@ -8,7 +8,9 @@ Recall that a cdga $A$ is a commutative differential graded algebra, meaning tha
\item it has an associative and unital multiplication: $\mu: A \tensor A \to A$ and
\item it is commutative: $x y =(-1)^{\deg{x}\cdot\deg{y}} y x$.
\end{itemize}
And all of the above structure is compatible with each other (e.g. the differential is a derivation of degree $1$, the maps are graded, \dots). We have a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules.
And all of the above structure is compatible with each other (e.g. the differential is a derivation of degree $1$, the maps are graded, \dots). The exact requirements are stated in the appendix on algebra. An algebra $A$ is augmented if it has a specified map (of algebras) $A \tot{\counit}\k$. Furthermore we adopt the notation $A^{\leq n}=\bigoplus_{k \leq n} A^k$ and similarly for $\geq n$.
There is a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules. We denote the subspace of elements of wordlength $n$ by $\Lambda^n V$ (note that this has nothing to do with the grading on $V$).
In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. maps $f: A \to B$ inducing an isomorphism on cohomology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.
@ -20,10 +22,6 @@ We furthermore have the following categorical properties of cdga's:
\item$\k$ and $0$ are the initial and final object.
\end{itemize}
In this chapter the ring $\k$ is assumed to be a field of characteristic zero. In particular the modules are vector spaces.
\todo{augmentations?}\todo{Kunneth from appendix}
\section{Cochain models for the $n$-disk and $n$-sphere}
@ -36,7 +36,7 @@ We assume the reader is familiar with category theory, basics from algebraic top
We will fix the following notations and categories.
\begin{itemize}
\item$\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-modules, tensor products over $\k$,\dots.
\item$\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-vector spaces, tensor products over $\k$,\dots.
\item$\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
\item$\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
\item$\Ab$: category of abelian groups and group homomorphisms.
@ -3,9 +3,8 @@ We will first define some basic cochain complexes which model the $n$-disk and $
$$ D(n)= ... \to0\to\k\to\k\to0\to ... $$
$$ S(n)= ... \to0\to\k\to0\to0\to ... $$
Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0\to D(n)$ induces an isomorphism in cohomology. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n))=\k\cdot[a]$. There is an injective function $i_n : S(n+1)\to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes:
Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0\to D(n)$ induces an isomorphism in cohomology. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n))=\k\cdot[a]$. There is an injective function $i_n : S(n+1)\to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes, where weak equivalences are quasi isomorphisms, fibrations are degreewise surjective and cofibrations are degreewise injective for positive degrees \cite[Example 1.6]{goerss2}.
\todo{Introduceer de model structuur}
\begin{claim}
The set $I =\{i_n : S(n+1)\to D(n)\I n \in\N\}$ generates all cofibrations and the set $J =\{j_n : 0\to D(n)\I n \in\N\}$ generates all trivial cofibrations.
@ -28,9 +28,16 @@ Consider the augmented cdga $V(n) = S(n) \oplus \k$, with trivial multiplication
}
\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:
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}
Now by \LemmaRef{cdga-homotopic-maps-equal-pin} we get a map from the set of homotopy classes $[A, V(n)]$ instead of the $\Hom$-set. It remains to prove that the map is an isomorphism. Surjectivity follows easily. Given a map $f: \pi^n(A)\to\k$, we can extend this to $A \to V(n)$ because the multiplication on $V(n)$ is trivial.
