Recall the following facts about cdga's over a ring $\k$:
\begin{itemize}
\item A map $f: A \to B$ in $\CDGA_\k$ is a \emph{quasi isomorphism} if it induces isomorphisms in cohomology.
\item The finite coproduct in $\CDGA_\k$ is the (graded) tensor products.
\item The finite product in $\CDGA_\k$ is the cartesian product (with pointwise operations).
\item The equalizer (resp. coequalizer) of $f$ and $g$ is given by the kernel (resp. cokernel) of $f - g$. Together with the (co)products this defines pullbacks and pushouts.
\item$\k$ and $0$ are the initial and final object.
\end{itemize}
In this chapter the ring $\k$ is assumed to be a field of characteristic zero.
\subsection{Cochain models for the $n$-disk and $n$-sphere}
@ -49,7 +49,7 @@ Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit}
Note that for a free cdga $\Lambda C$ there is a natural augmentation and the chain complexes of indecomposables $Q \Lambda C$ is naturally isomorphic to $C$. Consider the augmented cdga $V(n)= D(n)\oplus\k$, with trivial multiplication and where the term $\k$ is used for the unit and augmentation. There is a weak equivalence $A(n)\to V(n)$ (recall \DefinitionRef{minimal-model-sphere}).
\Lemma{cdga-dual-homotopy-groups}{
Let $A$ be an augemented cdga, then
Let $A$ be an augmented cdga, then
$$[A, V(n)]\tot{\iso}\Hom_\k(\pi^n(A), \k). $$
}
@ -68,7 +68,7 @@ We will denote the dual of a vector space as $V^\ast = \Hom_\k(V, \k)$.
@ -70,7 +70,7 @@ A differential graded module $(M, d)$ with $M_i = 0$ for all $i < 0$ is a \emph{
Finally we come to the definition of a differential graded algebra. This will be a graded algebra with a differential. Of course we want this to be compatible with the algebra structure, or stated differently: we want $\mu$ and $\eta$ to be chain maps.
\begin{definition}
A \emph{differential graded algebra (DGA)} is a graded algebra $A$ together with an differential $d$ such that in addition the \emph{Leibniz rule} holds:
A \emph{differential graded algebra (dga)} is a graded algebra $A$ together with an differential $d$ such that in addition the \emph{Leibniz rule} holds:
$$ d(x y)= d(x) y +(-1)^{\deg{x}} x d(y)\quad\text{ for all } x, y \in A. $$
@ -32,28 +32,28 @@ 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 \cite{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}. We do not assume $1$-connectedness here.
\Theorem{relative-hurewicz}{
(Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map
$$ h_i : \pi_i(X, A)\to H_i(X, A). $$
If furthermore $(X,A)$ is $n$-connected, then the map $h_i$ is an isomorphism for all $i \leq n +1$
If furthermore $(X,A)$ is $n$-connected ($n > 0$), then the map $h_i$ is an isomorphism for all $i \leq n +1$.
}
\Theorem{serre-les}{
(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence:
(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence (note that $X$ and $Y$ need not be $1$-connected):
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
$$\pi_i(f)\text{ is an isomorphism }\forall0 < i < r \iff H_i(f)\text{ is an isomorphism }\forall0 < i < r. $$
(Whitehead) For any map $f: X \to Y$ between $1$-connected spaces, $\pi_i(f)$ is an isomorphism $\forall0 < i < r$ if and only if $H_i(f)$ is an isomorphism $\forall0 < i < r$.
In particular we see that $f$ is a weak equivalence if and only if it induces an isomorphism on homology.
}
The following two theorems can be found in textbooks about homological algebra, such as [\cite{weibel}].
The following two theorems can be found in textbooks about homological algebra such as \cite{weibel, rotman}. Note that when the degrees are left out, $H(X; A)$ denotes the graded homology module with coefficients in $A$.
\Theorem{universal-coefficient}{
(Universal Coefficient Theorem)
For any space $X$ and abelian group $A$, there are natural short exact sequences
@ -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
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}
where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Geef algebraische versie voor ketencomplexen}
}
\subsection{Immediate results for rational homotopy theory}
@ -73,16 +73,18 @@ The following two theorems can be found in textbooks about homological algebra,
The latter two theorems have a direct consequence for rational homotopy theory. By taking $A =\Q$ we see that the torsion groups vanish. We have the immediate corollary.
\Corollary{rational-corollaries}{
We have the following natural isomorphisms
We have the following natural isomorphisms in homology
@ -4,24 +4,24 @@ We will first define some basic cochain complexes which model the $n$-disk and $
$$ D(n)= ... \to0\to\k\to\k\to0\to ... $$
$$ S(n)= ... \to0\to\k\to0\to0\to ... $$
Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0\to D(n)$ is a quasi isomorphism. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n))=\k\cdot[a]$. There is an injective function $i_n : S(n+1)\to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes:
Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0\to D(n)$ induces an isomorphism in cohomology. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n))=\k\cdot[a]$. There is an injective function $i_n : S(n+1)\to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes:
\begin{claim}
The set $I =\{i_n : S(n+1)\to D(n)\I n \in\N\}$ generates all cofibrations and the set $J =\{j_n : 0\to D(n)\I n \in\N\}$ generates all trivial cofibrations.
