In this thesis we will study rational homotopy theory. The subject was first considered by Serre in the 1950s, he was able to calculate the torsion free part of the homotopy of the spheres \cite{serre}. Despite the complicated structure of these homotopy groups, their torsion free parts have a nice and simple description.
Homotopy theory is the study of topological spaces and homotopy equivalences. These equivalences are weaker than isomorphism. An isomorphism is given by two maps $f : X \leftadj Y : g$, such that the both compositions are equal to identities. A homotopy equivalence weakens this by requiring the compositions to be homotopic to identities. Some properties of spaces, such as some kinds of connectedness, only depend on the homotopy type. Such properties are homotopy invariants. \todo{not happy with this yet...}
In order to investigate the torsion free part of any (abelian) group, one can tensor with the rationals to kill all torsion. This observation allows to define rational homotopy groups for any space.
Examples of homotopy invariants are homology groups $H_n(X)$ and homotopy groups $\pi_n(X)$. The latter is defined as the set of continuous maps $S^n \to X$ up to homotopy. Despite the easy definition, the groups $\pi_n(S^k)$ are very hard to calculate and much of it is even unknown as of today.
The fact that the rationals homotopy groups of the spheres are so simple led other mathematician believe that there could be a simple description for all of rational homotopy theory. The first to successfully give an algebraic model for rational homotopy theory was Quillen in the 1960s \cite{quillen}. His approach, however, is quite complicated. The equivalence he proves passes through four different categories. Not much later Sullivan gave an approach which resembles some ideas from de Rahm cohomology \cite{sullivan}.
In rational homotopy theory one ``localizes'' these invariants. Instead of considering $H_n(X)$ and $\pi_n(X)$, we consider the rational homology groups $H_n(X)\tensor\Q$ and the rational homotopy groups $\pi_n(X)\tensor\Q$. In fact, these groups are really $\Q$-vector spaces, and hence contain no torsion information. So rational homotopy theory is not able to see this information. This disadvantage is compensated by the fact that it is easier to calculate these invariants.
The first steps towards this theory were taken by Serre in the 1950s. In \cite{serre} he successfully calculated the torsion-free part of $\pi_n(S^k)$ for all $n$ and $k$. The outcome was remarkably easy and structured.
The fact that the rational homotopy groups of the spheres are so simple led other mathematician believe that there could be a simple description for all of rational homotopy theory. The first to successfully give an algebraic model for rational homotopy theory was Quillen in the 1960s \cite{quillen}. His approach, however, is quite complicated. The equivalence he proves passes through four different model categories. Not much later Sullivan gave an approach which resembles some ideas from de Rahm cohomology \cite{sullivan}, which is of a more geometric nature. The theory of Sullivan is the main subject of this thesis.
The most influential paper is from Bousfield and Gugenheim which combines Quillen's abstract machinery of model categories with the approach of Sullivan \cite{bousfield}. Being only a paper, it does not contain a lot of details, which might scare the reader at first.
The most influential paper is from Bousfield and Gugenheim which combines Quillen's abstract machinery of model categories with the approach of Sullivan \cite{bousfield}. Being only a paper, it does not contain a lot of details, which might scare the reader at first.
There is a much newer book by Félix, Halperin and Thomas \cite{felix}. This book covers much more than the paper from Bousfield and Gugenheim but does not use the theory of model categories. On one hand, this makes the proofs more elementary, on the other hand it may obscure some abstract constructions. This thesis will provide a middle ground. We will use model categories, but still provide a lot of detail.
There is a much newer book by Félix, Halperin and Thomas \cite{felix}. This book covers much more than the paper from Bousfield and Gugenheim but does not use the theory of model categories. On one hand, this makes the proofs more elementary, on the other hand it may obscure some abstract constructions. This thesis will provide a middle ground. We will use model categories, but still provide a lot of detail.
After some preliminaries this thesis will start with the work from Serre in \ChapterRef{Serre}. We will avoid the use of spectral sequences. The theorems stated in this chapter are not necessarily needed for the main theorems in this thesis. Nowadays there are more abstract tools to prove the needed results, but as Serre's theorems are nice in their own rights, they are included in this thesis.
After some preliminaries, this thesis will start with some of the work from Serre in \ChapterRef{Serre}. We will avoid the use of spectral sequences. The theorems are more specific than we actually need and there are easier, more abstract ways to prove what we need. But these theorems in their current form are nice on their own rights, and so they are included in this thesis.
The next chapter (\ChapterRef{Rationalization}) describes a way to localize a space, in the same way we can localize a ring. This technique allows us to consider ordinary homotopy equivalences between the localized spaces, instead of rational equivalences, which are harder to visualize.
The next chapter (\ChapterRef{Rationalization}) describes a way to localize a space directly, in the same way we can localize an abelian group. This technique allows us to consider ordinary homotopy equivalences between the localized spaces, instead of rational equivalences, which are harder to grasp.
The longest chapter is \ChapterRef{HomotopyTheoryCDGA}. In this chapter we will describe commutative differential graded algebras and their homotopy theory. One can think of these objects as rings which are at the same time cochain complexes. Not only will we describe a model structure on this category, we will also explicitly describe homotopy relations and homotopy groups.
The longest chapter is \ChapterRef{HomotopyTheoryCDGA}. In this chapter we will describe commutative differential graded algebras and their homotopy theory. One can think of these objects as rings which are at the same time cochain complexes. Not only will we describe a model structure on this category, we will also explicitly describe homotopy relations and homotopy groups.
In \ChapterRef{Adjunction} we define an adjunction between simplicial sets and commutative differential graded algebras. It is here that we see a construction similar to the de Rahm complex of a manifold.
In \ChapterRef{Adjunction} we define an adjunction between simplicial sets and commutative differential graded algebras. It is here that we see a construction similar to the construction of the de Rahm complex of a manifold.
\ChapterRef{MinimalModels} brings us back to the study of commutative differential graded algebras. In this chapter we study to so called minimal models. These models enjoy the property that homotopically equivalent minimal models are actually isomorphic. Furthermore their homotopy groups are easily calculated.
\ChapterRef{MinimalModels} brings us back to the study of commutative differential graded algebras. In this chapter we study to so called minimal models. These models enjoy the property that homotopically equivalent minimal models are actually isomorphic. Furthermore their homotopy groups are easily calculated.
The main theorem is proven in \ChapterRef{Equivalence}. The adjunction from \ChapterRef{Adjunction} turns out to induce an equivalence on (subcategories of) the homotopy categories. This unifies rational homotopy theory of spaces with the homotopy theory of commutative differential graded algebras.
The main theorem is proven in \ChapterRef{Equivalence}. The adjunction from \ChapterRef{Adjunction} turns out to induce an equivalence on (subcategories of) the homotopy categories. This unifies rational homotopy theory of spaces with the homotopy theory of commutative differential graded algebras.
Finally we will see some explicit calculations in \ChapterRef{Calculations}. These calculations are remarkable easy, once we have the main equivalence. To prove, for example, Serre's result on the rational homotopy groups of spheres, we construct a minimal model and read off their homotopy groups. We will also discuss related topics in \ChapterRef{Topics} which will conclude this thesis.
