Moves the section on homotopy groups to chapter 4

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -31,6 +31,9 @@ In this chapter the ring $\k$ is assumed to be a field of characteristic zero. I
 \section{Homotopy relations on \titleCDGA}
 \input{notes/Homotopy_Relations_CDGA}
 
+\section{Homotopy groups of cdga's}
+\input{notes/Homotopy_Groups_CDGA}
+
 
 \chapter{Polynomial Forms}
 \label{sec:cdga-of-polynomials}

thesis/notes/A_K_Quillen_Pair.tex

@@ -51,46 +51,8 @@ The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for
 }
 
 
-\section{Homotopy groups of cdga's}
-We are after an equivalence of homotopy categories, so it is natural to ask what the homotopy groups of $K(A)$ are for a cdga $A$. In order to do so, we will define homotopy groups of cdga's directly and compare the two notions.
-
-Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ such that $\counit \unit = \id$.
-
-\Definition{cdga-homotopy-groups}{
-	Define the \Def{augmentation ideal} of $A$ as $\overline{A} = \ker \counit$. Define the \Def{cochain complex of indecomposables} of $A$ as $QA = \overline{A} / \overline{A} \cdot \overline{A}$.
-
-	Now define the \Def{homotopy groups of a cdga} $A$ as
-	$$ \pi^i(A) = H^i(QA). $$
-}
-
-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 $\pi^n(\Lambda V) = V^n$.
-
-\Lemma{cdga-homotopic-maps-equal-pin}{
-	Let $f: A \to B$ be a map of augmented cdga's. Then there is an functorial induced map on the homotopy groups. Moreover if $g: A \to B$ is homotopic to $f$, then the induced maps are equal:
-	$$ f_\ast = g_\ast : \pi_\ast(A) \to \pi_\ast(B). $$
-}
-\Proof{
-	Let $\phi: A \to B$ be a map of algebras. Then clearly we get an induced map $\overline{A} \to \overline{B}$ as $\phi$ preserves the augmentation. By composition we get a map $\phi': \overline{A} \to Q(B)$ for which we have $\phi'(xy) = \phi'(x)\phi'(y) = 0$. So it induces a map $Q(\phi): Q(A) \to Q(B)$. By functoriality of taking homology we get $f_\ast : \pi^n(A) \to \pi^n(B)$.
-
-	Now if $f$ and $g$ are homotopic, then there is a homotopy $h: A \to \Lambda(t, dt) \tensor B$. By the Künneth theorem we have:
-	$$ {d_0}_\ast = {d_1}_\ast : H(\Lambda(t, dt) \tensor Q(B)) \to H(Q(B)). $$
-	This means that $f_\ast = {d_1}_\ast h_\ast = {d_0}_\ast h_\ast = g_\ast$. \todo{detail}
-}
-
-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. There is a weak equivalence $A(n) \to V(n)$ (recall \DefinitionRef{minimal-model-sphere}). This augmented cdga can be thought of as a specific model of the sphere. In particular the homotopy groups can be expressed as follows.
-
-\Lemma{cdga-dual-homotopy-groups}{
-	There is a natural bijection for any augmented cdga $A$
-	$$ [A, V(n)] \tot{\iso} \Hom_\k(\pi^n(A), \k). $$
-}
-\Proof{
-	Note that $Q(V(n))$ in degree $n$ is just $\k$ and $0$ in the other degrees, so its homotopy group consists of a single $\k$ in degree $n$. This establishes the map:
-	$$ \Phi: \Hom_\CDGA(A, V(n)) \to \Hom_\k(\pi^n(A), \k). $$
-
-	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}
-}
-
-We will denote the dual of a vector space as $V^\ast = \Hom_\k(V, \k)$.
+\section{Homotopy groups}
+The homotopy groups of cdga's are precisely the dual of the homotopy groups of their associated spaces.
 
