rational homotopy theory



Rational homotopy theory



As with many other parts of homotopy theory one can view rational homotopy theory in two ways:

  • either one looks at a subcategory of topological spaces for which a particular homotopy functor? gives exceptionally good results;

  • or alternatively one looks at fairly arbitrary nice spaces, but collects up and uses only the information available using some particular type of homotopy functor?.

From the first viewpoint, rational homotopy theory studies special topological spaces called rational spaces: simply connected spaces whose homotopy groups are vector spaces over the rational numbers.

The point of this is that

  • every (simply connected) topological space can be approximated by a rational space;

  • rational topological spaces are entirely encoded in terms of differential graded algebra.

Alternatively, looking at simply connected spaces, we can concentrate on what happens if we tensor their homotopy groups with the field of rationals. What information can be gleaned from these homotopy functors? The answer is ‘an incredible amount’ – namely all non-torsion information – and results in a very rich theory.

In these ways, rational homotopy theory connects and unifies two large areas of mathematics, 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 infinity-groupoids, so that rational homotopy theory induces a bridge between infinity-groupoids and differential graded algebra. It was observed essentially by Ezra Getzler that this bridge is nothing but higher Lie theory of L-infinity-algebras.


Rational homotopy types

Lie theoretic models for rational homotopy types

There are two main approaches in rational homotopy theory for encoding rational homotopy types in terms of Lie theoretic data:

  1. In the Sullivan approach 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 nn-simplex by the standard nn-simplex in n\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.

    This goes back to

    • Dennis Sullivan, Infinitesimal computations in topology Publications mathématiques de l’ I.H.É.S. tome 47 (1977), p. 269-331.
  2. In the Quillen approach 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.

    This goes back to (Quillen 69).

The connection between these two appoaches is discussed in

  • Martin Majewski, Rational homotopy models and uniqueness , AMS Memoir (2000):

the Sullivan dg-algebra of forms is dual to an L-infinity algebra and may be strictified to a dg-Lie algebra, and this is equivalent to the dg-Lie algebra obtained from Quillen’s construction.

Sullivan approach

Differential forms on topological spaces

A central tool in the study of rational topological spaces is an assignment that sends each topological space/simplicial set XX to a dg-algebra Ω poly (X)\Omega^\bullet_{poly}(X) that behaves like the deRham dg-algebra of a smooth manifold. Instead of consisting of smooth differential forms, Ω poly (X)\Omega^\bullet_{poly}(X) consists of piecewise linear polynomial differential forms , in a way described in detail now.

The construction of Ω poly \Omega^\bullet_{poly} is a special case of the following general construction:

Differential forms on presheaves

See differential forms on presheaves for more.

Let CC be any small category, write PSh(C)=[C op,Set]PSh(C) = [C^{op}, Set] for its category of presheaves and let

Ω C :C opdgAlg \Omega^\bullet_C : C^{op} \to dgAlg

be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of CC as being test spaces and the functor Ω C \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 CC is some subcategory of the category Diff of smooth manifolds, and Ω C \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 CC and Ω C \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 Ω C \Omega^\bullet_C becomes itself a presheaf on CC. For XPSh(C)X \in PSh(C) any other presheaf, we extend the notation and write

Ω C (X):=Hom PSh(C)(X,Ω C ) \Omega^\bullet_C(X) := Hom_{PSh(C)}(X, \Omega^\bullet_C)

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:UXp : U \to X with UCU \in C the operations in Ω C (U)\Omega^\bullet_C(U). This way we get a functor

Ω C :PSh(C)dgAlg op \Omega^\bullet_C : PSh(C) \to dgAlg^{op}

to the opposite category of that of dg-algebras. We may think of Ω C (X)\Omega^\bullet_C(X) as the deRham complex of the presheaf XX as seen by the functor Ω C :CdgAlg op\Omega^\bullet_C : C \to dgAlg^{op}.