Finally we will see some explicit calculations in \ChapterRef{Calculations}. These calculations are remarkable easy. To prove for instance Serre's result on the rational homotopy groups of spheres, we construct a minimal model and read off their homotopy groups. We will also discuss related topics in \ChapterRef{Topics} which will conclude this thesis.
\section{Preliminaries and Notation}
\section{Preliminaries and Notation}
@ -32,7 +36,7 @@ We assume the reader is familiar with category theory, basics from algebraic top
We will fix the following notations and categories.
We will fix the following notations and categories.
\begin{itemize}
\begin{itemize}
\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.\todo{$\k$ doesn't always seem to work...}
\item$\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-modules, tensor products over $\k$,\dots.\todo{$\k$ doesn't always seem to work...}
\item$\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
\item$\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
\item$\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
\item$\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
\item$\Ab$: category of abelian groups and group homomorphisms.
\item$\Ab$: category of abelian groups and group homomorphisms.
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.
In this chapter we aim to prove that the homotopy theory of rational spaces is the same as the homotopy theory of cdga's over $\k=\Q$. We will only work over the rationals in this chapter, so we will omit the subscript $\Q$ in many places. 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.
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.
@ -9,7 +9,7 @@ We will prove that $A$ preserves cofibrations and trivial cofibrations. We only
$A(i) : A(\Delta[n])\to A(\del\Delta[n])$ is surjective.
$A(i) : A(\Delta[n])\to A(\del\Delta[n])$ is surjective.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
Let $\phi\in A(\del\Delta[n])$ be an element of degree $k$, hence it is a map $\del\Delta[n]\to\Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is Kan and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
Let $\phi\in A(\del\Delta[n])$ be an element of degree $k$, hence it is a map $\del\Delta[n]\to\Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is a Kan complex and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
\begin{displaymath}
\begin{displaymath}
\xymatrix{
\xymatrix{
\del\Delta[n]\ar[r]^\phi\arcof[d]^i &\Apl^k \\
\del\Delta[n]\ar[r]^\phi\arcof[d]^i &\Apl^k \\
@ -46,8 +46,8 @@ The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for
\Corollary{minimal-model-adjunction}{
\Corollary{minimal-model-adjunction}{
There is an adjunction:
There is an adjunction:
$$ M : \Ho(\sSet_1)\leftadj\opCat{\Ho(\text{Minimal models}^1)} : RK, $$
$$ M : \Ho(\sSet_1)\leftadj\opCat{\Ho(\text{Minimal models}_1)} : RK, $$
where $M$ is given by $M(X)= M(A(X))$ and $RK$ is given by $RK(Y)= K(Y)$ (because minimal models are always cofibrant).
where $M$ is given by $M(X)= M(A(X))$ and $RK$ is given by $RK(Y)= K(Y)$ (note that minimal models are always cofibrant).
}
}
@ -64,7 +64,7 @@ The homotopy groups of augmented cdga's are precisely the dual of the homotopy g
The first isomorphism \refison{1} follows from the homotopy adjunction in \CorollaryRef{ak-homotopy-adjunction} (note that $KX$ is a Kan complex, since it is a simplicial group). The next two isomorphisms \refison{2} and \refison{3} are induced by the weak equivalences $A(n)\we A(S^n)$ and $A(n)\we V(n)$and\CorollaryRef{cdga_homotopy_properties}. Finally we get \refison{4} from \LemmaRef{cdga-dual-homotopy-groups}.
The first isomorphism \refison{1} follows from the homotopy adjunction in \CorollaryRef{ak-homotopy-adjunction} (note that $KX$ is a Kan complex, since it is a simplicial group). The next two isomorphisms \refison{2} and \refison{3} are induced by the weak equivalences $A(n)\we A(S^n)$ and $A(n)\we V(n)$by using\CorollaryRef{cdga_homotopy_properties}. Finally we get \refison{4} from \LemmaRef{cdga-dual-homotopy-groups}.
}
}
\Theorem{cdga-dual-homotopy-groups-iso}{
\Theorem{cdga-dual-homotopy-groups-iso}{
@ -73,9 +73,9 @@ The homotopy groups of augmented cdga's are precisely the dual of the homotopy g
\Proof{
\Proof{
We will prove that the map in the previous theorem preserves the group structure. We will prove this by endowing a certain cdga with a coalgebra structure, which induces the multiplication in both $\pi_n(KX)$ and $\pi^n(X)^\ast$.
We will prove that the map in the previous theorem preserves the group structure. We will prove this by endowing a certain cdga with a coalgebra structure, which induces the multiplication in both $\pi_n(KX)$ and $\pi^n(X)^\ast$.
Since every $1$-connected cdga admits a minimal model, we will assume that $X$ is a minimal model, generated by $V$ (filtered by degree). We first observe that $\pi^n(X)\iso\pi^n(\Lambda V(n))$, since elements of degree $n+1$ or higher do not influence the homology of $QX$.
Since every $1$-connected cdga admits a minimal model, we will assume that $X$ is a minimal model, generated by $V$ (filtered by degree). We first observe that $\pi^n(X)\iso\pi^n(\Lambda V^{\leq n}$, since elements of degree $n+1$ or higher do not influence the homology of $QX$.
Now consider the cofibration $i: \Lambda V(n-1)\cof\Lambda V(n)$ and its associated long exact sequence (\CorollaryRef{long-exact-cdga-homotopy}). It follows that $\pi^n(\Lambda V(n))\iso\pi^n(\coker(i))$. Now $\coker(i)$ is generated by elements of degree $n$ only (as algebra), i.e. $\coker(i)=(\Lambda W, 0)$ for some vector space $W = W^n$. Let $Y$ denote this space $Y =(\Lambda, 0)$. Define a comultiplication on generators $w \in W$:
Now consider the cofibration $i: \Lambda V(n-1)\cof\Lambda V(n)$ and its associated long exact sequence (\CorollaryRef{long-exact-cdga-homotopy}). It follows that $\pi^n(\Lambda V(n))\iso\pi^n(\coker(i))$. Now $\coker(i)$ is generated by elements of degree $n$ only (as algebra), i.e. $\coker(i)=(\Lambda W, 0)$ for some vector space $W = W^n$. Let $Y$ denote this space $Y =(\Lambda W, 0)$. Define a comultiplication on generators $w \in W$:
\[\Delta : Y \to Y \tensor Y : \quad w \mapsto1\tensor w \,+\, w \tensor1. \]
\[\Delta : Y \to Y \tensor Y : \quad w \mapsto1\tensor w \,+\, w \tensor1. \]
This will always define a map on free cga's, but in general might not respect the differential. But since the differential is trivial, this defines a map of cdga's. We have the following diagram:
This will always define a map on free cga's, but in general might not respect the differential. But since the differential is trivial, this defines a map of cdga's. We have the following diagram:
\[\xymatrix @R =0.4cm{
\[\xymatrix @R =0.4cm{
@ -89,7 +89,7 @@ The homotopy groups of augmented cdga's are precisely the dual of the homotopy g
For the upper triangle we note that $KY$ is in now a simplicial monoid (induced by the map $\Delta$), and we know from \cite{goerss} that the multiplication on the homotopy groups of a simplicial monoid is induced by the monoid structure.
