\documentclass[11pt,notitlepage]{article}
\usepackage{amsmath, amssymb}
\usepackage{graphicx, stmaryrd, datetime, amsthm, amscd, multicol, txfonts,
  ../picins, mathabx, dbnsymb, tensor, needspace, mathtools, import
}
\usepackage[normalem]{ulem} % For \sout.
%\usepackage{mathbbol} % Needed for \bbe.
\usepackage{bbm} % Needed for \bbe.
\usepackage[margin=0.25in,foot=0.2in]{geometry}
\usepackage[usenames,dvipsnames]{xcolor}
% 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{pdfcomment}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{insdljs,rangen}
\usepackage[greek,english]{babel}
% Cyrillic letters following http://tex.stackexchange.com/questions/14633/what-packages-will-let-me-use-cyrillic-characters-in-math-mode:
% ... a better solution at pensieve://2015-11/xtx/xtx.tex, except if fails here because of "too many fonts".
\usepackage[T2A,T1]{fontenc}
\makeatletter
\def\easycyrsymbol#1{\mathord{\mathchoice
  {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\sf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\ssf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
}}
\makeatother
\newcommand{\Zhe}{{\easycyrsymbol{\CYRZH}}}
\newcommand{\zhe}{{\easycyrsymbol{\cyrzh}}}

% From http://tex.stackexchange.com/questions/1429/latex-package-to-generate-qr-codes:
%\usepackage{pst-barcode}
%\usepackage{auto-pst-pdf}
% From http://tex.stackexchange.com/questions/89649/pgf-tikz-qr-code-generator
\usepackage[]{qrcode}

\usepackage[inline]{enumitem}
\setlist{nosep}
\setlist[itemize]{left=0pt .. 10pt}
\setlist[enumerate]{left=0pt .. 12pt}

\usepackage{twemojis} % For \twemoji{notebook}

% \llangle and \rrangle from https://www.latex4technics.com/?note=2dg7:
\makeatletter
\newsavebox{\@brx}
\newcommand{\llangle}[1][]{\savebox{\@brx}{\(\m@th{#1\langle}\)}%
  \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}
\newcommand{\rrangle}[1][]{\savebox{\@brx}{\(\m@th{#1\rangle}\)}%
  \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}
\makeatother

\def\sheeturl{{\url{https://drorbn.net/AP/People/Afeke/}}}
\def\myurl{http://www.math.toronto.edu/~drorbn}
%\def\arXiv#1{{\qrcode[height=1em,level=L,nolink]{arxiv.org/abs/#1}\,\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{\small #1}}}}
\def\bbs#1#2#3{{\href{http://drorbn.net/bbs/show?shot=#1-#2-#3.jpg}{BBS:\linebreak[0]#1-\linebreak[0]#2}}}
\def\mo#1{{\href{https://mathoverflow.net/questions/#1}{{\footnotesize m{\bf\sl o}:}\linebreak[0]{\small #1}}}}

%\paperwidth 8in \paperheight 10.5in
%\textwidth 8in \textheight 10.5in
%\oddsidemargin -0.75in \evensidemargin \oddsidemargin \topmargin -0.75in
%\headheight 0in \headsep 0in \footskip 0in
\parindent 0in
%\setlength{\topsep}{0pt}
%\pagestyle{empty}
\dmyydate

% Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top:
\def\imagetop#1{\vtop{\null\hbox{#1}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\Ad{\operatorname{Ad}}
\def\ad{\operatorname{ad}}
\def\adj{\operatorname{adj}}
\def\ann{{\operatorname{ann}}}
\def\Arg{\operatorname{Arg}}
\def\aug{\operatorname{aug}}
\def\Aut{\operatorname{Aut}}
\def\attr{\operatorname{\mathfrak{tr}}}
\def\atdiv{\operatorname{div}}
\def\coker{\operatorname{coker}}
\def\crit{\operatorname{crit}}
\def\der{\operatorname{der}}
\def\Emb{\operatorname{Emb}}
\def\End{\operatorname{End}}
\def\Fun{\operatorname{Fun}}
\def\gr{\operatorname{gr}}
\def\GRT{\operatorname{GRT}}
\def\hgt{\operatorname{ht}}
\def\Hom{\operatorname{Hom}}
\def\im{\operatorname{im}}
\def\Imm{\operatorname{Imm}}
\def\Inn{\operatorname{Inn}}
\def\lk{\operatorname{lk}}
\def\mal{\operatorname{mal}}
\def\Mal{\operatorname{Mal}}
\def\Mat{\operatorname{Mat}}
\def\Mod{\operatorname{Mod}}
\def\mor{\operatorname{mor}}
\def\obj{\operatorname{obj}}
\def\Out{\operatorname{Out}}
\def\rad{\operatorname{rad}}
\def\Rker{\operatorname{Rker}}
\def\sign{\operatorname{sign}}
\def\surjects{{\twoheadrightarrow}}
\def\tAut{\operatorname{tAut}}
\def\wt{\operatorname{wt}}

