@ -22,7 +22,7 @@ In this chapter we will calculate the rational homotopy groups of the spheres us
\end{cases}$$
where $x$ is a generator of degree $n$. Define $M_{S^n}=\Lambda(e)$ with $d(e)=0$ and $e$ of degree $n$. Notice that since $n$ is odd, we get $e^2=0$. By taking a representative for $x$, we can give a map $M_{S^n}\to A(S^n)$, which is a weak equivalence.
Clearly $M_{S^n}$ is minimal, and hence it is a minimal model for $S^n$. By \CorollaryRef{minimal-cdga-homotopy-groups} and the main equivalence we have
Clearly $M_{S^n}$ is minimal, and hence it is a minimal model for $S^n$. By \TheoremRef{main-theorem} we have
@ -35,10 +35,10 @@ In this chapter we will calculate the rational homotopy groups of the spheres us
\end{cases}$$
}
\Proof{
Again since we know the cohomology of the sphere, we can construct its minimal model. Define $M_{S^n}=\Lambda(e, f)$ with $d(e)=0, d(f)= e^2$ and $\deg{e}= n, \deg{f}=2n-1$. Let $x \in H^n(S^n; \Q)$ be a generator and notice that $x^2=0$. This means that for a representative $x' \in A(S^n)$ of $x$ there exists an element $y \in A(S^n)$ such that $dy = x'^2$. Mapping $e$and $f$ to $x'$ and$y$ respectively defines a quasi isomorphism $M_{S^n}\to A(S^n)$.
Again since we know the cohomology of the sphere, we can construct its minimal model. Define $M_{S^n}=\Lambda(e, f)$ with $d(e)=0, d(f)= e^2$ and $\deg{e}= n, \deg{f}=2n-1$. Let $[x]\in H^n(S^n; \Q)$ be a generator and $x \in A(S^n)$ its representative, then notice that $[x]^2=0$. This means that there exists an element $y \in A(S^n)$ such that $dy = x^2$. Mapping $e$to $x$ and $f$ to$y$ defines a quasi isomorphism $M_{S^n}\to A(S^n)$.
Again we can use \CorollaryRef{minimal-cdga-homotopy-groups} to directly conclude:
The generators $e$ and $f$ in the last proof are related by the so called \Def{Whitehead product}. The whitehead product is a bilinear map $\pi_p(X)\times\pi_q(X)\to\pi_{p+q-1}(X)$ satisfying a graded commutativity relation and a graded Jacobi relation, see \cite{felix}. If we define a \Def{Whitehead algebra} to be a graded vector space with such a map satisfying these relations, we can summarize the above two propositions as follows \cite{berglund}.
@ -48,7 +48,7 @@ The generators $e$ and $f$ in the last proof are related by the so called \Def{W
$$\pi_\ast(S^n)\tensor\Q=\text{the free whitehead algebra on 1 generator}. $$
}
Together with the fact that all groups $\pi_i(S^n)$ are finitely generated (this was proven by Serre \cite{serre}) we can conclude that $\pi_i(S^n)$ is a finite group unless $i=n$ or $i=2n-1$ when $n$ is even. The fact that $\pi_i(S^n)$ are finitely generated can be proven by the Serre-Hurewicz theorems (\TheoremRef{serre-hurewicz}) when taking the Serre class of finitely generated abelian groups (but this requires a weaker notion of a Serre class, and stronger theorems, than the one given in this thesis).
Together with the fact that all groups $\pi_i(S^n)$ are finitely generated (this was proven by Serre in \cite{serre}) we can conclude that $\pi_i(S^n)$ is a finite group unless $i=n$ or $i=2n-1$ when $n$ is even. The fact that $\pi_i(S^n)$ are finitely generated can be proven by the Serre-Hurewicz theorems (\TheoremRef{serre-hurewicz}) when taking the Serre class of finitely generated abelian groups (but this requires a weaker notion of a Serre class, and stronger theorems, than the one given in this thesis).
@ -10,7 +10,7 @@ Recall that a cdga $A$ is a commutative differential graded algebra, meaning tha
\end{itemize}
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 cohomology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. 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:
In this section we aim to prove that the homotopy theory of rational spaces is the same as the homotopy theory of cdga's over $\Q$. Before we prove the equivalence, we will show that $A$ and $K$ form a Quillen pair. This already provides an adjunction between the homotopy categories. Besides the equivalence of the homotopy categories we will also investigate homotopy groups on a cdga directly. The homotopy groups of a space will be dual to the homotopy groups of the associated cdga.
In this section we aim to prove that the homotopy theory of rational spaces is the same as the homotopy theory of cdga's over $\Q$. Before we prove the equivalence, we will show that $A$ and $K$ form a Quillen pair. This already provides an adjunction between the homotopy categories. Besides the equivalence of the homotopy categories we will also prove that the homotopy groups of a space will be dual to the homotopy groups of the associated cdga.
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.
@ -116,7 +116,7 @@ Before we prove the actual equivalence, we will discuss a theorem of Eilenberg a
Now the Tor group appearing in the theorem can be computed via a \emph{bar construction}. The explicit construction for cdga's can be found in \cite{bousfield}, but also in \cite{olsson} where it is related to the homotopy colimit of cdga's. We will not discuss the details of the bar construction. However it is important to know that the Tor group only depends on the cohomology of the dga's in use (see \cite[Corollary 7.7]{mccleary}), in other words: quasi isomorphic dga's (in a compatible way) will have isomorphic Tor groups. Since $C^\ast(-;\k)$ is isomorphic to $A(-)$, the above theorem also holds for our functor $A$. We can restate the theorem as follows.
