe to start with, then it can be recovered, up to weak equivalence, using this category fibered in groupoids using the simplicial nerve construction.
A gerbe $\sX$, is called neutral if it has a global section. In this case, by making a choice of a global section, one can define the fundamental group of $\sX$. Since $\sX$ is connected, a different choice gives a fundamental group which can be canonically identified with the previous one, modulo an inner automorphism.
This motivates the following definition by Giraud.
\begin{definition}\cite[1.1.3]{giraud}
For two sheaves of groups $F$ and $G$, let ${\rm Isex}(F,G)$ denote the set of isomorphisms from $F$ onto $G$ modulo the action of inner automorphisms of $F$ (acting on the left) and the action of inner automorphisms of $G$.
\end{definition}
Consider the pre-stack whose objects over $U$ are sheaves of groups over $U$ (small w.r.t to a fixed universe) with morphisms between $F$ and $G$ defined as elements of ${\rm Isex}(F,G)$. One can stackify this pre-stack (see \cite{laumon}) and objects of this stack are called bands. In particular, every sheaf of groups defines a band. Since every band is represented locally by a sheaf of groups, all those concepts related to sheaves of groups which are local in nature (e.g. exact sequence, epimorphism, kernel, center) also make sense for bands. It is a simple exercise to show that the center of a band is necessarily represented by a sheaf of groups. The 'fundamental group' of any gerbe $\sX$ (neutral or not) is always defined as a band. For a band $L$, a gerbe banded by $L$ (or simply an $L$-gerbe) will mean a gerbe together with an isomorphism of $L$ with the band defined by $\sX$. An equivalence of $L$-gerbes means an equivalence of the gerbes compatible with the given isomorphisms of their bands with $L$.
\begin{definition}\cite[3.1.1]{giraud}
For a band $L$ on a site $\sC$, let $H^2(\sC, L)$ or simply $H^2(L)$ denote the equivalence class of $L$-gerbes. The subset represented by neutral classes in $H^2(\sC, L)$ is denoted by $H^2(\sC, L)'$ or simply ($H^2(L)'$).
\end{definition}
\begin{remark}
Note that $H^2(L)'$ is non-empty if and only if $L$ can be represented by a sheaf of groups (see \cite[3.2.4]{giraud}), in which case it is a singleton set, as can be seen for e.g. by Theorem \ref{all-abelian} stated below.
\end{remark}
\begin{remark}
If a band $L$ is representable by a sheaf of abelian groups $A$, then $H^2(L)$ defined above is in canonical bijection with the $H^2(A)$ as defined by sheaf cohomology \cite[3.4]{giraud}.
\end{remark}
Given an exact sequence of sheaves of groups
$$ 1 \to A \xrightarrow{a} B \xrightarrow{b} C \to 1$$
one has a long exact sequence
$$ 1 \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C).$$
One of the goals of introducing the non-abelian $H^2$ is to extend this exact sequence on the right. We first note that $B$ acts on itself by inner automorphisms. Since $A$ is normal in $B$, this action also induces an action on $A$. Thus $B$ also acts on ${\mathsf {band}}(A)$, where ${\mathsf {band}}(A)$ denotes the band defined by $A$. This action factors through $C$.
We need the following definition to state the result extending the above long exact sequence to $H^2$:
\begin{definition}\cite[4.2.3]{giraud} \label{o}
For an epimorphism of bands $v: M \to N$, one defines a pointed set
$${\mathsf O}(v) := N(v)/R$$ where
\begin{enumerate}
\item $N(v)$ is the set of all triples $(K,L,u)$ where $K$ is a gerbe with band $L$ and $u:L\to M$ is a monomorphism which makes the following sequence exact
$$ 1 \to L \xrightarrow{u} M \xrightarrow{v} N \to 1.$$
\item $R$ is the equivalence relation defined by declaring $(K,L,u) \sim (K',L',u')$ if there exists a morphism of gerbes $\alpha: K\to K'$ such that the induced morphism $\alpha: L\to L'$ on bands makes the following diagram commute
$$\xymatrix{
L \ar[d]_\alpha \ar[r]^u & M \ar@{=}[d] \\
L\ar[r]^{u'} & M
}$$
\item ${\mathsf O}(v)'$ denotes the subset of ${\mathsf O}(v)$ defined by all those $(K,L,u)$ where $K$ is a neutral $L$-gerbe.
