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Sullivan model

Idea

A Sullivan model of a rational space $X$ is a particularly well-behaved commutative dg-algebra quasi-isomorphic to the dg-algebra of Sullivan forms on $X$. 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 simple dg-algebras that are equivalent to the dg-algebras of Sullivan differrential 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.

Formally, (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 $V$ is of finite type if in each degree it is finite dimensional. In this case we write $V^{*}$ for its degreewise dual.

• A Grassmann algebra is of finite type if it is the Grassmann algebra ${\wedge }^{•}{V}^{*}$ 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 $V$ a $\mathbb{N}$-graded vector space write ${\wedge }^{•}V$ for the Grassmann algebra over it. Equipped with the trivial differential $d=0$ this is a semifree dga $({\wedge }^{•}V,d=0)$.

With $k$ our ground field we write $(k,0)$ for the corresponding dg-algebra, the tensor unit for the standard monoidal structure on $\mathrm{dgAlg}$. This is the Grassmann algebra on the 0-vector space $(k,0)=({\wedge }^{•}0,0)$.

Definition

(Sullivan algebras)

A relatived Sullivan algebra is a morphism of dg-algebras that is an inclusion

$$(A,d)\to (A{\otimes }_k{\wedge }^{•}V,d\prime )$$(A,d) \to (A \otimes_k \wedge^\bullet V, d')

for $(A,d)$ some dg-algebra and for $V$ some graded vector space, such that

• there is a well ordered set? $J$

• indexing a basis $\{v_{\alpha }\in V\mid \alpha \in J\}$ of $V$;

• such that with $V_{<\beta }=\mathrm{span}(v_{\alpha }\mid \alpha <\beta )$ for all basis elements $v_{\beta }$ we have that

  $$d\prime v_{\beta }\in A\otimes {\wedge }^{•}{V}_{<\beta }.$$d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,.

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

$$(\alpha <\beta )\Rightarrow (\mathrm{deg}{v}_{\alpha }\le \mathrm{deg}{v}_{\beta })$$(\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta)

holds. For a Sullivan algebra $(k,0)\to ({\wedge }^{•}V,d)$ relative to the tensor unit we call the semifree dga $({\wedge }^{•}V,d)$ simply a Sullivan algebra. And a minimal Sullivan algebra if $(k,0)\to ({\wedge }^{•}V,d)$ 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 precisely that these morphisms are composites of pushouts of the generating cofibrations of the model structure on dg-algebras, which are the inclusions

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

where

$$S(n)=({\wedge }^{•}⟨c⟩,d=0)$$S(n) = (\wedge^\bullet \langle c \rangle, d= 0)

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

$$D(n)=({\wedge }^{•}(⟨b⟩\oplus ⟨c⟩),db=c,dc=0)$$D(n) = (\wedge^\bullet (\langle b \rangle \oplus \langle c \rangle), d b = c, d c = 0)

with $b$ an additional generator in degree $n-1$.

Therefore for $A\in \mathrm{dgAlg}$ dg-algebra, a pushout

$$\begin{array}{ccc}S(n)& \stackrel{\varphi }{\to }& A\\ \downarrow & & \downarrow \\ D(n)& \to & (B\otimes {\wedge }^{•}⟨b⟩,db=\varphi )\end{array}$$\array{ S(n) &\stackrel{\phi}{\to}& A \\ \downarrow && \downarrow \\ D(n) &\to& (B \otimes \wedge^ \bullet \langle b \rangle, d b = \phi) }

is precisely a choice $\varphi \in A$ of a $d_A$-closed element in degree $n$ and results in adjoining to $A$ the element $b$ whose differential is $db=\varphi$. This gives the condition in the above definition: the differential of any new element has to be one of the old elements.

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

Remark

($L_{\mathrm{\infty }}$-algebras)

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

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

$$\omega (D(v_1\vee v_2\vee \cdots v_n))=-(d\omega )(v_1,v_2,\cdots ,v_n)$$\omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n)

for all $\omega \in V$ and all $v_i\in V^{*}$.