For the upper triangle we note that $KY$ is in now a simplicial monoid (induced by the map $\Delta$), and we know from \cite{goerss} that the multiplication on the homotopy groups of a simplicial monoid is induced by the monoid structure.
For the bottom triangle we first note that the isomorphism $\pi^n(Y)\oplus\pi^n(Y)\iso\pi^n(Y \tensor Y)$ follows from \LemmaRef{Q-preserves-copord}. Now the induced map $QY \to Q(Y \tensor Y)\iso Q(Y)\oplus QY$ is defined as $w \mapsto(w, w)$, and so the dual is precisely addition.
For the bottom triangle we first note that the isomorphism $\pi^n(Y)\oplus\pi^n(Y)\iso\pi^n(Y \tensor Y)$ follows from \LemmaRef{Q-preserves-copord}. Now the induced map $QY \tot{Q \Delta} Q(Y \tensor Y)\iso Q(Y)\oplus QY$ is defined as $w \mapsto(w, w)$, and so the dual is precisely addition.
So the multiplication on the homotopy groups $\pi_n(KY)$ and $\pi^n(Y)^\ast$ are induced by the same map. So by the commutativity of the above diagram the natural bijection is a group isomorphism.
So the multiplication on the homotopy groups $\pi_n(KY)$ and $\pi^n(Y)^\ast$ are induced by the same map. So by the commutativity of the above diagram the natural bijection is a group isomorphism.
}
}
@ -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 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.
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(-;\Q)$ 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}{
\Corollary{A-preserves-htpy-pullbacks}{
Given the following pullback diagram of spaces
Given the following pullback diagram of spaces
@ -135,7 +135,7 @@ Another exposition of this corollary can be found in \cite[Section 8.4]{berglund
\section{Equivalence on rational spaces}
\section{Equivalence on rational spaces}
In this section we will prove that the adjunction in \CorollaryRef{minimal-model-adjucntion} is in fact an equivalence when restricted to certain subcategories. One of the restriction is the following.
In this section we will prove that the adjunction in \CorollaryRef{minimal-model-adjucntion} is in fact an equivalence when restricted to certain subcategories. One of the restrictions is the following.
\Definition{finite-type}{
\Definition{finite-type}{
A cdga $A$ is said to be of \Def{finite type} if $H(A)$ is finite dimensional in each degree. Similarly $X$ is of \Def{finite type} if $H^i(X; \Q)$ is finite dimensional for each $i$.
A cdga $A$ is said to be of \Def{finite type} if $H(A)$ is finite dimensional in each degree. Similarly $X$ is of \Def{finite type} if $H^i(X; \Q)$ is finite dimensional for each $i$.
@ -196,9 +196,9 @@ Now we wish to use the previous lemma as an induction step for minimal models. L
In particular if the vector space $V'$ is finitely generated, we can repeat this procedure for all basis elements (it does not matter in what order we do so, as $dv \in\Lambda V(n)$). So in this case, if $(\Lambda V(n), d)\to A(K(\Lambda V(n), d))$ is a weak equivalence, so is$(\Lambda V(n+1), d)\to A(K(\Lambda V(n+1), d))$
In particular if the vector space $V'$ is finitely generated, we can repeat this procedure for all basis elements (it does not matter in what order we do so, as $dv \in\Lambda V(n)$). So in this case where $V'$ is finite-dimensional, if $(\Lambda V(n), d)\to A(K(\Lambda V(n), d))$ is a weak equivalence, then by the above lemma$(\Lambda V(n+1), d)\to A(K(\Lambda V(n+1), d))$ is a weak equivalence as well.
Note that by \RemarkRef{finited-dim-minimal-model} every cdga of finite type has a minimal model in which the generating set is finite dimensional in each degree.
Note that by \RemarkRef{finited-dim-minimal-model} every cdga of finite type has a minimal model which is finite dimensional in each degree.
\Corollary{cdga-unit-we}{
\Corollary{cdga-unit-we}{
Let $(\Lambda V, d)$ be a $1$-connected minimal algebra with $V^i$ finite dimensional for all $i$. Then $(\Lambda V, d)\to A(K(\Lambda V, d))$ is a weak equivalence.
Let $(\Lambda V, d)$ be a $1$-connected minimal algebra with $V^i$ finite dimensional for all $i$. Then $(\Lambda V, d)\to A(K(\Lambda V, d))$ is a weak equivalence.
@ -209,10 +209,10 @@ Note that by \RemarkRef{finited-dim-minimal-model} every cdga of finite type has
Now $V(n)$ is finitely generated for all $n$ by assumption. By the inductive procedure above we see that $(\Lambda V(n), d)\to A(K(\Lambda V(n), d))$ is a weak equivalence for all $n$. Hence $(\Lambda V, d)\to A(K(\Lambda V, d))$ is a weak equivalence.
Now $V(n)$ is finitely generated for all $n$ by assumption. By the inductive procedure above we see that $(\Lambda V(n), d)\to A(K(\Lambda V(n), d))$ is a weak equivalence for all $n$. Hence $(\Lambda V, d)\to A(K(\Lambda V, d))$ is a weak equivalence.
}
}
Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by \CorollaryRef{serre-whitehead} (the Serre-Whitehead theorem). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X)))\to A(X)$ is a weak equivalence.
Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by the Serre-Whitehead theorem (\CorollaryRef{serre-whitehead}). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X)))\to A(X)$ is a weak equivalence.
\Lemma{}{
\Lemma{}{
The map $X \to K(M(X))$ is a weak equivalence for simply connected rational spaces $X$ of finite type.
The map $X \to K(M(X))$ is a weak equivalence for $1$-connected, rational spaces $X$ of finite type.
}
}
\Proof{
\Proof{
Recall that the map $X \to K(M(X))$ was defined to be the composition of the actual unit of the adjunction and the map $K(m_X)$. When applying $A$ we get the following situation, where commutativity is ensured by the adjunction laws:
Recall that the map $X \to K(M(X))$ was defined to be the composition of the actual unit of the adjunction and the map $K(m_X)$. When applying $A$ we get the following situation, where commutativity is ensured by the adjunction laws:
@ -220,13 +220,13 @@ Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply con
A(X) &\ar[l] A(K(A(X))) &\ar[l] A(K(M(X))) \\
A(X) &\ar[l] A(K(A(X))) &\ar[l] A(K(M(X))) \\
&\ar[lu]^\id A(X) \ar[u]&\arwe[l] M(X) \ar[u]
&\ar[lu]^\id A(X) \ar[u]&\arwe[l] M(X) \ar[u]
}\]
}\]
The map on the right is a weak equivalence by \CorollaryRef{cdga-unit-we}. Then by the 2-out-of-3 property we see that the above composition is indeed a weak equivalence. Since $A$ detects weak equivalences (Serre-Whitehead), we conclude that $X \to K(M(X))$ is a weak equivalence.
