@ -8,9 +8,9 @@ Recall that a cdga $A$ is a commutative differential graded algebra, meaning tha
\item it has an associative and unital multiplication: $\mu: A \tensor A \to A$ and
\item it has an associative and unital multiplication: $\mu: A \tensor A \to A$ and
\item it is commutative: $x y =(-1)^{\deg{x}\cdot\deg{y}} y x$.
\item it is commutative: $x y =(-1)^{\deg{x}\cdot\deg{y}} y x$.
\end{itemize}
\end{itemize}
And all of the above structure is compatible with each other (e.g. the differential is a derivative, the maps are graded, \dots). We have a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules.
And all of the above structure is compatible with each other (e.g. the differential is a derivation of degree $1$, the maps are graded, \dots). We have a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules.
In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. the maps $f: A \to B$ inducing an isomorphism on homology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. the maps $f: A \to B$ inducing an isomorphism on cohomology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
We furthermore have the following categorical properties of cdga's:
We furthermore have the following categorical properties of cdga's:
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.
In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will call a space \Def{simple} if it is connected and its fundamental group is abelian.
In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will call a space \Def{simple} if it is connected and its fundamental group is abelian.\todo{non-standard}
\Definition{rational-space}{
\Definition{rational-space}{
A simple space $X$ is a \emph{rational space} if
A simple space $X$ is a \emph{rational space} if
@ -30,7 +30,7 @@ Note that for a rational space $X$, the ordinary homotopy groups are isomorphic
Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence.
Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence.
The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1)=\Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences).
The theory of rational homotopy is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1)=\Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences).
\section{Classical results from algebraic topology}
\section{Classical results from algebraic topology}
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 remember the topological $n$-simplex defined as convex span.
\Definition{apl}{
\Definition{apl}{
For all $n \in\N$ define the following cdga:
For all $n \in\N$ define the following cdga:
$$(\Apl)_n =\frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i -1, \sum_{i=0}^n d x_i)}$$
$$(\Apl)_n =\frac{\Lambda(x_0, \ldots, x_n, d x_0, \ldots, d x_n)}{(\sum_{i=0}^n x_i -1, \sum_{i=0}^n d x_i)},$$
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 $\sum_{i=0}^n d x_i =0$.
where $\deg{x_i}=0$. 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 $\sum_{i=0}^n d x_i =0$.
}
}
Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n)\to\Apl_n$ 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 $d x_i$). This fact will be used throughout.
Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n)\to\Apl_n$ 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 $d x_i$). This fact will be used throughout.
Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
Consider the free cdga on one generator $\Lambda(t, dt)$, where $\deg{t}=0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
Consider the free cdga on one generator $\Lambda(t, dt)$\todo{same as $\Lambda D(0)$}, where $\deg{t}=0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
$$ d_0, d_1 : \Lambda(t, dt)\to\k$$
$$ d_0, d_1 : \Lambda(t, dt)\to\k$$
$$ d_0(t)=1, \qquad d_1(t)=0, $$
$$ d_0(t)=1, \qquad d_1(t)=0, $$
this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t)=0$ and $d_1(1-t)=1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt)\tensor X \to\k\tensor X \tot{\iso} X$.
this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t)=0$ and $d_1(1-t)=1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt)\tensor X \to\k\tensor X \tot{\iso} X$.
@ -12,7 +12,7 @@ this extends linearly and multiplicatively. Note that it follows that we have $d
such that $d_0 h = g$ and $d_1 h = f$.
such that $d_0 h = g$ and $d_1 h = f$.
}
}
In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt)\tensor X$ is a path object for $X$. We can easily see that it is a very good path object. First note that $\Lambda(t, dt)\tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y)\in X \oplus X$ take $t \tensor x +(1-t)\tensor y$). Secondly we note that $\Lambda(t, dt)=\Lambda(D(0))$ and hence $\k\to\Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to\Lambda(t, dt)\tensor X$ is a (necessarily trivial) cofibration.
In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt)\tensor X$ is a path object for $X$. We can easily see that it is a very good path object\todo{Refereer}. First note that $\Lambda(t, dt)\tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y)\in X \oplus X$ take $t \tensor x +(1-t)\tensor y$). Secondly we note that $\Lambda(t, dt)=\Lambda(D(0))$ and hence $\k\to\Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to\Lambda(t, dt)\tensor X$ is a (necessarily trivial) cofibration.
Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
@ -93,7 +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{bousfield}. 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:\todo{This is later defined as $A \tensor\Lambda(C(X))$, which is precisely the same}
$$ A \tot{i} E \tot{p} X, $$
$$ A \tot{i} E \tot{p} X, $$
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 \Def{Kan extensions}.
There is a general way to construct contravariant 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:
In this section we will prove the existence of rationalizations $X \to X_\Q$. We will do this in a cellular way. The $n$-spheres play an important role here, so their rationalizations will be discussed first. In this section $1$-connectedness of spaces will play an important role.
In this section we will prove the existence of rationalizations $X \to X_\Q$. We will do this in a cellular way. The $n$-spheres play an important role here, so their rationalizations will be discussed first. In this section $1$-connectedness of spaces will play an important role.
\section{Rationalization of \texorpdfstring{$S^n$}{Sn}}
\section{Rationalization of \texorpdfstring{$S^n$}{Sn}}
In this section we fix $n>0$. We will construct $S^n_\Q$ in stages $S^n(1), S^n(2), \ldots$, where at each stage we wedge a sphere and then glue a $n+1$-cell to ``invert'' some element in the $n$th homotopy group.