Definition

(Sullivan models)

For $X$ a simply connected? topological space $X$, a Sullivan (minimal) model for $X$ is

• a quasi-free dg-algebra $({\wedge }^{•}{V}^{*},d_V)$ which is a (minimal) Sullivan algebra

• such that there exists a quasi-isomorphism

  $$({\wedge }^{•}{V}^{*},d_V)\stackrel{\simeq }{\to }{\Omega }_{\mathrm{Sull}}^{•}(X).$$(\wedge^\bullet V^*, d_V) \stackrel{\simeq}{\to} \Omega^\bullet_{Sull}(X) \,.

Properties

Proposition

(cofibrations are relative Sullivan algebras)

The cofibration in the standard model structure on dg-algebras $C{\mathrm{dgAlg}}_{\mathbb{N}}$ are precisely the retracts of relative Sullivan algebras $(A,d)\to (A{\otimes }_k{\wedge }^{•}V,d\prime )$.

Accordingly, the cofibrant objects in $C\mathrm{dgAlg}$ are precisely the Sullivan algebras $({\wedge }^{•}V,d)$

Theorem

Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan algebras $({\wedge }^{•}V,d)$

$$({\wedge }^{•}V,d)\stackrel{\simeq }{\to }{\Omega }_{\mathrm{Sullivan}}^{•}(X).$$(\wedge^\bullet V , d) \stackrel{\simeq}{\to} \Omega^\bullet_{Sullivan}(X) \,.

And homotopy classes of morphisms on both sides are in bijection.

Proof

This appears for instance as corollary 1.26 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Write

$$({\Omega }_{\mathrm{Sul}}^{•}⊣K):{\mathrm{dgAlg}}^{\mathrm{op}}\stackrel{\stackrel{{\Omega }_{\mathrm{Sul}}^{•}}{\leftarrow }}{\underset{K}{\to }}\mathrm{sSet}$$(\Omega^\bullet_{Sul} \dashv K) : dgAlg^{op} \stackrel{\overset{\Omega^\bullet_{Sul}}{\leftarrow}}{\underset{K}{\to}} sSet

for the Quillen adjunction induced by forming Sullivan differential forms, as discussed above.

Theorem

Let $({\wedge }^{•}{V}^{*},d_V)$ be a simply connected Sullivan algebra of finite type. Then

• the unit of the adjunction $({\wedge }^{•}{V}^{*},d_V)\to {\Omega }_{\mathrm{Sul}}^{•}(K({\wedge }^{•},d_V)⟩)$ is a quasi-isomorphism;

• The elements of the homotopy groups of the rational space modeled by $({\wedge }^{•}{V}^{*},d_V)$ are the generators in $V$:

  there is an isomorphism of $\mathbb{N}$-graded vector spaces over $\mathbb{Q}$

  $${\pi }_{•}(⟨({\wedge }^{•}{V}^{*},d_V)⟩)\simeq V.$$\pi_\bullet(\langle (\wedge^\bullet V^*, d_V)\rangle) \simeq V.

Proof

This is recalled for instance as theorem 1.24 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Proposition

Sullivan mimimal models are unique up to isomorphism.

Proof

This appears for instance as prop 1.18 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Theorem

Rational homotopy types of simply connected spaces $X$ are in bijective corespondence with minimal Sullivan models $({\wedge }^{•}V,d)$

$$({\wedge }^{•}V,d)\stackrel{\simeq }{\to }{\Omega }_{\mathrm{Sullivan}}^{•}(X).$$(\wedge^\bullet V , d) \stackrel{\simeq}{\to} \Omega^\bullet_{Sullivan}(X) \,.

And homotopy classes of morphisms on both sides are in bijection.

Proof

This appears for instance as corollary 1.26 in

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Corollary

It follows that if (\wedge^^\bullet V^{*}, d) is a minimal Sullivan model for $X$, then the rational homotopy groups of $X$ can be read off from the generators $V$:

$${\pi }_{•}(X)\otimes \mathbb{Q}\simeq V.$$\pi_\bullet(X) \otimes \mathbb{Q} \simeq V \,.

Examples

The $n$-sphere

…

Complex projective space

…

Product spaces

…

Formal spaces

• formal dg-algebra

References

In

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Sullivan algebras and minimal algebras appear in def 1.10