@ -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:
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}.
}
\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 $w \in W$:
\[\Delta : \Lambda W \to\Lambda W \tensor\Lambda W : w \mapsto1\tensor w + w \tensor1. \]
Since the differential is trivial, this defines a map of cdga's. \todo{show the induced maps in $\pi$}
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 $w \in W$:
\[\Delta : Y \to Y \tensor Y : \quad w \mapsto1\tensor w \,+\, w \tensor1. \]
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:
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
$$\pi_n(KX)={V^n}^\ast. $$
}
Let $M =(\Lambda V, d)$ be a minimal algebra, then the homotopy groups of $KM$ are $\pi_n(KM)={V^n}^\ast$.
\Corollary{minimal-cdga-EM-space}{
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.
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 $K(\k^\ast, n)$.
}
@ -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 $\deg{v}= n \geq1$. 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}.
Joshua Moerman
9 years ago