\def\qq{blue}
\def\pp{blue!70!red!50}
\def\dexc{cyan!23}%{red!15!blue!40!green}
\def\back{green!35}
\def\vertexc{blue}
\def\halfc{blue}%Periwinkle BurntOrange JungleGreen
\def\thirdc{blue}
\def\edge{blue}
\def\vertex{red}%Periwinkle BurntOrange JungleGreen
\def\tzm{blue!80!red!15}
\newcommand{\sslash}{\mathbin{/\mkern-6mu/}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\C{\mathbb{C}}
\def\G{\mathbf{G}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \newcommand*{\Relbarfill@}{\arrowfill@\Relbar\Relbar\Relbar}
% \newcommand*{\xequal}[2][]{\ext@arrow 0055\Relbarfill@{#1}{#2}}
%%%%%%%%%%%%%%%%%%%%%% Defintions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\op}{\operatorname}
\newcommand{\oop}{\text{op}}
% \newcommand{\N}{\mathbb{N}}
% \newcommand{\R}{\mathbb{R}}
% \newcommand{\Q}{\mathbb{Q}}
% \newcommand{\Z}{\mathbb{Z}}
\newcommand{\K}{\mathbb{K}}
\renewcommand{\O}{\mathbb{O}}
\newcommand{\Etau}{{\text{E}_\tau}}
\newcommand{\E}{{\mathcal E}}
\newcommand{\F}{\mathbf{F}}
\renewcommand{\H}{\mathbb{H}}
\newcommand{\eps}{\epsilon}
\newcommand{\g}{\mathfrak{g}}
\newcommand{\im}{\op{im}}
%%%%%%%%%%%%%%%%%%%%%% Functions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\abs}[1]{\left\lvert#1\right\rvert}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\abracket}[1]{\left\langle#1\right\rangle}
\newcommand{\bbracket}[1]{\left[#1\right]}
\newcommand{\fbracket}[1]{\left\{#1\right\}}
\newcommand{\bracket}[1]{\left(#1\right)}
\newcommand{\vac}[1]{\left|#1\right\rangle}
\newcommand{\from}{\leftarrow}
\newcommand{\bl}{\textbf}
\newcommand{\mbf}{\mathbf}
\newcommand{\mbb}{\mathbb}
\newcommand{\mf}{\mathfrak}
\newcommand{\mc}{\mathcal}
\newcommand{\cinfty}{C^{\infty}}
\newcommand{\pa}{\partial}
\newcommand{\prm}{\prime}
\newcommand{\dbar}{\bar\pa}
\newcommand{\hotimes}{\hat\otimes}
\newcommand{\CE}{Chevalley-Eilenberg }
\newcommand{\suml}{\sum\limits}
\newcommand{\prodl}{\prod\limits}
\newcommand{\into}{\hookrightarrow}
\newcommand{\Ol}{\mathcal O_{loc}}
\newcommand{\mD}{{\mathcal D}}
\newcommand{\iso}{\cong}
\newcommand{\dpa}[1]{{\pa\over \pa #1}}
\newcommand{\BB}{\mathscr{B}}
\newcommand{\Kahler}{K\"{a}hler }
\newcommand{\MC}{Maurer-Cartan }
\newcommand{\CS}{\mathbb S}
\newcommand{\kahler}{K\"{a}hler }
\newcommand{\Zk}{\underline{\text{Alg}}_{\Z_2\times k}}
\renewcommand{\Im}{\op{Im}}
\renewcommand{\Re}{\op{Re}}
\newcommand{\Poincare}{Poincar\'{e} }
\newcommand{\PP}{{\mathbf P}}
\newcommand{\MM}{{\mathbf M}}
\newcommand{\DD}{\slashed{D}}
\newcommand{\MD}{\mathcal D}
\newcommand{\rep}[1]{\underline{\mathbf{#1}}}
%%%%%%%%%%%%%%%%%%%%%% Math Operators %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\mHom}{\operatorname{\mathcal{H}om}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Supp}{\operatorname{Supp}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Deg}{\operatorname{Deg}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\Ber}{\operatorname{Ber}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Or}{\operatorname{Or}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Mat}{\operatorname{Mat}}
\newcommand{\Obs}{\operatorname{Obs}}
\newcommand{\Isom}{\operatorname{Isom}}
\newcommand{\PV}{\operatorname{PV}}
\newcommand{\Der}{\operatorname{Der}}
\newcommand{\HW}{\operatorname{HW}}
\newcommand{\Eu}{\operatorname{Eu}}
