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rational homotopy theory

Context

Homotopy theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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:

  1. 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 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.

  2. 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 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.)

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Rational homotopy theory is mostly restricted to simply connected topological spaces. This is due to the existence of acyclic groups Γ\Gamma whose classifying space BΓB \Gamma is an “acyclic space” in that its ordinary cohomology vanishes in positive degrees. This means that Sullivan algebras do not distinguish classifying spaces of acyclic groups from contractible spaces. But by Hurewicz theorem asking that all spaces be simply connected precisely makes all “acyclic spaces” be contractible.

Generalities

Proposition

Let XX and YY by simply connected topological spaces. Then a continuous function f:XYf \colon X \to Y is a isomorphism on homotopy groups after tensor product with the rational numbers \mathbb{Q}

π (f):π (X)π (Y) \pi_\bullet(f) \otimes \mathbb{Q} \;\colon\; \pi_\bullet(X) \otimes \mathbb{Q} \overset{\simeq}{\longrightarrow} \pi_\bullet(Y) \otimes \mathbb{Q}

precisely if it induces an isomorphism on ordinary homology with rational numbers coefficients:

H (f,):H (X,)H (Y,). H_\bullet(f,\mathbb{Q}) \;\colon\; H_\bullet(X,\mathbb{Q}) \overset{\simeq}{\longrightarrow} H_\bullet(Y,\mathbb{Q}) \,.

This is due to (Serre 53).

Definition

A map ff satisfying the equivalent conditions of def. 1 is called a rational homotopy equivalence.

Sullivan approach

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.

First we discuss how to define an analog of the construction of the de Rham dg-algebra of a smooth manifold for topological spaces:

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:

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.

We first discuss this semi-formally in

and then in more detail in

Idea

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

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 opdgcAlg \Omega^\bullet_C : C^{op} \to dgcAlg

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)dgcAlg op \Omega^\bullet_C : PSh(C) \to dgcAlg^{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 :CdgcAlg op\Omega^\bullet_C : C \to dgcAlg^{op}.

By the general discussion at nerve and realization this functor has a right adjoint K C:dgcAlg opPSh(C)K_C : dgcAlg^{op} \to PSh(C), that sends a dg-algebra AA to the presheaf

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

The adjunction

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

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 Δ\Delta.

Therefore set in the above CΔC \coloneqq \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 opdgcAlg\Omega^\bullet_{deRham}( \Delta_{Diff}^{(-)} ) : \Delta^{op} \to dgcAlg 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 atdifferential forms on simplices.

Details

We discuss the definition of polynomial differential forms on topological spaces in more detail.

Definition

(smooth nn-simplex)

For nn \in \mathbb{N} the smooth n-simplex Δ smth n\Delta^n_{smth} is the smooth manifold with boundary and corners defined, up to isomorphism, as the following locus inside the Cartesian space n+1\mathbb{R}^{n+1}:

Δ smth n{(x 0,x 1,,x n) n+1|0x i1andi=0nx i=0} n+1. \Delta^n_{smth} \;\coloneqq\; \left\{ (x_0, x_1, \cdots, x_n) \in \mathbb{R}^{n+1} \;\vert\; 0 \leq x_i \leq 1 \;\text{and}\; \underoverset{i = 0}{n}{\sum} x_i \; = 0 \right\} \hookrightarrow \mathbb{R}^{n+1} \,.

For 0in0 \leq i \leq n the function

x i:Δ smth n x_i \;\colon\; \Delta^n_{smth} \to \mathbb{R}

which picks the iith component in the above definition is called the iith barycentric coordinate function.

For

f:[n 1][n 2] f \;\colon\; [n_1] \longrightarrow [n_2]

a morphism of finite non-empty linear orders [n]{0<1<<n}[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}, let

Δ smth(f):Δ smth n 1Δ smth n 2 \Delta_{smth}(f) \;\colon\; \Delta^{n_1}_{smth} \longrightarrow \Delta^{n_2}_{smth}

be the smooth function defined by x ix f(i)x_i \mapsto x_{f(i)}.

Definition

For definiteness, we write

dgcAlg ,0 dgcAlg_{\mathbb{Q}, \geq 0}

for the category of differential graded-commutative algebras over the complex numbers in non-negative \mathbb{Z}-degree, i.e. \mathbb{N}-graded.