\def\qed{{\hfill\text{$\Box$}}}
\def\remove{\!\setminus\!}
\def\FA{{F\!A}}
\def\FG{{F\!G}}
\def\FL{{F\!L}}
\def\St{S\!t}
\def\SL{{S\!L}}
\def\tr{\operatorname{tr}}

\def\sder{\operatorname{\mathfrak{sder}}}
\def\Span{\operatorname{span}}
\def\tder{\operatorname{\mathfrak{tder}}}
\def\uB{\text{\it u\!B}}
\def\uT{{u\calT}}
\def\vB{\text{\it v\!B}}
\def\vK{{v\calK}}
\def\vT{{v\calT}}
\def\wB{\text{\it w\!B}}
\def\wK{{w\calK}}
\def\wT{{w\calT}}
\def\PB{\text{\it P\!B}}
\def\PuB{\text{\it P\!u\!B}}
\def\PaB{\text{\it PaB}}
\def\PvB{\text{\it P\!v\!B}}
\def\PvT{\text{\it P\!v\!T}}
\def\PwB{\text{\it P\!w\!B}}

\def\barA{{\bar{A}}}
\def\barT{{\bar{T}}}
\def\bb1{{\mathbbm{1}}}
\def\bbA{{\mathbb A}}
\def\bbB{{\mathbb B}}
\def\bbC{{\mathbb C}}
\def\bbD{{\mathbb D}}
\def\bbe{\mathbbm{e}}
\def\bbE{\mathbb{E}}
\def\bbH{{\mathbb H}}
\def\bbi{\mathbb{i}}
\def\bbK{{\mathbb K}}
\def\bbN{{\mathbb N}}
\def\bbO{{\mathbb O}}
\def\bbP{{\mathbb P}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calB{{\mathcal B}}
\def\calC{{\mathcal C}}
\def\calD{{\mathcal D}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calL{{\mathcal L}}
\def\calM{{\mathcal M}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\calV{{\mathcal V}}
\def\calW{{\mathcal W}}
\def\calZ{{\mathcal Z}}
\def\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\frakl{{\mathfrak l}}
\def\frakn{{\mathfrak n}}
\def\frakt{{\mathfrak t}}

\def\krv{\mathfrak{krv}}
\def\RP{\bbR{\mathrm P}}

\def\eps{\epsilon}

\def\gray{\color{gray}}
\def\green{\color{green}}
\def\magenta{\color{magenta}}
\def\red{\color{red}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}

\newtheoremstyle{handoutstyle} % Modified from
    % https://tex.stackexchange.com/questions/37472/spacing-before-and-after-with-newtheoremstyle
  {0pt} % Space above
  {0pt} % Space below
  {} % Body font
  {} % Indent amount
  {\red} % Theorem head font
  {.} % Punctuation after theorem head
  {.5em} % Space after theorem head
  {} % Theorem head spec (can be left empty, meaning `normal')
\theoremstyle{handoutstyle}
\newtheorem{claim}{Claim}
\newtheorem{ex}[claim]{Exercise}
\newtheorem{lem}[claim]{Lemma}
\newtheorem{prop}[claim]{Proposition}
\newtheorem{rem}[claim]{Remark}
\newtheorem{thm}[claim]{Theorem}

\begin{document}
%\setlength{\jot}{0ex}
%\setlength{\abovedisplayskip}{0.5ex}
%\setlength{\belowdisplayskip}{0.5ex}
%\setlength{\abovedisplayshortskip}{0ex}
%\setlength{\belowdisplayshortskip}{0ex}

\parbox[b]{5.6in}{\small
  \href{http://www.math.toronto.edu/drorbn}{DBN}:
  \href{http://drorbn.net/AP}{AP}:
  \href{http://drorbn.net/AP/People}{People}:
  \href{http://drorbn.net/AP/People/Afeke}{Afeke}:
  {\LARGE\bf On the Equivariant Intersection Pairing}
  }
\hfill\parbox[b]{2.1in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated July 30, 2024; modified \today, \ampmtime
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/AP/People/Afeke}
}

