From 1b8ae2ff33110297758f5aa19336da192f609a2c Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Tue, 7 Oct 2014 17:16:33 +0200 Subject: [PATCH] Fixes some titles and cites and typos --- thesis/images/Serre_Whitehead_LES.png | Bin 0 -> 71928 bytes thesis/images/Topr_No_Coequalizer.png | Bin 0 -> 9997 bytes thesis/images/Topr_No_Equalizer.png | Bin 0 -> 18104 bytes thesis/notes/A_K_Quillen_Pair.tex | 3 +- thesis/notes/Basics.tex | 12 +++--- thesis/notes/CDGA_Basic_Examples.tex | 4 +- thesis/notes/CDGA_Of_Polynomials.tex | 5 ++- thesis/notes/Free_CDGA.tex | 3 +- thesis/notes/Minimal_Models.tex | 1 + thesis/notes/Model_Categories.tex | 2 +- thesis/notes/Model_Of_CDGA.tex | 5 ++- thesis/notes/Polynomial_Forms.tex | 8 +++- thesis/notes/Serre.tex | 7 ++-- thesis/preamble.tex | 13 ++++++- thesis/references.bib | 54 +++++++++++++++++++++++++- thesis/thesis.tex | 9 +++-- 16 files changed, 100 insertions(+), 26 deletions(-) create mode 100644 thesis/images/Serre_Whitehead_LES.png create mode 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z1BM>6Zvzoyl;y3_SPxSD+7UnWiL=H(Z8-{hBgvuv#t>ta<~`T-uVmcx zJ}8D5qcF49E%B{y$3%jrlAbwkt!``xsxit^Wv^9#%{oVBgKCW4+Riwq_3BE9p&Fw+ ziwr;KDI8?#vIVL!N_@5V86>&PM2Mjpqg3;}nrej@1JG0gh%o>$28b~LF$N&U0NOqP X0f5i-(v2@W00000NkvXXu0mjf+p*zE literal 0 HcmV?d00001 diff --git a/thesis/notes/A_K_Quillen_Pair.tex b/thesis/notes/A_K_Quillen_Pair.tex index db59d24..7a3dd1f 100644 --- a/thesis/notes/A_K_Quillen_Pair.tex +++ b/thesis/notes/A_K_Quillen_Pair.tex @@ -1,5 +1,6 @@ -\subsection{$A$ and $K$ form a Quillen pair} +\section{\texorpdfstring{$A$}{A} and \texorpdfstring{$K$}{K} form a Quillen pair} +\label{sec:a-k-quillen-pair} We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration. diff --git a/thesis/notes/Basics.tex b/thesis/notes/Basics.tex index 2b6b859..a730568 100644 --- a/thesis/notes/Basics.tex +++ b/thesis/notes/Basics.tex @@ -1,6 +1,6 @@ \section{Rational homotopy theory} -\label{sec:rational} +\label{sec:basics} In this section we will state the aim of rational homotopy theory. Moreover we will recall classical theorems from algebraic topology and deduce rational versions of them. @@ -32,7 +32,7 @@ We will later see that any space admits a rationalization. The theory of rationa \subsection{Classical results from algebraic topology} -We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as [May, Dold, ...]. Note that all spaces are assumed to be $1$-connected. +We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as \cite{may, dold}. Note that all spaces are assumed to be $1$-connected. \Theorem{relative-hurewicz}{ (Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map @@ -46,7 +46,7 @@ We will now recall known results from algebraic topology, without proof. One can where $F$ is the fibre of $f$. } -Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffith}). +Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffiths}). \Theorem{whitehead-homology}{ (Whitehead) For any map $f: X \to Y$ we have $$ \pi_i(f) \text{ is an isomorphism } \forall 0 < i < r \iff H_i(f) \text{ is an isomorphism } \forall 0 < i < r. $$ @@ -65,7 +65,7 @@ The following two theorems can be found in textbooks about homological algebra, (Künneth Theorem) For spaces $X$ and $Y$, there is a short exact sequence $$ 0 \to H(X; A) \tensor H(Y; A) \to H(X \times Y; A) \to \Tor(H(X; A), H(Y; A)) \to 0, $$ - where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. + where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Add algebraic version for (co)chain complexes} } \subsection{Immediate