Now the Tor group appearing in the theorem can be computed via a \emph{bar construction}. The explicit construction for cdga's can be found in \cite{bousfield}, but also in \cite{olsson} where it is related to the homotopy pushout of cdga's. We will not discuss the details of the bar construction. However it is important to know that the Tor group only depends on the cohomology of the dga's in use (see \cite[Corollary 7.7]{mccleary}), in other words: quasi isomorphic dga's (in a compatible way) will have isomorphic Tor groups. Since $C^\ast(-;\k)$ is isomorphic to $A(-)$, the above theorem also holds for our functor $A$. We can restate the theorem as follows.
\Corollary{A-preserves-htpy-pullbacks}{
Given the following pullback diagram of spaces
@ -146,7 +146,7 @@ and the second map is obtained by the map $A \to A(K(A))$ and using the bijectio
(Base case) Let $A =(\Lambda(v), 0)$ be a minimal model with one generator of degree $\deg{v}= n \geq1$. Then $A \we A(K(A))$.
}
\Proof{
By \CorollaryRef{minimal-cdga-homotopy-groups} we know that $K(A)$ is an Eilenberg-MacLane space of type $K(\Q^\ast, n)$. The cohomology of an Eilenberg-MacLane space with coefficients in $\Q$ is known:
By \CorollaryRef{minimal-cdga-homotopy-groups} we know that $K(A)$ is an Eilenberg-MacLane space of type $K(\Q^\ast, n)$. The cohomology of an Eilenberg-MacLane space with coefficients in $\Q$ is known (note that this is specific for $\Q$):
$$ H^\ast(K(\Q^\ast, n); \Q)=\Q[x], $$
that is, the free commutative graded algebra with one generator $x$. This can be calculated, for example, with spectral sequences \cite{griffiths}.
@ -157,8 +157,8 @@ and the second map is obtained by the map $A \to A(K(A))$ and using the bijectio
(Induction step) Let $A$ be a cofibrant, connected algebra. Let $B$ be the pushout in the following square, where $m \geq1$:
\begin{displaymath}
\xymatrix{
S(m+1) \arcof[d]\ar[r]\xypo& A \arcof[d]\\
T(m) \ar[r]& B
\LambdaS(m+1) \arcof[d]\ar[r]\xypo& A \arcof[d]\\
\Lambda D(m) \ar[r]& B
}
\end{displaymath}
Then if $A \to A(K(A))$ is a weak equivalence, so is $B \to A(K(B))$
@ -169,13 +169,13 @@ and the second map is obtained by the map $A \to A(K(A))$ and using the bijectio
Applying $A$ again gives the following cube of cdga's:
\begin{displaymath}
\xymatrix @=9pt{
S(m+1) \arcof[dd]\ar[rr]\arwe[rd]\xypo&& A \arcof'[d][dd] \arwe[rd]&\\
& A(K(S(m+1))) \ar[dd]\ar[rr]&& A(K(A)) \ar[dd]\\
T(m) \ar'[r][rr] \arwe[rd]&& B \ar[rd]&\\
& A(K(T(m))) \ar[rr]&& A(K(B))
\LambdaS(m+1) \arcof[dd]\ar[rr]\arwe[rd]\xypo&& A \arcof'[d][dd] \arwe[rd]&\\
Note that we have a weak equivalence in the top left corner, by the base case ($S(m+1)=(\Lambda(v), 0)$). The weak equivalence in the top right is by assumption. Finally the bottom left map is a weak equivalence because both cdga's are acyclic.
Note that we have a weak equivalence in the top left corner, by the base case ($\LambdaS(m+1)=(\Lambda(v), 0)$). The weak equivalence in the top right is by assumption. Finally the bottom left map is a weak equivalence because both cdga's are acyclic.
By \CorollaryRef{A-preserves-htpy-pullbacks} we know that the front face is a homotopy pushout. The back face is a homotopy pushout by \LemmaRef{htpy-pushout-reedy} and to conclude that $B \to A(K(B))$ is a weak equivalence, we use the cube lemma (\LemmaRef{cube-lemma}).
@ -76,9 +76,9 @@ Finally we come to the definition of a differential graded algebra. This will be
$$ d(x y)= d(x) y +(-1)^{\deg{x}} x d(y)\quad\text{ for all } x, y \in A. $$
\end{definition}
\todo{Define the notion of derivation?}
In general, a map which satisfies the above Leibniz rule is called a \Def{derivation}.
It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of dga's will be denoted by $\DGA_\k$, the category of commutative dga's (cdga's) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation.
It is not hard to see that the definition of a dga precisely defines the monoidal objects in the category of differential graded modules. The category of dga's will be denoted by $\DGA_\k$, the category of commutative dga's (cdga's) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation.
Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i =0$ for $i < 0$. Similarly if $M^i =0$ for all $i < 0$, then $M$ is a \emph{cochain algebra}.
