@ -52,10 +52,10 @@ The induced adjunction in the previous corollary is given by $LA(X) = A(X)$ for
\section{Homotopy groups}
\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}{
\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
$$\pi_n(KX)\iso\pi^n(X)^\ast. $$
$$\pi_n(KX)\iso\pi^n(X)^\ast. $$
}
}
\Proof{
\Proof{
@ -71,7 +71,20 @@ The homotopy groups of cdga's are precisely the dual of the homotopy groups of t
\todo{Group structure?}
\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 $w \in W$:
\[\Delta : \Lambda W \to\Lambda W \tensor\Lambda W : w \mapsto1\tensor w + w \tensor1. \]
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?}