Definition

(smooth differential forms on the smooth nn-simplex)

For kk \in \mathbb{N} then a smooth differential k-form on the smooth nn-simplex (def. 2) is a smooth differential form in the sense of smooth manifolds with boundary and corners. Explicitly this means the following.

Let

F n{(x 0,x 1,,x n) n+1|i=0nx i=0} n+1 F^n \;\coloneqq\; \left\{ (x_0, x_1, \cdots, x_n) \in \mathbb{R}^{n+1} \;\vert\; \underoverset{i = 0}{n}{\sum} x_i \; = 0 \right\} \hookrightarrow \mathbb{R}^{n+1}

be the affine plane in n+1\mathbb{R}^{n+1} that contains Δ smth n\Delta^n_{smth} in its defining inclusion from def. 2. This is a smooth manifold diffeomorphic to the Cartesian space n\mathbb{R}^{n}.

A smooth differential form on Δ smth n\Delta^n_{smth} of degree k$ is a collection of linear functions

kT xF n \wedge^k T_x F^n \longrightarrow \mathbb{R}

out of the kk-fold skew-symmetric tensor power of the tangent space of F nF^n at some point xx to the real numbers, for all xΔ smth nx \in \Delta^n_{smth} such that this extends to a smooth differential kk-form on F nF^n.

Write Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) for the graded real vector space defined this way. By definition there is then a canonical linear map

Ω (F n)Ω (Δ smth n) \Omega^\bullet(F^n) \longrightarrow \Omega^\bullet(\Delta^n_{smth})

from the de Rham complex of F nF^n and there is a unique structure of a differential graded-commutative algebra on Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) that makes is a homomorphism of dg-algebras form the de Rham algebra of F nF^n. This is the de Rham algebra of smooth differential forms on the smooth nn-simplex.

For f:[n 1][n 2]f \colon [n_1] \to [n_2] a homomorphism of finite non-empty linear orders with Δ smth(f):Δ smth n 1Δ smth n 2\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth} the corresponding smooth function according to def. 2, there is the induced homomorphism of differential graded-commutative algebras

(Δ smth(f)) *:Ω (Δ smth n 2)Ω (Δ smth n 1) (\Delta_{smth}(f))^\ast \;\colon\; \Omega^\bullet(\Delta^{n_2}_{smth}) \longrightarrow \Omega^\bullet(\Delta^{n_1}_{smth})

induced from the usual pullback of differential forms on F nF^n. This makes smooth differential forms on smooth simplices be a simplicial object in differential graded-commutative algebras (def. 3):

Ω (Δ smth ()):Δ opdgcAlg ,0. \Omega^\bullet(\Delta^{(-)}_{smth}) \;\colon\; \Delta^{op} \longrightarrow dgcAlg_{\mathbb{R}, \geq 0} \,.
Definition

For nn \in \mathbb{N} write

Ω poly (Δ n)Sym t 0,,t n,dt 0,,dt n/(t i1,dt i) \Omega_{poly}^{\bullet}(\Delta^n) \;\coloneqq\; Sym^\bullet_{\mathbb{Q}} \langle t_0, \cdots, t_n, d t_0, \cdots, d t_n\rangle/\left(\sum t_i -1, \sum d t_i \right)

for the quotient of the \mathbb{Z}-graded symmetric algebra over the rational numbers on n+1n+1 generators t it_i in degree 0 and n+1n+1 generators dt id t_i of degree 1.

In particular in degree 0 this are called the polynomial functions

Ω poly 0(Δ n)=[t 0,t 1,t n]/(it i=0) \Omega^0_{poly}(\Delta^n) \;=\; \mathbb{Q}[t_0, t_1, \cdots t_n]/\left( \underset{i}{\sum} t_i = 0 \right)

due to the canonical inclusion

Ω poly 0(Δ n)C (Δ smth n) \Omega^0_{poly}(\Delta^n) \hookrightarrow C^\infty(\Delta^n_{smth})

into the smooth functions on the nn-simplex according to def. 4, obtained by regarding the generator t it_i as the iith barycentric coordinate function.

Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the nn-simplex, is canonically isomorphic to the space Ω (Δ smth n)\Omega^\bullet(\Delta^n_{smth}) of smooth differential forms, according to def. 4:

Ω (Δ smth n)C (Δ smth n) Ω poly 0(Δ n)Ω poly (Δ n) \Omega^\bullet(\Delta^n_{smth}) \simeq C^\infty(\Delta^n_{smth}) \otimes_{\Omega^0_{poly}(\Delta^n)} \Omega^\bullet_{poly}(\Delta^n)

where moreover the generators dt id t_i are identified with the de Rham differential of the iith barycentric coordinate functions.