The map on the right is a weak equivalence by \CorollaryRef{cdga-unit-we}. Then by the 2-out-of-3 property we see that the above composition is indeed a weak equivalence. Since $A$ detects weak equivalences, we conclude that $X \to K(M(X))$ is a weak equivalence.
}
}
We have proven the following theorem.
We have proven the following theorem.
\Theorem{main-theorem}{
\Theorem{main-theorem}{
The functors $A$ and $K$ induce an equivalence of homotopy categories, when restricted to rational, $1$-connected objects of finite type. More formally, we have:
The functors $A$ and $K$ induce an equivalence of homotopy categories, when restricted to rational, $1$-connected objects of finite type. More formally, we have:
$$\Ho(\sSet_1^{\Q,f})\iso\Ho(\CDGA_{\Q,1,f}). $$
$$\Ho(\sSet_{\Q,1,f})\iso\Ho(\CDGA_{\Q,1,f}). $$
Furthermore, for any $1$-connected space $X$ of finite type, we have the following isomorphism of groups:
Furthermore, for any $1$-connected space $X$ of finite type, we have the following isomorphism of groups:
@ -48,7 +48,7 @@ The generators $e$ and $f$ in the last proof are related by the so called \Def{W
$$\pi_\ast(S^n)\tensor\Q=\text{the free whitehead algebra on 1 generator}. $$
$$\pi_\ast(S^n)\tensor\Q=\text{the free whitehead algebra on 1 generator}. $$
}
}
Together with the fact that all groups $\pi_i(S^n)$ are finitely generated (this was proven by Serre in \cite{serre}) we can conclude that $\pi_i(S^n)$ is a finite group unless $i=n$ and unless $i=2n-1$ for even $n$. 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 is proven by Serre in \cite{serre}) we can conclude that $\pi_i(S^n)$ is a finite group unless $i=n$ and unless $i=2n-1$ for even $n$. 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).
\section{Eilenberg-MacLane spaces}
\section{Eilenberg-MacLane spaces}
@ -69,12 +69,16 @@ The following result is already used in proving the main theorem. But using the
This means that $V$ is concentrated in degree $n$ and that the differential is trivial. Take a generator $x$ of degree $n$ such that $V =\Q\cdot x$ and conclude that the cohomology of the minimal model, and hence the rational cohomology of $K(\Z, n)$, is $H(\Lambda V, 0)=\Q[x]$.
This means that $V$ is concentrated in degree $n$ and that the differential is trivial. Take a generator $x$ of degree $n$ such that $V =\Q\cdot x$ and conclude that the cohomology of the minimal model, and hence the rational cohomology of $K(\Z, n)$, is $H(\Lambda V, 0)=\Q[x]$.
}
}
Note that both the result on the spheres and this result are very different in ordinary homotopy theory. The ordinary homotopy groups of the spheres are very hard to calculate and in many cases even unknown. Similarly the (co)homology of Eilenberg-MacLane spaces is complicated (but known). In rational homotopy theory, both are easy to calculate.
Another remarkable thing happens here, the odd spheres are rationally equivalent to Eilenberg-MacLane spaces. In a further section we will briefly see that this allows us to prove that $S^n_\Q$ is an H-space if and only if $n$ is odd.
\section{Products}
\section{Products}
% page 142 and 248
% page 142 and 248
Let $X$ and $Y$ be two $1$-connected spaces, we will determine the minimal model for $X \times Y$. We have the two projections maps $X \times Y \tot{\pi_X} X$ and $X \times Y \tot{\pi_Y} Y$ which induces maps of cdga's: $A(X)\tot{{\pi_X}_\ast} A(X \times Y)$ and $A(Y)\tot{{\pi_Y}_\ast} A(X \times Y)$. Because we are working with commutative algebras, we can multiply the two maps to obtain:
Let $X$ and $Y$ be two $1$-connected spaces, we will determine the minimal model for $X \times Y$. We have the two projections maps $X \times Y \tot{\pi_X} X$ and $X \times Y \tot{\pi_Y} Y$ which induces maps of cdga's: $A(X)\tot{{\pi_X}_\ast} A(X \times Y)$ and $A(Y)\tot{{\pi_Y}_\ast} A(X \times Y)$. Because we are working with commutative algebras, we can multiply the two maps to obtain:
This is different from the singular cochain complex where the Eilenberg-Zilber map is needed. However by passing to cohomology the multiplication is identified with the cup product. Hence, by applying the Künneth theorem, we see that $\mu$ is a weak equivalence.
This is different from the singular cochain complex where the Eilenberg-Zilber map is needed. By passing to cohomology the multiplication is identified with the cup product. Hence, by applying the Künneth theorem, we see that $\mu$ is a weak equivalence.
Now if $M_X =(\Lambda V, d_X)$ and $M_Y =(\Lambda W, d_Y)$ are the minimal models for $A(X)$ and $A(Y)$, we see that $M_X \tensor M_Y \we A(X)\tensor A(Y)$ is a weak equivalence (again by the Künneth theorem). Furthermore $M_X \tensor M_Y =(\Lambda V \tensor\Lambda W, d_X \tensor d_Y)$ is itself minimal, with $V \oplus W$ as generating space. As a direct consequence we see that
Now if $M_X =(\Lambda V, d_X)$ and $M_Y =(\Lambda W, d_Y)$ are the minimal models for $A(X)$ and $A(Y)$, we see that $M_X \tensor M_Y \we A(X)\tensor A(Y)$ is a weak equivalence (again by the Künneth theorem). Furthermore $M_X \tensor M_Y =(\Lambda V \tensor\Lambda W, d_X \tensor d_Y)$ is itself minimal, with $V \oplus W$ as generating space. As a direct consequence we see that
\begin{align*}
\begin{align*}
@ -105,11 +109,11 @@ for some element $\psi \in H^{+}(X; \Q) \tensor H^{+}(X; \Q)$. This means that t
Choose a subspace $V$ of $H^+(X; \Q)$ such that $H^+(X; \Q)= V \oplus H^+(X; \Q)\cdot H^+(X; \Q)$. In particular we get $V^1= H^1(X; \Q)$ and $H^2(X; \Q)= V^2\oplus H^1(X; \Q)\cdot H^1(X; \Q)$. Continuing with induction we see that the induced map $\phi : \Lambda V \to H^\ast(X; \Q)$ is surjective. One can prove (by induction on the degree and using the counitality) that the elements in $V$ are primitive, i.e. $\mu^\ast(v)=1\tensor v + v \tensor1$ for all $v \in V$. The free algebra also admits a comultiplication, by requiring that the generators are the primitive elements. It follows that the following diagram commutes:
Choose a subspace $V$ of $H^+(X; \Q)$ such that $H^+(X; \Q)= V \oplus H^+(X; \Q)\cdot H^+(X; \Q)$. In particular we get $V^1= H^1(X; \Q)$ and $H^2(X; \Q)= V^2\oplus H^1(X; \Q)\cdot H^1(X; \Q)$. Continuing with induction we see that the induced map $\phi : \Lambda V \to H^\ast(X; \Q)$ is surjective. One can prove (by induction on the degree and using the counitality) that the elements in $V$ are primitive, i.e. $\mu^\ast(v)=1\tensor v + v \tensor1$ for all $v \in V$. The free algebra also admits a comultiplication, by requiring that the generators are the primitive elements. It follows that the following diagram commutes:
