Adds group iso (stub)

thesis/notes/A_K_Quillen_Pair.tex

@@ -52,10 +52,10 @@ The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for
 
 
 \section{Homotopy groups}
-The homotopy groups of cdga's are precisely the dual of the homotopy groups of their associated spaces.
+The homotopy groups of augmented 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
+	Let $X$ be a cofibrant augmented cdga, then there is a natural bijection
 	$$ \pi_n(KX) \iso \pi^n(X)^\ast. $$
 }
 \Proof{
@@ -71,7 +71,20 @@ The homotopy groups of cdga's are precisely the dual of the homotopy groups of t
 	\todo{Group structure?}
 }
 
-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.
+\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 $.
+}
+\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 $w \in W$:
+	\[ \Delta : \Lambda W \to \Lambda W \tensor \Lambda W : w \mapsto 1 \tensor w + w \tensor 1. \]
+	Since the differential is trivial, this defines a map of cdga's. \todo{show the induced maps in $\pi$}
+}
+
+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?}
 
 \Corollary{minimal-cdga-homotopy-groups}{
 	For a minimal cdga $X = \Lambda V$ we get