@ -123,7 +123,7 @@ Note that taking homology of a differential graded module (or algebra) is functo
\section{Classical results}
We will give some classical known results of algebraic topology or homological algebra. Proofs of these theorems can be found in many places. \todo{cite at least 1 place}
We will give some classical known results of algebraic topology or homological algebra. Proofs of these theorems can be found in many places such as \cite{rotman, weibel}.
\begin{theorem}
(Universal coefficient theorem) Let $C$ be a chain complex and $A$ an abelian group, then there are natural short exact sequences for each $n$:
@ -136,10 +136,5 @@ The first statement generalizes to a theorem where $A$ is a chain complex itself
\begin{theorem}
(Künneth) Assume that $\k$ is a field and let $C$ and $D$ be (co)chain complexes, then there is a natural isomorphism (a linear graded map of degree $0$):
$$ H(C)\tensor H(D)\tot{\iso} H(C \tensor D), $$
where we understand both tensors as graded.
where we understand both tensors as graded. If $C$ and $D$ are algebras, this isomorphism is an isomorphism of algebras.
One can check that $\Apl\in\simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k\in\simplicial{\Mod{\k}}$, this is indeed a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
One can check that $\Apl\in\simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k$, this is a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
\Lemma{apl-contractible}{
$\Apl^k$ is contractible.
@ -49,7 +49,7 @@ One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace
\end{align*}
So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s =0$.
So we have an extra degeneracy $s: \Apl^k \to\Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl\to\ast$ is a weak equivalence.
So we have an extra degeneracy $s: \Apl^k \to\Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl^k\to\ast$ is a weak equivalence.
@ -3,10 +3,10 @@ Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit}
Although the model structure is completely induced, it might still be fruitful to discuss the right notion of a homotopy for augmented cdga's. Consider the following pullback of cdga's:
\[\xymatrix{
\Lambda(t, dt) \overline{\tensor} A \ar[r]\xypb\ar[d]&\Lambda(t, dt) \tensor A \ar[d]\\
\k\ar[r]&\k\tensor\Lambda(t, dt)
\Lambda(t, dt) \overline{\tensor} A \ar[r]\xypb\ar[d]&\Lambda(t, dt) \tensor A \ar[d]^{\id\tensor\counit}\\
\k\ar[r]&\Lambda(t, dt) \tensor\k
}\]
The pullback is the subspace of elements $x \tensor a$ in $\Lambda(t, dt)\tensor A$ such that $\counit(a)\cdot x\in\k$. Note that this construction is dual to a construction on topological spaces: in order to define a homotopy which is constant on the point $x_0$, we define the homotopy to be a map from a quotient ${X \times I}/{x_0\times I}$.
The pullback is the subspace of elements $x \tensor a$ in $\Lambda(t, dt)\tensor A$ such that $x \cdot\counit(a)\in\k$. Note that this construction is dual to a construction on topological spaces: in order to define a homotopy which is constant on the point $x_0$, we define the homotopy to be a map from a quotient ${X \times I}/{x_0\times I}$.
\Definition{homotopy-augmented}{
Two maps $f, g: A \to X$ between augmented cdga's are said to be \emph{homotopic} if there is a map
$$h : A \to\Lambda(t, dt)\overline{\tensor} X$$
@ -37,7 +37,7 @@ The second observation is that $Q$ is nicely behaved on tensor products and coke
@ -68,8 +68,8 @@ Furthermore we have the following lemma which is of homotopical interest.
}
\Proof{
First we define an augmented cdga $U(n)$ for each positive $n$ as $U(n)= D(n)\oplus\k$ with trivial multiplication and where the term $\k$ is used for the unit and augmentation. Notice that the map $U(n)\to\k$ is a trivial fibration. By the lifting property we see that the induced map
is surjective for each positive $n$. Note that maps from $X$ to $U(n)$ will send products to zero and that it is fixed on the augmentation. So there is a natural isomorphism $\Hom_\AugCDGA(X, U(n))\iso\Hom_\k(Q(X)^n, \k)$. Hence
is surjective for each positive $n$. Note that maps from $A$ to $U(n)$ will send products to zero and that it is fixed on the augmentation. So there is a natural isomorphism $\Hom_\AugCDGA(A, U(n))\iso\Hom_\k(Q(A)^n, \k)$. Hence
Now $Q(\Lambda(t, dt))= D(0)$ and hence it is acyclic, so when passing to homology, this term vanishes. In other words both maps ${d_i}_\ast : H(D(0))\oplus H(Q(A))\to H(Q(A))$ are the identity maps on $H(Q(A))$.
Now $Q(\Lambda(t, dt))= D(0)$ and hence it is acyclic, so when we pass to homology, this term vanishes. In other words both maps ${d_i}_\ast : H(D(0))\oplus H(Q(A))\to H(Q(A))$ are the identity maps on $H(Q(A))$.
}
Consider the augmented cdga $V(n)= S(n)\oplus\k$, with trivial multiplication and where the term $\k$ is used for the unit and augmentation. This augmented cdga can be thought of as a specific model of the sphere. In particular the homotopy groups can be expressed as follows.
@ -28,7 +28,7 @@ Consider the augmented cdga $V(n) = S(n) \oplus \k$, with trivial multiplication
}
\Proof{
Note that $Q(V(n))$ in degree $n$ is just $\k$ and $0$ in the other degrees, so its homotopy groups consists of a single $\k$ in degree $n$. This establishes the map:
Now by \LemmaRef{cdga-homotopic-maps-equal-pin} we get a map from the set of homotopy classes $[A, V(n)]$ instead of just maps. \todo{injective, surjective}
}
@ -47,7 +47,7 @@ In topology we know that a fibration induces a long exact sequence of homotopy g
First note that $j$ is also a cofibration. By \LemmaRef{Q-preserves-cofibs} the maps $Qg$ and $Qj$ are injective in positive degrees. By applying $Q$ we get two exact sequence (in positive degrees) as shown in the following diagram. By the fact that $Q$ preserves pushouts (\LemmaRef{Q-preserves-pushouts}) the cokernels coincide.