\end{claim}
The proof is omitted but can be found in different sources \todo{Cite sources}. In the next section we will prove a similar result for cdga's, so the reader can also refer to that proof.
As we do not directly need this claim, we omit the proof. However, in the next section we will prove a similar result for cdga's in detail.
$S(n)$ plays a another special role: maps from $S(n)$ to some cochain complex $X$ correspond directly to elements in the kernel of $\restr{d}{X^n}$. Any such map is null-homotopic precisely when the corresponding elements in the kernel is a coboundary. So there is a natural isomorphism: $\Hom(S(n), X)/~\iso H^n(X)$. So the cohomology groups can be considered as honest homotopy groups.
$S(n)$ plays a another special role: maps from $S(n)$ to some cochain complex $X$ correspond directly to elements in the kernel of $\restr{d}{X^n}$. Any such map is null-homotopic precisely when the corresponding elements in the kernel is a coboundary. So there is a natural isomorphism: $\Hom(S(n), X)/\simeq\iso H^n(X)$.
By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^n$ and $c b^n$ when $n$ is even, and $c^n b$ and $c^n$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule:
By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^n$ and $c b^n$ when $n$ is even, and it consists of linear combinations of $c^n b$ and $c^n$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule:
$$ d(b^n)= n \cdot c b^{n-1}$$
$$ d(c b^n)=0$$
$$ d(c^n b)= c^{n+1}$$
$$ d(c^n)=0$$
Those cocycles are in fact coboundaries (remember that $\k$ is a field of characteristic $0$):
Those cocycles are in fact coboundaries (using that $\k$ is a field of characteristic $0$):
A model category induces a homotopy category $\Ho(\cat{C})$, in which weak equivalences are isomorphisms and homotopic maps are equal. This category only depens on the category $\cat{C}$ and the class of weak equivalences.
A model category induces a homotopy category $\Ho(\cat{C})$, in which weak equivalences are isomorphisms and homotopic maps are equal. This category only depends on the category $\cat{C}$ and the class of weak equivalences.
.\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.}
In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
In this section we will define a model structure on cdga's over a field $\k$ of characteristic zero, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
\begin{proposition}
There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
@ -20,7 +18,7 @@ Note that with these classes, every cdga is a fibrant object.
[MC1] The category has all finite limits and colimits.
\end{lemma}
\begin{proof}
As discussed earlier \todo{really discuss this somewhere}products are given by direct sums and equalizers are kernels. Furthermore the coproducts are tensor products and coequalizers are quotients.
As discussed earlier products are given by direct sums and equalizers are kernels. Furthermore the coproducts are tensor products and coequalizers are quotients.
\end{proof}
\begin{lemma}
@ -34,7 +32,7 @@ Note that with these classes, every cdga is a fibrant object.
[MC3] All three classes are closed under retracts
\end{lemma}
\begin{proof}
\todo{Make some diagrams and write it out}
\todo{Diagrammen en uitschrijven}
\end{proof}
Next we will prove the factorization property [MC5]. We will do this by Quillen's small object argument. When proved, we get an easy way to prove the missing lifting property of [MC4]. For the Quillen's small object argument we use classes of generating cofibrations.
@ -53,7 +51,7 @@ Next we will prove the factorization property [MC5]. We will do this by Quillen'
The maps $i_n$ are trivial cofibrations and the maps $j_n$ are cofibrations.
\end{lemma}
\begin{proof}
Since $H(T(n))=\k$\todo{Note that this only hold when characteristic = 0} we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations. \todo{give a bit more detail}
Since $H(T(n))=\k$(as stated earlier this uses $\Char{\k}=0$) we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations and the freeness of $T(n)$ and $S(n)$. \todo{Iets meer detail?}
\end{proof}
\begin{lemma}
@ -69,14 +67,16 @@ As a consequence of the above two lemmas, the class generated by $I$ is containe
If $p: X \to Y$ has the RLP w.r.t. $I$ then $p$ is a fibration.
\end{lemma}
\begin{proof}
Easy\todo{Define a lift}.
Let $y \in Y^n$ an element of degree $n$, then we have the following commuting diagram:
\cdiagram{CDGA_Model_I_Fib}
where $g$ sends the generator $b$ to $y$ and $c$ to $dy$. By assumption there exists a lift $h$. Now $h(b)\in X^n$ is a preimage for $y$, proving that $p$ is surjective.
\end{proof}
\begin{lemma}
If $p: X \to Y$ has the RLP w.r.t. $J$ then $p$ is a trivial fibration.