\[\xymatrix{
\[\xymatrix{
\Lambda V \ar[r]^\phi\ar[d]^\Delta& H^\ast(X; \Q) \ar[d]^{\mu^\ast}\\
\Lambda V \ar[r]^-\phi\ar[d]^\Delta& H^\ast(X; \Q) \ar[d]^{\mu^\ast}\\
\Lambda V \tensor\Lambda V \ar[r]^{\phi\tensor\phi}& H^\ast(X; \Q) \tensor H^\ast(X; \Q) \\
\Lambda V \tensor\Lambda V \ar[r]^-{\phi\tensor\phi}& H^\ast(X; \Q) \tensor H^\ast(X; \Q) \\
}\]
}\]
We will now prove that $\phi$ is also injective. Suppose by induction that $\phi$ is injective on $\Lambda V^{<n}$. An element $w \in\Lambda V^{\leq n}$ can be written as $\sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_r^{k_r} a_{k_1\cdots k_r}$, where $\{v_1, \ldots, v_r\}$ is a basis for $V^n$ and $a_{k_1\cdots k_r}\in\Lambda V^{<n}$. Assume $\phi(w)=0$. Let $\pi : H^\ast(X; \Q)\to H^\ast(X; \Q)/\phi(\Lambda V^{<n})$is the (linear) projection map. Now consider the image of $(\pi\tensor\id)\mu^\ast(\phi(w)$ in the component $\im(\pi)^n \tensor H^\ast(X; \Q)$, it can be written as (here we use the above commuting square):
We will now prove that $\phi$ is also injective. Suppose by induction that $\phi$ is injective on $\Lambda V^{<n}$. An element $w \in\Lambda V^{\leq n}$ can be written as $\sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_r^{k_r} a_{k_1\cdots k_r}$, where $\{v_1, \ldots, v_r\}$ is a basis for $V^n$ and $a_{k_1\cdots k_r}\in\Lambda V^{<n}$. Assume $\phi(w)=0$. Let $\pi : H^\ast(X; \Q)\to H^\ast(X; \Q)/\phi(\Lambda V^{<n})$be the (linear) projection map. Now consider the image of $(\pi\tensor\id)\mu^\ast(\phi(w)$ in the component $\im(\pi)^n \tensor H^\ast(X; \Q)$, it can be written as (here we use the above commuting square):
As $\phi(w)=0$ and the elements $\pi(v_i)$ are linearly independent, we see that $\phi(\sum_{k_1, \ldots, k_r} k_i v_1^{k_1}\cdots v_i^{k_i -1}\cdots v_r^{k_r}a_{k_1\cdots k_r})=0$ for all $i$. By induction on the degree of $w$ (the base case being $\deg{w}= n$ is trivial), we conclude that
As $\phi(w)=0$ and the elements $\pi(v_i)$ are linearly independent, we see that $\phi(\sum_{k_1, \ldots, k_r} k_i v_1^{k_1}\cdots v_i^{k_i -1}\cdots v_r^{k_r}a_{k_1\cdots k_r})=0$ for all $i$. By induction on the degree of $w$ (the base case being $\deg{w}= n$ is trivial), we conclude that
\section{Localizations at primes and the arithmetic square}
\section{Quillen's approach}
In \ChapterRef{Serre} we proved some results by Serre to relate homotopy groups and homology groups modulo a class of abelian groups. Now the class of $p$-torsion groups and the class of $p$-divisible groups are also Serre classes. This suggests that we can also ``localize homotopy theory at primes''. Indeed the construction in section \ChapterRef{Rationalization} can be altered to give a $p$-localization $X_p$ of a space $X$. Recall that for the rationalization we constructed a telescope with a sphere for each $k > 0$. For the $p$-localization we only add a copy of the sphere for $k > 0$ relative prime to $p$.
Now that we have a bunch of localizations $X_\Q, X_2, X_3, X_5, \ldots$ we might wonder what homotopical information of $X$ we can recover from these localizations. In other words: can we go from local to global? The answer is yes in the following sense. Details can be found in \cite{may2} and \cite{sullivan}.
\Theorem{arithmetic-square}{
Let $X$ be a space, then $X$ is the homotopy pullback in
\[\xymatrix{
X \ar[r]\ar[d]&\prod_{p\text{ prime}} X_p \ar[d]\\
X_\Q\ar[r]& (\prod_{p\text{ prime}} X_p)_\Q
}\]
}
This theorem is known as \emph{the arithmetic square}, \emph{fracture theorem} or \emph{local-to-global theorem}.
As an example we find that if $X$ is an H-space, then so are its localizations. The converse also holds when certain compatibility requirements are satisfied \cite{sullivan}. In the previous section we were able to prove that $S^n_\Q$ is an H-space if and only if $n$ is odd. It turns out that the prime $p=2$ brings the key to Adams' theorem: for odd $n$ we have that $S^n_2$ is an H-space if and only if $n=1, 3$ or $7$. For the other primes $S^n_p$ is always an H-space for odd $n$. This observation leads to one approach to prove Adams' theorem.
\section{Quillen's approach to rational homotopy theory}
In this thesis we used Sullivan's approach to give algebraic models for rational spaces. However, Sullivan was not the first to give algebraic models. Quillen gave a dual approach in \cite{quillen}. By a long chain of homotopy equivalences his main result is
In this thesis we used Sullivan's approach to give algebraic models for rational spaces. However, Sullivan was not the first to give algebraic models. Quillen gave a dual approach in \cite{quillen}. By a long chain of homotopy equivalences his main result is
The first category is the one of differential graded Lie algebras over $\Q$ and the second is cocommutative (coassociative) differential graded coalgebras. Quillen's approach does not need the finite dimensionality assumptions and is hence more general.
The first category is the one of differential graded Lie algebras over $\Q$ and the second is cocommutative (coassociative) differential graded coalgebras. Quillen's approach does not need the finite dimensionality assumptions and is hence more general.
Minimal models in these categories also exist, as shown in \cite{neisendorfer}. They are defined analogously, we require the object to be cofibrant (or fibrant in the coalgebra case) and that the differential is zero in the chain complex of indecomposables. Of course the meaning of indecomposable depends on the category.
Minimal models in these categories also exist, as shown in \cite{neisendorfer}. They are defined analogously, we require the object to be cofibrant (or fibrant in the case of coalgebras) and that the differential is zero in the chain complex of indecomposables. Of course the meaning of indecomposable depends on the category.
Despite the generality of Quillen's approach, the author of this thesis \todo{ok?}prefers the approach by Sullivan as it provides a single, elegant functor $A: \sSet\to\CDGA$. Moreover cdga's are easier to manipulate, as commutative ring theory is a more basic subject than Lie algebras or coalgebras.
