diff --git a/presentation/Makefile b/presentation/Makefile new file mode 100644 index 0000000..734a023 --- /dev/null +++ b/presentation/Makefile @@ -0,0 +1,20 @@ + +.PHONY: thesis fast dirs + +# We don want to pollute the root dir, so we use a build dir +# http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f +thesis: dirs + xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex + xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex + cp build/Rational_Homotopy_Theory.pdf ./ + +fast: dirs + xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex + cp build/Rational_Homotopy_Theory.pdf ./ + +haltfast: dirs + xelatex -file-line-error -output-directory=build --halt-on-error Rational_Homotopy_Theory.tex + cp build/Rational_Homotopy_Theory.pdf ./ + +dirs: + mkdir -p build diff --git a/presentation/Rational_Homotopy_Theory.tex b/presentation/Rational_Homotopy_Theory.tex new file mode 100644 index 0000000..59c058a --- /dev/null +++ b/presentation/Rational_Homotopy_Theory.tex @@ -0,0 +1,185 @@ +\documentclass[14pt]{beamer} + +\definecolor{todocolor}{rgb}{1, 0.3, 0.2} +\newcommand{\td}[1]{\colorbox{todocolor}{*\footnote{TODO: #1}}} + +\input{preamble} +\usepackage{tabularx} +\renewcommand{\tabularxcolumn}[1]{p{#1}} + +\newcommand{\Frame}[2]{ + \begin{frame}{#1}#2\end{frame} +} + +\title{Rational Homotopy Theory} +\author{Joshua Moerman} +\institute[Radboud Universiteit Nijmegen]{Supervisor: Ieke Moerdijk} +\date{} + +\begin{document} + +\AtBeginSection[]{ + \begin{frame} + \tableofcontents[currentsection] + \end{frame} +} + +\Frame{}{ + \titlepage +} + + +\section{Introduction to homotopy theory} +\Frame{Homotopy theory}{ + \begin{center} + Study of space or shapes \\ + with ``weak equivalences'' + + \bigskip + \td{plaatje} + \end{center} +} + +\Frame{Important spaces}{ +\td{plaatjes} +\begin{align*} + S^1 &= ... \\[1em] + S^2 &= ... \\[1em] + S^3 &= \cdots \\[1em] + &\cdots +\end{align*} +} + +\Frame{Important tool}{ + Fundamental group: + + \[ \pi_1(X) = \text{maps } S^1 \to X \text{ up to homotopy} \] + + \bigskip + \td{plaatje} +} + +\Frame{Important tools}{ + Homotopy groups: + + \begin{align*} + \pi_1(X) &= \text{maps } S^1 \to X \text{ up to homotopy} \\[1em] + \pi_2(X) &= \text{maps } S^2 \to X \text{ up to homotopy} \\[1em] + \pi_3(X) &= \text{maps } S^3 \to X \text{ up to homotopy} \\[1em] + &\cdots + \end{align*} +} + +\Frame{Torsion-free}{ + Serre proved in 1950s: + \begin{align*} + \text{odd } k: \quad \pi_n(S^k) \tensor \Q &= + \begin{cases} + \Q &\text{ if } n = k \\ + 0 &\text{ otherwise } + \end{cases} \\[1em] + \text{even } k: \quad \pi_n(S^k) \tensor \Q &= + \begin{cases} + \Q &\text{ if } n = k, 2k-1 \\ + 0 &\text{ otherwise } + \end{cases} \\ + \end{align*} +} + + +\section{Rational homotopy theory} +\Frame{Rational homotopy theory}{ + \begin{center} + Study of spaces\\ + with ``rational equivalences'' \\ + and ``rational homotopy groups'' + + \pause + \bigskip + or + \bigskip + + Study of \emph{rational} spaces \\ + with weak equivalences \\ + and ordinary homotopy groups + \end{center} +} + +\Frame{Rational spaces}{ + $X$ is \emph{rational} if $\pi_n(X)$ is a $\Q$-vector space + + \bigskip + \td{plaatje telescoop} +} + + +\section{The main equivalence} +\Frame{Main equivalence}{ + \begin{theorem} + \begin{center} + Homotopy theory of rational spaces \\ + = \\ + Homotopy theory of commutative differential graded algebras + \end{center} + \end{theorem} +} + +\Frame{Main equivalence (precise version)}{ + \begin{theorem} + \[ \Ho(\Top_{\Q, 1, f}) \simeq \opCat{\Ho(\CDGA_{\Q, 1, f})} \] + \end{theorem} +} + +\Frame{What is a cdga?