\end{enumerate}
\end{definition}
Now one has the following general result.
\begin{theorem}\cite[4.2.8, 4.2.10]{giraud}
Given an exact sequence of sheaves of groups $ 1 \to A \xrightarrow{f} B \xrightarrow{g} C \to 1$, we have the following long exact sequence
$$1 \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C)\xrightarrow{d} {\mathsf O}(g) \to H^2(B) \to H^2(C) .$$
where the map $d$ is as defined in \cite[4.7.2.4]{giraud} and exactness of the sequence is defined similar to that in the case of pointed sets with subsets ${\mathsf O}(g)'$ and $H^2(B)'$, $H^2(C)'$ playing the role of base points. When the action of $C$ on ${\mathsf {band}}(A)$ is trivial, on has a canonical bijection ${\mathsf O}(g) \cong H^2(A)$.
\end{theorem}
Another important result of Giraud we need is the following.
\begin{theorem}\cite[3.3.3]{giraud} \label{all-abelian}
Let $L$ be a band and $C$ be its center. Then one has a canonical action of $H^2(C)$ on $H^2(L)$ which is free and transitive.
\end{theorem}
This theorem, loosely speaking, says that the non-abelian cohomology set $H^2(L)$ is essentially all abelian as it "comes from" its center. Note however that if $H^2(L)$ has no class represented by a neutral gerbe, then there is no canonical bijection between $H^2(L)$ and $H^2(C)$. \\
\noindent The following is a direct consequence of the above theorem and the definition of
\begin{lemma} \label{o-trivial}
Let $1 \to A\xrightarrow{f} B \xrightarrow{g} C \to 1$ be an exact sequence of sheaves of groups. Assume that $H^2(A)$ is trivial. Then ${\mathsf O}(g)' = {\mathsf O}(g)$.
\end{lemma}
\begin{proof}
Let $(K, L, u)$ be any triple where $L$ is a band which fits in the exact sequence
$$ 1 \to L \xrightarrow{u} {\mathsf {band}}(B) \to {\mathsf {band}}(C) \to 1$$
and $K$ is an $L$-gerbe. To prove the theorem it is enough to show that $K$ is neutral. However we note that center $Z(L)$ coincides with $Z(A)$, the center of $A$. The result then directly follows from the above theorem.
\end{proof}
\section{Strong $\A^1$-invariance of the center and applications}
The goal of this section is to prove the theorems mentioned in the introduction. We start by proving the following.
\begin{thm} \label{center} Let $G$ be a strongly $\A^1$-invariant sheaf of groups on $Sm/k$. Then $Z(G)$, the center of $G$ is also strongly $\A^1$-invariant.
\end{thm}
\begin{proof}
Since $G$ is strongly $\A^1$-invariant, $BG$ is $\A^1$-local. By choosing a simplicially fibrant model for $BG$ we may further assume, without loss of generality, that $BG$ is $\A^1$-fibrant. To show $Z(G)$ is strongly $\A^1$-invariant, we need to show $BZ(G)$ is $\A^1$-local. Let $BZ(G) \xrightarrow{u} \sX$ be an $\A^1$-fibrant replacement (in the category of pointed spaces). Thus $u$ is a trivial $\A^1$-cofibration. To prove the theorem, it suffices to show that $u$ is a simplicial weak equivalence. Equivalently, it suffices to show that the map on sheaves of fundamental groups
$$ \pi_1(BZ(G)) \longrightarrow \pi_1(\sX) (=: H)$$ is an isomorphism. Since $BG$ is $\A^1$-fibrant and $u$ is a trivial $\A^1$-cofibration, we have a factorization $h$ as below
$$\xymatrix{
BZ(G) \ar[d]_-u \ar[r]& BG \\
\sX \ar[ru]_-{\exists h} &
}$$
which gives a commutative diagram of the maps induces on the fundamental groups
$$\xymatrix{
Z(G) \ar[d]_-{u_*} \ar[r] & G \\
H \ar[ru]_-{h_*} &
}$$
\end{proof}
In the above diagram, if we show that the image of $h_*$ is $Z(G)$, then it will follow that $Z(G)$ is a retract of the strongly $\A^1$-invariant sheaf $H$ and hence is itself strongly $\A^1$-invariant. Thus it suffices to show that image of $h_*$ is contained in $Z(G)$.
