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Sullivan model

Contents

Idea

A Sullivan model of a rational space XX is a particularly well-behaved commutative dg-algebra quasi-isomorphic to the dg-algebra of Sullivan forms on XX. These Sullivan algebras are precisely the cofibrant objects in the standard model structure on dg-algebras.

Sullivan models are a central tool in rational homotopy theory.

Definition

Sullivan models are particularly well-behaved differential graded-commutative algebras that are equivalent to the dg-algebras of [[piecewise polynomial differential forms on topological spaces]. Conversely, every rational space can be obtained from a dg-algebra and the minimal Sullivan algebras provide convenient representatives that correspond bijectively to rational homtopy types under this correspondence.

Abstractly, (relative) Sullivan models are the (relative) cell complexes in the standard model structure on dg-algebras.

We now describe this in detail. First some notation and preliminaries:

Definition

(finite type)

  • A graded vector space VV is of finite type if in each degree it is finite dimensional. In this case we write V *V^* for its degreewise dual.

  • A Grassmann algebra is of finite type if it is the Grassmann algebra V *\wedge^\bullet V^* on a graded vector space of finite type

    (the dualization here is just convention, that will help make some of the following constructions come out nicely).

  • A CW-complex is of finite type if it is built out of finitely many cells in each degree.

For VV a \mathbb{N}-graded vector space write V\wedge^\bullet V for the Grassmann algebra over it. Equipped with the trivial differential d=0d = 0 this is a semifree dgc-algebra ( V,d=0)(\wedge^\bullet V, d=0).

With kk our ground field we write (k,0)(k,0) for the corresponding dg-algebra, the tensor unit for the standard monoidal structure on dgAlgdgAlg. This is the Grassmann algebra on the 0-vector space (k,0)=( 0,0)(k,0) = (\wedge^\bullet 0, 0).

Definition

(Sullivan algebras)

A relatived Sullivan algebra is a homomorphism of differential graded-commutative algebras that is an inclusion of the form

(A,d)(A k V,d) (A,d) \hookrightarrow (A \otimes_k \wedge^\bullet V, d')

for (A,d)(A,d) 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} \,.

This is called a minimal relative Sullivan algebra if in addition the condition

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

holds. For a Sullivan algebra (k,0)( V,d)(k,0) \to (\wedge^\bullet V, d) relative to the tensor unit we call the semifree dgc-algebra ( V,d)(\wedge^\bullet V,d) simply a Sullivan algebra, and we call it a minimal Sullivan algebra if (k,0)( V,d)(k,0) \to (\wedge^\bullet V, d) is a minimal relative Sullivan algebra.

See also the section Sullivan algebras at model structure on dg-algebras.

Remark

The special condition on the ordering in the relative Sullivan algebra says that these morphisms are composites of pushouts of the generating cofibrations for the model structure on dg-algebras, which are the inclusions

S(n)D(n), S(n) \hookrightarrow D(n) \,,

where

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

is the dg-algebra on a single generator in degree nn with vanishing differential, and where

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

with bb an additional generator in degree n1n-1.

Therefore for AdgcAlgA \in dgcAlg, a pushout

S(n) ϕ A D(n) (A b,db=ϕ) \array{ S(n) &\stackrel{\phi}{\to}& A \\ \downarrow && \downarrow \\ D(n) &\to& (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.

Notice that it follows in particular that the cofibrations in dgAlg projdgAlg_{proj} are precisely all the retracts of relative Sullivan algebra inclusions.

Remark

(L L_\infty-algebras)

Because they are semifree dgas, Sullivan dg-algebras ( V,d)(\wedge^\bullet V,d) are (at least for degreewise finite dimensional VV) Chevalley-Eilenberg algebras of L-∞-algebras.

The co-commutative differential co-algebra encoding the corresponding L-∞-algebra is the free cocommutative algebra V *\vee^\bullet V^* on the degreewise dual of VV with differential D=d *D = d^*, i.e. the one given by the formula

ω(D(v 1v 2v n))=(dω)(v 1,v 2,,v n) \omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n)

for all ωV\omega \in V and all v iV *v_i \in V^*.

Definition

(Sullivan models)

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

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

to the dg-algebra of piecewise polynomial differential forms.

Properties

As cofibrations

Proposition

(cofibrations are relative Sullivan algebras)

The cofibrations in the projective model structure on differential graded-commutative algebras (dgcAlg ) proj(dgcAlg_{\mathbb{N}})_{proj} are precisely the retracts of relative Sullivan algebra inclusions (def. 2).

Accordingly, the cofibrant objects in (dgcAlg ) proj(dgcAlg_{\mathbb{N}})_proj{} are precisely the retracts of Sullivan algebras.

Proposition

Minimal Sullivan models are unique up to isomorphism.

e.g Hess 06, prop 1.18.

Rationalization

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

(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 theorem).

Then: On the full subcategory Ho(Top ,1 fin)Ho(Top_{\mathbb{Q}, \geq 1}^{fin}) of simply connected rational topological spaces of finite type this adjunction restricts to an equivalence of categories

Ho(Top ,>1 fin)Ω poly 𝕃K polyHo((dgcAlg ,>1 fin) op). Ho(Top_{\mathbb{Q}, \gt 1}^{fin}) \underoverset {\underset{\mathbb{R} \Omega^\bullet_{poly}}{\longrightarrow}} {\overset{\mathbb{L} K_{poly} }{\longleftarrow}} {\simeq} Ho( (dgcAlg_{\mathbb{Q}, \gt 1}^{fin})^{op} ) \,.

In particular the adjunction unit

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

exhibits the rationalization of XX.

This is a central theorem of rational homotopy theory, see for instance 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:

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.

Examples

The nn-sphere

See at rational n-sphere.

Complex projective space

Product spaces

Formal spaces

Free loop spaces

References

Revised on February 22, 2017 14:29:46 by Urs Schreiber (83.208.22.80)