This defines a canonical inclusion

Ω poly (Δ n)Ω (Δ smth n) \Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n_{smth})

and there is uniquely the structure of a differential graded-commutative algebra on Ω poly (Δ n)\Omega^\bullet_{poly}(\Delta^n) that makes this a homomorphism of dg-algebras. This is the dg-algebra of polynomial differential forms.

For f:[n 1][n 1]f \colon [n_1] \to [n_1] a morphism of finite non-empty linear orders, let

Ω poly (f):Ω poly (Δ n 2)Ω poly (Δ n 1) \Omega^\bullet_{poly}(f) \;\colon\; \Omega^\bullet_{poly}(\Delta^{n_2}) \to \Omega^\bullet_{poly}(\Delta^{n_1})

be the morphism of dg-algebras given on generators by

Ω poly (f):t i f(j)=it j. \Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j \,.

This yields a simplicial differential graded-commutative algebra

Ω poly (Δ ()):Δ opcdgcAlg k \Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgcAlg_k

which is a sub-simplicial object of that of smooth differential form

Ω poly (Δ ())Ω (Δ smth ()). \Omega^\bullet_{poly}(\Delta^{(-)}) \hookrightarrow \Omega^\bullet(\Delta_{smth}^{(-)}) \,.
Definition

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):

Ω poly :ΔdgcAlg ,0 op. \Omega^\bullet_{poly} \;\colon\; \Delta \longrightarrow dgcAlg_{\mathbb{Q}, \geq 0}^{op} \,.

By the general discussion at nerve and realization, this induces a pair of adjoint functors between the opposite category of differential graded-commutative algebras (dgcAlg ,0) op(dgcAlg_{\mathbb{Q}, \geq 0})^{op} (def. 3) and the category sSet of simplicial sets:

(Ω poly 𝒦 poly):(dgcAlg ,0) opK polyΩ poly sSet. (\Omega^\bullet_{poly} \dashv \mathcal{K}_{poly}) \;\colon\; (dgcAlg_{\mathbb{Q}, \geq 0})^{op} \underoverset {\underset{K_{poly}}{\longrightarrow}} {\overset{\Omega^\bullet_{poly}}{\longleftarrow}} {\bot} sSet \,.

Here the left adjoint is the left Kan extension of Ω poly\Omega \bullet_{poly} along the Yoneda embedding ΔsSet\Delta \hookrightarrow sSet, which we denote by the same symbols.

This adjunction is an algebraic analog of the singular simplicial complex construction, which we briefly recall (for detailed exposition see at geometry of physics -- homotopy types):

Definition

For nonn \on \mathbb{N}, write Δ top n\Delta^n_{top} for the n-simplex canonically regarded as a topological space, i.e. the topological space underlying the smooth simplex Δ smth n\Delta^n_{smth} from def. 2. As in def. 5 this is a cosimplicial object

Δ top ():ΔTop \Delta^{(-)}_{top} \;\colon\; \Delta \longrightarrow Top

now in the category Top of topological spaces.

This also induces a nerve and realization adjunction

(||Sing):TopSing||sSet ({\vert -\vert} \dashv Sing) \;\colon\; Top \underoverset {\underset{Sing}{\longrightarrow}} {\overset{{\vert -\vert}}{\longleftarrow}} {\bot} sSet

where the left adjoint SingSing is the singular simplicial complex functor and the right adjoint ||{\vert- \vert} is the geometric realization functor.

Definition

Combining the functors from def. 6 and def. 7 we finally obtain a functorial association of differential forms to a topological space

Ω pwpoly :TopSingsSetΩ poly (dgcAlg ,0) op \Omega^\bullet_{pwpoly} \;\colon\; Top \overset{Sing}{\longrightarrow} sSet \overset{\Omega^\bullet_{poly}}{\longrightarrow} (dgcAlg_{\mathbb{Q}, \geq 0})^{op}

(foring “piecewise polynomial differential forms”) and a functorial operation that turns every differential graded-commutative algebra into a topological space

Spec:(dgcAlg ,0) opK polysSet||Top. Spec \;\colon\; (dgcAlg_{\mathbb{Q}, \geq 0})^{op} \overset{K_{poly}}{\longrightarrow} sSet \overset{\vert - \vert}{\longrightarrow} Top \,.