}{ + \begin{definition} + a cdga $A$ is + \begin{itemize} + \item a $\Q$-vector space + \item with a multiplication $A \tensor A \tot{\mu} A$ + \item with a differential $A \tot{d} A$ such that $d^2 = 0$ + \item with a grading $A = \bigoplus_{n \in \N} A^n$ + \item it is commutative: $ x y = (-1)^{\deg{x}\cdot\deg{y}} y x $ + \end{itemize} + \end{definition} +} + +\Frame{Free cdga's}{ + As always: there is a free guy: $\Lambda(...)$ + + For example + \[ \Lambda(t, dt) \text{ with } \deg{t} = 0 \] + is just polynomials in $t$, with its differential $dt$ +} + +\newcommand{\Dict}[1]{ + \noindent + \begin{tabularx}{\textwidth}{ X X } + {\bf rational spaces} & {\bf cdga's} \\[1em] + #1 + \end{tabularx} +} + +\Frame{Dictionary}{ + \Dict{ + $S^n$ with $n$ odd + & $\Lambda(e)$ with $\deg{e} = n$ \\[1em] + + $S^n$ with $n$ even + & $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em] + + Eilenberg-MacLane space $K(\Q, n)$ + & $\Lambda(e)$ with $\deg{e} = n$ + } +} + +\Frame{Dictionary}{ + \Dict{ + homotopy $$h: X \times I \to Y$$ + & homotopy $$h: A \to B \tensor \Lambda(t, dt)$$ \\[1em] + + $f: X \to Y$ weak equivalence if $\pi_n(f): \pi_n(X) \iso \pi_n(Y)$ + & $f: A \to B$ weak equivalence if $H(f): H(X) \iso H(Y)$ + } +} + +\end{document} diff --git a/presentation/preamble.tex b/presentation/preamble.tex new file mode 100644 index 0000000..6c37f48 --- /dev/null +++ b/presentation/preamble.tex @@ -0,0 +1,160 @@ + +% normally included with amsart +% \usepackage{amsmath, amsthm} + +% font with unicode support, does not work with classicthesis +% \usepackage{fontspec} + +% clickable tocs +\usepackage{hyperref} + +% floating figures +\usepackage{float} + +% for multiple cites +\usepackage{cite} + +% fancy diagrams +\usepackage{tikz} +\usetikzlibrary{matrix, arrows, decorations} +\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} + +% simple diagrams +% \usepackage[all,cmtip]{xy} + +\usepackage{graphicx} +% \graphicspath{ {./images/} } +\usepackage{caption} +\usepackage{subcaption} + +% for the fib arrow +\usepackage{amssymb} + +% Some basic objects +\newcommand{\N}{\mathbb{N}} % natural numbers +\newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers +\newcommand{\Z}{\mathbb{Z}} % integers +\newcommand{\R}{\mathbb{R}} % reals +\DeclareRobustCommand{\Q}{\mathbb{Q}} % rationals +\renewcommand{\k}{\mathrm{I\!k}} % default ground ring + +% Basic category stuff +\newcommand{\cat}[1]{\mathbf{#1}} % the category of ... +\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category +\newcommand{\Hom}{\mathbf{Hom}} +\newcommand{\id}{\mathbf{id}} +\newcommand{\Ho}{\cat{Ho}} + +% Categories +\newcommand{\Set}{\cat{Set}} % sets +\newcommand{\Top}{\cat{Top}} % topological spaces +\newcommand{\Grp}{\cat{Grp}} % groups +\newcommand{\Ab}{\cat{Ab}} % abelian groups +\newcommand{\DELTA}{\boldsymbol{\Delta}}% the simplicial cat +\newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects +\newcommand{\sSet}{\simplicial{\Set}} % simplicial sets +\newcommand{\Mod}[1]{\cat{{#1}Mod}} % modules over a ring +\newcommand{\Alg}[1]{\cat{{#1}Alg}} % algebras over a ring +\newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring +\newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring +\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}} % differential graded modules over a ring +\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}} % differential