results for rational homotopy theory} @@ -75,7 +75,7 @@ The latter two theorems have a direct consequence for rational homotopy theory. \Corollary{rational-corollaries}{ We have the following natural isomorphisms $$ H(X) \tensor \Q \tot{\iso} H(X; \Q), $$ - $$ H^n(X; \Q) \tot{\iso} \Hom(H(X); \Q), $$ + $$ H^n(X; \Q) \tot{\iso} \Hom(H_n(X); \Q), $$ $$ H(X \times Y) \tot{\iso} H(X) \tensor H(Y). $$ } @@ -85,5 +85,5 @@ The long exact sequence for a Serre fibration also has a direct consequence for $$ \cdots \tot{\del} \pi_i(F) \tensor \Q \tot{i_\ast} \pi_i(X) \tensor \Q \tot{f_\ast} \pi_i(Y) \tensor \Q \tot{\del} \cdots, $$ } -In the next sections we will prove the rational Hurewicz and rational Whitehead theorems. These theorems are due to Serre [Serre]. +In the next sections we will prove the rational Hurewicz and rational Whitehead theorems. These theorems are due to Serre \cite{serre}. diff --git a/thesis/notes/CDGA_Basic_Examples.tex b/thesis/notes/CDGA_Basic_Examples.tex index d35c2ad..41adac5 100644 --- a/thesis/notes/CDGA_Basic_Examples.tex +++ b/thesis/notes/CDGA_Basic_Examples.tex @@ -1,5 +1,7 @@ -\section{Cochain models for the $n$-disk and $n$-sphere} +\section{Cochain models for the \texorpdfstring{$n$}{n}-disk and \texorpdfstring{$n$}{n}-sphere} +\label{sec:cdga-basic-examples} + We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b) \in D(n)^{n+1}$. $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ which differential vanishes (i.e. $da = 0$). In other words: $$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$ diff --git a/thesis/notes/CDGA_Of_Polynomials.tex b/thesis/notes/CDGA_Of_Polynomials.tex index a8d038b..7fab3ee 100644 --- a/thesis/notes/CDGA_Of_Polynomials.tex +++ b/thesis/notes/CDGA_Of_Polynomials.tex @@ -1,15 +1,16 @@ \section{CDGA of Polynomials} +\label{sec:cdga-of-polynomials} \newcommand{\Apl}[0]{{A_{PL}}} We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span. -\begin{definition} +\Definition{apl}{ For all $n \in \N$ define the following cdga: $$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n dx_i)} $$ So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n dx_i = 0$. -\end{definition} +} Note that the inclusion $\Lambda(x_1, \ldots, x_n, dx_1, \ldots, dx_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $dx_i$). This fact will be used throughout. diff --git a/thesis/notes/Free_CDGA.tex b/thesis/notes/Free_CDGA.tex index b9ee7fc..114aace 100644 --- a/thesis/notes/Free_CDGA.tex +++ b/thesis/notes/Free_CDGA.tex @@ -1,5 +1,6 @@ -\subsection{The free cdga} +\section{The free cdga} +\label{sec:free-cdga} Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation. diff --git a/thesis/notes/Minimal_Models.tex b/thesis/notes/Minimal_Models.tex index be13b31..0607db9 100644 --- a/thesis/notes/Minimal_Models.tex +++ b/thesis/notes/Minimal_Models.tex @@ -1,5 +1,6 @@ \section{Minimal models} +\label{sec:minimal-models} In this section we will discuss the so called minimal models. These are cdga's with the property that a quasi isomorphism between them is an actual isomorphism. diff --git a/thesis/notes/Model_Categories.tex b/thesis/notes/Model_Categories.tex index d68c0a9..d3b0418 100644 --- a/thesis/notes/Model_Categories.tex +++ b/thesis/notes/Model_Categories.tex @@ -1,6 +1,6 @@ \section{Model categories} -\label{sec:model_cats} +\label{sec:model_categories} \newcommand{\W}{\mathfrak{W}} \newcommand{\Fib}{\mathfrak{Fib}} diff --git a/thesis/notes/Model_Of_CDGA.tex b/thesis/notes/Model_Of_CDGA.tex index dc4adfb..2e46421 100644 --- a/thesis/notes/Model_Of_CDGA.tex +++ b/thesis/notes/Model_Of_CDGA.tex @@ -1,5 +1,6 @@ -\section{Model structure on $\CDGA_\k$} +\section{Model structure on \texorpdfstring{$\CDGA_\k$}{CDGA}} +\label{sec:model-of-cdga} \TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.