Sullivan models

Definition

(Sullivan algebras)

A relative Sullivan algebra is a homomorphism of differential graded-commutative algebras in non-negative degrees, hence a morphism in dgcAlg 0dgcAlg_{\geq 0}, that is an inclusion of the form

(A,d)(A k V,d) (A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d')
aa1 a \mapsto a \otimes 1

for (A,d)dgcAlg 0(A,d) \in dgcAlg_{\geq 0} any dgc-algebra and for VV some graded vector space, such that

  1. there is a well ordered set JJ indexing a linear basis {v αV|αJ}\{v_\alpha \in V| \alpha \in J\} of VV;

  2. writing V <βspan(v α|α<β)V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta) then for all basis elements v βv_\beta we have that

dv βA V <β. d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,.

(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:

(α<β)(degv αdegv β) (\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta)

If AdgcAlg 0A \in dgcAlg_{\geq 0} is such that the unique homomorphism

(,d=0)A (\mathbb{Q}, d = 0) \hookrightarrow A

is a (minimal) relative Sullivan algebra in the above sense, then AA is simply called a (minimal) Sullivan algebra. In particular this means that A=( ,d)A = (\wedge^\bullet, d) is a semifree dgc-algebra.

Example

Let 𝔤\mathfrak{g} be a finite dimensional Lie algebra and write

CE(𝔤)( 𝔤 *,d 𝔤=[,] *) CE(\mathfrak{g}) \coloneqq (\wedge^\bullet \mathfrak{g}^\ast, d_\mathfrak{g} = [-,-]^\ast)

for its Chevalley-Eilenberg algebra. This CE-algebra is a Sullivan algebra in the sense of def. 9, precisely if 𝔤\mathfrak{g} is a nilpotent Lie algebra.

Example

For nn \in \mathbb{N} let

S(n)( c,d=0) S(n) \coloneqq (\wedge^\bullet \langle c \rangle, d = 0)

be the dgc-algebra on a single generator in degree nn with vanishing differential.

For n1n \geq 1 let

D(n)( (bc),db=c,dc=0) D(n) \coloneqq (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0)

be the dgc-algebra generated by an additional generator in degree n1n-1 such that the differential takes this to the previous generator.

Then the canonical inclusions

i 0:(,d=0)S(0) i_0 \;\colon\; (\mathbb{Q},d = 0) \hookrightarrow S(0)

and for n1n \geq 1

i n:S(n)D(n) i_n \;\colon\; S(n) \hookrightarrow D(n)

are relative Sullivan algebras according to 9.

These are to be called the generating cofibrations for the projective model structure on dgc-algebras below in theorem 4.

Moreover, the inclusions

(,d=0)D(n) (\mathbb{Q},d = 0) \hookrightarrow D(n)

for n1n \geq 1 are relative Sullivan algebras.

These are to be called the generating acyclic cofibrations for the projective model structure on dgc-algebras below in theorem 4.

(Hess 06, p. 6)

The examples in 2 are trivial, but they generate all examples of relative Sullivan algebras:

Remark

The special condition on the ordering in the relative Sullivan algebra in def. 9 says that these morphisms are transfinite compositions of pushouts of the generating cofibrations in def. 2:

For AdgcAlg 0A \in dgcAlg_{\geq 0} any dgc-algebra, then a pushout of the form

S(n) ϕ A i n D(n) (A b,db=ϕ) \array{ S(n) &\stackrel{\phi}{\longrightarrow}& A \\ {}^{\mathllap{i_n}}\downarrow && \downarrow \\ D(n) &\longrightarrow& (A \otimes \wedge^\bullet \langle b \rangle, d b = \phi) }

is precisely a choice ϕA\phi \in A of a d Ad_A-closed element in degree nn and results in adjoining to AA the element bb whose differential is db=ϕd b = \phi.

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.

Definition

(Sullivan models)

For XX a simply connected topological space XX, a Sullivan (minimal) model for XX is a Sullivan (minimal) algebra ( V *,d V)(\wedge^\bullet V^\ast, d_V) equipped with a quasi-isomorphism

( V *,d V)Ω pwpoly (X) (\wedge^\bullet V^*, d_V) \stackrel{\simeq}{\longrightarrow} \Omega^\bullet_{pwpoly}(X)

to the dg-algebra of piecewise polynomial differential forms.

