and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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
it is simply connected in that the 1st homotopy group vanishes, $\pi_1 X = 0$ (more generally we may use nilpotent topological spaces here)
and the following equivalent conditions are satisfied
the collection of homotopy groups form a $\mathbb{Q}$-vector space,
the reduced homology of $X$, $\tilde H_*(X,\mathbb{Z})$ is a $\mathbb{Q}$-vector space,
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
Equivlently, $\ell$ is a rationalization of $X$ if it induces an isomorphism on the rationalized homotopy groups, i.e. when the morphism
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:
it induces an isomorphism on rationalized homotopy groups in that $\pi_*(\phi) \otimes \mathbb{Q}$ is an isomorphism;
it induces an isomorphism on rationalized homology groups in that $H_*(\phi,\mathbb{Q})$ is an isomorohism;
it induces an isomorphism on rationalized cohomology groups in that $H^*(\phi,\mathbb{Q})$ is an isomorphism;
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.
One of the central theorems of rational homotopy theory says:
Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan models $(\wedge^\bullet V,d)$
And homotopy classes of morphisms on both sides are in bijection.
This appears for instance as corollary 1.26 in
The rational n-sphere $(S^n)_0$ can be written as
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
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
…
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
of rational $n$-spheres.
Moreover, Lie groups are formal homotopy types, whose Sullivan model has a quasi-isomorphism to its cochain cohomology.
With $G$ as above, let $\mathcal{B}G$ be the corresponding classifying space. Then
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 $(H^\bullet(\mathcal{B}G,\mathbb{R}), d=0)$.
We may think of $\mathcal{B}G$ as the action groupoid $*// G$. The above discussion generalizes to more general such quotients.
…
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
The corresponding quotient space is also called a biquotient.
…
See