\newcommand{\HH}{\operatorname{H}}
\newcommand{\Jet}{\operatorname{Jet}}
\newcommand{\Res}{\operatorname{Res}}
\newcommand{\Jac}{\operatorname{Jac}}
\newcommand{\A}{\mathcal A}
\renewcommand{\S}{\mathcal S}
\renewcommand{\L}{\mathcal L}
\newcommand{\Pin}{\operatorname{Pin}}
\newcommand{\Spin}{\operatorname{Spin}}
\renewcommand{\P}{\text{P}}
\newcommand{\ii}{\mathrm{i}}
\newcommand{\jj}{\mathrm{j}}
\newcommand{\kk}{\mathrm{k}}
\newcommand{\Poin}{\operatorname{Poin}}
\newcommand{\poin}{\operatorname{poin}}
\newcommand{\Stab}{\operatorname{Stab}}
\newcommand{\Obj}{\operatorname{Obj}}
\newcommand{\Set}{\mathsf{Set}}
\newcommand{\GSet}[1]{{#1\op{-\mathsf{Set}}}}
\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\Group}{\mathsf{Grp}}
\newcommand{\Groupoid}{\mathsf{Grpd}}
\newcommand{\Ab}{\mathsf{Ab}}
\newcommand{\Ring}{\mathsf{Ring}}
\newcommand{\CRing}{\mathsf{CRing}}
\newcommand{\Top}{\mathsf{Top}}
\newcommand{\hTop}{h\mathsf{Top}}
\newcommand{\pTop}{\mathsf{Top}_{\star}}
\newcommand{\hpTop}{h\mathsf{Top}_{\star}}
\newcommand{\TopP}{\mathsf{TopP}}
\newcommand{\pTopP}{\mathsf{TopP}_{\star}}
\newcommand{\hTopP}{h\mathsf{TopP}}
\newcommand{\hpTopP}{h\mathsf{TopP}_{\star}}
\newcommand{\Chain}{\mathsf{Ch}_\bullet}
\newcommand{\hChain}{h\mathsf{Ch}_\bullet}
\newcommand{\Ch}{\mathsf{Ch}}
\newcommand{\Tp}{\underline{\mathbf{\mathscr T}}}
\newcommand{\hTp}{\underline{h\mathbf{\mathscr T}}}
\newcommand{\pTp}{\underline{\mathbf{\mathscr T}_{\star}}}
\newcommand{\hpTp}{\underline{h\mathbf{\mathscr T}_{\star}}}
\newcommand{\TpP}{\underline{\mathbf{\mathscr{T}P}}}
\newcommand{\pTpP}{\underline{\mathbf{\mathscr{T}P}_{\star}}}
\newcommand{\hTpP}{\underline{h\mathbf{\mathscr{T}P}}}
\newcommand{\hpTpP}{\underline{h\mathbf{\mathscr{T}P}_{\star}}}
\newcommand{\ChainR}[1]{\op{\underline{Ch_{\bullet}(#1)}}}
\newcommand{\CGWH}{\mathsf{CGWH}}
\newcommand{\CG}{\mathsf{CG}}
\newcommand{\wH}{\mathsf{wH}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Tor}{\operatorname{Tor}}
\newcommand{\colim}[1]{\underset{#1}{\op{colim}} }
\newcommand{\Lim}{{\mathbf{lim}}}
\newcommand{\Colim}{{\mathbf{colim}}}
\newcommand{\Forget}{\operatorname{Forget}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Gr}{\operatorname{Gr}}
\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\Cat}{\mathsf{Cat}}
\newcommand{\Cov}{{\text{Cov}}}
\newcommand{\Fun}{\mathsf{Fun}}
\newcommand{\Orb}{Orb}
\newcommand{\kTop}{\op{\mathsf{CGTop}}}
\newcommand{\kHTop}{\op{\mathsf{CGHaus}}}
\newcommand{\CW}{\op{\mathsf{CW}}}
\newcommand{\hCW}{\op{\mathsf{hCW}}}
\newcommand{\Map}{\operatorname{Map}}
\newcommand{\Ve}{Vect}
\def\rel{\;\operatorname{rel}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\dprime}{{\prime \prime}}
\newcommand{\nn}{node{$\bullet$}}
\newcommand{\snn}{node{\footnotesize{$\bullet$}}}
\newcommand{\MCG}{\operatorname{MCG}} % mapping class group
\newcommand{\Dehn}[1]{\mathbf{D}_{#1}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\PSL}{\operatorname{PSL}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Theorems type enviroments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{thm-defn}{Theorem/Definition}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{lem-defn}[thm]{Lemma/Definition}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{prob}[thm]{Problem}
