@ -109,7 +109,68 @@ Choose a subspace $V$ of $H^+(X; \Q)$ such that $H^+(X; \Q) = V \oplus H^+(X; \Q
\Lambda V \tensor\Lambda V \ar[r]^{\phi\tensor\phi}& H^\ast(X; \Q) \tensor H^\ast(X; \Q) \\
}\]
We will now prove that $\phi$ is in fact injective. Suppose by induction that $\phi$ is injective on $\Lambda V^{<n}$. An element $w \in\Lambda V^{\leq n}$ can be written as $\Sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_r^{k_r} a_{k_1\cdots k_r}$, where $\{v_1, \ldots, v_r\}$ is a basis for $V^n$ and $a_{k_1\cdots k_r}\in\Lambda V^{<n}$. Let $\pi : H^\ast(X; \Q)\to H^\ast(X; \Q)/\phi(\Lambda V^{<n})$ is the (linear) projection map. Now consider the image of $(\pi\tensor\id)\mu^\ast(\phi(w))$ in the component $\im(\pi)\tensor H^\ast(X; \Q)$, it can be written as (here we use the above commuting square):
We will now prove that $\phi$ is also injective. Suppose by induction that $\phi$ is injective on $\Lambda V^{<n}$. An element $w \in\Lambda V^{\leq n}$ can be written as $\sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_r^{k_r} a_{k_1\cdots k_r}$, where $\{v_1, \ldots, v_r\}$ is a basis for $V^n$ and $a_{k_1\cdots k_r}\in\Lambda V^{<n}$. Assume $\phi(w)=0$. Let $\pi : H^\ast(X; \Q)\to H^\ast(X; \Q)/\phi(\Lambda V^{<n})$ is the (linear) projection map. Now consider the image of $(\pi\tensor\id)\mu^\ast(\phi(w))$ in the component $\im(\pi)^n\tensor H^\ast(X; \Q)$, it can be written as (here we use the above commuting square):
As $\phi(w)=0$ and the elements $\pi(v_i)$ are linearly independent, we see that $\phi(\sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_i^{k_i -1}\cdots v_r^{k_r}a_{k_1\cdots k_r})=0$ for all $i$. By induction on the degree of $w$ (the base case being $\deg{w}= n$ is trivial), we conclude that
\[\sum_{k_1, \ldots, k_r} v_1^{k_1}\cdots v_i^{k_i -1}\cdots v_r^{k_r}a_{k_1\cdots k_r}=0\text{ for all } i\]
This means that either all $k_i =0$, in which case $w \in\Lambda V^{<n}$ and so $w =0$ by induction, or all $a_{k_1, \ldots, k_r}=0$, in which case we have $w =0$. This proves that $\phi$ is injective.
We have proven that $\phi : \Lambda V \to H^\ast(H; \Q)$ is an isomorphism. So the cohomology of an H-space is free as cga. Now we can choose cocycles in $A(X)$ which represent the cohomology classes. More precisely for $v_i^{(n)}\in V^n$ we choose $w_i^{(n)}\in A(X)^n$ representing it. This defines a map, which sends $v_i^{(n)}$ to $w_i^{(n)}$. Since $w_i^{(n)}$ are cocycles, this is a map of cdga's:
\[ m : (\Lambda V, 0)\to A(X)\]
Now by the calculation above, this is a weak equivalence. Furthermore $(\Lambda V, 0)$ is minimal. We have proven the following lemma:
\Lemma{H-spaces-minimal-models}{
Let $X$ be a $0$-connected H-space of finite type. Then its minimal model is of the form $(\Lambda V, 0)$. In particular we see:
\[ H(X; \Q)=\Lambda V \qquad\pi_\ast(X)\tensor\Q= V^\ast\]
}
This allows us to directly relate the rational homotopy groups (recall that $\pi_n(X)\tensor X ={V^n}^\ast$) to the cohomology groups.
\Corollary{spheres-not-H-spaces}{
The spheres $S^n$ are not H-spaces if $n$ is even.
