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\begin{document}

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\section*{rational topological space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{rational_sphere}{Rational $n$-sphere}\dotfill \pageref*{rational_sphere} \linebreak
\noindent\hyperlink{rational_disk}{Rational $n$-disk}\dotfill \pageref*{rational_disk} \linebreak
\noindent\hyperlink{rational_compact_liegroups}{Rational compact Lie-groups}\dotfill \pageref*{rational_compact_liegroups} \linebreak
\noindent\hyperlink{RationalClassifyingSpace}{Rational classifying spaces of compact Lie groups}\dotfill \pageref*{RationalClassifyingSpace} \linebreak
\noindent\hyperlink{quotient_spaces}{Quotient spaces}\dotfill \pageref*{quotient_spaces} \linebreak
\noindent\hyperlink{biquotient_spaces}{Biquotient spaces}\dotfill \pageref*{biquotient_spaces} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{rational topological space} is a [[topological space]] all whose (reduced) [[integral cohomology|integral homology]] groups are [[vector space]]s over the [[rational numbers]] $\mathbb{Q}$.

Every [[simply connected topological space|simply connected]] [[topological space]] has a [[rationalization]] and passing to that rationalization amounts to forgetting all [[torsion]] information in the homology groups and the [[homotopy group]]s of that space. So rational spaces are a way to approximate homotopical and cohomological characteristics of topological spaces. The idea is that comparatively little information (though sometimes crucial information) is lost by passing to rationalizations, while there are powerful tools to handle and compute with rational spaces. In particular, there is a precise sense in which rational spaces are modeled by graded commutative [[differential graded algebra|differential graded cochain algebra]]s. This is the topic of [[rational homotopy theory]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A [[topological space]] is called \emph{rational} if

\begin{enumerate}%
\item it is [[simply connected topological space|simply connected]] in that the 1st [[homotopy group]] vanishes, $\pi_1 X = 0$ (more generally we may use [[nilpotent topological spaces]] here)


\item and the following equivalent conditions are satisfied

\begin{enumerate}%
\item the collection of [[homotopy group]]s form a $\mathbb{Q}$-[[vector space]],


\item the [[reduced cohomology|reduced homology]] of $X$, $\tilde H_*(X,\mathbb{Z})$ is a $\mathbb{Q}$-[[vector space]],


\item the [[reduced cohomology|reduced homology]] of the [[loop space]] $\Omega X$ of $X$, $\tilde H_*(\Omega X,\mathbb{Z})$ is a $\mathbb{Q}$-[[vector space]].



\end{enumerate}


\end{enumerate}
A morphism $\ell : X \to Y$ of simply connected [[topological space]] is called a \textbf{[[rationalization]]} of $X$ if $Y$ is a rational topological space and if $\ell$ induces an [[isomorphism]] in rational homology

\begin{displaymath}
H_*(\ell,\mathbb{Q}) : H_*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_*(Y,\mathbb{Q})
  \,.
\end{displaymath}
Equivlently, $\ell$ is a rationalization of $X$ if it induces an [[isomorphism]] on the rationalized [[homotopy group]]s, i.e. when the morphism

\begin{displaymath}
\pi_* \ell \otimes \mathbb{Q} : \pi_* X \otimes \mathbb{Q}
   \to \pi_* Y \otimes Q \simeq \pi_* Y
\end{displaymath}
is an [[isomorphism]].

A continuous map $\phi : X \to Y$ between simply connected space is a \textbf{rational homotopy equivalence} if the following equivalent conditions are satisfied:

\begin{enumerate}%
\item it induces an isomorphism on rationalized homotopy groups in that $\pi_*(\phi) \otimes \mathbb{Q}$ is an isomorphism;


\item it induces an isomorphism on rationalized homology groups in that $H_*(\phi,\mathbb{Q})$ is an isomorohism;


\item it induces an isomorphism on rationalized [[cohomology group]]s in that $H^*(\phi,\mathbb{Q})$ is an isomorphism;


\item it induces a weak [[homotopy equivalence]] on rationalizations $X_0, Y_0$ in that $\phi_0 : X_0 \to Y_0$ is a [[weak homotopy equivalence]].



\end{enumerate}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

One of the central theorems of [[rational homotopy theory]] says:

\begin{utheorem}
Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal [[Sullivan model]]s $(\wedge^\bullet V,d)$

\begin{displaymath}
(\wedge^\bullet V , d) \stackrel{\simeq}{\to}
  \Omega^\bullet_{Sullivan}(X)
  \,.
\end{displaymath}
And homotopy classes of morphisms on both sides are in bijection.

