Rational homotopy theory is the homotopy theory of rational topological spaces, hence of rational homotopy types: simply connected topological spaces whose homotopy groups are vector spaces over the rational numbers.
Much of the theory is concerned with rationalization, the process that sends a general homotopy type to its closest rational approximation, in a precise sense. On the level of homotopy groups this means to retain precisely the non-torsion subgroups of the homotopy groups.
This way rational homotopy theory connects homotopy theory and differential graded algebra. Akin to the Dold-Kan correspondence, the Sullivan construction in rational homotopy theory connects the conceptually powerful perspective of homotopy theory with the computationally powerful perspective of differential graded algebra.
Moreover, via the homotopy hypothesis the study of topological spaces is connected to that of ∞-groupoids, so that rational homotopy theory induces a bridge (the Sullivan construction) between ∞-groupoids and differential graded algebra. It was observed essentially by (Henriques 08, Getzler 09) that this bridge is Lie integration (see there) in the ∞-Lie theory of L-∞ algebras.
There are two established approaches in rational homotopy theory for encoding rational homotopy types in terms of Lie theoretic data:
In the Sullivan approach (Sullivan 77) a 1-connected rational space, in its incarnation as a simplicial set, is turned into something like a piecewise smooth space by realizing each abstract -simplex by the standard -simplex in ; and then a dg-algebra of differential forms on this piecewise smooth space is formed by taking on each simplex the dg-algebra of ordinary rational polynomial differential forms and gluing these dg-algebras all together.
In the Quillen approach (Quillen 69) the loop space of the rational space/simplicial set is formed and its H-space structure strictified to a simplicial group, of which then a dg-Lie algebra (a strict L-infinity-algebra) is formed by mimicking the construction of the Lie algebra of a Lie group from the primitive elements of its completed group ring: the group ring of the simplicial group here is a simplicial ring, whose degreewise primitive elements hence yield a simplicial Lie algebra. The Moore complex functor maps this to the dg-Lie algebra functor that models the rational homotopy type in the Quillen approach.
The connection between these two appoaches is discussed in (Majewski 00): The Sullivan dg-algebra of forms is the formal dual (the Chevalley-Eilenberg algebra) of an L-infinity algebra that may be rectified (see at model structure for L-infinity algebras) to a dg-Lie algebra, and that is the one from Quillen’s construction.
(Beware that – while both rational homotopy types as well as -algebras are presented by formal duals of dg-algebras (via Sullivan construction and via forming Chevalley-Eilenberg algebras, respectively) – the class of weak equivalences in the former case strictly includes that in the latter. See this remark at model structure for L-∞ algebras.)
Rational homotopy theory is mostly restricted to simply connected topological spaces. This is due to the existence of acyclic groups whose classifying space is an “acyclic space” in that its ordinary cohomology vanishes in positive degrees. This means that Sullivan algebras do not distinguish classifying spaces of rational groups from contractible spaces. But by Hurewicz theorem asking that all spaces be simply connected precisely makes all “acyclic spaces” be contractible.
This is due to (Serre 53).
We review here the Sullivan approach to rational homotopy theory, where rational topological spaces are modeled by differential graded-commutative algebras over the rational numbers with good (cofibrant) representatives being Sullivan algebras which are formal duals to L-infinity algebras.
These dg-algebras of “piecewise polynomial” differential forms on a topological space are typically extremely large and unwieldy. Much more tractable dg-algebras are the minimal Sullivan algebra, which we discuss next:
The relation between these algebras is that the Sullivan algebras are the cofibrant resolutions of the larger dg-algebras. In order to make this precise, we next recall some basics of topological and algebraic homotopy theory in
Finally we state and discuss the main theorem, that the construction of dg-algebras of [piecewise polynomial differential forms]] on a topological space exhibits an equivalence between the homotopy theory of simply connected rational topological spaces of finite type and that of minimal Sullivan algebras:
A central tool in the study of rational topological spaces is an assignment that sends each topological space/simplicial set to a dg-algebra that behaves like the deRham dg-algebra of a smooth manifold. Instead of consisting of smooth differential forms, consists of piecewise linear polynomial differential forms , in a way described in detail now.
We first discuss this semi-formally in
and then in more detail in
The construction of is a special case of the following general construction:
be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of as being test spaces and the functor as assigning to each test space its deRham dg-algebra.
An example of this construction that is natural from the point of view of differential geometry appears in the study of diffeological spaces, where is some subcategory of the category Diff of smooth manifolds, and is the restriction of the ordinary assignment of differential forms to this. But in the application to topological spaces, in the following, we need a choice for and that is non-standard from the point of view of differential geometry. Still, it follows the same general pattern.
for the hom-set of presheaves. One checks that this set naturally inherits the structure of a dg-algebra itself, where all operations are given by applying “pointwise” for each with the operations in . This way we get a functor
is an example for the adjunction induced from a dualizing object.
