nLab rational topological space

Idea

A rational topological space is a topological space all whose (reduced) integral homology groups are vector spaces over the rational numbers $\mathbb{Q}$.

Every simply connected topological space has a rationalization and passing to that rationalization amounts to forgetting all torsion in the homology- and homotopy-groups of that space.

Thereby rational spaces serve as approximations to the homotopy type of general topological spaces in rational homotopy theory.

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 the homotopy types of (nilpotent, finite type) rational spaces are modeled simply by differential graded cochain algebras?: This is the fundamental theorem of dg-algebraic rational homotopy theory.

Definition

A topological space is called rational if

1. it is simply connected in that the 1st homotopy group vanishes, $\pi_1 X = 0$ (more generally we may use nilpotent topological spaces here)

2. and the following equivalent conditions are satisfied

   1. the collection of homotopy groups form a $\mathbb{Q}$-vector space,

   2. the reduced homology of $X$, $\tilde H_*(X,\mathbb{Z})$ is a $\mathbb{Q}$-vector space,

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

A morphism $\ell : X \to Y$ of simply connected topological space is called a rationalization of $X$ if $Y$ is a rational topological space and if $\ell$ induces an isomorphism in rational homology

$$H_*(\ell,\mathbb{Q}) : H_*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_*(Y,\mathbb{Q}) \,.$$

Equivlently, $\ell$ is a rationalization of $X$ if it induces an isomorphism on the rationalized homotopy groups, i.e. when the morphism

$$\pi_* \ell \otimes \mathbb{Q} : \pi_* X \otimes \mathbb{Q} \to \pi_* Y \otimes \mathbb{Q} \simeq \pi_* Y$$

is an isomorphism.

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

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

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

3. it induces an isomorphism on rationalized cohomology groups in that $H^*(\phi,\mathbb{Q})$ is an isomorphism;

4. 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.

Properties

One of the central theorems of rational homotopy theory says:

Theorem

Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan models $(\wedge^\bullet V,d)$

$$(\wedge^\bullet V , d) \stackrel{\simeq}{\to} \Omega^\bullet_{Sullivan}(X) \,.$$

And homotopy classes of morphisms on both sides are in bijection.

Proof

This appears for instance as corollary 1.26 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Examples

Rational $n$-sphere

The rational n-sphere $(S^n)_0$ can be written as

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

where…

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

$$(\wedge^\bullet \langle c\rangle, d = 0) \to \Omega^\bullet_{Sullivan}(S^{2k + 1})$$

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

$$d c_{2k} = 0$$

$$d c_{4k-1} = c_{2k} \wedge c_{2k} \,.$$

Rational $n$-disk

The rational $n$-disk is computed as the cone of the rational sphere:

$$D^{n+1}_{\mathbb{Q}} = C(S^n_{\mathbb{Q}}) = (S^n_{\mathbb{Q}}\times I)/(S^n_{\mathbb{Q}}\times\{0\}).$$

Since the cochain complex of the $n$-disk consists of two generators $b$ and $c$ of degree $n$ and $(n+1)$, respectively, such that $\mathrm{d} b = c$, the cdga corresponding to the rational $n$-disk consists of linear combinations of $b^k$ and $c b^k$ when $n$ is even, with differentials:

$$\mathrm{d}(b^k) \;=\; k \cdot c b^{k-1}$$

$$\mathrm{d} (c b^k) \;=\; 0 \,,$$

and of linear combinations of $c^k b$ and $c^k$ when $n$ is odd, with differentials

$$\mathrm{d}(c^k b)=c^{k+1}$$

$$d(c^k)=0 \,.$$

Rational compact Lie-groups

For $G$ a 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 polynomials on $\mathfrak{g}$.

Such $G$ is rationally equivalent to the product

$$\prod_{i = 1}^{rank G} S^{2 k_i -1}$$

of rational $n$-spheres.

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

Rational classifying spaces of compact Lie groups

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

$$H^\bullet(\mathcal{B}G, \mathbb{Q}) \simeq \mathbb{Q}[P_{k_1}, P_{k_2}, \cdots] \,,$$

where $P_{k_i}$ is an invariant polynomial generator in degre $2 k_i$.

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

Quotient spaces

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

…

Biquotient spaces

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

$$(g_1, g_2) : h \mapsto g_1 h g_2^{-1} \,.$$

The corresponding quotient space is also called a biquotient.

…

See

• Vitali Kapovitch, A note on rational homotopy of biquotients (pdf)

Related concepts

• rational topological G-space