Joshua Moerman
10 years ago
4 changed files with 149 additions and 1 deletions
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\section{Rational homotopy theory} |
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\label{sec:rational} |
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In this section we will state the aim of rational homotopy theory. Moreover we will recall classical theorems from algebraic topology and deduce rational versions of them. |
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In the following definition \emph{space} is to be understood as a topological space or a simplicial set. We will restrict to simply connected spaces. |
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\Definition{rational-space}{ |
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A space $X$ is a \emph{rational space} if |
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$$ \pi_i(X) \text{ is a $\Q$-vectorspace } \quad\forall i > 0. $$ |
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} |
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\Definition{rational-homotopy-groups}{ |
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We define the \emph{rational homotopy groups} of a space $X$ as: |
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$$ \pi_i(X) \tensor \Q \quad \forall i > 0.$$ |
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} |
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Note that for a rational space $X$, the homotopy groups are isomorphic to the rational homotopy groups, i.e. $\pi_i(X) \tensor \Q \iso \pi_i(X)$. |
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\Definition{rational-homotopy-equivalence}{ |
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A map $f: X \to Y$ is a \emph{rational homotopy equivalence} if $\pi_i(f) \tensor \Q$ is a linear isomorphism for all $i > 0$. |
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} |
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\Definition{rationalization}{ |
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A map $f: X \to X_0$ is a \emph{rationalization} if $X_0$ is rational and $f$ is a rational homotopy equivalence. |
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} |
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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 iff $f$ is a weak equivalence. |
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We will later see that any space admits a rationalization. The theory of rational homotopy theory is then the study of the homotopy category $\Ho_\Q(\Top) \iso \Ho(\Top_\Q)$, which is on its own turn equivalent to $\Ho(\sSet_\Q) \iso \Ho_\Q(\sSet)$. |
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\subsection{Classical results from algebraic topology} |
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We will now recall known results from algebraic topology, without proof. One can find many of these results in basic text books, such as [May, Dold, ...]. Note that all spaces are assumed to be $1$-connected. |
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\Theorem{relative-hurewicz}{ |
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(Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map |
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$$ h_i : \pi_i(X, A) \to H_i(X, A). $$ |
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If furhtermore $(X,A)$ is $n$-connected, then the map $h_i$ is an isomorphism for all $i \leq n + 1$ |
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} |
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\Theorem{serre-les}{ |
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(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence: |
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$$ \cdots \tot{\del} \pi_i(F) \tot{i_\ast} \pi_i(X) \tot{f_\ast} \pi_i(Y) \tot{\del} \cdots \to \pi_0(Y) \to \ast, $$ |
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where $F$ is the fibre of $f$. |
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} |
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Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffith}). |
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\Theorem{whitehead-homology}{ |
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(Whitehead) For any map $f: X \to Y$ we have |
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$$ \pi_i(f) \text{ is an isomorphism } \forall 0 < i < r \iff H_i(f) \text{ is an isomorphism } \forall 0 < i < r. $$ |
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In particular we see that $f$ is a weak equivalence iff it induces an isomorphism on homology. |
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} |
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The following two theorems can be found in textbooks about homological algebra, such as [Weibel]. |
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\Theorem{universal-coefficient}{ |
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(Universal Coefficient Theorem) |
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For any space $X$ and abelian group $A$, there are natural short exact sequcenes |
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$$ 0 \to H_n(X) \tensor A \to H_n(X; A) \to \Tor(H_{n-1}(X), A) \to 0, $$ |
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$$ 0 \to \Ext(H_{n-1}(X), A) \to H^n(X; A) \to \Hom(H_n(X), A) \to 0. $$ |
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} |
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\Theorem{kunneth}{ |
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(Künneth Theorem) |
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For spaces $X$ and $Y$, there is a short exact sequence |
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$$ 0 \to H(X; A) \tensor H(Y; A) \to H(X \times Y; A) \to \Tor(H(X; A), H(Y; A)) \to 0, $$ |
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where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. |
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} |
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\subsection{Immediate results for rational homotopy theory} |
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The latter two theorems have a direct consequence for rational homotopy theory. By taking $A = \Q$ we see that the torsion groups vanish. We have the immediate corollary. |
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\Corollary{rational-corollaries}{ |
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We have the following natural isomorphisms |
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$$ H(X) \tensor \Q \tot{\iso} H(X; \Q), $$ |
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$$ H^n(X; \Q) \tot{\iso} \Hom(H(X); \Q), $$ |
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$$ H(X \times Y) \tot{\iso} H(X) \tensor H(Y). $$ |
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} |
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The long exact sequence for a Serre fibration also has a direct consequence for rational homotopy theory. |
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\Corollary{rational-les}{ |
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Let $f: X \to Y$ be a Serre fibration, then there is a natural long exact sequence of rational homotopy groups: |
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$$ \cdots \tot{\del} \pi_i(F) \tensor \Q \tot{i_\ast} \pi_i(X) \tensor \Q \tot{f_\ast} \pi_i(Y) \tensor \Q \tot{\del} \cdots, $$ |
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} |
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In the next sections we will prove the rational Hurewicz and rational Whitehead theorems. These theorems are due to Serre [Serre]. |
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\section{Serre theorems mod $C$} |
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In this section we will prove the Whitehead and Hurewicz theorems in a rational context. Serre proved these results in [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. |
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\Lemma{whitehead-decomposition}{ |
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(Whitehead Decomposition) |
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For a space X, we have a decomposition in fibrations: |
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$$ \cdots \fib X(n+1) \fib X(n) \fib X(n) \fib \cdots \fib X(1) = X, $$ |
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such that: |
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\begin{itemize} |
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\item $K(\pi_n(X), n-1) \cof X(n+1) \fib X(n)$ is a fiber sequence, |
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\item There is a space $X'_n$ weakly equivelent to $X(n)$ such that $X(n+1) \ cof X'_n \fib K(\pi_n(X), n)$ is a fiber sequence, and |
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\item $\pi_i(X(n)) = 0$ for all $i < n$ and $\pi_i(X(n)) \iso \pi_i(X)$ for all $i \leq n$. |
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\end{itemize} |
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} |
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\Theorem{absolute-serre-hurewicz}{ |
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(Absolute Serre-Hurewicz Theorem) |
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Let $C$ be a Serre-class of abelian groups. Let $X$ a $1$-connected space. |
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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 $i \leq n$. |
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} |
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\Theorem{relative-serre-hurewicz}{ |
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(Relative Serre-Hurewicz Theorem) |
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Let $C$ be a Serre-class of abelian groups. Let $A \subset X$ be $1$-connected spaces ($A \neq \emptyset$). |
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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 $i \leq n$. |
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} |
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\Theorem{serre-whitehead}{ |
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(Serre-Whitehead Theorem) |
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Let $C$ be a Serre-class of abelian groups. Let $f: X \to Y$ be a map between $1$-connected spaces such that $\pi_2(f)$ is surjective. |
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Then $\pi_i(f)$ is a $C$-iso for all $i<n$ $\iff$ $H_i(f)$ is a $C$-iso for all $i<n$. |
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} |
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\Corollary{serre-whitehead}{ |
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Let $f: X \to Y$ be a map between $1$-connected spaces such that $\pi_2(f)$ is surjective. |
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Then $f$ is a rational equivalence $\iff$ $H_i(f; \Q)$ is an isomorphism for all $i$. |
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} |
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