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.)
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.
The construction of is a special case of the following general construction:
See differential forms on presheaves for more.
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.
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
…See Sullivan model…
Moreover, this restricts to an equivalence between simply connected rational homotopy types and (minimal) Sullivan algebras.
For a homotopy type, then
1.its rational vector space of the rational homotopy group in degree is spanned by the generators of degree of its Sullivan model;
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 prop. 1: 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 .
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 original articles are
Bousfield, 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 includes
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