See differential forms on presheaves for more.
For the purpose of rational homotopy theory, consider the following special case of the above general discussion of differential forms on presheaves.
Recall that by the homotopy hypothesis theorem, Top is equivalent to sSet. In the sense of space and quantity, a simplicial set is a “generalized space modeled on the simplex category”: a presheaf on .
Therefore set in the above .
Now, a simplicial set has no smooth structure in terms of which one could define differential forms globally, but of course each abstract -simplex may be regarded as the standard -simplex in Diff, and as such it supports smooth differential forms .
The functor obtained this way is almost the one that – after fed into the above procedure – is used in rational homotopy theory.
The only difference is that for the purposes needed here, it is useful to cut down the smooth differential forms to something smaller. Let be the dg-algebra of polynomial differential forms on the standard -simplex. Notice that this recovers all differential forms after tensoring with smooth functions:
For more details see atdifferential forms on simplices.
We discuss the definition of polynomial differential forms on topological spaces in more detail.
For the function
which picks the th component in the above definition is called the th barycentric coordinate function.
a morphism of finite non-empty linear orders , let
be the smooth function defined by .
For definiteness, we write
(smooth differential forms on the smooth -simplex)
A smooth differential form on of degree k$ is a collection of linear functions
Write for the graded real vector space defined this way. By definition there is then a canonical linear map
from the de Rham complex of and there is a unique structure of a differential graded-commutative algebra on that makes is a homomorphism of dg-algebras form the de Rham algebra of . This is the de Rham algebra of smooth differential forms on the smooth -simplex.
In particular in degree 0 this are called the polynomial functions
due to the canonical inclusion
Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the -simplex, is canonically isomorphic to the space of smooth differential forms, according to def. 4:
where moreover the generators are identified with the de Rham differential of the th barycentric coordinate functions.
This defines a canonical inclusion
be the morphism of dg-algebras given on generators by
which is a sub-simplicial object of that of smooth differential form
Consider the simplicial differential graded-commutative algebra of polynomial differential forms from def. 5, equivalently a cosimplicial object in the opposite category of differential graded-commutative algebras (def. 3):
By the general discussion at nerve and realization, this induces a pair of adjoint functors between the opposite category of differential graded-commutative algebras (def. 3) and the category sSet of simplicial sets:
writing then for all basis elements we have that
(See remark 1 below for what this means.)
Such a relative Sullivan algebra if called minimal if in addition the degrees of these basis elements increase monotonicly:
If is such that the unique homomorphism
is a (minimal) relative Sullivan algebra in the above sense, then is simply called a (minimal) Sullivan algebra. In particular this means that is a semifree dgc-algebra.
be the dgc-algebra on a single generator in degree with vanishing differential.
be the dgc-algebra generated by an additional generator in degree such that the differential takes this to the previous generator.
Then the canonical inclusions
are relative Sullivan algebras according to 9.
Moreover, the inclusions
for are relative Sullivan algebras.
The examples in 2 are trivial, but they generate all examples of relative Sullivan algebras:
is precisely a choice of a -closed element in degree and results in adjoining to the element whose differential is .
This gives the condition in the above definition: the differential of any new element has to be a sum of wedge products of the old elements.
to the dg-algebra of piecewise polynomial differential forms.
e.g Hess 06, prop 1.18.
We briefly recall classical statement of the equivalene of the homotopy theories of topological spaces and of simplicial sets (simplicial homotopy theory), i.e. the “homotopy hypothesis”. For full exposition see at geometry of physics -- homotopy types.
The singular nerve and realization adjunction from def. 7 is a Quillen equivalence between the classical model structure on topological spaces (theorem 1) and the classical model structure on simplicial sets (theore 2):
a fibration if it is degreewise a surjection
These classes of morphisms make the category of differential graded-commutative algebras over the rational numbers and in non-negative degree into a model category, to be called the projective model structure on differential graded-commutative algebras, .
The adjunction of def. 6 is a Quillen adjunction with respect to the classical model structure on simplicial sets on the left (theorem 1), and the opposite model structure of the projective model structure on differential graded-commutative algebras on the right (theorem 4):
(subcategories of nilpotent objects of finite type)
for the full subcategory on the differential graded-commutative algebras which are equivalent to minimal Sullivan models (def. 9) for which the graded vector space is of finite type, i.e. is degreewise of finite dimension over .
In particular for such spaces the adjunction unit
exhibits the rationalization of .
e. g. Hess 06, corollary 1.26.
