nLab rational topological space




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 information in the homology groups and the homotopy groups 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 cochain algebras. This is the topic of rational homotopy theory.


A topological space is called rational if

  1. it is simply connected in that the 1st homotopy group vanishes, π 1X=0\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 XX, H˜ *(X,)\tilde H_*(X,\mathbb{Z}) is a \mathbb{Q}-vector space,

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

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

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

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

π *:π *Xπ *Yπ *Y \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 ϕ:XY\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 π *(ϕ)\pi_*(\phi) \otimes \mathbb{Q} is an isomorphism;

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

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

  4. it induces a weak homotopy equivalence on rationalizations X 0,Y 0X_0, Y_0 in that ϕ 0:X 0Y 0\phi_0 : X_0 \to Y_0 is a weak homotopy equivalence.


One of the central theorems of rational homotopy theory says:


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

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

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


This appears for instance as corollary 1.26 in


Rational nn-sphere

The rational n-sphere (S n) 0(S^n)_0 can be written as

(S n) 0:=( k1S k n)( k2D k n+1), (S^n)_0 := \left( \vee_{k\geq 1} S^n_k \right) \cup \left( \coprod_{k \geq 2} D^{n+1}_k \right) \,,


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

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

that picks any representative of the degree nn-cohomology of S nS^{n} is a quasi-isomorphism.

For n=2kn = 2k with k1k \geq 1 there is a second generator c 4k+1c_{4k+1} with differential

dc 2k=0 d c_{2k} = 0
dc 4k1=c 2kc 2k. d c_{4k-1} = c_{2k} \wedge c_{2k} \,.

Rational nn-disk

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

D n+1=C(S n)=(S n×I)/(S n×{0}). 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 nn-disk consists of two generators bb and cc of degree nn and (n+1)(n+1), respectively, such that db=c\mathrm{d} b = c, the cdga corresponding to the rational nn-disk consists of linear combinations of b kb^k and cb kc b^k when nn is even, with differentials:

d(b k)=kcb k1 \mathrm{d}(b^k) \;=\; k \cdot c b^{k-1}
d(cb k)=0, \mathrm{d} (c b^k) \;=\; 0 \,,

and of linear combinations of c kbc^k b and c kc^k when nn is odd, with differentials

d(c kb)=c k+1 \mathrm{d}(c^k b)=c^{k+1}
d(c k)=0. d(c^k)=0 \,.

Rational compact Lie-groups

For GG a compact Lie group with Lie algebra 𝔤\mathfrak{g}, let {μ k i} i=1 rankG\{\mu_{k_i}\}_{i=1}^{rank G} be generators of its Lie algebra cohomology with degμ k i=2k i1deg \mu_{k_i} = 2 k_i-1. Accordingly there are generators {P k i} i\{P_{k_i}\}_i of invariant polynomials on 𝔤\mathfrak{g}.

Such GG is rationally equivalent to the product

i=1 rankGS 2k i1 \prod_{i = 1}^{rank G} S^{2 k_i -1}

of rational nn-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 GG as above, let G\mathcal{B}G be the corresponding classifying space. Then

H (G,)[P k 1,P k 2,], H^\bullet(\mathcal{B}G, \mathbb{Q}) \simeq \mathbb{Q}[P_{k_1}, P_{k_2}, \cdots] \,,

where P k iP_{k_i} is an invariant polynomial generator in degre 2k i2 k_i.

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

Quotient spaces

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

Biquotient spaces

Let HH be a compact Lie group and GH×HG \subset H \times H a closed subgroup of the direct product group. This GG acts on HH by left and right multiplication

(g 1,g 2):hg 1hg 2 1. (g_1, g_2) : h \mapsto g_1 h g_2^{-1} \,.

The corresponding quotient space is also called a biquotient.


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

Last revised on September 19, 2023 at 18:08:49. See the history of this page for a list of all contributions to it.