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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.
Two algebraic models of rational homotopy types exist, via differential graded-commutative algebras (Sullivan model) and via dg-Lie algebras (Quillen model).
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.
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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 $n$-simplex by the standard $n$-simplex in $\mathbb{R}^n$; 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 $L_\infty$-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 $X$ to a dg-algebra $\Omega^\bullet_{poly}(X)$ that behaves like the deRham dg-algebra of a smooth manifold. Instead of consisting of smooth differential forms, $\Omega^\bullet_{poly}(X)$ consists of piecewise linear polynomial differential forms , in a way described in detail now.
The construction of $\Omega^\bullet_{poly}$ is a special case of the following general construction:
See differential forms on presheaves for more.
Let $C$ be any small category, write $PSh(C) = [C^{op}, Set]$ for its category of presheaves and let
be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of $C$ as being test spaces and the functor $\Omega^\bullet_C$ 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 $C$ is some subcategory of the category Diff of smooth manifolds, and $\Omega^\bullet_C$ 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 $C$ and $\Omega^\bullet_C$ that is non-standard from the point of view of differential geometry. Still, it follows the same general pattern.
After postcomposing with the forgetful functor that sends each dg-algebra to its underlying set, the functor $\Omega^\bullet_C$ becomes itself a presheaf on $C$. For $X \in PSh(C)$ any other presheaf, we extend the notation and write
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 $p : U \to X$ with $U \in C$ the operations in $\Omega^\bullet_C(U)$. This way we get a functor
to the opposite category of that of dg-algebras. We may think of $\Omega^\bullet_C(X)$ as the deRham complex of the presheaf $X$ as seen by the functor $\Omega^\bullet_C : C \to dgAlg^{op}$.
By general abstract nonsense this functor has a right adjoint $K_C : dgAlg^{op} \to PSh(C)$, that sends a dg-algebra $A$ to the presheaf
The adjunction
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 $\Delta$.
Therefore set in the above $C := \Delta$.
Now, a simplicial set has no smooth structure in terms of which one could define differential forms globally, but of course each abstract $k$-simplex $\Delta[k]$ may be regarded as the standard $k$-simplex $\Delta^k_{Diff}$ in Diff, and as such it supports smooth differential forms $\Omega^\bullet_{deRham}(\Delta^k_{Diff})$.
The functor $\Omega^\bullet_{deRham}( \Delta_{Diff}^{(-)} ) : \Delta^{op} \to dgAlg$ 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 $\Omega^\bullet_{poly}(\Delta^k_{Diff})$ be the dg-algebra of polynomial differential forms on the standard $k$-simplex. Notice that this recovers all differential forms after tensoring with smooth functions:
For more details see
…See Sullivan model…
So we have a functor $\Omega^\bullet_{polynomial} \colon \Delta \to dgAlg^{op}$. Feeding that into the above general machinery produces a pair of adjoint functors
between simplicial sets and dg-algebras.
This is a Quillen adjunction with respect to the standard model structure on simplicial sets on the left, and the standard model structure on dg-algebras on the right.
Moreover, this restricts to an equivalence between simply connected rational homotopy types and (minimal) Sullivan algebras.
(Bousfield-Gugenheim 76, section 8) Review includes (Hess 06, p. 9).
For $X$ a homotopy type, then
1.its rational vector space of the rational homotopy group in degree $n$ is spanned by the generators of degree $n$ of its Sullivan model;
We discuss the minimal Sullivan models of rational n-spheres.
The minimal Sullivan model of a sphere $S^{2k+1}$ of odd dimension is the dg-algebra with a single generator $\omega_{2k+1}$ in degre $2k+1$ and vanishing differential
The minimal Sullivan model of a sphere $S^{2k}$ of even dimension, for $k \geq 1$. is the dg-algeba with a generator $\omega_{2k}$ in degree $2k$ and another generator $\omega_{4k-1}$ in degree $4k+1$ with the differential defined by
One may understand this form prop. 1: an $n$-sphere has rational cohomology concentrated in degree $n$. 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 $\omega_{2k}\wedge \omega_{2k}$ needs to be canceled in cohomology. That is accomplished by the second generator $\omega_{4k-1}$.
Again by prop. 1, this now implies that the rational homotopy groups of spheres are concentrated, in degree $2k+1$ for the odd $(2k+1)$-dimensional spheres, and in degrees $2k$ and $4k-1$ in for the even $2k$-dimensional spheres.
For instance the 4-sphere has rational homotopy in degree 4 and 7. The one in degree 7 being represented by the quaternionic Hopf fibration.
The following sequence of six consecutive functors, each of which is a Quillen equivalence, takes one from a 1-connected rational space $X$ to a dg-Lie algebra.
One starts with the singular simplicial set
and throws away all the simplices except the basepoint in degrees $0$ and $1$, 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 $S(X)$, obtaining an a simplicial group
Then one forms its group ring
and completes it with respect to powers of its augmentation ideal, obtaining a “reduced, complete simplicial Hopf algebra”,
which happens to be cocommutative, since the group ring is cocommutative. Taking degreewise primitives, one then gets a reduced simplicial Lie algebra
At the next stage, the normalized chains functor is applied, to get Quillen’s dg-Lie algebra model of $X$:
Finally, to get a cocommutative dg coalgebra model for $X$, 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” $\Omega X$ which is the loop space of $X$. 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
Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR, pdf)
Dennis Sullivan, Infinitesimal computations in topology Publications mathématiques de l’ I.H.É.S. tome 47 (1977), p. 269-331. (pdf)
Bousfield, Gugenheim, On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)
Survey and review includes
Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)
Yves Félix, Steve Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.
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:
The above description of the Quillen approach draws on blog comments by Kathryn Hess here and by David Ben-Zvi here.
Discussion from the point of view of (∞,1)-category theory and E-∞ algebras is in
An extension of rational homotopy theory to describe (some) non-simply connected spaces is given, using derived algebraic geometry, in
See in particular Cor. 2.4.11, Cor. 2.5.3 and Cor. 2.5.4, and the MathOverflow answer MO/79309/2503 by Denis-Charles Cisinski.