\newtheorem*{claim}{Claim}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{notn}[thm]{Notation}
\theoremstyle{remark}
\newtheorem{eg}[thm]{Example}
\newtheorem{ex}[thm]{Exercise}
\newtheorem{rmk}[thm]{Remark}
\newtheorem*{remk}{Remark}
\newtheorem{warn}[thm]{Warning}
\def\Br{\operatorname{Br}}
\def\<{\langle}
\def\>{\rangle}
\newcommand{\Conf}{\operatorname{Conf}}
%=========================================================
% Notation
%=========================================================
%=========================================================Bary
\newcommand\Bary[3]{% n, a, b, c
\draw[blue, thick]
(#1)to($(#2)!.5!(#3)$)
(#2)to($(#1)!.5!(#3)$)
(#3)to($(#2)!.5!(#1)$)
coordinate (#1#2);
\draw[orange,thin](o)edge(#1)edge(#2)edge(#3)
edge($(#2)!.5!(#3)$)edge($(#1)!.5!(#3)$)edge($(#2)!.5!(#1)$)
edge($(#3)!.6666!(#1#2)$);
\draw[blue,very thick] ($(#3)!.6666!(#1#2)$) (#1)to(#2)to(#3)to(#1);
\draw[blue!50](#1)\nn(#2)\nn(#3)
\nn($(#2)!.5!(#3)$)\nn($(#1)!.5!(#3)$)\nn($(#2)!.5!(#1)$)
\nn($(#3)!.6666!(#1#2)$)\nn;
}
\newcommand\bary[3]{% n, a, b, c
\draw[dashed,cyan]
(#1) to($(#2)!.5!(#3)$)
(#2) to($(#1)!.5!(#3)$)
(#3) to($(#2)!.5!(#1)$)
coordinate (#1#2);
\draw[orange,thin](o)edge(#1)edge(#2)edge(#3)
edge($(#2)!.5!(#3)$)edge($(#1)!.5!(#3)$)edge($(#2)!.5!(#1)$)
edge($(#3)!.6666!(#1#2)$);
\draw[dashed,thick,cyan] ($(#3)!.6666!(#1#2)$) (#1)to(#2)to(#3)to(#1);
\draw[cyan!50](#1)\nn(#2)\nn(#3)
\nn($(#2)!.5!(#3)$)\nn($(#1)!.5!(#3)$)\nn($(#2)!.5!(#1)$)
\nn($(#3)!.6666!(#1#2)$)\nn;
}
\newcommand\bbb[3]{% n, a, b, c
\draw[blue, thick]
(#1)to($(#2)!.5!(#3)$)
(#2)to($(#1)!.5!(#3)$)
(#3)to($(#2)!.5!(#1)$)
coordinate (#1#2);
\draw[blue,very thick] ($(#3)!.6666!(#1#2)$) (#1)to(#2)to(#3)to(#1);
\draw[cyan](#1)\nn(#2)\nn(#3)
\nn($(#2)!.5!(#3)$)\nn($(#1)!.5!(#3)$)\nn($(#2)!.5!(#1)$)
\nn($(#3)!.6666!(#1#2)$)\nn;
}
\newcommand\bb[3]{% n, a, b, c
\draw[blue, thick]
(#1)to($(#2)!.5!(#3)$)coordinate (#2#3)
(#2)to($(#1)!.5!(#3)$)coordinate (#3#1)
(#3)to($(#2)!.5!(#1)$)coordinate (#1#2);
\draw[blue!50,very thick] ($(#3)!.6666!(#1#2)$)coordinate (q) (#1)to(#2)to(#3)to(#1);
\bbb{#1}{#1#2}{q}\bbb{#2}{#1#2}{q}
\bbb{#2}{#2#3}{q}\bbb{#3}{#2#3}{q}
\bbb{#3}{#3#1}{q}\bbb{#1}{#3#1}{q}
}
\newcommand\BBc[3]{% n, a, b, c
\draw[blue, thick]
(#1)to($(#2)!.5!(#3)$)coordinate (#2#3)
(#2)to($(#1)!.5!(#3)$)coordinate (#3#1)
(#3)to($(#2)!.5!(#1)$)coordinate (#1#2);
\draw[blue!50,very thick] ($(#3)!.6666!(#1#2)$)coordinate (#1#2#3) (#1)to(#2)to(#3)to(#1);
\bb{#1}{#1#2}{#1#2#3}\bb{#2}{#1#2}{#1#2#3}
\bb{#2}{#2#3}{#1#2#3}\bb{#3}{#2#3}{#1#2#3}
\bb{#3}{#3#1}{#1#2#3}\bb{#1}{#3#1}{#1#2#3}
}
\newcommand\BFour[3]{% n, a, b, c
\draw[blue, thick]
(#1)to($(#2)!.5!(#3)$)coordinate (#2#3)
(#2)to($(#1)!.5!(#3)$)coordinate (#3#1)
(#3)to($(#2)!.5!(#1)$)coordinate (#1#2);
\draw[blue!50,very thick] ($(#3)!.6666!(#1#2)$)coordinate (#1#2#3) (#1)to(#2)to(#3)to(#1);
\BBc{#1}{#1#2}{#1#2#3}\BBc{#2}{#1#2}{#1#2#3}
\BBc{#2}{#2#3}{#1#2#3}\BBc{#3}{#2#3}{#1#2#3}
\BBc{#3}{#3#1}{#1#2#3}\BBc{#1}{#3#1}{#1#2#3}
}
%%%%%%%%%% banana space
\english
\nosectionnumbers
\pdef\subsection*{\section}
% \newcounter{figure}
% \newenvironment{figure}[1][]{\refstepcounter{figure}\addtocounter{figure}{-1}\def\caption##1{\stepcounter{figure}\begin{center}Figure \thefigure. ##1\end{center}}}{}
\def\varprojlim{\mathop{\underleftarrow{\lim}}}
\def\varinjlim{\mathop{\underrightarrow{\lim}}}