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Sullivan model in nLab

Contents

Context

Rational homotopy theory

differential graded objects

and

rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)

dg-Algebra

Rational spaces

PL de Rham complex

Sullivan models

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 homotopy 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

(of 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 ℕ\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 α∈V|α∈J}\{v_\alpha \in V| \alpha \in J\} of VV;

  2. writing V <β≔span(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

d′v β∈A⊗∧ •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 α≤degv 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.

(e.g. Hess 06, def. 1.10, remark 1.11)

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

Properties

As cofibrations

Proposition

Minimal Sullivan models are unique up to isomorphism.

e.g Hess 06, prop 1.18.

Rationalization

Theorem

Consider the derived adjunction

Ho(Top)≃Ho(sSet)⊥⟶ℝΩ poly •⟵𝕃K polyHo((dgcAlg ℚ,≥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 of the PL de Rham complex-Quillen adjunction

(dgcAlg ℚ,≥0proj) op⊥⟶K poly⟵Ω 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 ℚ,≥1 fin)Ho(Top_{\mathbb{Q}, \geq 1}^{fin}) of nilpotent rational topological spaces of finite type this adjunction restricts to an equivalence of categories

Ho(Top ℚ,>1 fin)≃⟶ℝΩ poly •⟵𝕃K polyHo((dgcAlg ℚ,>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 derived adjunction unit

X⟶K poly(Ω pwpoly •(X)) X \longrightarrow K_{poly}(\Omega^\bullet_{pwpoly}(X))

exhibits the rationalization of XX.

This is the fundamental theorem of dgc-algebraic rational homotopy theory, see there for more.

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.

Relation to nilpotent L ∞L_\infty-algebras

Under the formal duality between L ∞ L_\infty -algebras and their Chevalley-Eilenberg dgc-algebras, the connected Sullivan models correspond bijectively to connective nilpotent L ∞ L_\infty -algebras (Berglund 2015, Thm. 2.3).

Relation to Whitehead products

See at the co-binary Sullivan differential is the dual Whitehead product.

Examples

References

Original articles:

Review and application:

Dual interpretation as nilpotent L ∞ L_\infty -algebras:

Last revised on June 21, 2023 at 16:40:38. See the history of this page for a list of all contributions to it.