By general abstract nonsense this functor has a right adjoint K C:dgAlg opPSh(C)K_C : dgAlg^{op} \to PSh(C), that sends a dg-algebra AA to the presheaf

K C(A):UHom dgAlg(Ω C (U),A). K_C(A) : U \mapsto Hom_{dgAlg}(\Omega^\bullet_C(U), A) \,.

The adjunction

Ω C :PSh(C):dgAlg op:K C \Omega^\bullet_C : PSh(C) \stackrel{\leftarrow}{\to} : dgAlg^{op} : K_C

is an example for the adjunction induced from a dualizing object.

Piecewise linear differential forms

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 Δ\Delta.

Therefore set in the above C:=ΔC := \Delta.

Now, a simplicial set has no smooth structure in terms of which one could define differential forms globally, but of course each abstract kk-simplex Δ[k]\Delta[k] may be regarded as the standard kk-simplex Δ Diff k\Delta^k_{Diff} in Diff, and as such it supports smooth differential forms Ω deRham (Δ Diff k)\Omega^\bullet_{deRham}(\Delta^k_{Diff}).

The functor Ω deRham (Δ Diff ()):Δ opdgAlg\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 Ω poly (Δ Diff k)\Omega^\bullet_{poly}(\Delta^k_{Diff}) be the dg-algebra of polynomial differential forms on the standard kk-simplex. Notice that this recovers all differential forms after tensoring with smooth functions:

Ω (Δ Diff k)=C (Δ Diff k) Ω poly 0(Δ Diff k)Ω poly (Δ Diff k). \Omega^\bullet(\Delta^k_{Diff}) = C^\infty(\Delta^k_{Diff}) \otimes_{\Omega^0_{poly}(\Delta^k_{Diff})} \Omega^\bullet_{poly}(\Delta^k_{Diff}) \,.

For more details see

So we have a functor Ω polynomial :ΔdgAlg op\Omega^\bullet_{polynomial} : \Delta \to dgAlg^{op}. Feeding that into the above general machinery produces a pair of adjoint functors

Ω poly :SSetdgAlg op:K poly. \Omega^\bullet_{poly} : SSet \stackrel{\leftarrow}{\to} dgAlg^{op} : K_{poly} \,.

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.


The original proof in the literature is apparently the one in section 8 of

  • Bousfield, Gugenheim, On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)

A review is on page 9


Sullivan models

See Sullivan model.

Quillen approach

The following sequence of six consecutive functors, each of which is a Quillen equivalence, take one from a 1-connected rational space XX to a dg-Lie algebra.

One starts with the singular simplicial set

S(X) S(X)

and throws away all the simplices except the basepoint in degrees 00 and 11. Then one applies the Kan loop group functor (the simplicial analogue of the based loop space functor) to S(X)S(X), obtaining an honest simplicial group

GS(X). G S(X).

Then one takes the group ring

[GS(X)] \mathbb{Q}[G S(X)]

and completes it with respect to powers of its augmentation ideal, obtaining a “reduced, complete simplicial Hopf algebra”,

^[GS(X)], \hat \mathbb{Q}[G S(X)],

which happens to be cocommutative, since the group ring is cocommutative. Taking degreewise primitives, one then gets a reduced simplicial Lie algebra

Prim(^[GS(X)]). Prim(\hat \mathbb{Q}[G S(X)]).

At the next stage, the normalized chains functor is applied, to get Quillen’s dg-Lie algebra model of XX:

N (Prim(^[GS(X)])). N^\bullet(Prim(\hat \mathbb{Q}[G S(X)])).

Finally, to get a a cocommutative dg coalgebra model for XX, 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” ΩX\Omega X which is the loop space of XX. 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.


A useful introduction to rational homotopy theory is

A standard textbook is

  • Y. Félix, S. Halperin and J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

Early original articles include:

  • Dan Quillen, Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (JSTOR)
  • Dennis Sullivan, Infinitesimal computations in topology Publications mathématiques de l’ I.H.É.S. tome 47 (1977), p. 269-331. (pdf)

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

  • B. Toën, Champs affines, Selecta Math. (N.S.) 12 (2006), no. 1, 39-135.

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.

Revised on November 20, 2015 13:28:52 by Urs Schreiber (