First note that $j$ is also a cofibration. By \LemmaRef{Q-preserves-cofibs} the maps $Qg$ and $Qj$ are injective in positive degrees. By applying $Q$ we get two exact sequence (in positive degrees) as shown in the following diagram. By the fact that $Q$ preserves pushouts (\CorollaryRef{Q-preserves-pushouts}) the cokernels coincide.
H(\Lambda(t, dt) \tensor A) \ar[r]^-{d_i}& H(\k\tensor A)
}\]
Now we know that $H(d_0)= H(d_1) : H(\Lambda(t, dt))\to\k$ as $\Lambda(t, dt)$ is acyclic and the induced map send $1$ to $1$. So the two bottom maps in the diagram are equal as well. Now we conclude $H(f)= H(d_1)H(h)= H(d_0)H(h)= H(g)$.
Now we know that $H(d_0)= H(d_1) : H(\Lambda(t, dt))\to\k$ as $\Lambda(t, dt)$ is acyclic and the induced map sends$1$ to $1$. So the two bottom maps in the diagram are equal as well. Now we conclude $H(f)= H(d_1)H(h)= H(d_0)H(h)= H(g)$.
In this section we will discuss the so called minimal models. These cdga's enjoy the property that we can easily prove properties inductively. Moreover it will turn out that weakly equivalent minimal models are actually isomorphic.
\begin{definition}
\Definition{minimal-algebra}{
A cdga $(A, d)$ is a \Def{Sullivan algebra} if
\begin{itemize}
\item$A =\Lambda V$ is free as a commutative graded algebra, and
@ -17,7 +17,7 @@ In this section we will discuss the so called minimal models. These cdga's enjoy
\begin{itemize}
\item$d$ is decomposable, i.e. $\im(d)\subset\Lambda^{\geq2}V$.
\end{itemize}
\end{definition}
}
\begin{definition}
Let $(A, d)$ be any cdga. A \Def{(minimal) Sullivan model} is a (minimal) Sullivan algebra $(M, d)$ with a weak equivalence:
@ -35,7 +35,9 @@ We will often say \Def{minimal model} or \Def{minimal algebra} to mean minimal S
As $A$ is $1$-reduced we have $\deg{x}, \deg{y}\geq2$ and so by the above $\deg{x}, \deg{y}\leq n-1$. Conclude that $d(V(k))\subset\Lambda(V(n-1))$.
}
The above definition is the same as in \cite{felix} without assuming connectivity. We find some different definitions of (minimal) Sullivan algebras in the literature. For example we find a definition using well orderings in \cite{hess}. The decomposability of $d$ also admits a different characterization (at least in the connected case). The equivalence of the definitions is expressed in the following two lemmas.\todo{to prove or not to prove}
Minimal models admit very nice homotopy groups. Note that for a minimal algebra $\Lambda V$ there is a natural augmentation and the the differential is decomposable. Hence $Q \Lambda V$ is naturally isomorphic to $(V, 0)$. In particular the homotopy groups are simply given by $\pi^n(\Lambda V)= V^n$.
\DefinitionRef{minimal-algebra} is the same as in \cite{felix} without assuming connectivity. We find some different definitions of (minimal) Sullivan algebras in the literature. For example we find a definition using well orderings in \cite{hess}. The decomposability of $d$ also admits a different characterization (at least in the connected case). The equivalence of the definitions is expressed in the following two lemmas.\todo{to prove or not to prove}
\Lemma{}{
A cdga $(\Lambda V, d)$ is a Sullivan algebra if and only if there exists a well order $J$ such that $V$ is generated by $v_j$ for $j \in J$ and $d v_j \in\Lambda V_{<j}$.
@ -47,13 +49,10 @@ The above definition is the same as in \cite{felix} without assuming connectivit
It is clear that induction will be an important technique when proving things about (minimal) Sullivan algebras. We will first prove that minimal models always exist for $1$-connected cdga's and afterwards prove uniqueness.
\todo{at the moment this is just cut n pasted. Rewrite to make sense in this context}
Minimal models admit very nice homotopy groups. Note that for a minimal algebra $\Lambda V$ there is a natural augmentation and the the differential is decomposable. Hence $Q \Lambda V$ is naturally isomorphic to $(V, 0)$. In particular the homotopy groups are simply given by $\pi^n(\Lambda V)= V^n$.
\section{Existence}
\begin{theorem}
Let $(A, d)$ be an$0$-connected cdga, then it has a Sullivan model $(\Lambda V, d)$. Furthermore if $(A, d)$ is $r$-connected with $r \geq1$ then $V^i =0$ for all $i \leq r$ and in particular $(\Lambda V, d)$ is minimal.
Let $(A, d)$ be a $0$-connected cdga, then it has a Sullivan model $(\Lambda V, d)$. Furthermore if $(A, d)$ is $r$-connected with $r \geq1$ then $V^i =0$ for all $i \leq r$ and in particular $(\Lambda V, d)$ is minimal.
\end{theorem}
\begin{proof}
Start by setting $V(0)= H^{\geq1}(A)$ and $d =0$. This extends to a morphism $m_0 : (\Lambda V(0), 0)\to(A, d)$.