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

\begin{multicols}{2}

\includegraphics[width=\linewidth]{Conventions.png}

Let $D_n$ be the $n$-punctured disk and let $\pi\colon\widetilde{D_n}\to D_n$ be its universal Abelian cover. The group $\bbZ^n=\langle t_1,\dots,t_n\rangle$ acts on $\widetilde{D_n}$ by deck transformations. We take as a given the ``standard'' (anti-symmetric) intersection pairing $\langle\cdot,\cdot\rangle \colon H_1(\widetilde{D_n})\otimes H_1(\widetilde{D_n}) \to \bbZ$ and define the {\em equivariant} intersection pairing
\begin{equation}
  \langle\cdot,\cdot\rangle_e \colon H_1(\widetilde{D_n})\otimes H_1(\widetilde{D_n}) \to \bbZ\bbZ^n = \bbZ[t_1^{\pm1},\dots,t_n^{\pm1}]
\end{equation}
by
\begin{equation} \langle a,b\rangle_e = \sum_{k\in\bbZ^n}\langle t^ka,b\rangle t^k,
\end{equation}
where $t^k\coloneqq t_1^{k_1}t_2^{k_2}\dots t_n^{k_n}$.

Let $p\in D_n$ be a basepoint and let $\tilde{p}$ be a fixed lift of $p$ to $\widetilde{D_n}$ (namely, fix $\tilde{p}\in\widetilde{D_n}$ is such that $\pi(\tilde{p})=p$). Given a curve $\alpha$ in $D_n$ which begins and ends at $p$, we denote by $\tilde{\alpha}$ the unique lift of $\alpha$ to $\widetilde{D_n}$ for which $\tilde{\alpha}(0)=\tilde{p}$. We will have then a unique $w(\alpha)\in\bbZ^n$ for which $\widetilde{\alpha}(1)=t^{w(\alpha)}\tilde{p}$.

Every element $a\in H_1(\widetilde{D_n})$ can be written as a finite sum $a=\sum_i f_i\widetilde{\alpha_i}$, where $f_i\in\bbZ\bbZ^n$ and where the $\alpha_i$'s are curves in $D_n$ which begin and end at $p$. Similarly write $b=\sum_j g_j\widetilde{\beta_j}$. Without loss of generality, all the $\alpha_i$ ``lean to the left'' near their ends, and all the $\beta_j$ ``lean to the right'' near their ends (see the figure). Also without loss of generality the interiors $\alpha_i^o$ and $\beta_j^o$ of $\alpha_i$ and $\beta_j$ avoid the basepoint $p$.

\begin{claim} Suppose also that for every $i$ and $j$ the intersection $\alpha_i^o\cap\beta_j^o$ is transverse, and if $q\in\alpha_i^o\cap\beta_j^o$, denote by $\eps_q$ the sign of that intersection point. Then
\begin{equation}
  \langle a,b\rangle_e =
  \sum_{i,j}\overline{f_i}g_j \sum_{q\in\alpha_i^o\cap\beta_j^o} \eps_q t^{-w\left(\widetilde{\alpha_i\#_q\beta_j}\right)},
\end{equation}
where $f\mapsto\bar{f}$ is $\bbZ$-linear automorphism of $\bbZ\bbZ^n$ which extends $t^k\mapsto t^{-k}$ and where $\alpha_i\#_q\beta_j$ is the curve in $D_n$ that follows $\alpha_i$ from its beginning up to the point $q$, and then follows $\beta_j$ backwards from $q$ to the beginning of $\beta_j$.
\end{claim}

\begin{rem}
If for all $i$ and $j$ the curves $\widetilde{\alpha_i}$ and $\widetilde{\beta_j}$ are closed, we can write
\begin{equation}
  \langle a,b\rangle_e \!=\! \sum_{i,j}\overline{f_i}g_j \langle\widetilde{\alpha_i},\widetilde{\beta_j}\rangle_e
  \quad\text{and}\quad
  \langle\widetilde{\alpha_i},\widetilde{\beta_j}\rangle_e = \hspace{-3mm} \sum_{q\in\alpha_i^o\cap\beta_j^o} \hspace{-2mm} \eps_q t^{-w\left(\widetilde{\alpha_i\#_q\beta_j}\right)},
\end{equation}
but in general, $\langle\widetilde{\alpha_i},\widetilde{\beta_j}\rangle_e$ may not make sense as a pairing of homology classes because $\widetilde{\alpha_i}$ and $\widetilde{\beta_j}$ may not be homology classes.
\end{rem}

\begin{rem} Rather than ``$\alpha_i$ leans left and $\beta_j$ leans right near $p$'' we could say ``the intersections of $\alpha_i$ and $\beta_j$ with every small circle around $p$ are not interlaced''.
\end{rem}

\end{multicols}

\end{document}

\endinput