Proposition

Minimal Sullivan models (def. 10) are unique up to isomorphism.

e.g Hess 06, prop 1.18.

Homotopy theory

Topological homotopy theory

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.

Theorem

Say that a continuous function, hence a morphism in the category Top of topological spaces is

These classes of morphisms make the category Top into a model category, the classical model structure on topological spaces, to be denoted Top QuillenTop_{Quillen}.

Theorem

Say that a morphism of simplicial sets is

These classes of morphisms make the category sSet of simplicial sets into a model category, the classical model structure on simplicial sets, to be denoted sSet QuillensSet_{Quillen}.

Theorem

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):

Top Quillen QuillenSing||sSet Quillen Top_{Quillen} \underoverset {\underset{Sing}{\longrightarrow}} {\overset{{\vert -\vert}}{\longleftarrow}} {\simeq_{Quillen}} sSet_{Quillen}

Algebraic homotopy theory

Theorem

Say that a homomorphism of differential graded-commutative algebras in non-negative degrees is

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, (dcgAlg ,0) proj(dcgAlg_{\mathbb{Q} ,\geq 0})_{proj}.

This is a cofibrantly generated model category, with generating (acyclic) cofibrations the morphisms from example 2.

(Hess 06, p. 6)

The rationalization adjunction

Theorem

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):

(dgcAlg ,0proj) opK polyΩ poly sSet Quillen (dgcAlg_{\mathbb{Q}, \geq 0}_{proj})^{op} \underoverset {\underset{K_{poly}}{\longrightarrow}} {\overset{\Omega^\bullet_{poly}}{\longleftarrow}} {\bot} sSet_{Quillen}

This is due to (Bousfield-Gugenheim 76, section 8) Review includes (Hess 06, p. 9).

Definition

(subcategories of nilpotent objects of finite type)

Write

  1. Ho(Top ,nil,fin)Ho(Top)Ho(Top_{\mathbb{Q},nil,fin}) \hookrightarrow Ho(Top) for the full subcategory on those topological spaces XX which are

    1. rational: their homotopy groups are uniquely divisible;

    2. nilpotent: their fundamental group is a nilpotent group;

    3. finite type: the \mathbb{Q}-vector space H (X,)H_\bullet(X,\mathbb{Q}) are of finite dimension.

  2. Ho(dgcAlg ,nil fin)Ho(dgcAlg^{fin}_{\mathbb{Q},nil}) for the full subcategory on the differential graded-commutative algebras which are equivalent to minimal Sullivan models ( V,d)(\wedge^\bullet V, d) (def. 9) for which the graded vector space VV is of finite type, i.e. is degreewise of finite dimension over \mathbb{Q}.

Bousfield-Gugenheim 76, p. viii

Theorem

Consider the adjunction of derived functors

Ho(Top)Ho(sSet)Ω poly 𝕃K polyHo((dgcAlg ,0) op) Ho(Top) \simeq Ho(sSet) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\bot} Ho( (dgcAlg_{\mathbb{Q}, \geq 0})^{op} )

induced from the Quillen adjunction from theorem 5.

On the full subcategories of nilpotent objects of finite type, def. 11, this adjunction restricts to an equivalence of categories

Ho(Top ,nil fin)Ω poly 𝕃K polyHo((dgcAlg ,nil fin) op). Ho(Top_{\mathbb{Q}, nil}^{fin}) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\simeq} Ho( (dgcAlg_{\mathbb{Q}, nil}^{fin})^{op} ) \,.

In particular for such spaces the adjunction unit

XK poly(Ω pwpoly (X)) X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X))

exhibits the rationalization of XX.

Bousfield-Gugenheim 76, p. viii

e. g. Hess 06, corollary 1.26.

Corollary

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:

H (X,)H (Ω pwpoly (X),d dR). H^\bullet(X,\mathbb{Q}) \;\simeq\; H^\bullet( \Omega^\bullet_{pwpoly}(X), d_{dR} ) \,.

But more is true, also the rationalization of the homotopy groups may be read off from any minimal Sullivan model:

Theorem

Let ( V *,d V)(\wedge^\bullet V^*, d_V) be a minimal Sullivan model of a simply connected rational topological space XX. Then there is an isomorphism

π (X)V \pi_\bullet(X) \simeq V

between the homotopy groups of XX and the generators of the minimal Sullivan model.

e.g. Hess 06, theorem 1.24.

