\documentclass[11pt,notitlepage]{article}
\usepackage{graphicx, multicol, qrcode, import, amsmath,
  stmaryrd, % for \sslash
  mathbbol, % for \bbe.
  mathtools % for \coloneqq
}
\usepackage[nodayofweek]{datetime} \yyyymmdddate
\usepackage[margin=0.25in,foot=0.2in]{geometry}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
\usepackage{amsthm}
\newtheorem{definition}{Definition}
\usepackage[all]{xy}

\def\sheeturl{{\url{http://drorbn.net/ap/2022-11/AlexanderLagrangians4Handlebodies}}}
\def\myurl{http://www.math.toronto.edu/~drorbn}

\def\blue{\color{blue}}
\def\green{\color{green}}
\def\red{\color{red}}

\def\bbe{\mathbb{e}}
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calV{{\mathcal V}}
\def\calX{{\mathcal X}}
\def\calY{{\mathcal Y}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\AL{{\it AL}}
\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{\small #1}}}}
\def\Hom{\operatorname{Hom}}
\def\interior{\operatorname{int}}
\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}


\begin{document}
\parindent 0in

\parbox[b]{4.8in}{
  {\small Dror Bar-Natan:} {\large\bf Alexander Lagrangians for Handlebodies}
  }
\hfill\parbox[b]{2.9in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated 2022/11/21; modified \today, \ampmtime
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/ap/2022-11/AlexanderLagrangians4Handlebodies}
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols}{2}

Let $H$ be a genus $g$ handlebody in $\bbR^3$, let $\Sigma=\partial H$, and let $p\in\Sigma$ be a basepoint (think ``$H$ is a tubular neighborhood of a pinched tangle $T$''). Let $C=(\interior H)^c$ and let $\tau\colon\pi_1(C)\to H^1(H;\bbZ)\cong\bbZ^g$ be induced by Alexander (?) duality. Let $\tilde{\Sigma}\subset\tilde{C}$ be the $\tau$-covers of $\Sigma\subset C$, respectively, with covering projections $\phi$. Let $\tilde{p}=\phi^{-1}(p)$; it is a copy of $\bbZ^g$. Let $R \coloneqq \bbZ H^1(H;\bbZ) \cong \bbZ[T_i^{\pm 1}]_{i=1}^g$ be the group ring of $H^1(H;\bbZ)$, the ring of Laurent polynomials in variables $T_1,\ldots,T_g$, and note that $\Omega_g \coloneqq H_1(\tilde{\Sigma},\tilde{p})$ and $H_1(\tilde{C},\tilde{p})$ are $R$-modules and that $\Omega_g$ does not depend on the embedding of $H$.

\begin{definition} $A(H) \coloneqq H_1(\tilde{C})$, the Alexander module of $H$, and $\AL(H)\coloneqq\ker i_*\colon \Omega_g\to H_1(\tilde{C},\tilde{p})$, the Alexander Lagrangian of $H$.
\end{definition}

Note that if $H$ is a tubular neighborhood of a knot $K$ then $A(H)$ is the standard Alexander module of $K$. Note that $i_*$ is always surjective so we have the exact sequence
\[ \xymatrix{
  0 \ar[r] &
  \AL(H) \ar[r] &
  \Omega_g \ar[r]^-{i_*} &
  H_1(\tilde{C},\tilde{p}) \ar[r] &
  0.
} \]
The following, coming from $\tilde{p}\to\tilde{C}\to(\tilde{C},\tilde{p})$, is also exact, and non-canonically split:
\[ \xymatrix@C=5mm{
  0 \ar[r] &
  H_1(\tilde{C}) \ar[r] \ar@2{-}[d] &
  H_1(\tilde{C},\tilde{p}) \ar[r] &
  H_0(\tilde{p}) \ar[r] \ar@2{-}[d] &
  H_0(\tilde{C}) \ar[r] \ar@2{-}[d] &
  0 \\
  & A(H) & & R \ar[r]^-\epsilon & \bbZ &
} \]
where $\epsilon$ is the augmentation map.
And so with $R_0=\ker\epsilon$, $A(H)\oplus R_0 \cong H_1(\tilde{C},\tilde{p}) \cong \Omega_g/\AL(H)$, where the first isomorphism is non-canonical. In the knot case the isomorphism is canonical.

\end{multicols}

\end{document}