39 lines
2 KiB
TeX
39 lines
2 KiB
TeX
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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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