Despite the generality of Quillen's approach, the author of this thesis prefers the approach by Sullivan as it provides a single, elegant functor $A: \sSet\to\CDGA$. Moreover cdga's are easier to manipulate, as commutative ring theory is a more basic subject than Lie algebras or coalgebras.
\section{Nilpotency}
\section{Nilpotency}
@ -36,6 +19,21 @@ In many locations in this thesis we assumed simply connectedness of objects (bot
In short, a nilpotent group is a group which is constructed by finitely many extensions of abelian groups. A space is called nilpotent if its fundamental group is nilpotent and the action of $\pi_1$ on $\pi_n$ satisfies a related requirement.
In short, a nilpotent group is a group which is constructed by finitely many extensions of abelian groups. A space is called nilpotent if its fundamental group is nilpotent and the action of $\pi_1$ on $\pi_n$ satisfies a related requirement.
Now the base cases in our proofs become more complicated, as we need another inductive argument (on these extensions of abelian groups) in the base case.
Many theorems remain valid when assuming nilpotent spaces instead of simply connected spaces. However the proofs are complicated, as they need another inductive argument on these extensions of abelian groups in the base case.
\todo{note $\Q$-completion?}
\section{Localizations at primes}
In \ChapterRef{Serre} we proved some results by Serre to relate homotopy groups and homology groups modulo a class of abelian groups. Now the class of $p$-torsion groups and the class of $p$-divisible groups are also Serre classes. This suggests that we can also ``localize homotopy theory at primes''. Indeed the construction in \ChapterRef{Rationalization} can be altered to give a $p$-localization $X_p$ of a space $X$. Recall that the rationalization was constructed as a telescope with a copy of the sphere for each $k > 0$. The $k$th copy was used in order to divid by $k$. For the $p$-localization we only add a copy of the sphere for $k > 0$ relative prime to $p$.
Now that we have a bunch of localizations $X_\Q, X_2, X_3, X_5, \ldots$ we might wonder what homotopical information of $X$ we can recover from these localizations. In other words: can we go from local to global? The answer is yes in the following sense. Details can be found in \cite{may2} and \cite{sullivan}.
\Theorem{arithmetic-square}{
Let $X$ be a space, then $X$ is the homotopy pullback in
\[\xymatrix{
X \ar[r]\ar[d]&\prod_{p\text{ prime}} X_p \ar[d]\\
X_\Q\ar[r]& (\prod_{p\text{ prime}} X_p)_\Q
}\]
}
This theorem is known as \emph{the arithmetic square}, \emph{fracture theorem} or \emph{local-to-global theorem}.
As an example we find that if $X$ is an H-space, then so are its localizations. The converse also holds when certain compatibility requirements are satisfied \cite{sullivan}. In the previous section we were able to prove that $S^n_\Q$ is an H-space if and only if $n$ is odd. It turns out that the prime $p=2$ brings the key to Adams' theorem: for odd $n$ we have that $S^n_2$ is an H-space if and only if $n=1, 3$ or $7$. For the other primes $S^n_p$ is always an H-space for odd $n$. This observation leads to one approach to prove Adams' theorem.
@ -13,7 +13,7 @@ In this section we will discuss the so called minimal models. These cdga's enjoy
such that $d(V(k))\subset\Lambda V(k-1)$.
such that $d(V(k))\subset\Lambda V(k-1)$.
\end{itemize}
\end{itemize}
An cdga $(A, d)$ is a \Def{minimal Sullivan algebra} if in addition
A cdga $(A, d)$ is a \Def{minimal Sullivan algebra} if in addition
\begin{itemize}
\begin{itemize}
\item$d$ is decomposable, i.e. $\im(d)\subset\Lambda^{\geq2}V$.
\item$d$ is decomposable, i.e. $\im(d)\subset\Lambda^{\geq2}V$.
\end{itemize}
\end{itemize}
@ -24,28 +24,28 @@ In this section we will discuss the so called minimal models. These cdga's enjoy
$$(M, d)\we(A, d). $$
$$(M, d)\we(A, d). $$
\end{definition}
\end{definition}
We will often say \Def{minimal model} or \Def{minimal algebra} to mean minimal Sullivan model or minimal Sullivan algebra. Note that a minimal algebra is naturally augmented by the freeness. This will be used implicitly. In many cases we can take the degree of the elements in $V$ to induce the filtration, as seen in the following lemma.
We will often say \Def{minimal model} or \Def{minimal algebra} to mean minimal Sullivan model or minimal Sullivan algebra. Note that a minimal algebra is naturally augmented as it is free as an algebra. This will be used implicitly. In many cases we can take the degree of the elements in $V$ to induce the filtration, as seen in the following lemma.
\Lemma{1-reduced-minimal-model}{
\Lemma{1-reduced-minimal-model}{
Let $(A, d)$ be a cdga which is $1$-reduced, such that $A =\Lambda V$ is free as cga. Then the differential $d$ is decomposable if and only if $(A, d)$ is a Sullivan algebra filtered by degree.
Let $(A, d)$ be a cdga which is $1$-reduced, such that $A =\Lambda V$ is free as cga. Then the differential $d$ is decomposable if and only if $(A, d)$ is a Sullivan algebra filtered by degree.
}
}
\Proof{
\Proof{
Let $V$ be filtered by degree: $V(k)= V^{\leq k}$. Now $d(v)\in\Lambda V^{< k}$ for any $v \in V^k$. For degree reasons $d(v)$ is a product, so $d$ is decomposable.
Let $V$ be filtered by degree: $V(k)= V^{\leq k}$\todo{is this notation introduced?}. Now $d(v)\in\Lambda V^{< k}$ for any $v \in V^k$. For degree reasons $d(v)$ is a product, so $d$ is decomposable.
For the converse take $V(n)=\bigoplus_{k=0}^n V^k$ (note that $V^0= V^1=0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v)= x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees:
For the converse take $V(n)=V^{\leq n}$ (note that $V^0= V^1=0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v)= x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees:
$$\deg{x}+\deg{y}=\deg{xy}=\deg{dv}=\deg{v}+1= n +1. $$
$$\deg{x}+\deg{y}=\deg{xy}=\deg{dv}=\deg{v}+1= n +1. $$
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 $d(V(k))\subset\Lambda(V(n-1))$.
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 $d(V(k))\subset\Lambda(V(n-1))$.
}
}
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 $\pi^n(\Lambda V)= V^n$.
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}
\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. The first can be easily proven by choosing subspaces with bases $V'_k =\langle v_j \rangle_{j \in J_k}$ such that $V(k)= V(k-1)\oplus V'_k$ for each degree. Then choose some well order on $J_k$ to define a well order on $J =\bigcup_k J_k$. The second lemma is a more refined version of \LemmaRef{1-reduced-minimal-model}. Since we will not need these equivalent definitions, the details are left out.
\Lemma{}{
\Lemma{sullivan-hess}{
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 $d v_j \in\Lambda V_{<j}$.
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 $d v_j \in\Lambda V_{<j}$.