@ -86,7 +85,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m
}
\end{displaymath}
By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ such that $p \circ\phi= g$. We will do this by induction. Note that the induction step proves precisely that $(\Lambda V(k), d)\to(\Lambda V(k+1), d)$ is a cofibration.
By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$which commutes with the differential such that $p \circ\phi= g$. We will do this by induction. Note that the induction step proves precisely that $(\Lambda V(k), d)\to(\Lambda V(k+1), d)$ is a cofibration.
\begin{itemize}
\item Suppose $\{v_\alpha\}$ is a basis for $V(0)$. Define $V(0)\to X$ by choosing preimages $x_\alpha$ such that $p(x_\alpha)= g(v_\alpha)$ ($p$ is surjective). Define $\phi(v_\alpha)= x_\alpha$.
\item Suppose $\phi$ has been defined on $V(n)$. Write $V(n+1)= V(n)\oplus V'$ and let $\{v_\alpha\}$ be a basis for $V'$. Then $dv_\alpha\in\Lambda V(n)$, hence $\phi(dv_\alpha)$ is defined and
@ -132,17 +131,17 @@ Now if the map $f$ is a weak equivalence, both maps $\phi$ and $\psi$ are surjec
\begin{proof}
Since both $M$ and $M'$ are minimal, they are cofibrant and so the weak equivalence is a strong homotopy equivalence (\CorollaryRef{cdga_homotopy_properties}). And so the induced map $\pi^n(\phi) : \pi^n(M)\to\pi^n(M')$ is an isomorphism (\LemmaRef{cdga-homotopic-maps-equal-pin}).
Since $M$ (resp. $M'$) is free as a cga's, it is generated by some graded vector space $V$ (resp. $V'$). By an earlier remark \todo{where?}the homotopy groups were easy to calculate and we conclude that $\phi$ induces an isomorphism from $V$ to $V'$:
Since $M$ (resp. $M'$) is free as a cga's, it is generated by some graded vector space $V$ (resp. $V'$). By an earlier remark the homotopy groups were easy to calculate and we conclude that $\phi$ induces an isomorphism from $V$ to $V'$:
\[\pi^\ast(\phi) : V \tot{\iso} V'. \]
Conclude that $\phi=\Lambda\phi_0$\todo{why?} is an isomorphism.
By induction on the degree one can prove that $\phi$ needs to be surjective and hence is a fibration. By the lifting property we can find a right inverse $\psi$, which is then injective and a weak equivalence. Now the above argument also proves that $\psi$ is surjective. Conclude that $\psi$ is an isomorphism and $\phi$, being its right inverse, is an isomorphism as well.
\end{proof}
\Theorem{unique-minimal-model}{
Let $m: (M, d)\we(A, d)$ and $m': (M', d')\we(A, d)$ be two minimal models for $A$. Then there is an isomorphism $\phi(M, d)\tot{\iso}(M', d')$ such that $m' \circ\phi\eq m$.
}
\begin{proof}
By the previous lemmas we have $[M', M]\iso[M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ\phi\eq m$. On homology we get $H(m')\circ H(\phi)= H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
By \LemmaRef{minimal-model-bijection} we have $[M', M]\iso[M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ\phi\eq m$. On homology we get $H(m')\circ H(\phi)= H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
\end{proof}
The assignment to $X$ of its minimal model $M_X =(\Lambda V, d)$ can be extended to morphisms. Let $X$ and $Y$ be two cdga's and $f: X \to Y$ be a map. By considering their minimal models we get the following diagram.
@ -11,9 +11,9 @@ As this thesis considers different categories, each with its own homotopy theory
\begin{definition}
A \Def{model category} is a category $\cat{C}$ together with three subcategories:
\begin{itemize}
\itemthe class of \Def{weak equivalences}$\W$,
\itemthe class of \Def{fibrations}$\Fib$ and
\itemthe class of \Def{cofibrations}$\Cof$,
\itema class of \Def{weak equivalences}$\W$,
\itema class of \Def{fibrations}$\Fib$ and
\itema class of \Def{cofibrations}$\Cof$,
\end{itemize}
such that the following five axioms hold:
\begin{itemize}
@ -23,11 +23,7 @@ As this thesis considers different categories, each with its own homotopy theory
\item[MC4] In any commuting square as follows where $i \in\Cof$ and $p \in\Fib$,
\cdiagram{Model_Liftproblem}
there exist a lift $h: B \to Y$ if either
\begin{itemize}
\item[a)]$i \in\W$ or
\item[b)]$p \in\W$.
\end{itemize}
there exist a lift $h: B \to Y$ if either (a) $i \in\W$ or (b) $p \in\W$.
\item[MC5] Any map $f : A \to B$ can be factored in two ways:
\begin{itemize}
\item[a)] as $f = pi$, where $i \in\Cof\cap\W$ and $p \in\Fib$ and
@ -59,14 +55,14 @@ The fourth axiom actually characterizes the classes of (trivial) fibrations and
\Lemma{model-cats-characterization}{
Let $\cat{C}$ be a model category.
\begin{itemize}
\item The cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. trivial fibrations.
\item The fibrations in $\cat{C}$ are the maps with a RLP w.r.t. trivial cofibrations.
\item The trivial cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. fibrations.
\item The trivial fibrations in $\cat{C}$ are the maps with a RLP w.r.t. cofibrations.
\item\smallThe cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. trivial fibrations.
\item\smallThe fibrations in $\cat{C}$ are the maps with a RLP w.r.t. trivial cofibrations.
\item\smallThe trivial cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. fibrations.