\end{utheorem}
\begin{proof}
This appears for instance as corollary 1.26 in

\begin{itemize}%
\item [[Kathryn Hess]], \emph{Rational homotopy theory: a brief introduction} (\href{http://arxiv.org/abs/math.AT/0604626}{arXiv})

\end{itemize}
\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{rational_sphere}{}\subsubsection*{{Rational $n$-sphere}}\label{rational_sphere}

The \textbf{[[rational n-sphere]]} $(S^n)_0$ can be written as

\begin{displaymath}
(S^n)_0  :=
  \left(
     \vee_{k\geq 1} S^n_k
  \right) 
  \cup
  \left(
    \coprod_{k \geq 2} D^{n+1}_k
  \right)
  \,,
\end{displaymath}
where\ldots{}

For $n = 2 k +1$ odd, a [[Sullivan model]] for the $n$-sphere is the very simple [[dg-algebra]] with a single generator $c$ in degree $n$ and vanishing differential, i.e. the morphism

\begin{displaymath}
(\wedge^\bullet \langle c\rangle, d = 0)
  \to 
  \Omega^\bullet_{Sullivan}(S^{2k + 1})
\end{displaymath}
that picks any representative of the degree $n$-cohomology of $S^{n}$ is a [[quasi-isomorphism]].

For $n = 2k$ with $k \geq 1$ there is a second generator $c_{4k+1}$ with differential

\begin{displaymath}
d c_{2k} = 0
\end{displaymath}
\begin{displaymath}
d c_{4k-1} = c_{2k} \wedge c_{2k}
 \,.
\end{displaymath}
\hypertarget{rational_disk}{}\subsubsection*{{Rational $n$-disk}}\label{rational_disk}

\ldots{}

\hypertarget{rational_compact_liegroups}{}\subsubsection*{{Rational compact Lie-groups}}\label{rational_compact_liegroups}

For $G$ a [[compact space|compact]] [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$, let $\{\mu_{k_i}\}_{i=1}^{rank G}$ be generators of its [[Lie algebra cohomology]] with $deg \mu_{k_i} = 2 k_i-1$. Accordingly there are generators $\{P_{k_i}\}_i$ of [[invariant polynomial]]s on $\mathfrak{g}$.

Such $G$ is rationally equivalent to the product

\begin{displaymath}
\prod_{i = 1}^{rank G} S^{2 k_i -1}
\end{displaymath}
of rational $n$-spheres.

Moreover, Lie groups are [[formal homotopy type]]s, whose [[Sullivan model]] has a [[quasi-isomorphism]] to its [[cochain cohomology]].

\hypertarget{RationalClassifyingSpace}{}\subsubsection*{{Rational classifying spaces of compact Lie groups}}\label{RationalClassifyingSpace}

With $G$ as above, let $\mathcal{B}G$ be the corresponding [[classifying space]]. Then

\begin{displaymath}
H^\bullet(\mathcal{B}G, \mathbb{Q}) \simeq \mathbb{Q}[P_{k_1},  P_{k_2}, \cdots]
  \,,
\end{displaymath}
where $P_{k_i}$ is an [[invariant polynomial]] generator in degre $2 k_i$.

Indeed, also these classifying spaces are [[formal homotopy type]]s and hence a [[Sullivan model]] for $\mathcal{B}G$ is given by $(H^\bullet(\mathcal{B}G,\mathbb{R}), d=0)$.

\hypertarget{quotient_spaces}{}\subsubsection*{{Quotient spaces}}\label{quotient_spaces}

We may think of $\mathcal{B}G$ as the [[action groupoid]] $*// G$. The above discussion generalizes to more general such quotients.

\ldots{}

\hypertarget{biquotient_spaces}{}\subsubsection*{{Biquotient spaces}}\label{biquotient_spaces}

Let $H$ be a [[compact Lie group]] and $G \subset H \times H$ a [[closed subspace|closed]] [[subgroup]] of the [[direct product group]]. This $G$ acts on $H$ by left and right multiplication

\begin{displaymath}
(g_1, g_2) : h \mapsto g_1 h g_2^{-1}
  \,.
\end{displaymath}
The corresponding [[quotient space]] is also called a \emph{[[biquotient]]}.

\ldots{}

See

\begin{itemize}%
\item Vitali Kapovitch, \emph{A note on rational homotopy of biquotients} (\href{http://www.math.utoronto.ca/vtk/biquotient.pdf}{pdf})

\end{itemize}
[[!redirects rational topological spaces]]

[[!redirects rational space]] [[!redirects rational spaces]] [[!redirects rational topological space]]



\end{document}