 \Theorem{cdga-dual-homotopy-groups}{
 	Let $X$ be a cofibrant augmented cdga, then
@@ -120,6 +82,7 @@ We get a particularly nice result for minimal cdga's, because the functor $Q$ is
 	For a cdga with one generator $X = \Lambda(v)$ with $d v = 0$ and $\deg{v} = n$. We conclude that $KX$ is a $K(\k^\ast, n)$-space.
 }
 
+
 \section{The Eilenberg-Moore theorem}
 Before we prove the actual equivalence, we will discuss a theorem of Eilenberg and Moore. The theorem tells us that the singular cochain complex of a pullback along a fibration is nice in a particular way. The theorem and its proof (using spectral sequences) can be found in \cite[Theorem 7.14]{mccleary}.
 

thesis/notes/Homotopy_Groups_CDGA.tex

@@ -0,0 +1,38 @@
+
+As the eventual goal is to compare the homotopy theory of spaces with the homotopy theory of cdga's, it is natural to investigate an analogue of homotopy groups in the category of cdga's. In topology we can only define homotopy groups on pointed spaces, dually we will consider augmented cdga's in this section. Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ such that $\counit \unit = \id$.
+
+\Definition{cdga-homotopy-groups}{
+	Define the \Def{augmentation ideal} of $A$ as $\overline{A} = \ker \counit$. Define the \Def{cochain complex of indecomposables} of $A$ as $QA = \overline{A} / \overline{A} \cdot \overline{A}$.
+
+	Now define the \Def{homotopy groups of a cdga} $A$ as
+	$$ \pi^i(A) = H^i(QA). $$
+}
+
+This construction is functorial and, as the following lemma shows, homotopy invariant.
+
+\Lemma{cdga-homotopic-maps-equal-pin}{
+	Let $f: A \to B$ be a map of augmented cdga's. Then there is an functorial induced map on the homotopy groups. Moreover if $g: A \to B$ is homotopic to $f$, then the induced maps are equal:
+	$$ f_\ast = g_\ast : \pi_\ast(A) \to \pi_\ast(B). $$
+}
+\Proof{
+	Let $\phi: A \to B$ be a map of algebras. Then clearly we get an induced map $\overline{A} \to \overline{B}$ as $\phi$ preserves the augmentation. By composition we get a map $\phi': \overline{A} \to Q(B)$ for which we have $\phi'(xy) = \phi'(x)\phi'(y) = 0$. So it induces a map $Q(\phi): Q(A) \to Q(B)$. By functoriality of taking homology we get $f_\ast : \pi^n(A) \to \pi^n(B)$.
+
+	Now if $f$ and $g$ are homotopic, then there is a homotopy $h: A \to \Lambda(t, dt) \tensor B$. By the Künneth theorem we have:
+	$$ {d_0}_\ast = {d_1}_\ast : H(\Lambda(t, dt) \tensor Q(B)) \to H(Q(B)). $$
+	This means that $f_\ast = {d_1}_\ast h_\ast = {d_0}_\ast h_\ast = g_\ast$. \todo{detail}
+}
+
+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.
+
+\Lemma{cdga-dual-homotopy-groups}{
+	There is a natural bijection for any augmented cdga $A$
+	$$ [A, V(n)] \tot{\iso} \Hom_\k(\pi^n(A), \k). $$
+}
+\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:
+	$$ \Phi: \Hom_\CDGA(A, V(n)) \to \Hom_\k(\pi^n(A), \k). $$
+
+	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}
+}
+
+From now on the dual of a vector space will be denoted as $V^\ast = \Hom_\k(V, \k)$. So the above lemma states that there is a bijection $[A, V(n)] \iso \pi^n(A)^\ast$.

thesis/notes/Minimal_Models.tex

@@ -154,6 +154,12 @@ Now by \LemmaRef{minimal-model-bijection} we get a bijection ${m_Y}_\ast^{-1} :
 	where weakly equivalent cdga's are sent to \emph{isomorphic} minimal models.
 }
 
+
+\section{Homotopy groups of minimal models}
+\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{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 $S^n$.
 \Definition{minimal-model-sphere}{