It follows that the cochain cohomology of the cochain complex of piecewise polynomial differential forms on any topological, hence equivalently that of any of its Sullivan models, coincides with its ordinary cohomology with coefficients in the rational numbers:
Let be a minimal Sullivan model of a simply connected rational topological space . Then there is an isomorphism
between the homotopy groups of and the generators of the minimal Sullivan model.
e.g. Hess 06, theorem 1.24.
The need to restrict to simply connected topological spaces in theorem 6 is due to the existence of acyclic groups. This are discrete groups such that their classifying space has trival ordinary cohomology in positive degree
Therefore, by corollary 1, its dg-algebra of piecewise polynomial differential forms do not distinguish such spaces from contractible topological spaces. But, unless is in fact the trivial group, is not contractible, instead it is the Eilenberg-MacLane space with nontrivial fundamental group . However, by the Hurewicz theorem, this fundamental group is the only obstruction to contractibility.
We discuss the minimal Sullivan models of rational n-spheres.
The minimal Sullivan model of a sphere of even dimension, for . is the dg-algeba with a generator in degree and another generator in degree with the differential defined by
One may understand this form theorem 7: an -sphere has rational cohomology concentrated in degree . Hence its Sullivan model needs at least one closed generator in that degree. In the odd dimensional case one such is already sufficient, since the wedge square of that generator vanishes and hence produces no higher degree cohomology classes. But in the even degree case the wedge square needs to be canceled in cohomology. That is accomplished by the second generator .
Again by theorem 7, this now implies that the rational homotopy groups of spheres are concentrated, in degree for the odd -dimensional spheres, and in degrees and in for the even -dimensional spheres.
One starts with the singular simplicial set
and throws away all the simplices except the basepoint in degrees and , to get a reduced simplicial set. Then one applies the Kan loop group functor (the simplicial analogue of the based loop space functor, see here) to , obtaining an a simplicial group
Then one forms its group ring
which happens to be cocommutative, since the group ring is cocommutative. Taking degreewise primitives, one then gets a reduced simplicial Lie algebra
Finally, to get a cocommutative dg coalgebra model for , one uses a slight generalization of a functor first defined by Koszul for computing the homology of a Lie algebra, which always gives rise to a cocommutative dg coalgebra.
One may think of this procedure as doing the following: we are taking the Lie algebra of the “group” which is the loop space of . From a group we pass to the enveloping algebra, i.e. the algebra of distributions supported at the identity, completed. The topological analog of distributions are chains (dual to functions = cochains), so Quillen’s completed chains construction is exactly the completed enveloping algebra. From the (completed) enveloping algebra we recover the Lie algebra as its primitive elements.
The left derived functor of the Quillen left adjoint (thorem 5) preserves homotopy pullbacks of objects of finite type (each rational homotopy group is a finite dimensional vector space over the ground field).
In other words in the induced pair of adjoint (∞,1)-functors
the left adjoint preserves (∞,1)-categorical pullbacks of objects of finite type.
This is effectively a restatement of a result that appears effectively below proposition 15.8 in HalperinThomas and is reproduced in some repackaged form as Hess 06, theorem 2.2. We recall the model category-theoretic context that allows to rephrase this result in the above form.
Let be the pullback diagram category.
At model structure on functors it is discussed that composition with the Quillen pair induces a Quillen adjunction
We need to show that for every fibrant and cofibrant pullback diagram there exists a weak equivalence
here is a fibrant replacement of in .
Every object is cofibrant. It is fibrant if all three objects , and are fibrant and one of the two morphisms is a fibration. Let us assume without restriction of generality that it is the morphism that is a fibration. So we assume that and are three Kan complexes and that is a Kan fibration. Then sends to the ordinary pullback in , and so the left hand side of the above equivalence is
Recall that the Sullivan algebras are the cofibrant objects in , hence the fibrant objects of . Therefore a fibrant replacement of may be obtained by
first choosing a Sullivan model
then choosing factorizations in of the composites of this with and into cofibrations follows by weak equivalences.
The result is a diagram
that in exhibits a fibrant replacement of . The limit over that in is the colimit
in . So the statement to be proven is that there exists a weak equivalence
This is precisely the statement of that quoted result Hess 06, theorem 2.2.
There are various variants of homotopy theory, such as stable homotopy theory or equivariant homotopy theory. Several of these have their coresponding rational models in terms of rational chain complexes equipped with extra structure. This includes the following:
The original articles are
Aldridge Bousfield, V. K. A. M. Gugenheim, On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)
Survey and review includes
Martin Majewski, Rational homotopy models and uniqueness , AMS Memoir (2000):
Review that makes the L-infinity algebra aspect completely manifest is in
More on the relation to Lie theory is in:
An extension of rational homotopy theory to describe (some) non-simply connected spaces is given, using derived algebraic geometry, in