This is equivalent to showing that for every smooth $k$-scheme $U$ and an element $g\in G(U)$, the map $h_{|U}$ is homotopic to the composite of
$$ \sX_{|U} \xrightarrow{h_{|U}} BG_{|U} \xrightarrow{x\mapsto gxg^{-1}} BG_{|U}.$$
\noindent But note that the base change functor from $\Delta^{op}(\Sh(k)) \to \Delta^{op}(\Sh(U))$ preserves trivial $\A^1$-cofibrations since it is a left quillen functor. Moreover it also preserves $\A^1$-fibrations. Thus
$$BZ(G)_{|U} \xrightarrow{\nu_{|U}} \sX_{|U} $$ is a trivial $\A^1$-cofibration. Since $Z(G)$ is the center of $G$, the map $BZ(G)_{|U} \xrightarrow{\nu_{|U}} BG_{|U}$ is homotopic (in fact equal) to the composite
$$ BZ(G)_{|U} \to BG_{|U} \xrightarrow{x\mapsto gxg^{-1}} BG_{|U}.$$
The proof now follows from commutative diagram below, using the fact that $\nu_{\scriptscriptstyle{ |U}}$ is an $\A^1$-weak equivalence and the fact that $BG_{|U}$ is $\A^1$-local by Lemma \ref{projective-trick}.
$$\xymatrix{
BZ(G)_{|U} \ar[d]_-{\nu_{\scriptscriptstyle{ |U}}} \ar[r] & BG_{|U} \ar[r]^{x\mapsto gxg^{-1}} & BG_{|U}\\
\sX_{|U} \ar[ru]_-{h_{|U}} & &
}$$
\begin{lemma}\label{projective-trick}
$U\in Sm/k$. Then the restriction functor $\Delta^{op}(\Sh(k)) \to \Delta^{op}(\Sh(U))$ takes $\A^1$-fibrant objects to $\A^1$-local objects.
\end{lemma}
\begin{proof}
Let $\sY\in \Delta^{op}(\Sh(k))$ be an $\A^1$-fibrant object. Then $\sY_{|U}$ has BG property, since BG property is defined in terms of Nisnevich distinguished triangles and every Nisnevich distinguished triangle in the category $Sm/U$ is also a Nisnevich disntinguished triangle in $Sm/k$.
Moreover, since $\sY(V\times \A^1) \to \sY(V)$ is a weak equivalence for every $V/U$. Thus by arguments given as in \cite[A.6]{morel}, $\sY_{|U}$ is $\A^1$-local.
\end{proof}
\begin{proof}[Proof of \ref{main theorem}]
Let $K$ denote the kernel of the the epimorphism $G\to H$. Thus we have a short exact sequence of Nisnevich sheaves of groups
$$ 1 \to K \to G \to H \to 1. $$
\noindent \underline{Step 1}:
For every smooth $k$-scheme $U$, this gives us an exact sequence (see \cite[3.3.1]{giraud}) of pointed cohomology sets
$$ 1 \to H^0(U,K) \to H^0(U, G) \to H^0(U, H) \to H^1(U, K) \to H^1(U, G) \to H^1(U, H)$$
Using functoriality of the above exact sequence in the case when $U$ is Hensel local, we deduce that the $\A^1$-invariance of $H$ implies (in fact is equivalent to) strong $\A^1$-invariance of $K$. By Theorem \ref{center}, $Z(K)$, the center of $K$ is strictly $\A^1$-invariant sheaf. \\
\noindent \underline{Step 2}: To show strong $\A^1$-invariance of $H$, it is enough to show that for all henselian local essentially smooth schemes $U/k$, $H^1(U\times \A^1_k,H)$ is trivial. By \cite[4.7.2.4]{giraud}, we have an exact sequence of pointed sets
$$ \to H^1(U\times \A^1_k, G) \to H^1(U\times \A^1_k, H) \to O(\phi) $$
where $\phi$ denotes restriction of $G\to H$ to the over-category $(Sm/k)/U\times\A^1$ which will be denoted by $Sm_k/U\times \A^1$ for simplicity and $O(\phi)$ is as defined in \ref{o}. It is enough to show $O(\phi)$ is trivial. This follows from
Lemma \ref{o-trivial} and the strict $\A^1$-invariance of $Z(K)$.
