@ -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.\todo{$\k$ doesn't always seem to work...}
\item$\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-modules, 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.
@ -34,7 +34,6 @@ Recall that the tensor product of modules distributes over direct sums. This def
The tensor product extends to graded maps. Let $f: A \to B$ and $g:X \to Y$ be two graded maps, then their tensor product $f \tensor g: A \tensor B \to X \tensor Y$ is defined as:
The sign is due to \emph{Koszul's sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map
$$\tau : A \tensor B \to B \tensor A : a \tensor b \mapsto(-1)^{\deg{a}\deg{b}} b \tensor a. $$
@ -71,7 +71,7 @@ The following two theorems can be found in textbooks about homological algebra s
\[\footnotesize\xymatrix @C=0.3cm{
0 \ar[r]& H(X; A) \tensor H(Y; A) \ar[r]& H(X \times Y; A) \ar[r]&\Tor_{\ast-1}(H(X; A), H(Y; A)) \ar[r]& 0
},\]
where $H(X; A)$ and$H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded.
where the $H(X; A)$,$H(X; A)$ and their tensor product are considered as graded modules. The Tor group is graded as $\Tor_n(A, B)=\bigoplus_{i+j=n}(A_i, B_j)$.
}
\section{Consequences for rational homotopy theory}
@ -25,7 +25,7 @@ Those cocycles are in fact coboundaries (using that $\k$ is a field of character
$$ c b^k =\frac{1}{k} d(b^{k+1})$$
$$ c^k = d(b c^{k-1})$$
There are no additional cocycles in $\Lambda D(n)$ besides the constants and $c$. So we conclude that $\Lambda D(n)$ is acyclic as an \todo{augmented?} algebra. In other words $\Lambda(j_n): \k\to\Lambda D(n)$ is a quasi isomorphism.
There are no additional cocycles in $\Lambda D(n)$ besides the constants and $c$. So we conclude that $\Lambda D(n)$ is acyclic as an augmented algebra. In other words $\Lambda(j_n): \k\to\Lambda D(n)$ is a quasi isomorphism.
The situation for $\Lambda S(n)$ is easier as it has only one generator (as algebra). For even $n$ this means it is given by polynomials in $a$. For odd $n$ it is an exterior algebra, meaning $a^2=0$. Again the sets $\Lambda(I)=\{\Lambda(i_n) : \Lambda S(n+1)\to\Lambda D(n)\I n \in\N\}$ and $\Lambda(J)=\{\Lambda(j_n) : \k\to\Lambda D(n)\I n \in\N\}$ play an important role.
By the Serre-Hurewicz theorem (\TheoremRef{serre-hurewicz}, with $\C$ the class of uniquely divisible groups) we see that $S^n_\Q$ is indeed rational. Then by the Serre-Whitehead theorem (\TheoremRef{serre-whitehead}, with $\C$ the class of torsion groups) the inclusion map $S^n \to S^n_\Q$ is a rationalization.
By the Serre-Hurewicz theorem (\CorollaryRef{rational-hurewicz-2}) we see that $S^n_\Q$ is indeed rational. Then by the Serre-Whitehead theorem (\CorollaryRef{rational-whitehead}) the inclusion map $S^n \to S^n_\Q$ is a rationalization.
\Corollary{rationalization-Sn}{
The inclusion $S^n \to S^n_\Q$ is a rationalization.
@ -23,8 +23,8 @@ Serre gave weaker axioms for his classes and proves some of the following lemmas
\item The class $\C$ of all uniquely divisible groups. Note that these groups can be given a unique $\Q$-vector space structure (and conversely every $\Q$-vector space is uniquely divisible).
\end{itemize}
}
\todo{refer to Moerdijk? for $H(\Z_p)=$ torsion.}
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:
The first three axioms of Serre classes are easily checked. For the group homology we find a calculation of the group homology of cyclic groups in \cite{moerdijk}. The group homology itself is also a torsion group, this result extends to all torsion groups. 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 \Def{$\C$-isomorphism} if both the kernel and cokernel lie in $\C$.
@ -80,7 +80,7 @@ In the following arguments we will consider fibrations and need to compute homol
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 as the construction \todo{which?} 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 which is in $\C$ by the axioms:
We prove this by induction on $n$. The base case $n =1$ follows from group homology. By considering the nerve of $G$ we can construct$K(G,1)$. This construction can be related to the bar construction as found in \cite{moerdijk}. This then identifies the homology of the Eilenberg-MacLane space with the group homology which is in $\C$ by the axioms:
$$ H_i(K(G, 1); \Z)\iso H_i(G; \Z)\in\C. $$
Suppose we have proven the statement for $n$. If we consider the case of $n+1$ we can use the path fibration to relate it to the case of $n$:
@ -189,13 +189,18 @@ For the main theorem we need the following construction. \todo{referentie}
}
Combining this lemma and \TheoremRef{serre-hurewicz} we get the following corollary for rational homotopy theory:
\Corollary{rational-hurewicz}{
(Rational Hurewicz Theorem)
Let $X$ be a $1$-connected space. If $\pi_i(X)\tensor\Q=0$ for all $i < n$, then $H_i(X; \Q)=0$ for all $i < n$. Furthermore we have an isomorphism for all $i \leq n$:
$$\pi_i(X)\tensor\Q\tot{\iso} H_i(X; \Q)$$
}
By using the class of $\Q$-vector spaces we get a dual theorem.
\Corollary{rational-hurewicz-2}{
(Rational Hurewicz Theorem 2)
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.
Joshua Moerman
10 years ago