Remark

The need to restrict to simply connected topological spaces in theorem 6 is due to the existence of acyclic groups. This are discrete groups Γ\Gamma such that their classifying space BΓB \Gamma has trival ordinary cohomology in positive degree

H (BΓ,). H^\bullet(B \Gamma, \mathbb{Q}) \simeq \mathbb{Q} \,.

Therefore, by corollary 1, its dg-algebra of piecewise polynomial differential forms do not distinguish such spaces from contractible topological spaces. But, unless Γ=1\Gamma = 1 is in fact the trivial group, BΓB \Gamma is not contractible, instead it is the Eilenberg-MacLane space K(Γ,1)K(\Gamma,1) with nontrivial fundamental group π 1(BΓ)Γ\pi_1(B \Gamma) \simeq \Gamma. However, by the Hurewicz theorem, this fundamental group is the only obstruction to contractibility.

Examples

Rational nn-spheres

We discuss the minimal Sullivan models of rational n-spheres.

The minimal Sullivan model of a sphere S 2k+1S^{2k+1} of odd dimension is the dg-algebra with a single generator ω 2k+1\omega_{2k+1} in degre 2k+12k+1 and vanishing differential

dω 2k+1=0. d \omega_{2k+1} = 0 \,.

The minimal Sullivan model of a sphere S 2kS^{2k} of even dimension, for k1k \geq 1. is the dg-algeba with a generator ω 2k\omega_{2k} in degree 2k2k and another generator ω 4k1\omega_{4k-1} in degree 4k+14k+1 with the differential defined by

dω 2k=0 d \omega_{2k}= 0
dω 4k+1=ω 2kω 2k. d \omega_{4k+1} = \omega_{2k}\wedge \omega_{2k} \,.

One may understand this form theorem 7: an nn-sphere has rational cohomology concentrated in degree nn. 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 ω 2kω 2k\omega_{2k}\wedge \omega_{2k} needs to be canceled in cohomology. That is accomplished by the second generator ω 4k1\omega_{4k-1}.

Again by theorem 7, this now implies that the rational homotopy groups of spheres are concentrated, in degree 2k+12k+1 for the odd (2k+1)(2k+1)-dimensional spheres, and in degrees 2k2k and 4k14k-1 in for the even 2k2k-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.

Quillen approach

We briefly review Quillen’s approach to rational homotopy theory (Quillen 69), see for instance (Griffith-Morgan 13, chapter 17) .

The following sequence of six consecutive functors, each of which is a Quillen equivalence, takes 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, 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)S(X), obtaining an a simplicial group

GS(X). G S(X).

Then one forms its 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 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.

Properties

Preservation of homotopy pullbacks

Theorem

The left derived functor of the Quillen left adjoint Ω poly :sSetdgcAlg 0\Omega^\bullet_{poly} \colon sSet \to dgcAlg^{\geq 0}_{\mathbb{Q}} (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

(Ω poly || poly):(dgcAlg 0op) Grpd (\Omega^\bullet_{poly} \dashv {\vert -\vert_{poly}}) : (dgcAlg^{\geq 0}_\mathbb{Q}^{op})^\circ \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} \infty Grpd

the left adjoint preserves (∞,1)-categorical pullbacks of objects of finite type.

Proof

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 C={acb}C = \{a \to c \leftarrow b\} be the pullback diagram category.

The homotopy limit functor is the right derived functor lim C\mathbb{R} lim_C for the Quillen adjunction (described in detail at homotopy Kan extension)

[C,sSet] injlim CconstsSet. [C,sSet]_{inj} \stackrel{\overset{const}{\leftarrow}}{\underset{lim_C}{\to}} sSet \,.

At model structure on functors it is discussed that composition with the Quillen pair Ω K\Omega^\bullet \dashv K induces a Quillen adjunction

([C,Ω bullet][C,K]):[C,dgcAlg op][C,K][C,Ω ][C,sSet]. ([C,\Omega^bullet] \dashv [C,K]) : [C, dgcAlg^{op}] \stackrel{\overset{[C,\Omega^\bullet]}{\leftarrow}}{\underset{[C,K]}{\to}} [C,sSet] \,.

