\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.
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.
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.
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}
\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}{
\Theorem{relative-hurewicz}{
(Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map
(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 $f$.
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}{
\Theorem{whitehead-homology}{
(Whitehead) For any map $f: X \to Y$ we have
(Whitehead) For any map $f: X \to Y$ we have
$$\pi_i(f)\text{ is an isomorphism }\forall0 < i < r \iff H_i(f)\text{ is an isomorphism }\forall0 < i < r. $$
$$\pi_i(f)\text{ is an isomorphism }\forall0 < i < r \iff H_i(f)\text{ is an isomorphism }\forall0 < i < r. $$
@ -65,7 +65,7 @@ The following two theorems can be found in textbooks about homological algebra,
(Künneth Theorem)
(Künneth Theorem)
For spaces $X$ and $Y$, there is a short exact sequence
For spaces $X$ and $Y$, there is a short exact sequence
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}
\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}{
\Corollary{rational-corollaries}{
We have the following natural isomorphisms
We have the following natural isomorphisms
$$ H(X)\tensor\Q\tot{\iso} H(X; \Q), $$
$$ H(X)\tensor\Q\tot{\iso} H(X; \Q), $$
$$ H^n(X; \Q)\tot{\iso}\Hom(H(X); \Q), $$
$$ H^n(X; \Q)\tot{\iso}\Hom(H_n(X); \Q), $$
$$ H(X \times Y)\tot{\iso} H(X)\tensor H(Y). $$
$$ H(X \times Y)\tot{\iso} H(X)\tensor H(Y). $$
}
}
@ -85,5 +85,5 @@ The long exact sequence for a Serre fibration also has a direct consequence for
\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:
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:
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.
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.
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 $\sum_{i=0}^n dx_i =0$.
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.
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.
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.
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.
\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.}
\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.
[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}
\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:
$$ A \tot{i} E \tot{p} X, $$
$$ A \tot{i} E \tot{p} X, $$
where $i$ is the obvious inclusion $i(a)= a \tensor1$ and $p$ maps (products of) generators $a \tensor b_x$ with $b_x \in T(\deg{x})$ to $f(a)\cdot x \in X$.
where $i$ is the obvious inclusion $i(a)= a \tensor1$ and $p$ maps (products of) generators $a \tensor b_x$ with $b_x \in T(\deg{x})$ to $f(a)\cdot x \in X$.
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:
Given a category $\cat{C}$ and a functor $F: \DELTA\to\cat{C}$, then define the following on objects:
\begin{align*}
\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}
\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:
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:
\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}{
\Lemma{whitehead-tower}{
(Whitehead tower)
(Whitehead tower)
@ -24,7 +25,7 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational
\end{itemize}
\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.
Joshua Moerman
10 years ago