\end{proof}
Let $\sF$ be a sheaf on $Sm/k$. For $U\in Sm/k$ we say $\alpha, \beta \in \sF(U)$ are naive $\A^1$-homotopic if there exists a $\gamma \in \sF(U\times \A^1)$ such that
$$ \alpha = \sigma_0^*(\gamma) \ \ \text{and} \ \ \beta = \sigma_1^*(\gamma)$$
where $\sigma_0, \sigma_1 : U\to U\times \A^1$ are the sections defined by $0$ and $1$ respectively. As in \cite[2.9]{CAA}, let $S(\sF)$ denote the sheaf associated to the presheaf
$$ U \mapsto \frac{\sF(U)}{\sim} $$
where $\sim$ denotes the equivalence relation generated by naive $\A^1$-homotopies. There is a canonical epimorphism
$ \sF \to S(\sF)$. Let
$$S^{\infty}(\sF) := \lim_{n \to \infty} S^n(\sF)$$
The following lemma is straightforward to check.
\begin{lemma} \label{caa} (see \cite[2.13]{CAA}) The canonical morphism $\sF \to S^{\infty}(\sF)$ is an epimorphism and is a universal map from $\sF$ to an $\A^1$-invariant sheaf. Moreover if $\sF$ is a sheaf of groups, then so is $S^{\infty}(\sF)$.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{hurewicz}]
By lemma \ref{caa}, the map $$ \pi_1^{\A^1}(\sX)\xrightarrow{h} H_1^{\A^1}(\sX)$$
factors uniquely through
$$\pi_1^{\A^1}(\sX) \xrightarrow{s} S^{\infty}\left( \pi_1^{\A^1}(\sX)^{ab} \right).$$
Since the map $s$ is an epimorphism, Theorem \ref{main theorem} implies that $S^{\infty}\left( \pi_1^{\A^1}(\sX)^{ab} \right)$ is strongly $\A^1$-invariant sheaf of abelian groups. Thus $s$ is a universal map to strictly $\A^1$-invariant sheaf of abelian groups. However by \cite[6.35]{morel}, so is $h$. Thus the induced map from $S^{\infty}\left( \pi_1^{\A^1}(\sX)^{ab} \right) \to H_1^{\A^1}(\sX)$ must be an isomorphism. In particular $h$ must be an epimorphism.
\end{proof}
\section{Hurewicz map for $\P^1_k$}
In this section we reserve the notation $H$ to denote the Hurewicz map for $\P^1$, i.e. $ \pi_1^{\A^1}(\P^1) \xrightarrow{H} H^{\A^1}_1(\P^1)$. The goal of this section is to prove the following propositions
\begin{proposition}\label{hurewicz-p1}
The kernel of the Hurewicz map for $\P^1_k$,
$$ \pi_1^{\A^1}(\P^1) \xrightarrow{H} H^{\A^1}_1(\P^1)$$
is equal to the commutator subgroup of $\pi_1^{\A^1}(\P^1)$.
\end{proposition}
As a consequence of the explicit computation of the Hurewicz map we obtain the following :
\begin{proposition}\label{etah}
The sequence of Nisnevich sheaves $$ 0 \to h\underline{K}_2^{MW} \to \underline{K}_2^{MW} \xrightarrow{\eta} \underline{K}_1^{MW}$$ is exact.