We need to show that for every fibrant and cofibrant pullback diagram F[C,sSet]F \in [C,sSet] there exists a weak equivalence

Ω lim CFlim CΩ (F)^, \Omega^\bullet \circ lim_C F \;\; \simeq \;\; lim_C \widehat{\Omega^\bullet(F)} \,,

here Ω (F)^\widehat{\Omega^\bullet(F)} is a fibrant replacement of Ω (F)\Omega^\bullet(F) in dgcAlg opdgcAlg^{op}.

Every object f[C,sSet] injf \in [C,sSet]_{inj} is cofibrant. It is fibrant if all three objects F(a)F(a), F(b)F(b) and F(c)F(c) are fibrant and one of the two morphisms is a fibration. Let us assume without restriction of generality that it is the morphism F(a)F(c)F(a) \to F(c) that is a fibration. So we assume that F(a),F(b)F(a), F(b) and F(c)F(c) are three Kan complexes and that F(a)F(b)F(a) \to F(b) is a Kan fibration. Then lim Clim_C sends FF to the ordinary pullback lim CF=F(a)× F(c)F(b)lim_C F = F(a) \times_{F(c)} F(b) in sSetsSet, and so the left hand side of the above equivalence is

Ω (F(a)× F(c)F(b)). \Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.

Recall that the Sullivan algebras are the cofibrant objects in dgcAlgdgcAlg, hence the fibrant objects of dgcAlg opdgcAlg^{op}. Therefore a fibrant replacement of Ω (F)\Omega^\bullet(F) may be obtained by

  • first choosing a Sullivan model ( V,d V)Ω (c)(\wedge^\bullet V, d_V) \stackrel{\simeq}{\to} \Omega^\bullet(c)

  • then choosing factorizations in dgcAlgdgcAlg of the composites of this with Ω (F(c))Ω (F(a))\Omega^\bullet(F(c)) \to \Omega^\bullet(F(a)) and Ω (F(c))Ω (F(b))\Omega^\bullet(F(c)) \to \Omega^\bullet(F(b)) into cofibrations follows by weak equivalences.

The result is a diagram

( U *,d U) ( V *,d V) ( W *,d W) Ω (F(a)) Ω (F(c)) Ω (F(b)) \array{ (\wedge^\bullet U^*, d_U) &\leftarrow& (\wedge^\bullet V^*, d_V) &\hookrightarrow& (\wedge^\bullet W^* , d_W) \\ \downarrow^{\simeq} && \downarrow^{\simeq} && \downarrow^{\simeq} \\ \Omega^\bullet(F(a)) &\stackrel{}{\leftarrow}& \Omega^\bullet(F(c)) &\stackrel{}{\to}& \Omega^\bullet(F(b)) }

that in dgcAlg opdgcAlg^{op} exhibits a fibrant replacement of Ω (F)\Omega^\bullet(F). The limit over that in dgcAlg opdgcAlg^{op} is the colimit

( U *,d U) ( V *,d V)( W *,d W) (\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W)

in dgcAlgdgcAlg. So the statement to be proven is that there exists a weak equivalence

( U *,d U) ( V *,d V)( W *,d W)Ω (F(a)× F(c)F(b)). (\wedge^\bullet U^* , d_U) \otimes_{(\wedge^\bullet V^* , d_V)} (\wedge^\bullet W^* , d_W) \simeq \Omega^\bullet(F(a) \times_{F(c)} F(b)) \,.

This is precisely the statement of that quoted result Hess 06, theorem 2.2.

Further variants of rational homotopy theory

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:

References

Precursors include

  • Jean-Pierre Serre, Groupes d’homotopy et classes de groupes abelians, Ann. of Math. 58 (1953) 258-294

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. (Numdam, pdf)

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

  • Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. Volume 74, Number 2 (1978), 429-460. (euclid)

  • Flavio da Silveira, Rational homotopy theory of fibrations, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (pdf)

Survey and review includes

  • 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):

  • Kathryn Hess, Rational homotopy theory: a brief introduction, Interactions between homotopy theory and algebra, 175-202. Contemp. Math 436 (arXiv:math.AT/0604626)

  • Phillip Griffiths, John Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (2013)

Review that makes the L-infinity algebra aspect completely manifest is in

  • Urtzi Buijs, Yves Félix, Aniceto Murillo, section 2 of L L_\infty-rational homotopy of mapping spaces (arXiv:1209.4756), published as L L_\infty-models of based mapping spaces, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.

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 March 30, 2017 09:40:17 by Urs Schreiber (192.41.134.32)