\item\smallThe trivial fibrations in $\cat{C}$ are the maps with a RLP w.r.t. cofibrations.
\end{itemize}
}
This means that once we choose weak equivalences and fibrations for a category $\cat{C}$, the third class is determined, and vice versa. The classes of fibrations behave nice with respect to pullbacks and dually cofibrations behave nice with pushouts:
This means that once we choose the weak equivalences and the fibrations for a category $\cat{C}$, the cofibrations are determined, and vice versa. The classes of fibrations behave nice with respect to pullbacks and dually cofibrations behave nice with pushouts:
\Lemma{model-cats-pushouts}{
Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively.
@ -81,12 +77,6 @@ This means that once we choose weak equivalences and fibrations for a category $
Let $\cat{C}$ be a model category. Let $f: A \cof B$ and $g:A' \cof B'$ be two (trivial) cofibrations, then the induced map of the coproducts $f+g: A+A' \to B+B'$ is also a (trivial) cofibration. Dually: the product of two (trivial) fibrations is a (trivial) fibration.
}
\TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski):
Of course the most important model category is the one of topological spaces. We will be interested in the standard model structure on topological spaces, which has weak homotopy equivalences as weak equivalences. Equally important is the model category of simplicial sets.
\Example{top-model-structure}{
@ -94,14 +84,14 @@ Of course the most important model category is the one of topological spaces. We
\begin{itemize}
\item Weak equivalences: maps inducing isomorphisms on all homotopy groups.
\item Fibrations: Serre fibrations, i.e. maps with the right lifting property with respect to the inclusions $D^n \cof D^n \times I$.
\item Cofibrations: maps $S^{n-1}\cof D^n$and transfinite compositions of pushouts and coproducts thereof.
\item Cofibrations: the smallest class of maps containing$S^{n-1}\cof D^n$which is closed under transfinite compositions, pushouts, coproducts and retracts.
\end{itemize}
}
\Example{sset-model-structure}{
The category $\sSet$ of simplicial sets has the following model structure.
\begin{itemize}
\item Weak equivalences:
\item Weak equivalences: maps inducing isomorphisms on all homotopy groups.
\item Fibrations: Kan fibrations, i.e. maps with the right lifting property with respect to the inclusions $\Lambda_n^k \cof\Delta[n]$.
\item Cofibrations: all monomorphisms.
\end{itemize}
@ -251,6 +241,8 @@ For arbitrary categories and classes of weak equivalences, such a localization n
In \cite{dwyer} it is proven that this indeed defines a localization of $\cat{C}$ with respect to $\W$. It is good to note that $\Ho(\Top)$ does not depend on the class of cofibrations or fibrations.
Note that whenever we have a full subcategory $\cat{C'}\subset\cat{C}$, where $\cat{C}$ is a model category, there is a subcategory of the homotopy category: $\Ho(\cat{C'})\subset\Ho(\cat{C})$. There is no need for a model structure on the subcategory.
\Example{ho-top}{
The category $\Ho(\Top)$ has as objects just topological spaces and homotopy classes between cofibrant replacements (note that every objects is already fibrant). Moreover, if we restrict to the full subcategory of CW complexes, maps are precisely homotopy classes between objects.
@ -310,7 +302,7 @@ In category theory we know that colimits (and limits) are unique up to isomorphi
}\]
The diagrams are pointwise weakly equivalent. But the induced map $S^n \to\ast$ on the pushout is clearly not. In this section we will briefly indicate what homotopy pushouts are (and dually we get homotopy pullbacks).
One direct way to obtain a homotopy pushout is by the use of \emph{Reedy categories}\cite{hovey}. In this case the diagram category is endowed with a model structure, which gives a notion of cofibrant diagram. In such diagrams the ordinary pushout is the homotopy pushout. The key result is the following.
One direct way to obtain a homotopy pushout is by the use of \emph{Reedy categories}\cite{hovey}. In this case the diagram category is endowed with a model structure, which gives a notion of cofibrant diagram. In such diagrams the ordinary pushout is the homotopy pushout.
\Lemma{htpy-pushout-reedy}{
Consider the following pushout diagram. The if all objects are cofibrant and the map $f$ is a cofibration, then the homotopy pushout is given by the ordinary pushout.
@ -320,7 +312,7 @@ One direct way to obtain a homotopy pushout is by the use of \emph{Reedy categor
}\]
}
There are other ways to obtain homotopy pushouts. A very general way is given by the \emph{bar construction}\cite{riehl}.\todo{do we need this?}
There are other ways to obtain homotopy pushouts. A very general way is given by the \emph{bar construction}\cite{riehl}.
The important property of homotopy pushout we use in this thesis is the uniqueness (up to homotopy). In particular we need the following fact.
@ -335,4 +327,4 @@ The important property of homotopy pushout we use in this thesis is the uniquene
If the three maps $A^\ast\to B^\ast$ are weak equivalences, then so is the map $P \to Q$.
}
We get similar theorems for the dual case of homotopy pullbacks.
If we combine this lemma with \LemmaRef{htpy-pushout-reedy} we obtain precisely Lemma 5.2.6 in \cite{hovey}. We get similar theorems for the dual case of homotopy pullbacks.
@ -10,7 +10,7 @@ In this section we will define a model structure on cdga's over a field $\k$ of
\end{itemize}
\end{proposition}
We will prove the different axioms in the following lemmas. First observe that the classes as defined above are indeed closed under multiplication and contain all isomorphisms.
