In this section we will prove that the rational cohomology of an H-space is free as commutative graded algebra. We will also give its minimal model and relate it to the homotopy groups. In some sense H-spaces are homotopy generalizations of topological monoids. In particular topological groups (and hence Lie groups) are H-spaces.
\Definition{H-space}{
An H-space is a pointed topological space $x_0\in X$ with a map $\mu: X \times X \to X$, such that $\mu(x_0, -), \mu(-, x_0) : X \to X$ are homotopic to $\id_X$.
An \Def{H-space} is a pointed topological space $x_0\in X$ with a map $\mu: X \times X \to X$, such that $\mu(x_0, -), \mu(-, x_0) : X \to X$ are homotopic to $\id_X$.
}
Let $X$ be an H-space, then we have the induced map $\mu^\ast: H^\ast(X; \Q)\to H^\ast(X; \Q)\tensor H^\ast(X; \Q)$ on cohomology. Because homotopic maps are sent to equal maps in cohomology, we get $H^\ast(\mu(x_0, -))=\id_{H^\ast(X; \Q)}$. Now write $H^\ast(\mu(x_0, -))=(\counit\tensor\id)\circ H^\ast(\mu)$, where $\counit$ is the augmentation induced by $x_0$, to conclude that for any $h \in H^{+}(X; \Q)$ the image is of the form
@ -30,7 +30,7 @@ Note that for a rational space $X$, the ordinary homotopy groups are isomorphic
Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence.
The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{Quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1)=\Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences).
The theory of rational homotopy theory is the study of simple spaces with rational equivalences. Quillen defines a model structure on simply connected simplicial sets with rational equivalences as weak equivalences \cite{quillen}. This means that there is a homotopy category $\Ho^\Q(\sSet_1)$. However we will later prove that every simply connected space has a rationalization, so that $\Ho^\Q(\sSet_1)=\Ho(\sSet^\Q_1)$ are equivalent categories. This means that we do not need the model structure defined by Quillen, but we can simply restrict ourselves to rational spaces (with ordinary weak equivalences).
\section{Classical results from algebraic topology}
\chapter{Serre theorems mod \texorpdfstring{$\C$}{C}}
\label{sec: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.
\Definition{serre-class}{
A class $\C\subset\Ab$ is a \Def{Serre class} if
\begin{itemize}
\item for all exact sequences $0\to A \to B \to C \to0$ with $A$ and $C$ in $\C$, $B$ also belongs to $\C$,
\item$\C$ is closed under taking direct sums (both finite and infinite).
\item for all exact sequences $0\to A \to B \to C \to0$ if two abelian groups are in $\C$, then so is the third,
\item for all $A \in\C$ the tensor product $A \tensor B$ is in $\C$ for any abelian group $B$,
\item for all $A \in\C$ the Tor group $\Tor(A, B)$ is in $\C$ for any abelian group $B$, and
\item for all $A \in\C$ the group-homology $H_i(A; \Z)$ is in $\C$ for all positive $i$.
\end{itemize}
}
@ -23,37 +24,18 @@ Serre gave weaker axioms for his classes and proves some of the following lemmas
\end{itemize}
}
The most important properties we need of a Serre class are the following:
\Lemma{Serre-properties}{
Let $\C\subset\Ab$ be a Serre class. Then we have:
\begin{enumerate}
\item If $A \in\C$, then $A \tensor B \in\C$ for all $B$.
\item If $A \in\C$, then $H(A)\in\C$ ($H(A)$ is the group homology of $A$).
\end{enumerate}
}
As noted by Hilton in \cite{hilton} we think of Serre classes as a generalized 0. This means that we can also express some kind of generalized injective and surjectivity. Here we only need the notion of a $\C$-isomorphism:
\Definition{serre-class-maps}{
Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a $\C$-isomorphism if both the kernel and cokernel lie in $\C$.
Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a \Def{$\C$-isomorphism} if both the kernel and cokernel lie in $\C$.
}
Note that the map $0\to C$ is a $\C$-isomorphism for any $C \in\C$. \todo{Er missen nog wat eigenschappen voor tensors}
\Lemma{serre-class-rational-iso}{
Let $\C$ be the Serre class of all torsion groups. Then
$f$ is a $\C$-iso $\iff$$f \tensor\Q$ is an isomorphism.