} @@ -92,7 +93,7 @@ We can use Quillen's small object argument with these sets. The argument directl [MC5a] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a trivial cofibration and $p$ a fibration. \end{corollary} -The previous factorization can also be described explicitly as seen in \cite{bous}. Let $f: A \to X$ be a map, define $E = A \tensor \bigtensor_{x \in X}T(\deg{x})$. Then $f$ factors as: +The previous factorization can also be described explicitly as seen in \cite{bousfield}. Let $f: A \to X$ be a map, define $E = A \tensor \bigtensor_{x \in X}T(\deg{x})$. Then $f$ factors as: $$ A \tot{i} E \tot{p} X, $$ where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products of) generators $a \tensor b_x$ with $b_x \in T(\deg{x})$ to $f(a) \cdot x \in X$. diff --git a/thesis/notes/Polynomial_Forms.tex b/thesis/notes/Polynomial_Forms.tex index 2bb7684..d0597b2 100644 --- a/thesis/notes/Polynomial_Forms.tex +++ b/thesis/notes/Polynomial_Forms.tex @@ -1,7 +1,8 @@ -\subsection{Polynomial Forms} +\section{Polynomial Forms} +\label{sec:polynomial-forms} -There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via Kan extensions. +There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via \Def{Kan extensions}. Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects: \begin{align*} @@ -24,6 +25,9 @@ In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get: \cimage[scale=0.5]{Apl_Extension} + +\subsection{The cochain complex of polynomial forms} + In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description: \begin{align*} diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex index 4de4d4e..6b38e77 100644 --- a/thesis/notes/Serre.tex +++ b/thesis/notes/Serre.tex @@ -1,7 +1,8 @@ -\section{Serre theorems mod $C$} +\section{Serre theorems mod \texorpdfstring{$C$}{C}} +\label{sec:serre} -In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in [Serre]. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups. +In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in \cite{serre}. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups. \Lemma{whitehead-tower}{ (Whitehead tower) @@ -24,7 +25,7 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational \end{itemize} } -Serre gave weaker axioms for his classes and proves the following lemmas only using these weaker axioms. However the classes we are interested in do satisfy the above (stronger) requirements. One should think of such Serre class as a class of groups we want to \emph{invert}. We will be interested in the first two of the following examples. +Serre gave weaker axioms for his classes and proves some of the following lemmas only using these weaker axioms. However the classes we are interested in do satisfy the above (stronger) requirements. One should think of such Serre class as a class of groups we want to \emph{invert}. We will be interested in the first two of the following examples. \Example{serre-classes}{ We give three Serre classes without proof. diff --git a/thesis/preamble.tex b/thesis/preamble.tex index c04a5c9..98e606c 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -12,6 +12,12 @@ % floating figures \usepackage{float} +% for appendices +\usepackage[toc,page]{appendix} + +% for multiple cites +\usepackage{cite} + \usepackage{tikz} \usetikzlibrary{matrix, arrows, decorations} \tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} @@ -130,11 +136,13 @@ \newtheorem{example}[theorem]{Example} \newcommand{\EnvTemp}[4]{ - \begin{#1}\label{{#2}:{#3}} + \begin{#1}\label{#2:#3} {#4} \end{#1} } +\newcommand{\RefTemp}[3]{{#1}~\ref{#2:#3}} + \newcommand{\Theorem}{\EnvTemp{theorem}{thm}} \newcommand{\Proposition}{\EnvTemp{proposition}{prop}} \newcommand{\Lemma}{\EnvTemp{lemma}{lem}} @@ -147,7 +155,8 @@ \newcommand{\Notation}{\EnvTemp{notation}{not}} \newcommand{\Example}{\EnvTemp{example}{eg}} -\newcommand{\LemmaRef}[1]{Lemma~\ref{lem:{#1}}} +\newcommand{\TheoremRef}{\RefTemp{Theorem}{thm}} +\newcommand{\LemmaRef}{\RefTemp{Lemma}{lem}} % headings for a table \newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}} diff --git a/thesis/references.bib b/thesis/references.bib index 3e66d97..4c62975 100644 --- a/thesis/references.bib +++ b/thesis/references.bib @@ -1,4 +1,4 @@ -@book{bous, +@book{bousfield, title={On PL de Rham theory and rational homotopy type}, author={Bousfield, Aldridge Knight and Gugenheim, Victor KAM}, volume={179}, @@ -6,6 +6,13 @@ publisher={American Mathematical Soc.} } +@book{dold, + title={Lectures on Algebraic Topology}, + author={Dold, A}, + year={1972}, + publisher={Springer} +} + @article{dwyer, title={Homotopy theories and model categories}, author={Dwyer, William G and Spalinski, Jan}, @@ -22,7 +29,7 @@ publisher={Springer} } -@article{goerss, +@book{goerss, title={Simplicial Homotopy Theory}, author={Goerss, PG and Jardine, JF}, publisher={Birkh{\"a}user}, @@ -46,6 +53,16 @@ publisher={Providence, RI; American Mathematical Society; 1999} } +@inproceedings{kreck, + title={A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres}, + author={Klaus, Stephan and Kreck, Matthias}, + booktitle={Mathematical Proceedings of the Cambridge Philosophical Society}, + volume={136}, + pages={617--623}, + year={2004}, + organization={Cambridge Univ Press} +} + @book{loday, title={Algebraic operads}, author={Loday, Jean-Louis and Vallette, Bruno}, @@ -53,3 +70,36 @@ year={2012}, publisher={Springer} } + +@book{may, + title={A concise course in algebraic topology}, + author={May, J Peter}, + year={1999}, + publisher={University of Chicago Press} +} + +@article{quillen, + title={Rational homotopy theory}, + author={Quillen, Daniel}, + journal={Annals of Mathematics}, + pages={205--295}, + year={1969}, + publisher={JSTOR} +} + +@article{serre, + title={Groupes d'homotopie et classes de groupes abeliens}, + author={Serre, Jean-Pierre}, + journal={Annals of Mathematics}, + pages={258--294}, + year={1953}, + publisher={JSTOR} +} + +@book{spanier, + title={Algebraic topology}, + author={Spanier, Edwin H}, + volume={55}, + year={1994}, + publisher={Springer} +} diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 3080fef..aa4b5fa 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -26,17 +26,20 @@ Some general notation: \todo{leave this out, or define somewhere else?} \newcommand{\myinput}[1]{\include{#1}} \myinput{notes/Basics} -\myinput{notes/Algebra} +\myinput{notes/Serre} \myinput{notes/Free_CDGA} \myinput{notes/CDGA_Basic_Examples} -\myinput{notes/Serre} -\myinput{notes/Model_Categories} \myinput{notes/Model_Of_CDGA} \myinput{notes/CDGA_Of_Polynomials} \myinput{notes/Polynomial_Forms} \myinput{notes/A_K_Quillen_Pair} \myinput{notes/Minimal_Models} +\begin{appendices} + \myinput{notes/Algebra} + \myinput{notes/Model_Categories} +\end{appendices} + % \listoftodos \nocite{*}