For injectivity suppose $\phi, \psi: A \to V(n)$ be two maps such that $\pi^n(\phi)=\pi^n(\psi)$. We will first define a chain homotopy $D: A^\ast\to V(n)^{\ast-1}$, for this we only need to specify the map $D^n: A^{n+1}\to V(n)^n =\Q$. Decompose the vector space $A^{n+1}$ as $A^{n+1}=\im d \oplus V$ for some $V$. Now set $D^n(v)=0$ for all $v \in V$ and $D^n(db)=\phi(b)-\psi(b)$. We should check that $D$ is well defined. Note that for cycles we get $\phi(c)=\psi(c)$, as $H(Q(\phi))= H(Q(\psi))$. So if $db = dc$, then we get $D(db)=\phi(b)-\psi(b)=\phi(c)-\psi(c)= D(dc)$, i.e. $D$ is well defined. We can now define a map of augmented cdga's:
\begin{align*}
h : X &\to\Lambda(t, dt) \overline{\tensor} V(n) \\
x &\mapsto dt \tensor D(x) + 1 \tensor\phi(x) - t \tensor\phi(x) + t \tensor\psi(x)
\end{align*}
This map commutes with the differential by the definition of $D$. Now we see that $d_0 h =\psi$ and $d_1 h =\phi$. Hence the two maps represent the same class, and we have proven the injectivity.
}
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 $[A, V(n)]\iso\pi^n(A)^\ast$.
@ -201,8 +201,6 @@ By using the class of $\Q$-vector spaces we get a dual theorem.
Let $X$ be a $1$-connected space. The homotopy groups $\pi_i(X)$ are $\Q$-vector spaces for all $i > 0$ if and only if $H_i(X)$ are $\Q$-vector spaces for all $i > 0$.
}
\todo{$\pi$ is $\Q$ local iff $H$ is}
\TheoremRef{serre-whitehead} also applies verbatim to rational homotopy theory. However we would like to avoid the assumption that $\pi_2(f)$ is surjective. In \cite{felix} we find a way to work around this.
\Corollary{rational-whitehead}{
@ -217,13 +215,13 @@ By using the class of $\Q$-vector spaces we get a dual theorem.
\[\xymatrix @=0.4cm{
Y \arwe[rr]^-\lambda\ar[dr]_-q && Y \times_K MK \arfib[dl]^-{\overline{q}}\\
& K &
}\]\todo{$MK =$?}
Now $\overline{q}\lambda f$ is homotopic to the constant map, so there is a homotopy $h: \overline{q}\lambda f \eq\ast$ which we can lift against the fibration $\overline{q}$ to a homotopy $h' : \lambda f \eq f_1$ with $\overline{q} f_1=\ast$. In other words $f_1$ lands in the fiber of $\overline{q}$.
}\]
Where $MK$ is the Moore path space and $\overline{q}$ is induced by the map sending a path to its endpoint.\todo{read}Now $\overline{q}\lambda f$ is homotopic to the constant map, so there is a homotopy $h: \overline{q}\lambda f \eq\ast$ which we can lift against the fibration $\overline{q}$ to a homotopy $h' : \lambda f \eq f_1$ with $\overline{q} f_1=\ast$. In other words $f_1$ lands in the fiber of $\overline{q}$.
The important observation is that by the long exact sequence $\pi_\ast(i)\tensor\Q$ and $H_\ast(i; \Q)$ are isomorphisms (here we use that $\Gamma\tensor\Q=0$ and that tensoring with $\Q$ is exact). So by the above square $\pi_\ast(f_1)\tensor\Q$ is an isomorphism if and only if $\pi_\ast(f)\tensor\Q$ is (and similarly for homology). Finally we note that $\pi_2(f_1)$ is surjective\todo{?}, so \TheoremRef{serre-whitehead} applies and the result also holds for $f$.
The important observation is that by the long exact sequence $\pi_\ast(i)\tensor\Q$ and $H_\ast(i; \Q)$ are isomorphisms (here we use that $\Gamma\tensor\Q=0$ and that tensoring with $\Q$ is exact). So by the above square $\pi_\ast(f_1)\tensor\Q$ is an isomorphism if and only if $\pi_\ast(f)\tensor\Q$ is (and similarly for homology). Finally we note that $\pi_2(f_1)$ is surjective, so \TheoremRef{serre-whitehead} applies and the result also holds for $f$.
Joshua Moerman
9 years ago