%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]{C_*(\Homeo(#1))}

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

\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{\tensor}{\otimes}

\newcommand{\Tensor}{\bigotimes}

\newcommand{\selfarrow}{\ensuremath{\smash{\tikz[baseline]{\draw[->] (0,0.2) .. controls (0.6,0.8) and (0.6,-0.6) .. (0,0);}}}}

\newcommand{\bdy}{\partial}

\newcommand{\compose}{\circ}

\newcommand{\eset}{\emptyset}

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

\newtheorem{property}{Property}

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

% packages

\usepackage{tikz}

\usetikzlibrary{shapes}

\usetikzlibrary{backgrounds}

\usetikzlibrary{decorations,decorations.pathreplacing}

\usetikzlibrary{fit,calc,through}

\usepackage[all,color]{xy}

\SelectTips{cm}{}