We will prove the different axioms in the following lemmas. First observe that the classes as defined above are indeed closed under composition and contain all isomorphisms.
Note that with these classes, every cdga is a fibrant object.
@ -32,7 +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}
For the class of weak equivelances and fibrations this follows easily from basic category theory. For cofibrations we consider the following diagram where the horizontal compositions are identities:
For the class of weak equivalences and fibrations this follows easily from basic category theory. For cofibrations we consider the following diagram where the horizontal compositions are identities:
\[\xymatrix{
A' \ar[r]\ar[d]^g & A \ar[r]\arcof[d]^f & A' \ar[d]^g \\
B' \ar[r]& B \ar[r]& B'
@ -50,11 +50,10 @@ Next we will prove the factorization property [MC5]. We will do this by Quillen'
\begin{definition}
Define the following objects and sets of maps:
\begin{itemize}
\item$S(n)$ is the CDGA generated by one element $a$ of degree $n$ such that $da =0$.
\todo{Andere letters, of $\Lambda$}
\item$T(n)$ is the CDGA generated by two element $b$ and $c$ of degree $n$ and $n+1$ respectively, such that $db = c$ (and necessarily $dc =0$).
\item$I =\{ i_n: \k\to T(n)\I n \in\N\}$ is the set of units of $T(n)$.
\item$J =\{ j_n: S(n+1)\to T(n)\I n \in\N\}$ is the set of inclusions $j_n$ defined by $j_n(a)= b$.
\item$\Lambda S(n)$ is the cdga generated by one element $a$ of degree $n$ such that $da =0$.
\item$\Lambda D(n)$ is the CDGA generated by two element $b$ and $c$ of degree $n$ and $n+1$ respectively, such that $db = c$ (and necessarily $dc =0$).
\item$I =\{ i_n: \k\to\Lambda D(n)\I n \in\N\}$ is the set of units of $\Lambda D(n)$.
\item$J =\{ j_n: \Lambda S(n+1)\to\Lambda D(n)\I n \in\N\}$ is the set of inclusions $j_n$ defined by $j_n(a)= b$.
\end{itemize}
\end{definition}
@ -62,7 +61,7 @@ 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 $H(T(n))=\k$ (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?}
Since $H(\Lambda D(n))=\k$ (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 $\Lambda D(n)$ and $\Lambda S(n)$.
\end{proof}
\begin{lemma}
@ -92,13 +91,13 @@ As a consequence of the above two lemmas, the class generated by $I$ is containe
\todo{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.
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.\todo{Definieer wat ``small'' betkent en geef een referentie}
\begin{lemma}
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}
This follows from Quillen's small object argument.\todo{Definieer wat ``small'' betkent en geef een referentie}
This follows from Quillen's small object argument.
Furthermore we have $F_!\circ\Delta[-]\iso F$. In short we have the following:
@ -36,7 +36,7 @@ In our case we take the opposite category, so the definition of $A$ is in terms
where the addition, multiplication and differential are defined pointwise. Conclude that we have the following contravariant functors (which form an adjoint pair):
\begin{align*}
A(X) &= \Hom_\sSet(X, \Apl) & X \in\sSet\\
K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in\CDGA_\k.
K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in\CDGA_\k
\end{align*}
@ -62,7 +62,7 @@ Let $x$ be a $k$-simplex of $\Delta[n]$, i.e. $x: \Delta[k] \to \Delta[n]$. Then
Note that $\oint_n(v): \Delta[n]\to\k$ is just a map, we can extend this linearly to chains on $\Delta[n]$ to obtain $\oint_n(v): \Z\Delta[n]\to\k$, in other words $\oint_n(v)\in C_n$. By linearity of $\int_n$ and $x^\ast$, we have a linear map $\oint_n: \Apl_n \to C_n$.
Next we will show that $\oint=\{\oint_n\}_n$ is a simplicial map and that each $\oint_n$ is a chain map, in other words $\oint : \Apl\to C_n$ is a simplicial chain map (of complexes). Let $\sigma: \Delta[n]\to\Delta[k]$, and $\sigma^\ast: \Apl_k \to\Apl_n$ its induced map. We need to prove $\oint_n \circ\sigma^\ast=\sigma^\ast\circ\oint_k$. We show this as follows:
Next we will show that $\oint=\{\oint_n\}_n$ is a simplicial map and that each $\oint_n$ is a chain map, in other words $\oint : \Apl\to C$ is a simplicial chain map (of complexes). Let $\sigma: \Delta[n]\to\Delta[k]$, and $\sigma^\ast: \Apl_k \to\Apl_n$ its induced map. We need to prove $\oint_n \circ\sigma^\ast=\sigma^\ast\circ\oint_k$. We show this as follows:
For it to be a chain map, we need to prove $d \circ\oint_n =\oint_n \circ d$. This is precisely the same calculation as \emph{Stokes' theorem}. \todo{prove this?}
For it to be a chain map, we need to prove $d \circ\oint_n =\oint_n \circ d$. This is precisely the same calculation as \emph{Stokes' theorem}. \todo{Reference}
We now proved that $\oint$ is indeed a simplicial chain map. Note that $\oint_n$ need not to preserve multiplication, so it fails to be a map of cochain algebras. However $\oint(1)=1$ and so the induced map on homology sends the class of $1$ in $H(\Apl_n)=\k\cdot[1]$ to the class of $1$ in $H(C_n)=\k\cdot[1]$. We have proven the following lemma.