}
In fact we have that $S^n_\Q$ is an H-space if and only if $n$ is odd. The only if part is precisely the corollary above, the if part follows from the fact that $S^n_\Q$ is the loop space $K(\Q^\ast, n)$ for odd $n$.
\todo{$SO(n)$?}
We will end this chapter with some related topics.
\section{Localizations at primes and the arithmetic square}
In \ChapterRef{Serre} we proved some results by Serre to relate homotopy groups and homology groups modulo a class of abelian groups. Now the class of $p$-torsion groups and the class of $p$-divisible groups are also Serre classes. This suggests that we can also ``localize homotopy theory at primes''. Indeed the construction in section \ChapterRef{Rationalization} can be altered to give a $p$-localization $X_p$ of a space $X$. Recall that for the rationalization we constructed a telescope with a sphere for each $k > 0$. For the $p$-localization we can construct a telescope only for $k > 0$ relative prime to $p$.
Now that we have a bunch of localizations $X_\Q, X_2, X_3, X_5, \ldots$ we might wonder what homotopical information of $X$ we can reconstruct from these localizations. In other words: can we go from local to global? The answer is yes in the following sense. For details we refer to \cite{may2} and \cite{sullivan}.
\Theorem{arithmetic-square}{
Let $X$ be a space, then $X$ is the homotopy pullback in
\[\xymatrix{
X \ar[r]\ar[d]&\prod_{p\text{ prime}} X_p \ar[d]\\
X_\Q\ar[r]& (\prod_{p\text{ prime}} X_p)_\Q
}\]
}
This theorem is known as \emph{the arithmetic square}, \emph{fracture theorem} or \emph{local-to-global theorem}.
As an example we find that if $X$ is an H-space, then so are its localizations. The converse also holds when certain compatibility requirements are satisfied \cite{sullivan}. In the previous section we were able to prove that $S^n_\Q$ is an H-space if and only if $n$ is odd. It turns out that the prime $p=2$ brings the key to Adams's theorem: for odd $n$ we have that $S^n_2$ is an H-space if and only if $n=1, 3$ or $7$. For the other primes $S^n_p$ is always an H-space for odd $n$. This observation leads to one approach to prove Adams' theorem.
\section{Quillen's approach to rational homotopy theory}
In this thesis we used Sullivan's approach to give algebraic models for rational spaces. However, Sullivan was not the first to give algebraic models. Quillen gave a dual approach in \cite{quillen}. By a long chain of homotopy equivalences his main result is
The first category is the one of differential graded Lie algebras over $\Q$ and the second is cocommutative (coassociative) differential graded coalgebras. Quillen's approach does not need the finite dimensionality assumptions and is hence more general.
Minimal models in these categories also exist, as shown in \cite{neisendorfer}. They are defined analogously, we require the object to be cofibrant (of fibrant in the coalgebra case) and that the differential is zero in the chain complex of indecomposables. Of course the meaning of indecomposable depends on the category.
Despite the generality of Quillen's approach, the author of this thesis \todo{ok?} preferes the approach by Sullivan as it provides a single, elegant functor $A: \sSet\to\CDGA$. Moreover cdga's are easier to manipulate as commutative ring theory is a more basic subject than Lie algebras or coalgebras.
\section{Nilpotency}
In many localtions in this thesis we assumed simply connectedness of objects (both spaces an cdga's). The assumption was often use to prove the base case of some inductive argument. In \cite{bousfield} the main equivalence is proven for so called nilpotent spaces (which is more general than $1$-connected spaces).
In short, a nilpotent group is a group which is constructed by finitely many extensions of abelian groups. A space is called nilpotent if its fundamental group is nilpotent and the action of $\pi_1$ on $\pi_n$ satisfies a related requirement.
Now the base cases in our proofs become more complicated, as we need another inductive argument (on these extensions of abelian groups) in the base case.
\chapter{Serre theorems mod \texorpdfstring{$\C$}{C}}
\Chapter{Serre theorems mod \texorpdfstring{$\C$}{C}}{Serre}
In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in \cite{serre}. In his paper he considered homology groups `modulo a class of abelian groups'. In our case of rational homotopy theory, this class will be the class of torsion groups.
Joshua Moerman
10 years ago