graded algebras over a ring +\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}} % chain complexes +\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}} % cochain complexes +\DeclareRobustCommand{\DGA}{\cat{DGA}} % cochain algebras +\DeclareRobustCommand{\CDGA}{\cat{CDGA}} % commutative cochain algebras +\DeclareRobustCommand{\AugCDGA}{\cat{CDGA^\ast}}% augmentedcommutative cochain algebras + +\newcommand{\cof}{\hookrightarrow} % cofibration +\newcommand{\fib}{\twoheadrightarrow} % fibration +\newcommand{\we}{\tot{\simeq}} % weak equivalence + +% for use in xy diagrams +\newcommand{\arcof}{\ar@{^{(}->}} +\newcommand{\artcof}{\ar@{^{(}->}|\simeq} +\newcommand{\arfib}{\ar@{->>}} +\newcommand{\artfib}{\ar@{->>}|\simeq} +\newcommand{\arwe}{\ar|-\simeq} +\newcommand{\ariso}{\ar|-\iso} + +% adjunction symbol for xymatrices +\newcommand{\xyadj}{\raisebox{0.2\height}{\scalebox{0.5}{$\perp$}}} + +% pushout and pullback for xymatrices (makes empty arrow with text) +\newcommand{\xypo}{\ar@{}[dr]|(.75){\scalebox{1.2}{$\ulcorner$}}} +\newcommand{\xypb}{\ar@{}[dr]|(.25){\scalebox{1.2}{$\lrcorner$}}} + +%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}} +\newcommand{\leftadj}{\rightleftarrows} + +% Notation and operators +\newcommand{\I}{\,\mid\,} % seperator in set notation +\newcommand{\del}{\partial} % boundary +\newcommand{\iso}{\cong} % isomorphic +\newcommand{\eq}{\sim} % homotopic +\newcommand{\ison}[1]{\stackrel{(#1)}{\iso}} % isos to refer to +\newcommand{\refison}[1]{{\small $(#1)$}} % ref +\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name +\newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}} % left arrow with name +\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name +\newcommand{\unit}{\eta} +\newcommand{\counit}{\epsilon} +\DeclareMathOperator*{\im}{im} +\DeclareMathOperator*{\coker}{coker} +\DeclareMathOperator*{\colim}{colim} +\DeclareMathOperator*{\Tor}{Tor} +\DeclareMathOperator*{\Ext}{Ext} +\DeclareMathOperator*{\tensor}{\otimes} +\DeclareMathOperator*{\bigtensor}{\bigotimes} +\renewcommand{\deg}[1]{{|{#1}|}} +\newcommand{\Char}[1]{char({#1})} +\newcommand{\RH}{\widetilde{H}} % reduced homology +\DeclareRobustCommand{\C}{\mathcal{C}} % Serre mod C class +\newcommand{\Apl}[0]{{A_{PL}}} % Apl simplicial set of polynomials + +\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA$}{CDGA}} + +% restriction of a function +\newcommand\restr[2]{{% we make the whole thing an ordinary symbol + \left.\kern-\nulldelimiterspace % automatically resize the bar with \right + #1 % the function + \vphantom{\big|} % pretend it's a little taller at normal size + \right|_{#2} % this is the delimiter + }} + +% Todos in the margin +\newcommand{\todo}[1]{ + \addcontentsline{tdo}{todo}{\protect{#1}} + $\ast$ \marginpar{\tiny $\ast$ #1} +} +% Big todos in text +\newcommand{\TODO}[1]{ + \addcontentsline{tdo}{todo}{\protect{#1}} + {\tiny $\ast$ #1} +} +% TODO item, as itemize does not work +\newcommand{\titem}[0]{\\-\qquad} +% List of todos +\makeatletter + \newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}} + \newcommand\l@todo[2]{ + \par\noindent \textit{#2}, \parbox{10cm}{#1}\par + } +\makeatother + +% simple way to center an image +\newcommand{\cimage}[2][]{ + \begin{center} + \includegraphics[#1]{#2} + \end{center} +} + +% simple way to center a diagram +\newcommand{\cdiagrambase}[1]{ + \begin{displaymath} + \input{#1} + \end{displaymath} +} +\newcommand{\cdiagram}[1]{ + \cdiagrambase{diagrams/#1} +}