Adds prove of pi_n = pi^n as groups

thesis/notes/A_K_Quillen_Pair.tex

@@ -37,10 +37,10 @@ Since $A$ is a left adjoint, it preserves all colimits and by functoriality it p
 	$A$ preserves all cofibrations and all trivial cofibrations and hence is a left Quillen functor.
 \end{corollary}
 
-\begin{corollary}
+\Corollary{ak-homotopy-adjunction}{
 	$A$ and $K$ induce an adjunction on the homotopy categories:
 	$$ LA : \Ho(\sSet) \leftadj \opCat{\Ho(\CDGA)} : RK. $$
-\end{corollary}
+}
 
 The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for $X \in \sSet$ (note that every simplicial set is already cofibrant) and $RK(Y) = K(Y^{cof})$ for $Y \in \CDGA$. By the use of minimal models, and in particular the functor $M$. We get the following adjunction between $1$-connected objects:
 
@@ -59,40 +59,47 @@ The homotopy groups of augmented cdga's are precisely the dual of the homotopy g
 	$$ \pi_n(KX) \iso \pi^n(X)^\ast. $$
 }
 \Proof{
-	First note that $KX$ is a Kan complex (because it is a simplicial group). Using the homotopy adjunction and the lemma above we get:
+	We will prove the following chain of natural isomorphisms:
 	\begin{align*}
-		\pi_n(KX) &= [S^n, KX]      \\
+		\pi_n(KX) = [S^n, KX] &\ison{1} [X, A(S^n)] \\
-		          &\iso [X, A(S^n)] \\
+		          \ison{2} [X, A(n)] &\ison{3} [X, V(n)] \ison{4} \pi^n(X)^\ast
-		          &\iso [X, A(n)]   \\
-		          &\iso [X, V(n)]   \\
-		          &\iso \pi^n(X)^\ast
 	\end{align*}
-	\todo{Prove all isomorphisms.}
+	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}.
-	\todo{Group structure?}
 }
 
 \Theorem{cdga-dual-homotopy-groups-iso}{
-	Let $X$ be a $1$-connected cofibrant augmented cdga, then there is a natural group isomorphism $ \pi_n(KX) \iso \pi^n(X)^\ast $.
+	Let $X$ be a $1$-connected cofibrant augmented cdga, then the above bijection is a group isomorphism $\pi_n(KX) \iso \pi^n(X)^\ast$.
 }
 \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 $\pi^n(X)^\ast$.
 
 	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 $QX$.
 	
-	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 $W = W^n$. 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 $Y = (\Lambda, 0)$. Define a comultiplication on generators $w \in W$:
-	\[ \Delta : \Lambda W \to \Lambda W \tensor \Lambda W : w \mapsto 1 \tensor w + w \tensor 1. \]
+	\[ \Delta : Y \to Y \tensor Y : \quad w \mapsto 1 \tensor w \,+\, w \tensor 1. \]
-	Since the differential is trivial, this defines a map of cdga's. \todo{show the induced maps in $\pi$}
+	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:
+	\[ \xymatrix @R = 0.4cm{
+		\pi_n(KY) \times \pi_n(KY) \ar[d]^-\iso \ar[rd]^-\mu       & \\
+		\pi_n(KY \times KY) \ar@{-->}[dd]^-\iso \ar[r]^-{\Delta^\ast}& \pi_n(KY) \ar@{-->}[dd]^-\iso \\
+		                                                           &               \\
+		\pi^n(Y \tensor Y)^\ast \ar[d]^-\iso \ar[r]^-{\Delta^\ast} & \pi^n(Y)^\ast \\
+		\pi^n(Y)^\ast \oplus \pi^n(Y)^\ast \ar[ru]_-{+} \\
+	}\]
+	The middle part commutes by the naturality of the isomorphisms described in \TheoremRef{cdga-dual-homotopy-groups}. The only thing we need to prove is that the upper triangle and bottom triangle commute.\
+
+	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.
+
+	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.
 }
 
-We get a particularly nice result for minimal cdga's, because the functor $Q$ is the left inverse of the functor $\Lambda$ and the differential is decomposable. \todo{this remark has already been made earlier?}
+Recall that for a minimal model $M = (\Lambda V, d)$ the homotopy groups equal $\pi^n(M) = V^n$. So in particular we know the homotopy groups of the space $KM$.
 
 \Corollary{minimal-cdga-homotopy-groups}{
-	For a minimal cdga $X = \Lambda V$ we get
+	Let $M = (\Lambda V, d)$ be a minimal algebra, then the homotopy groups of $KM$ are $\pi_n(KM) = {V^n}^\ast$.
-	$$ \pi_n(KX) = {V^n}^\ast. $$
-}
 
-\Corollary{minimal-cdga-EM-space}{
+	In particular, for a cdga with only one generator $M = \Lambda(v)$ with $d v = 0$ and $\deg{v} = n$, we conclude that $KM$ is an Eilenberg-MacLane space of type $K(\k^\ast, n)$.
-	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.
 }
 
 
@@ -135,13 +142,11 @@ For the equivalence of rational spaces and cdga's we need that the unit and coun
 where the first of the two maps is given by the composition $X \to K(A(X)) \tot{K(m_X)} K(M(X))$,
 and the second map is obtained by the map $A \to A(K(A))$ and using the bijection from \LemmaRef{minimal-model-bijection}: $[A, A(K(A))] \iso [A, M(K(A))]$. By the 2-out-of-3 property the map $A \to M(K(A))$ is a weak equivalence if and only if the ordinary unit $A \to A(K(A))$ is a weak equivalence.
 
-\todo{state all theorems we need but do not prove}
-
 \Lemma{}{
 	(Base case) Let $A = (\Lambda(v), 0)$ be a minimal model with one generator of degree $\deg{v} = n \geq 1$. Then $A \we A(K(A))$.
 }
 \Proof{
-	By \CorollaryRef{minimal-cdga-EM-space} 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:
 	$$ H^\ast(K(\Q^\ast, n); \Q) = \Q[x], $$
 	that is, the free commutative graded algebra with one generator $x$. This can be calculated, for example, with spectral sequences \cite{griffiths}.
 

thesis/preamble.tex

@@ -95,6 +95,8 @@
 \newcommand{\del}{\partial}				% boundary
 \newcommand{\iso}{\cong}				% isomorphic
 \newcommand{\eq}{\sim}					% homotopic
+\newcommand{\ison}[1]{\stackrel{(#1)}{\iso}} % isos to refer to
+\newcommand{\refison}[1]{{\small $(#1)$}}    % ref
 \newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}}	% arrow with name
 \newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}}	% left arrow with name
 \newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}}	% mapsto with name