\end{lemma}
\begin{proof}
As $p$ has the RLP w.r.t. $J$, it also has the RLP w.r.t. $I$. From the previous lemma it follows that $p$ is a fibration. To show that $p$ is a weak equivalence ... \todo{write out}
\todo{Even bewijzen}
\end{proof}
We can use Quillen's small object argument with these sets. The argument directly proves the following lemma. Together with the above lemmas this translates to the required factorization.
@ -85,7 +85,7 @@ We can use Quillen's small object argument with these sets. The argument directl
A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is in the class generated by $I$ and $p$ has the RLP w.r.t. $I$.
\end{lemma}
\begin{proof}
Quillen's small object argument. \todo{small = finitely generated?}
Quillen's small object argument. \todo{Definieer wat ``small'' betkent en geef een referentie}
\end{proof}
\begin{corollary}
@ -106,3 +106,5 @@ where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products o
\begin{corollary}
[MC5b] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a cofibration and $p$ a trivial fibration.
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)
For a space X, we have a decomposition in fibrations:
\item$K(\pi_n(X), n-1)\cof X(n+1)\fib X(n)$ is a fiber sequence,
\item There is a space $X'_n$ weakly equivalent to $X(n)$ such that $X(n+1)\ cof X'_n \fib K(\pi_n(X), n)$ is a fiber sequence, and
\item$X(n)$ is $(n-1)$-connected and $\pi_i(X(n))\iso\pi_i(X)$ for all $i \geq n$.
\end{itemize}
}
\Definition{serre-class}{
A class $\C\subset\Ab$ is a \Def{Serre class} if
\begin{itemize}
@ -24,7 +12,7 @@ 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 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.
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{ignore}. We will be interested in the first two of the following examples.
\Example{serre-classes}{
We give three Serre classes without proof.
@ -36,15 +24,10 @@ Serre gave weaker axioms for his classes and proves some of the following lemmas
}
\Definition{serre-class-maps}{
Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is
\begin{itemize}
\item\Def{$\C$-injection} if $\ker(f)\in\C$,
\item\Def{$\C$-surjection} if $\coker(f)\in\C$, and
\item\Def{$\C$-iso} if it is both a $\C$-injection and $\C$-surjection.
\end{itemize}
Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a $\C$-isomorphism if both the kernel and cokernel lie in $\C$.
}
Note that the map $0\to C$ is a $\C$-iso for any $C \in\C$. \todo{Add some stuff about tensors}\todo{Five lemma mod $\C$}
Note that the map $0\to C$ is a $\C$-isomorphism for any $C \in\C$. \todo{Er missen nog wat eigenschappen voor tensors}
\Lemma{serre-class-rational-iso}{
Let $\C$ be the Serre class of all torsion groups. Then
@ -76,20 +59,32 @@ In the following arguments we will consider fibrations and need to compute homol
The morphism in the middle is a $\C$-iso by induction. We will prove that the left morphism is a $\C$-iso which implies by the five lemma that the right morphism is one as well.
\todo{finish proof}
\todo{Bewijs afmaken}
}
\Lemma{homology-em-space}{
Let $\C$ be a Serre class and $C \in\C$. Then for all $n > 0$ and all $i$ we have $\RH_i(K(C, n))\in\C$.
Let $\C$ be a Serre class and $C \in\C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(C, n))\in\C$.
}
\Proof{
We prove this by induction on $n$. The base case $n =1$ follows from group homology.
For the induction we can use the loop space and \LemmaRef{kreck}.
\todo{finish proof}
\todo{Bewijs afmaken}
}
For the main theorem we need the following construction. \todo{Geef de constructie}
@ -19,21 +19,21 @@ We assume the reader is familiar with category theory, basics from algebraic top
Some notation:
\begin{itemize}
\item$\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section).
\item$\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section). Modules, tensor products, \dots are understood as $\k$-modules, tensor products over $\k$, \dots. If ambiguitity can occur notation will be explicit.
\item$\cat{C}$ will denote an arbitrary category.
\item$\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category $\cat{C}$.
\item$\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. We may leave out the subscript $\cat{C}$.
\item$\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ is occasionally left out if the category is clear from the context.
\end{itemize}
Some categories:
\begin{itemize}
\item$\Top$: category of topological spaces and continuos maps.
\item$\Top$: category of topological spaces and continuous maps.
\item$\Ab$: category of abelian groups and group homomorphisms.
\item$\DELTA$: category of simplices (i.e. finite, non-empty ordinals) and order preserving maps.
\item$\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$).
\item$\Ch{\k}, \CoCh{\k}$: category of non-negatively graded chain (resp. cochain) complexes and chain maps.
\item$\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (these are cochain complexes with a multiplication) and graded algebra maps. As a shorthand we will refer to such an object as \emph{dga}.
\item$\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (cdga's).
\item$\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
Joshua Moerman
10 years ago