\end{proposition}
\begin{remark}
We do not know if there is an elementary way to prove the above proposition, using generator and relations. In particular we do not know if
$$ 0 \to h\underline{K}_n^{MW} \to \underline{K}_n^{MW} \xrightarrow{\eta} \underline{K}_{n-1}^{MW}$$
is exact for every $n\geq 1$. However, as pointed out to us by O. R{\"o}ndigs, it is possible that the above short exact sequence is induced by a cofiber sequence given in \cite[Prop. 11]{roendigs}.
\end{remark}
The most difficult part of the computation in the above propositions is the universality of the Hurewicz map and the computation of $\pi_1^{\A^1}(\P^1)$ itself, both of which has been elegantly done in \cite[6.35, 7.3]{morel}. We first restate Morel's computation of $\pi_1^{\A^1}(\P^1)$ as it will also help us to build notation for use in subsequent calculation.
Let $F^{\A^1}(1) := \pi_1^{\A^1}(\P^1)$. First we recall the following two maps defined by Morel:
\begin{enumerate}
\item A map $\theta: \G_m \to F^{\A^1}(1)$ which is a result of an $\A^1$-equivalence $\P^1_k \sim \Sigma(\G_m)$.
\item A map $\underline{K}_2^{MW} \to F^{\A^1}(1)$ which is a result of applying $\pi_1$ to the map $\A^2-0 \to \P^1$ and a theorem of Morel which shows $\underline{K}_2^{MW} \cong \pi_1^{\A^1}(\A^2-0)$.
\end{enumerate}
In what follows, we will freely use standard notation for denoting elements of $K_*^{MW}$ used in \cite[Chapter 3]{morel}, e.g. $\left< -1\right>, h, [U]$ etc.
\begin{theorem}\cite[7.29]{morel} \label{fa1} As a sheaf of sets $F^{\A^1}(1)$ is a product $K_2^{MW}\times \G_m$ and the following describes the structure of $F^{\A^1}(1)$ completely:
\begin{enumerate}
\item[(i)] The sequence $$ 1\to \underline{K}_2^{MW} \to F^{\A^1}(1) \xrightarrow{\gamma} \G_m \to 1$$ is exact and is a central extension.
\item[(ii)] For two units $U,V$ in any field extension $F$ of $k$, the following hold
$$ \theta(U)\theta(V)^{-1} = [-U][-V]\theta(U^{-1}V)^{-1}$$
$$ \theta(U)^{-1}\theta(V) = [U^{-1}][-V] \theta(U^{-1}V).$$
\end{enumerate}
\end{theorem}
The following is the main calculation in the proof of Proposition \ref{hurewicz-p1}.
\begin{lemma}\label{hk2} For any essentially smooth field extension $F/k$, the commutator subgroup of $F^{\A^1}(1)(F)$ is equal to
$h K_2^{MW}(F)$.
\end{lemma}
\begin{proof}
Since $K_2^{MW}(F)$ is in the center of $F^{\A^1}(1)(F)$, we have that the commutator subgroup
$$[F^{\A^1}(1)(F), F^{\A^1}(1)(F)] = \left< \theta(U)\theta(V)\theta(U)^{-1}\theta(V)^{-1} | \ U, V \in F\right>$$
where the angle brackets on RHS denote subgroup generated by the elements within.