In this section we fix $n>0$. We will construct $S^n_\Q$ in stages $S^n(1), S^n(2), \ldots$, where at each stage we wedge a sphere and then glue an$n+1$-cell to ``invert'' some element in the $n$th homotopy group.
\todo{plaatje}
\todo{plaatje en tekst leesbaarder maken}
We start the construction with $S^n(1)= S^n$. Assume we constructed $S^n(r)=\bigvee_{i=1}^{r} S^{n}\cup_{h}\coprod_{i=1}^{r-1} D^{n+1}$, where $h$ is a specific attaching map. Assume furthermore the following two properties. Firstly, the inclusion $i_r : S^n \to S^n(r)$ of the terminal sphere is a weak equivalence. Secondly, the inclusion $i_1 : S^n \to S^n(r)$ of the initial sphere induces the multiplication $\pi_n(S^n)\tot{\times r!}\pi_n(S^n(r))$ under the identification of $\pi_n(S^n)=\pi_n(S^n(r))=\Z$.
We start the construction with $S^n(1)= S^n$. Assume we constructed $S^n(r)=\bigvee_{i=1}^{r} S^{n}\cup_{h}\coprod_{i=1}^{r-1} D^{n+1}$, where $h$ is a specific attaching map. Assume furthermore the following two properties. Firstly, the inclusion $i_r : S^n \to S^n(r)$ of the terminal sphere is a weak equivalence. Secondly, the inclusion $i_1 : S^n \to S^n(r)$ of the initial sphere induces the multiplication $\pi_n(S^n)\tot{\times r!}\pi_n(S^n(r))$ under the identification of $\pi_n(S^n)=\pi_n(S^n(r))=\Z$.
@ -25,7 +25,7 @@ Now to finish the construction we define the \Def{rational sphere} as $S^n_\Q =
Moreover we note that the generator $1\in\pi_n(S^n)$ is sent to $1\in\pi_n(S^n_\Q)$ via the inclusion $S^n \to S^n_\Q$ of the initial sphere. However the other homotopy groups are harder to calculate as we have generally no idea how the induced maps will look like. But in the case of $n=1$, the other trivial homotopy groups of $S^1$ are trivial.
Moreover we note that the generator $1\in\pi_n(S^n)$ is sent to $1\in\pi_n(S^n_\Q)$ via the inclusion $S^n \to S^n_\Q$ of the initial sphere. However the other homotopy groups are harder to calculate as we have generally no idea what the induced maps are. But in the case of $n=1$, the other homotopy groups of $S^1$ are trivial.
\Corollary{rationalization-S1}{
\Corollary{rationalization-S1}{
The inclusion $S^1\to S^1_\Q$ is a rationalization.
The inclusion $S^1\to S^1_\Q$ is a rationalization.
@ -81,7 +81,7 @@ Having rational cells we wish to replace the cells in a CW complex $X$ by the ra
Any simply connected CW complex admits a rationalization.
Any simply connected CW complex admits a rationalization.
}
}
\Proof{
\Proof{
Let $X$ be a CW complex. We will define $X_\Q$ with induction on the dimension of the cells. Since $X$ is simply connected we can start with $X^0_\Q= X^1_\Q=\ast$. Now assume that the rationalization $X^k \tot{\phi^k} X^k_\Q$ is already defined. Let $A$ be the set of $k+1$-cells and $f_\alpha : S^k \to X^{k+1}$ be the attaching maps. Then by \LemmaRef{SnQ-extension} these extend to $g_\alpha=(\phi^k \circ f_\alpha)' : S^k_\Q\to X^k_\Q$. This defines $X^{k+1}_\Q$ as the pullback in the following diagram.
Let $X$ be a CW complex. We will define $X_\Q$ with induction on the dimension of the cells. Since $X$ is simply connected we can start with $X^0_\Q= X^1_\Q=\ast$. Now assume that the rationalization $X^k \tot{\phi^k} X^k_\Q$ is already defined. Let $A$ be the set of $k+1$-cells and $f_\alpha : S^k \to X^{k+1}$ be the attaching maps. Then by \LemmaRef{SnQ-extension} these extend to $g_\alpha=(\phi^k \circ f_\alpha)' : S^k_\Q\to X^k_\Q$. This defines $X^{k+1}_\Q$ as the pushout in the following diagram.
\begin{displaymath}
\begin{displaymath}
\xymatrix{
\xymatrix{
@ -137,7 +137,7 @@ There are others ways to obtain a rationalization. One of them relies on the obs
is a rationalization
is a rationalization
}
}
Any simply connected space can be decomposed into a Postnikov tower $X \to\ldots\fib P_2(X)\fib P_1(X)\fib P_0(X)$\cite[Chapter 22.4]{may}. Furthermore if $X$ is a simply connected CW complex, $P_{n}(X)$ can be constructed from $P_{n-1}(X)$ as the pushout in
Any simply connected space can be decomposed into a Postnikov tower $X \to\ldots\fib P_2(X)\fib P_1(X)\fib P_0(X)$\cite[Chapter 22.4]{may}. Furthermore if $X$ is a simply connected CW complex, $P_{n}(X)$ can be constructed from $P_{n-1}(X)$ as the pullback in
Joshua Moerman
10 years ago