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

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

Definition

(Sullivan algebras)

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

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

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

1. there is a well ordered set $J$ indexing a linear basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;

2. writing $V_{\lt \beta} \coloneqq span(v_\alpha | \alpha \lt \beta)$ then for all basis elements $v_\beta$ we have that

$$d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,.$$

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

$$(\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta)$$

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

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

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

where

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

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

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

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

Therefore for $A \in dgcAlg$, a pushout

$$\array{ S(n) &\stackrel{\phi}{\to}& A \\ \downarrow && \downarrow \\ D(n) &\to& (A \otimes \wedge^ \bullet \langle b \rangle, d b = \phi) }$$

is precisely a choice $\phi \in A$ of a $d_A$-closed element in degree $n$ and results in adjoining to $A$ the element $b$ whose differential is $d b = \phi$. 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 $dgAlg_{proj}$ are precisely all the retracts of relative Sullivan algebra inclusions.

Remark

($L_\infty$-algebras)

Because they are semifree dgas, Sullivan dg-algebras $(\wedge^\bullet V,d)$ 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 $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula

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

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 Sullivan (minimal) algebra $(\wedge^\bullet V^\ast, d_V)$ equipped with a quasi-isomorphism

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

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 $(dgcAlg_{\mathbb{N}})_{proj}$ are precisely the retracts of relative Sullivan algebra inclusions (def. ).

Accordingly, the cofibrant objects in $(dgcAlg_{\mathbb{N}})_proj{}$ 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 derived adjunction

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

induced from the of the PL de Rham complex-Quillen adjunction

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

(this theorem).

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

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

In particular the derived adjunction unit

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

exhibits the rationalization of $X$.

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

$$\pi_\bullet(X) \simeq V$$

between the homotopy groups of $X$ and the generators of the minimal Sullivan model.

e.g. Hess 06, theorem 1.24.

Relation to nilpotent $L_\infty$-algebras

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

Relation to Whitehead products

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

Examples

Examples of Sullivan models in rational homotopy theory:

• Sullivan model of an n-sphere

• Sullivan model of a spherical fibration

• Sullivan model of complex projective space

• Sullivan model of a classifying space

• Sullivan model of mapping space

• Sullivan model of free loop space

• Sullivan model of a suspension

• Sullivan model of a finite G-quotient

Related concepts

• formal dg-algebra

• minimal dg-module

• minimal fibration

• equivariant Sullivan model

References

Original articles:

• Dennis Sullivan, Differential forms and the topology of manifolds, in Proc. International Conf.: Manifolds Tokyo (1973), Univ. Tokyo Press (1975) 37-49 [pdf, pdf]

• Alan J. Deschner, Sullivan’s theory of minimal models, MSc thesis, Univ. British Columbia (1976) [doi:10.14288/1.0080132, pdf]

• Dennis Sullivan, Infinitesimal computations in topology, Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam:PMIHES_1977__47__269_0)

• Aldridge Bousfield, Victor Gugenheim, On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)

• Phillip Griffiths, John Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (2013) (doi:10.1007/978-1-4614-8468-4)

Review and application:

• Steve Halperin, Lectures on minimal models, Mem. Soc. Math. Franc. no 9/10 (1983) (doi:10.24033/msmf.294)

• Yves Félix, Stephen Halperin, Jean-Claude Thomas, Chapter II of: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)

• Kathryn Hess, around def 1.10 of Rational homotopy theory: a brief introduction (arXiv:math.AT/0604626)

• Luc Menichi, Rational homotopy – Sullivan models, in: Free Loop Spaces in Geometry and Topology, IRMA Lect. Math. Theor. Phys., EMS (2015) [arXiv:1308.6685, doi:10.4171/153]

• Yves Félix, Steve Halperin, Rational homotopy theory via Sullivan models: a survey, Notices of the International Congress of Chinese Mathematicians Volume 5 (2017) Number 2 (arXiv:1708.05245, doi:10.4310/ICCM.2017.v5.n2.a3)

Dual interpretation as nilpotent $L_\infty$-algebras:

• Alexander Berglund, Def. 2.1 in: Rational homotopy theory of mapping spaces via Lie theory for $L_\infty$ algebras, Homology, Homotopy and Applications, Volume 17 (2015) Number 2 (arXiv:1110.6145, doi:10.4310/HHA.2015.v17.n2.a16)