@ -95,9 +95,9 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis
We can apply our two functors to it, and use the natural transformation $\oint$ to obtain the following cube:
\cdiagram{Apl_C_Quasi_Iso_Cube}
Note that $A(\Delta[n])\we C^\ast(\Delta[n])$ by \CorollaryRef{apl-c-quasi-iso}, $A(X)\we C^\ast(X)$ by assumption and $A(\del\Delta[n])\we C^\ast(\del\Delta[n])$ by induction. Secondly note that both $A$ and $C^\ast$ send injective maps to surjective maps, so we get fibrations on the right side of the diagram. Finally note that the front square and back square are pullbacks, by adjointness of $A$ and $C^\ast$. Apply the cube lemma (\LemmaRef{cube-lemma}, \cite[Lemma 5.2.6]{hovey}) to conclude that also $A(X')\we C^\ast(X')$.
Note that $A(\Delta[n])\we C^\ast(\Delta[n])$ by \CorollaryRef{apl-c-quasi-iso}, $A(X)\we C^\ast(X)$ by assumption and $A(\del\Delta[n])\we C^\ast(\del\Delta[n])$ by induction. Secondly note that both $A$ and $C^\ast$ send injective maps to surjective maps, so we get fibrations on the right side of the diagram. Finally note that the front square and back square are pullbacks, by adjointness of $A$ and $C^\ast$. Apply the cube lemma (\LemmaRef{cube-lemma}) to conclude that also $A(X')\we C^\ast(X')$.
This proves $A(X)\we C^\ast(X)$ for any simplicial set with finitely many non-degenerate simplices. We can extend this to simplicial sets of finite dimension by attaching many simplices at once. For this observe that both $A$ and $C^\ast$ send coproducts to products and that cohomology commutes with products:
This proves $A(X)\we C^\ast(X)$ for any simplicial set with finitely many non-degenerate simplices. We can extend this to simplicial sets of finite dimension by attaching many simplices at once. For this we observe that both $A$ and $C^\ast$ send coproducts to products and that cohomology commutes with products:
@ -71,7 +71,9 @@ The \Def{rational disk} is now defined as cone of the rational sphere: $D^{n+1}_
Since $[\frac{1}{(k-1)!}f]= k[\frac{1}{k}f]\in\pi_n(X)$ we can define $f'$ accordingly on the $n+1$-cells. Since our inclusion $i: S^n \cof S^n_\Q$ is in the first sphere, we get $f = f' \circ i$.
Let $f''$ be any map such that $f''i = f$. Then \todo{finish proof}
Let $f''$ be any map such that $f''i = f$. Then $f''$ also represents $\alpha$ and all the functions $\frac{1}{2}f''$, $\frac{1}{6}f''$,\dots are hence homotopic to $\frac{1}{2}f$, $\frac{1}{6}f$,\dots. So indeed $f$ is homotopic to $f''$.
Now if $g$ is homotopic to $f$. We can extend the homotopy $h$ in a similar way to the rational sphere. Hence the extensions are homotopic.
}
\section{Rationalizations of arbitrary spaces}
@ -99,8 +101,7 @@ Having rational cells we wish to replace the cells in a CW complex $X$ by the ra
\Proof{
Let $Y \tot{f} X$ be a CW approximation and let $Y \tot{\phi} Y_\Q$ be the rationalization of $Y$. Now we define $X_\Q$ as the double mapping cylinder (or homotopy pushout):
$$ X_\Q= X \cup_f (Y \times I)\cup_{\phi} Y_\Q. $$
\todo{bewijs afmaken met excision?}
with the obvious inclusion $\psi: X \to X_\Q$. By excision we see that $H_\ast(X_\Q, Y_\Q)\iso H_\ast(X \cup_f (Y \times I), Y \times{1})=0$. So by the long exact sequence of the inclusion we get $H_\ast(X_\Q)\iso H_\ast(Y_\Q)$, which proves by the rational Hurewicz theorem that $X_\Q$ is a rational space. At last we note that $H_\ast(X_\Q, X; \Q)\iso H_\ast(Y_\Q, Y; \Q)=0$, since $\phi$ was a rationalization. This proves that $H_\ast(\psi; \Q)$ is an isomorphism, so by the rational Whitehead theorem, $\psi$ is a rationalization.
}
\Theorem{}{
@ -116,7 +117,7 @@ Having rational cells we wish to replace the cells in a CW complex $X$ by the ra
Moreover, $f'$ is determined up to homotopy and homotopic maps have homotopic extensions.
}
The extension property allows us to define a rationalization of maps. Given $f : X \to Y$, we can consider the composite $if : X \to Y \to Y_\Q$. Now this extends to $(if)' : X_\Q\to Y_\Q$. Note that this construction is not functorial, since there are choices of homotopies involved. When passing to the homotopy category, however, this construction \emph{is} functorial and has an universal property.
We will note prove that above theorem (it is analogue to \LemmaRef{SnQ-extension}), but refer to \cite{felix}. The extension property allows us to define a rationalization of maps. Given $f : X \to Y$, we can consider the composite $if : X \to Y \to Y_\Q$. Now this extends to $(if)' : X_\Q\to Y_\Q$. Note that this construction is not functorial, since there are choices of homotopies involved. When passing to the homotopy category, however, this construction \emph{is} functorial and has an universal property.
We already mentioned in the first section that for rational spaces the notions of weak equivalence and rational equivalence coincide. Now that we always have a rationalization we have:
Joshua Moerman
10 years ago