$${
\begin{split}
\theta(U)\theta(V) & = \left<-1\right>[U][V]\theta(UV) & ...(\text{by \cite[7.31]{morel}}) \\
\theta(V)\theta(U) & = \left<-1\right>[U][V]\theta(UV) & \\
\theta(U)\theta(V)\theta(U)^{-1}\theta(V)^{-1} & = \left<-1\right>[U][V]\theta(UV) \cdot \big(-\left<-1\right>[V][U]\theta(UV)^{-1}\big) & ...(\text{by \cite[7.29(ii)]{morel}}) \\
& = \left<-1\right>\big([U][V] - [V][U]) & ...(\because K_2^{MW} \text{is in the center})\\
& = [U][V]\big(\left<-1\right> + \left<-1\right>^2\big) & ...(\text{using \cite[3.7(3)]{morel}}) \\
& = h(h-1)[U][V] & ...(\because h=1+\left<-1\right>)
\end{split}
}$$
Thus we have
$${
\begin{split}
[F^{\A^1}(1)(F), F^{\A^1}(1)(F)] & = \left< h(h-1)[U][V] \ | \ U, V \in F\right> & \\
& = h(h-1)K_2^{MW}(F) & ...( \text{using \cite[3.6(1)]{morel}})\\
& = h\cdot K_2^{MW}(F) & ...(\because h-1=\left<-1\right> \text {is a unit by \cite[3.5(4)]{morel}})
\end{split}
}$$
\end{proof}
\begin{proof}[Proof of Proposition \ref{hurewicz-p1}]
Recall that $H$ is a universal map from $\pi_1^{\A^1}(\P^1)$ to a strongly $\A^1$-invariant sheaf of abelian groups. Thus
it is enough to show that the abelianisation of $\pi_1^{\A^1}(\P^1)$ is strongly $\A^1$-invariant. $\pi_1^{\A^1}(\P^1)^{ab}$ is a quotient of a strongly $\A^1$-invariant sheaf therefore by Theorem \ref{main theorem}, we need to show that $\pi_1^{\A^1}(\P^1)^{ab}$ is homotopy invariant. However by \eqref{fa1} and Lemma \ref{hk2}, $\pi_1^{\A^1}(\P^1)^{\rm ab}$ fits in the following exact sequence
$$ 0 \to \frac{\underline{K}_2^{MW}}{h\underline{K}_2^{MW}} \to \pi_1^{\A^1}(\P^1)^{ab} \to \G_m \to 0.$$
Thus it is enough to show that $\frac{\underline{K}_2^{MW}}{h\underline{K}_2^{MW}}$ is $\A^1$-invariant or equivalently $h\underline{K}_2^{MW}$ is strongly $\A^1$-invariant.
However this follows from \cite[3.32]{morel}.
\end{proof}
The following lemmas give explicit description of the Hurewicz morphism $H$.
\begin{lemma}\label{explicit-h} There exists an isomorphism $\phi : H_1^{\A^1}(\P^1)\cong \underline{K}_1^{MW}$ such that
$$ \phi \circ H([U][V]\theta(W)) = \eta[U^{-1}][V] + [W] $$
where $U,V,W$ are sections of $\G_m$ on an object in $\G_m$.
\end{lemma}
\begin{proof}
Since there is an $\A^1$-equivalence $\P^1_k \cong \Sigma \G_m $, we have
$$ H_1^{\A^1}(\P^1_k) \cong H_1^{\A^1}(\Sigma \G_m) \cong \tilde{H}_0^{\A^1}(\G_m)$$
where the second isomorphism is due to $\A^1$-suspension theorem for homology \cite[6.30]{morel}. In the above statement $\G_m$ is considered as a sheaf of sets pointed by $1$.