}
\Proof{
First note that if $C \in\C$ then $C \tensor\Q=0$.
Then consider the exact sequence
$$0\to\ker(f)\to A \tot{f} B \to\coker(f)\to0$$
and tensor this sequence with $\Q$. In this tensored sequence the kernel and cokernel vanish if and only if $f \tensor\Q$ is an isomorphism.
}
Note that the maps $0\to C$ and $C \to0$ are $\C$-isomorphisms for any $C \in\C$. More importantly the 5-lemma also holds for $\C$-isos and whenever $f$, $g$ and $g \circ f$ are maps such that two of them are $\C$-iso, then so is the third.
In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more elementary proof using cellular homology.
In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more geometric proof.
\Lemma{kreck}{
Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $1$-connected spaces and $F$ its fiber. If $\RH_i(F)\in\C$ for all $i < n$, then
Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $0$-connected spaces and $F$ its fiber. If $\RH_i(F)\in\C$ for all $i < n$, then
\begin{itemize}
\item$H_i(E, F)\to H_i(B, b_0)$ is a $\C$-iso for $i \leq n+1$ and
\item$H_i(E)\to H_i(B)$ is a $\C$-iso for all $i \leq n$.
@ -95,17 +77,18 @@ In the following arguments we will consider fibrations and need to compute homol
}
\Lemma{homology-em-space}{
Let $\C$ be a Serre class and $C\in\C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(C, n))\in\C$.
Let $\C$ be a Serre class and $G\in\C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(G, n))\in\C$.
}
\Proof{
We prove this by induction on $n$. The base case $n =1$ follows from group homology.
We prove this by induction on $n$. The base case $n =1$ follows from group homology as the construction of $K(G, 1)$ can be used to obtain a projective resolution of $\Z$ as $\Z[G]$-module \todo{reference}. This then identifies the homology of the Eilenberg-MacLane space with the group homology, we get for $i>0$ an isomorphism
$$ H_i(K(G, 1); \Z)\iso H_i(G; \Z)\in\C. $$
For the induction we can use the loop space and \LemmaRef{kreck}.
\todo{Bewijs afmaken}
Suppose we have proven the statment for $n$. If we consider the case of $n+1$ we can use the path fibration to relate it to the case of $n$:
$$\Omega K(G,n+1)\to P K(G, n+1)\fib K(G, n+1)$$
Now $\Omega K(G, n+1)= K(G, n)$, and we can apply \LemmaRef{kreck} as the reduced homology of the fiber is in $\C$ by induction hypothesis. Conclude that the homology of $P K(G, n+1)$ is $\C$-isomorphic to the homology of $K(G, n)$. Since $\RH_\ast(P K(G, n+1))=0$, we get $\RH_\ast(K(G, n+1))\in\C$.
}
For the main theorem we need the following construction. \todo{Geef de constructie}
For the main theorem we need the following construction. \todo{Geef de constructie of referentie}
\Lemma{whitehead-tower}{
(Whitehead tower)
We can decompose a $0$-connected space $X$ into fibrations:
@ -124,7 +107,9 @@ For the main theorem we need the following construction. \todo{Geef de construct
If $\pi_i(X)\in C$ for all $i<n$, then $H_i(X)\in C$ for all $i<n$ and the Hurewicz map $h: \pi_i(X)\to H_i(X)$ is a $\C$-isomorphism for all $i \leq n$.
}
\Proof{
We will prove the lemma by induction on $n$. Note that the base case ($n =1$) follows from the $1$-connectedness. For the induction step assume that $H_i(X)\in\C$ for all $i<n-1$ and that $h_{n-1}: \pi_{n-1}(X)\to H_{n-1}(X)$ is a $\C$-iso. Now given is that $\pi_{n-1}(X)\in\C$ and hence $H_{n-1}(X)\in\C$. \todo{kromme zin}
We will prove the lemma by induction on $n$. Note that the base case ($n =1$) follows from the $1$-connectedness.
For the induction step we may assume that $H_i(X)\in\C$ for all $i<n-1$ and that $h_{n-1}: \pi_{n-1}(X)\to H_{n-1}(X)$ is a $\C$-iso by induction hypothesis. Furthermore the theorem assumes that $\pi_{n-1}(X)\in\C$ and hence we conclude $H_{n-1}(X)\in\C$.