}
}
\Lemma{}{
\Lemma{minimal-hess}{
Let $(\Lambda V, d)$ be a Sullivan algebra with $V^0=0$, then $d$ is decomposable if and only if there is a well order $J$ as above such that $i < j$ implies $\deg{v_i}\leq\deg{v_j}$.
Let $(\Lambda V, d)$ be a Sullivan algebra with $V^0=0$, then $d$ is decomposable if and only if there is a well order $J$ as above such that $i < j$ implies $\deg{v_i}\leq\deg{v_j}$.
}
}
@ -57,18 +57,18 @@ It is clear that induction will be an important technique when proving things ab
Let $(A, d)$ be a $1$-connected cdga, then it has a minimal model $(\Lambda V, d)$.
Let $(A, d)$ be a $1$-connected cdga, then it has a minimal model $(\Lambda V, d)$.
\end{theorem}
\end{theorem}
\begin{proof}
\begin{proof}
We construct the model and by induction on the degree. The resulting filtration will be on degree, so that the minimality follows from \LemmaRef{1-reduced-minimal-model}. We start with$V^0= V^1=0$ and $V^2= H^2(A)$ and$d(V^2)=0$, this extends to a map of cdga's $m_2 : \Lambda V^{\leq2}\to A$.
We construct the model and by induction on the degree. The resulting filtration will be on degree, so that the minimality follows from \LemmaRef{1-reduced-minimal-model}. We start by setting$V^0= V^1=0$ and $V^2= H^2(A)$. At this stage the differential is trivial, i.e.$d(V^2)=0$. Sending the cohomology classes to their representatives extends to a map of cdga's $m_2 : \Lambda V^{\leq2}\to A$.
Suppose $m_k : \Lambda V^{\leq k}\to A$ is constructed. We will add elements in degree $k+1$ and extend $m_k$ to $m_{k+1}$ to assert surjectivity and injectivity of $H(m_{k+1})$. Let $\{[a_\alpha]\}_{\alpha\in I}$ be a basis for the cokernel of $H(m_k) : H^{k+1}(\Lambda V^{\leq k})\to H^{k+1}(A)$ and $b_\alpha\in A^{k+1}$ be a representing cycle for $a_\alpha$. Let $\{[z_\beta]\}_{\beta\in J}$ be a basis for the kernel of $H(m_k) : H^{k+2}(\Lambda V^{\leq k})\to H^{k+2}(A)$, note that $m_k(b_\beta)$ is a boundary, so that there are elements $c_\beta$ such that $m_k(b_\beta)= d c_\beta$.
Suppose $m_k : \Lambda V^{\leq k}\to A$ is constructed. We will add elements in degree $k+1$ and extend $m_k$ to $m_{k+1}$ to assert surjectivity and injectivity of $H(m_{k+1})$.\, Let $\{[a_\alpha]\}_{\alpha\in I}$ be a basis for the cokernel of $H(m_k) : H^{k+1}(\Lambda V^{\leq k})\to H^{k+1}(A)$ and $b_\alpha\in A^{k+1}$ be a representing cycle for $a_\alpha$.\, Let $\{[z_\beta]\}_{\beta\in J}$ be a basis for the kernel of $H(m_k) : H^{k+2}(\Lambda V^{\leq k})\to H^{k+2}(A)$, note that $m_k(b_\beta)$ is a boundary, so that there are elements $c_\beta$ such that $m_k(b_\beta)= d c_\beta$.
Define $V^{k+1}=\bigoplus_{\alpha\in I}\k\cdot v_\alpha\oplus\bigoplus_{\beta\in J}\k\cdot v'_\beta$ and extend $d$ and $m_{k+1}$ by defining
Define $V^{k+1}=\bigoplus_{\alpha\in I}\k\cdot v_\alpha\oplus\bigoplus_{\beta\in J}\k\cdot v'_\beta$ and extend $d$ and $m_{k+1}$ by defining
Now clearly $d^2=0$ on the generators, so this extends to a derivation on $\Lambda V^{k+1}$, similarly $m_{k+1}$ commutes with $d$ on the generators and hence extends to a chain map.
Now clearly $d^2=0$ on the generators, so this extends to a derivation on $\Lambda V^{\leqk+1}$, similarly $m_{k+1}$ commutes with $d$ on the generators and hence extends to a chain map.
This finished the construction of $V$ and $m : \Lambda V \to A$. Now we will prove that $H(m)$ is an isomorphism. We will do so by proving surjectivity and injectivity by induction on $k$.
This finished the construction of $V$ and $m : \Lambda V \to A$. Now we will prove that $H(m)$ is an isomorphism. We will do so by proving surjectivity and injectivity by induction on $k$.
Start by noting that $H^i(m_2)$ is corrective for $i \leq2$. now assume $H^i(m_k)$ is surjective for $i \leq k$. Since $\im H(m_k)\subset\im H(m_{k+1})$ we see that $H^i(m_{k+1})$ is surjective for $i < k+1$. By construction it is also surjective in degree $k+1$. So $H^i(m_k)$ is surjective for all $i \leq k$ for all $k$.
Start by noting that $H^i(m_2)$ is surjective for $i \leq2$. now assume $H^i(m_k)$ is surjective for $i \leq k$. Since $\im H(m_k)\subset\im H(m_{k+1})$ we see that $H^i(m_{k+1})$ is surjective for $i < k+1$. By construction it is also surjective in degree $k+1$. So $H^i(m_k)$ is surjective for all $i \leq k$ for all $k$.
For injectivity we note that $H^i(m_2)$ is injective for $i \leq3$, since $\Lambda V^{\leq2}$ has no elements of degree $3$. Assume $H^i(m_k)$ is injective for $i \leq k+1$ and let $[z]\in\ker H^i(m_{k+1})$. Now if $\deg{z}\leq k$ we get $[z]=0$ by induction and if $\deg{z}= k+2$ we get $[z]=0$ by construction. Finally if $\deg{z}= k+1$, then we write $z =\sum\lambda_\alpha v_\alpha+\sum\lambda'_\beta v'_\beta+ w$ where $v_\alpha, v'_\beta$ are the generators as above and $w \in\Lambda V^{\leq k}$. Now $d z =0$ and so $\sum\lambda'_\beta v'_\beta+ dw =0$, so that $\sum\lambda'_\beta[z_\beta]=0$. Since $\{[z_\beta]\}$ was a basis, we see that $\lambda'_\beta=0$ for all $\beta$. Now by applying $m_k$ we get $\sum\lambda_\alpha[b_\alpha]= H(m_k)[w]$, so that $\sum\lambda_\alpha[a_\alpha]=0$ in the cokernel, recall that $\{[a_\alpha]\}$ formed a basis and hence $\lambda_\alpha=0$ for all $\alpha$. Now $z = w$ and the statement follows by induction. Conclude that $H^i(m_{k+1})$ is injective for $i \leq k+2$.
