Rational homotopy theory
A rational topological space is a topological space all whose (reduced) integral homology groups are vector spaces over the rational numbers .
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
A morphism of simply connected topological space is called a rationalization of if is a rational topological space and if induces an isomorphism in rational homology
Equivlently, is a rationalization of if it induces an isomorphism on the rationalized homotopy groups, i.e. when the morphism
is an isomorphism.
A continuous map 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 is an isomorphism;
it induces an isomorphism on rationalized homology groups in that is an isomorohism;
it induces an isomorphism on rationalized cohomology groups in that is an isomorphism;
it induces a weak homotopy equivalence on rationalizations in that is a weak homotopy equivalence.
One of the central theorems of rational homotopy theory says:
Rational homotopy types of simply connected spaces are in bijective corespondence with minimal Sullivan models
And homotopy classes of morphisms on both sides are in bijection.
This appears for instance as corollary 1.26 in
The rational n-sphere can be written as
For odd, a Sullivan model for the -sphere is the very simple dg-algebra with a single generator in degree and vanishing differential, i.e. the morphism
that picks any representative of the degree -cohomology of is a quasi-isomorphism.
For with there is a second generator with differential
Rational compact Lie-groups
For a compact Lie group with Lie algebra , let be generators of its Lie algebra cohomology with . Accordingly there are generators of invariant polynomials on .
Such is rationally equivalent to the product
of rational -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 as above, let be the corresponding classifying space. Then
where is an invariant polynomial generator in degre .
Indeed, also these classifying spaces are formal homotopy types and hence a Sullivan model for is given by .
We may think of as the action groupoid . The above discussion generalizes to more general such quotients.
Let be a compact Lie group and a closed subgroup of the product. This acts on by left and right multiplication
- Vitali Kapovitch, A note on rational homotopy of biquotients (pdf)