It remains to show that $h_n$ is a $\C$-iso. Use the Whitehead tower from \LemmaRef{whitehead-tower} to obtain $\cdots\fib X(3)\fib X(2)= X$. Note that each $X(j)$ is also $1$-connected and that $X(2)= X(1)= X$.
@ -142,13 +127,31 @@ For the main theorem we need the following construction. \todo{Geef de construct
\Theorem{relative-serre-hurewicz}{
(Relative Serre-Hurewicz Theorem)
Let $\C$ be a Serre class. Let $A \subset X$ be $1$-connected spaces such that $\pi_2(A)\to\pi_2(B)$ is surjective.
Let $\C$ be a Serre class. Let $A \subset X$ be $1$-connected spaces such that $\pi_2(A)\to\pi_2(X)$ is surjective.
If $\pi_i(X, A)\in\C$ for all $i<n$, then $H_i(X, A)\in\C$ for all $i<n$ and the Hurewicz map $h: \pi_i(X, A)\to H_i(X, A)$ is a $\C$-isomorphism for all $i \leq n$.
}
\Proof{
Note that we can assume $A \neq\emptyset$. We will prove by induction on $n$, the base case again follows by $1$-connectedness.
\todo{Bewijs afmaken}
Let $P X$ be that path space on $X$ and $Y \subset P X$ be the subspace of paths of which the endpoint lies in $A$. Now we get a fibration (of pairs) by sending the path to its endpoint:
$$ p: (P X, Y)\fib(X, A), $$
with $\Omega X$ as its fiber. We get long exact sequences of homotopy groups of the triples $\Omega X \subset Y \subset P X$ and $\ast\in A \subset X$:
The outer vertical maps are isomorphisms (again by a long exact sequence argument), hence the center vertical map is an isomorphism. Furthermore $\pi_i(P X)=0$ as it is a path space, hence $\pi_{i-1}(Y)\iso\pi_i(P X, Y)\iso\pi_i(X, A)$. By assumption we have $\pi_1(X, A)=\pi_2(X, A)=0$. So $Y$ is $1$-connected. Furthermore $\pi_{i-1}(Y)\in\C$ for all $i < n$.
Now we can use the previous Serre-Hurewicz theorem to conclude $H_{i-1}(Y)\in\C$ for all $i < n$ and $\pi_{n-1}(Y)\tot{h} H_{n-1}(Y)$ is an $\C$-iso. We are in the following situation:
\[
\xymatrix{
\pi_{n-1}(Y) \ar[d]^{\C\text{-iso}}&\ar[l]_\iso\pi_n(P X, Y) \ar[r]^\iso\ar[d]&\pi_n(X, A) \ar[d]\\
H_{n-1}(Y) &\ar[l]_\iso H_n(P X, Y) \ar[r]^{\C\text{-iso}}& H_n(X, A)
}
\]
The horizontal maps on the left are isomorphisms by long exact sequences, this gives us that the middle vertical map is a $\C$-iso. The horizontal maps on the right are $\C$-isos by the above and a relative version of \LemmaRef{kreck}. Now we conclude that $\pi_n(X, A)\to H_n(X, A)$ is alao a $\C$-iso and that $H_i(X, A)\in\C$ for all $i < n$.
}
\Theorem{serre-whitehead}{
@ -170,7 +173,20 @@ For the main theorem we need the following construction. \todo{Geef de construct
Where (1) $\iff$ (2) and (3) $\iff$ (4) hold by exactness and (2) $\iff$ (3) by the Serre-Hurewicz theorem.
}
In the case of rational homotopy theory we get the following corollary.
\section{For rational equivalences}
\Lemma{serre-class-rational-iso}{
Let $\C$ be the Serre class of all torsion groups. Then
$f$ is a $\C$-iso $\iff$$f \tensor\Q$ is an isomorphism.
}
\Proof{
First note that if $C \in\C$ then $C \tensor\Q=0$.
Then consider the exact sequence
$$0\to\ker(f)\to A \tot{f} B \to\coker(f)\to0$$
and tensor this sequence with $\Q$. In this tensored sequence the kernel and cokernel vanish if and only if $f \tensor\Q$ is an isomorphism.
Joshua Moerman
9 years ago