By the definition of $\A^1$-homology groups, $\tilde{H}_0^{\A^1}(\G_m)$ is the strictly $\A^1$-invariant sheaf of abelian groups generated by the pointed sheaf $\G_m$. By \cite[3.2]{morel}, this must be isomorphic to $\underline{K}_1^{MW}$. Now recall that we have the following commutative diagram
$$\xymatrix{
\G_m \ar[r] \ar[d]_{U\mapsto [U]}\ar@/^2pc/[rr]^\theta & \pi_1(\Sigma \G_m) \ar[r] & \pi_1^{\A^1}(\P^1_k)\ar[d]^H \\
\underline{K}_1^{MW} & \tilde{H}_0^{\A^1}(\G_m)\ar[l]^-{\sim} & H_1^{\A^1}(\Sigma \G_m) \ar[l]^-{\sim} \ar@/^2pc/[ll]^\phi
}$$
The diagram commutes because all morphisms in this diagram are a result of some universal property. The commutativity of the diagram gives us the formula
\begin{equation}\phi(H([W])) = [W]
\end{equation}
Now, using the following equality proved in \cite[7.29(1)]{morel}
$$ \theta(U^{-1})^{-1} \theta(U^{-1}V) \theta(V)^{-1} = [U][V] $$
we get
$${
\begin{split}
\phi(H([U][V]\theta(W))) & = \phi( H (\theta(U^{-1})^{-1} \theta(U^{-1}V) \theta(V)^{-1}\theta(W))) & \\
& = -[U^{-1}] + [U^{-1}V] -[V] + [W] & ...(\text{using eqn(1) above}) \\
& = \eta[U^{-1}][V] + [W] & ...(\text{using \cite[3.1(2)]{morel}})
\end{split}
}$$
\end{proof}
To state the next lemma, recall that $\underline{K}_2^{MW}$ is the free strongly $\A^1$-invariant sheaf of abelian groups generated by the pointed sheaf of sets $\G_m\wedge \G_m$. Thus any automorphism of $\G_m\wedge \G_m$ gives rise to an automorphism of $\underline{K}_2^{MW}$. In particular, the automorphism of $\underline{K}_2^{MW}$ induced by
$${
\begin{split}
\G_m\wedge \G_m & \longrightarrow \G_m\wedge \G_m \\
(U, V) & \mapsto (U^{-1}, V)
\end{split}
}$$ will be denoted by $\tau$.
\begin{lemma} \label{hurewicz-decoded}
Let $\tau: \underline{K}_2^{MW} \to \underline{K}_2^{MW}$ denote the automorphism which sends $[U][V] \mapsto [U^{-1}][V]$. Then the restriction of the Hurewicz map $H$ to $\underline{K}_2^{MW}$ coincides with the composition
$$ \underline{K}_2^{MW} \xrightarrow{\tau} \underline{K}_2^{MW} \xrightarrow{\cdot \eta} \underline{K}_1^{MW}$$
\end{lemma}
\begin{proof}
This follows directly from the explicit formula for $H$ in the above lemma.
\end{proof}
\begin{proof}[Proof of Proposition \ref{etah}]
By universality of $H$, as noted before, the map $F^{\A^1}(1)\xrightarrow{\gamma} \G_m$ factors through $H$. Thus
$$\Ker(H) \subset \Ker(\gamma) = \underline{K}_2^{MW}.$$
$\Ker(H)$ must coincide with the kernel of restriction of $H$ to $\underline{K}_2^{MW}$. Now the proposition follows from Lemmas \ref{hk2} and \ref{hurewicz-decoded}.
\end{proof} | 161,570 |
Student Council
The main objective of this activity is to develop the students’ communication skills and seek their contributions and opinions in school events. Any student starting Grade 4 can join the student council. Members are elected by their classmates so that each class has a representative in the school council to present their requests. Through this council, students learn how to express concerns and think of solutions as well as develop their leadership skills.
Competitions
The Quran competition is a major event at AlHoda schools. It is supervised by the Azhar section and top students are granted tuition scholarships for the following academic year.
The Azhar primary section organizes a yearly ** competition where students are asked to create ** based on a specific theme.
The Sports Dept. also organizes inter-school competitions in soccer, basketball, volleyball, table tennis, swimming, wave skating and remotely controlled car racing, as well as internal and external camps.
Awards
- Awards are given for recognition of the following:
- Attitude & Good Behavior
- Extra Effort
- Advanced Progress
- Outstanding Achievement
- Exceptional knowledge of Quran
Trips
Each class will go on field trips according to the school calendar. Teachers will attend the field trip with their class. Field trips are educational in nature.
Parties/Celebrations
Parties are a time for celebration of special events. Only Classroom teachers can organize parties for special events. Parties are made meaningful; the lesson for the day usually ties into the celebration.
The school celebrates the following events:
- Eid Al-Fitr
- Eid Ul-Adha
- Al Hegra
- Al Mawled Annabawi (Prophet Mohamed’s Birthday, peace be upon him)
- End of year performance | 74,963 |