For injectivity we note that $H^i(m_2)$ is injective for $i \leq3$, since $\Lambda V^{\leq2}$ has no elements of degree $3$. Assume $H^i(m_k)$ is injective for $i \leq k+1$ and let $[z]\in\ker H^i(m_{k+1})$. Now if $\deg{z}\leq k$ we get $[z]=0$ by induction and if $\deg{z}= k+2$ we get $[z]=0$ by construction. Finally if $\deg{z}= k+1$, then we write $z =\sum\lambda_\alpha v_\alpha+\sum\lambda'_\beta v'_\beta+ w$ where $v_\alpha, v'_\beta$ are the generators as above and $w \in\Lambda V^{\leq k}$. Now $d z =0$ and so $\sum\lambda'_\beta v'_\beta+ dw =0$, so that $\sum\lambda'_\beta[z_\beta]=0$. Since $\{[z_\beta]\}$ was a basis, we see that $\lambda'_\beta=0$ for all $\beta$. Now by applying $m_k$ we get $\sum\lambda_\alpha[b_\alpha]= H(m_k)[w]$, so that $\sum\lambda_\alpha[a_\alpha]=0$ in the cokernel, recall that $\{[a_\alpha]\}$ formed a basis and hence $\lambda_\alpha=0$ for all $\alpha$. Now $z = w$ and the statement follows by induction. Conclude that $H^i(m_{k+1})$ is injective for $i \leq k+2$.
@ -76,14 +76,14 @@ It is clear that induction will be an important technique when proving things ab
\end{proof}
\end{proof}
\Remark{finited-dim-minimal-model}{
\Remark{finited-dim-minimal-model}{
The above construction will construct a $r$-reduced minimal model for an $r$-connected cdga $A$.
The above construction will construct an$r$-reduced minimal model for an $r$-connected cdga $A$.
Moreover if $H(A)$ is finite dimensional in each dimension, then so is the minimal model $\Lambda V$. This follows inductively. First notice that $V^2$ is clearly finite dimensional. Now assume that $\Lambda V^{<k}$ is finite dimension in each degree, then both the cokernel and kernel are, so we adjoin only finitely many elements in $V^k$.
Moreover if $H(A)$ is finite dimensional in each degree, then so is the minimal model $\Lambda V$. This follows inductively. First notice that $V^2$ is clearly finite dimensional. Now assume that $\Lambda V^{<k}$ is finite dimensional in each degree, then both the cokernel and kernel are, so we adjoin only finitely many elements in $V^k$.
}
}
\section{Uniqueness}
\section{Uniqueness}
Before we state the uniqueness theorem we need some more properties of minimal models. In this section we will use some general facts about model categories.
Before we state the uniqueness theorem we need some more properties of minimal models. In fact we will prove that Sullivan algebras are cofibrant. This allows us to use some general facts about model categories.
\begin{lemma}
\begin{lemma}
Sullivan algebras are cofibrant and the inclusions induced by the filtration are cofibrations.
Sullivan algebras are cofibrant and the inclusions induced by the filtration are cofibrations.
@ -131,7 +131,7 @@ As minimal models are cofibrant \RemarkRef{cdga-weak-eq-bijection} immediately i
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 $V'$:
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 $V'$:
\[\pi^\ast(\phi) : V \tot{\iso} V'. \]
\[\pi^\ast(\phi) : V \tot{\iso} V'. \]
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.
By induction on the degree one can prove that $\phi$is surjective and hence it 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 applies to $\psi$ and so$\psi$ is surjective. Conclude that $\psi$ is an isomorphism and $\phi$, being its right inverse, is an isomorphism as well.
\end{proof}
\end{proof}
\Theorem{unique-minimal-model}{
\Theorem{unique-minimal-model}{
@ -150,15 +150,14 @@ The assignment to $X$ of its minimal model $M_X = (\Lambda V, d)$ can be extende
\end{displaymath}
\end{displaymath}
Now by \LemmaRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} : [M_X, Y]\iso[M_X, M_Y]$. This gives a map $M(f)={m_Y}_\ast^{-1}(f m_X)$ from $M_X$ to $M_Y$. Of course this does not define a functor of cdga's as it is only well defined on homotopy classes. However it is clear that it does define a functor on the homotopy category of cdga's.
Now by \LemmaRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} : [M_X, Y]\iso[M_X, M_Y]$. This gives a map $M(f)={m_Y}_\ast^{-1}(f m_X)$ from $M_X$ to $M_Y$. Of course this does not define a functor of cdga's as it is only well defined on homotopy classes. However it is clear that it does define a functor on the homotopy category of cdga's.
\Corollary{}{
\Corollary{minimal-model-equivalence}{
The assignment $X \mapsto M_X$ defines a functor $M: \Ho(\CDGA^1_\Q)\to\Ho(\CDGA^1_\Q)$. Moreover, since the minimal model is weakly equivalent, $M$ gives an equivalence of categories:
The assignment $X \mapsto M_X$ defines a functor $M: \Ho(\CDGA_{\k,1})\to\Ho(\CDGA_{\k,1})$. Moreover, since the minimal model is weakly equivalent, $M$ gives an equivalence of categories:
where weakly equivalent cdga's are sent to \emph{isomorphic} minimal models.
}
}
\section{The minimal model of the sphere}
\section{The minimal model of the sphere}
We know from singular cohomology that the cohomology ring of a $n$-sphere is $\Z[X]/(X^2)$. This allows us to construct a minimal model for $S^n$.
We know from singular cohomology that the cohomology ring of a $n$-sphere is $\Z[X]/(X^2)$, i.e. the cga with 1 generator $X$ in degree $n$ such that $X^2=0$. This allows us to construct a minimal model for $S^n$.
\Definition{minimal-model-sphere}{
\Definition{minimal-model-sphere}{
Define $A(n)$ to be the cdga defined as
Define $A(n)$ to be the cdga defined as
$$ A(n)=\begin{cases}
$$ A(n)=\begin{cases}
@ -167,4 +166,8 @@ We know from singular cohomology that the cohomology ring of a $n$-sphere is $\Z
\end{cases}. $$
\end{cases}. $$
}
}
\todo{prove $HA(n)\tot{\iso} HA(S^n)$}
To prove that this indeed defines minimal models, we first note that $A(n)$ indeed has the same cohomology groups. All we need to prove is that there is an actual weak equivalence $A(n)\to A(S^n)$.
For the odd case, we can choose a representative $y \in A(S^n)$ for the generator $X$. Sending $e$ to $y$ defines a map $\phi: A(n)\to A(S^n)$. Note that since $\deg{y}$ is odd we have $y^2=0$ by commutativity of $A(S^n)$, so indeed $\phi$ is a map of algebras. Both $e$ and $y$ are cocycles, so $\phi$ is a chain map. Finally we see that $H(\phi)$ sends $[e]$ to $X$, hence this is an isomorphism.
For the even case, we need to choose two elements in $A(S^n)$. Again let $y \in A(S^n)$ be a representative for $X$. Now since $X^2=0$ there is an element $c \in A(S^n)$ such that $y^2= d c$. Sending $e$ to $y$ and $f$ to $c$ defines a map of cdga's $\phi : A(n)\to A(S^n)$. And $H(\phi)$ sends the class $[e]$ to $X$.
Joshua Moerman
10 years ago