%auto-ignore

%!TEX root = pnas.tex

\def\bc{{\mathcal B}}

\def\btc{{\mathcal{BT}}}

\newcommand{\HC}{\operatorname{Hoch}}

\newcommand{\HH}{\operatorname{HH}}

\newcommand{\CM}[2]{C_*(\Maps(#1 \to #2))}

\newcommand{\CD}[1]{C_*(\Diff(#1))}

\newcommand{\CH}[1]{CH_*(#1)}

\newcommand{\cl}[1]{\underrightarrow{#1}}

\newcommand{\clh}[1]{\underrightarrow{#1}_{{}_{{}_{{}_h}}}}

\newcommand{\Set}{\text{\textbf{Set}}}

\newcommand{\Vect}{\text{\textbf{Vect}}}

\newcommand{\Kom}{\text{\textbf{Kom}}}

\newcommand{\Cat}{\mathcal{C}}

\newcommand{\cell}{\mathfrak{D}}

\newcommand{\into}{\hookrightarrow}

\newcommand{\onto}{\twoheadrightarrow}

\newcommand{\iso}{\cong}

\newcommand{\quism}{\underset{\text{q.i.}}{\simeq}}

\newcommand{\htpy}{\simeq}

\newcommand{\actsOn}{\circlearrowright}

\newcommand{\xto}[1]{\xrightarrow{#1}}

\newcommand{\isoto}{\xto{\iso}}

\newcommand{\quismto}{\xrightarrow[\text{q.i.}]{\iso}}

\newcommand{\diffeoto}{\xrightarrow[\text{diffeo}]{\iso}}

\newcommand{\htpyto}{\xrightarrow[\text{htpy}]{\htpy}}

\newcommand{\directSum}{\oplus}

\newcommand{\DirectSum}{\bigoplus}

\newcommand{\ot}{\otimes}

\newcommand{\tensor}{\otimes}

\newcommand{\Tensor}{\bigotimes}

\newcommand{\Bord}{\operatorname{Bord}}

\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\clip (0,0.36) rectangle (0.39,-0.16); \draw[->] (0,0.2) .. controls (0.5,0.6) and (0.5,-0.4) .. (0,0);}}}}

\newcommand{\bd}{\partial}

\newcommand{\bdy}{\partial}

\newcommand{\du}{\sqcup}

\newcommand{\compose}{\circ}

\newcommand{\eset}{\emptyset}

\def\spl{_\pitchfork}

\newcommand{\id}{\boldsymbol{1}}

\newtheorem{prop}{Proposition}

\newtheorem{property}[prop]{Property}

\newtheorem{thm}[prop]{Theorem}

\newtheorem{lem}[prop]{Lemma}

\newtheorem{defn}[prop]{Definition}

\newtheorem{axiom}[prop]{Axiom}

\newenvironment{rem}{\noindent\textsl{Remark.}}{}

% \mathrlap -- a horizontal \smash--------------------------------

% For comparison, the existing overlap macros:

% \def\llap#1{\hbox to 0pt{\hss#1}}

% \def\rlap#1{\hbox to 0pt{#1\hss}}

\def\clap#1{\hbox to 0pt{\hss#1\hss}}

\def\mathllap{\mathpalette\mathllapinternal}

\def\mathrlap{\mathpalette\mathrlapinternal}

\def\mathclap{\mathpalette\mathclapinternal}

\def\mathllapinternal#1#2{%

\llap{$\mathsurround=0pt#1{#2}$}}

\def\mathrlapinternal#1#2{%

\rlap{$\mathsurround=0pt#1{#2}$}}

\def\mathclapinternal#1#2{%

\clap{$\mathsurround=0pt#1{#2}$}}

% tricky way to iterate macros over a list

\def\semicolon{;}

\def\applytolist#1{

    \expandafter\def\csname multi#1\endcsname##1{

        \def\multiack{##1}\ifx\multiack\semicolon

            \def\next{\relax}

        \else

            \csname #1\endcsname{##1}

            \def\next{\csname multi#1\endcsname}

        \fi

        \next}

    \csname multi#1\endcsname}

% \def\cA{{\cal A}} for A..Z

\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}}

\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;

% \def\bbA{{\mathbb A}} for A..Z

\def\bbc#1{\expandafter\def\csname bb#1\endcsname{{\mathbb #1}}}

\applytolist{bbc}QWERTYUIOPLKJHGFDSAZXCVBNM;

% \def\bA{{\mathbf A}} for A..Z

% \DeclareMathOperator{\pr}{pr} etc.

\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}

\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat};

% \todo, \nn and \noop

\newcommand{\todo}[1]{\textbf{\color[rgb]{.8,.2,.5}\small TODO: #1}}

\def\nn#1{{{\color[rgb]{.2,.5,.6} \small [[#1]]}}}

\long\def\noop#1{}

% references

\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}}

\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}}

\newcommand{\euclid}[1]{\href{http://projecteuclid.org/euclid.cmp/#1}{{\tt at Project Euclid: #1}}}

\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}}

\newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})}

% figures

\newcommand{\mathfig}[2]{\ensuremath{\hspace{-3pt}\begin{array}{c}%

  \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{diagrams/#2}}%

\end{array}\hspace{-3pt}}}

% packages

\usepackage{tikz}

\usetikzlibrary{shapes}

\usetikzlibrary{backgrounds}

\usetikzlibrary{decorations,decorations.pathreplacing}

\usetikzlibrary{fit,calc,through}

\usepackage[all,color]{xy}

\SelectTips{cm}{}

\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref}

\usepackage{xcolor}

\definecolor{dark-red}{rgb}{0.7,0.25,0.25}

\definecolor{dark-blue}{rgb}{0.15,0.15,0.55}

\definecolor{medium-blue}{rgb}{0,0,0.65}

\hypersetup{

    colorlinks, linkcolor={dark-red},

    citecolor={dark-blue}, urlcolor={medium-blue}

}