Sullivan minimal model

2020 Mathematics Subject Classification: Primary: 57D99 Secondary: 55D9958A10 [MSN][ZBL]

The theory of minimal models began with the work of D. Quillen [Qu]. A simply-connected topological space $X$ (cf. also Simply-connected domain) is called rational if its homotopy groups are rational vector spaces (cf. also Homotopy group; Vector space). The rationalization functor associates to each simply-connected space $X$ a mapping $X \to X_0$, such that $X_0$ is rational and $\pi^*(f) \otimes \QQ$ is an isomorphism. The interest of this construction is that the homotopy category of rational spaces has an algebraic nature. More precisely, in [Qu], D. Quillen established an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algebras (cf. also Lie algebra, graded).

In [Su], D. Sullivan associated to each space $X$ a commutative differential graded algebra (CDGA), $A_\text{PL}(X)$, which is linked to the cochain algebra $C^*(X; \QQ)$ by a chain of differential graded algebra quasi-isomorphisms (i.e. morphisms inducing isomorphisms in cohomology). This, in particular, gave a solution to Thom's problem of constructing commutative cochains over the rationals. The $A_{\text{PL}}$-functor together with its adjoint, the realization functor of a commutative differential graded algebra, induce an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces with finite Betti numbers and the homotopy category of rational commutative differential graded algebras, $(A, d)$, such that $H^0(A, d) = \QQ$, $(A, d)=0$, and $\dim H^p(A, d) < \infty$ for each $p$.

The correspondence

\begin{array}{c} \text{comutative differential graded algebra}\\ \Updownarrow\\ \text{Spaces} \end{array}

behaves well with respect to fibrations and cofibrations (cf. also Fibration). Rational homotopy invariants of a space are most easily obtained by means of constructions in the category of commutative differential graded algebras. This procedure has been made very powerful with the Sullivan minimal models.

Let $(A, d)$ be a commutative differential graded algebra such that $H^0(A, d) = \QQ$, $H^1(A, d) = 0$, and $\dim H^p(A, d) < \infty$ for each $p$. There exists then a quasi-isomorphism of commutative differential graded algebras $\varphi : (\wedge V, d) \to (A, d)$, where $\wedge V$ denotes the free commutative algebra on the graded vector space of finite type $V$, and $d(V) \subset \wedge^{\ge 2} V$. The cochain algebra $(\wedge V, d)$ is called the Sullivan minimal model of $(A, d)$; it is unique up to isomorphism.

The Sullivan minimal model of $A_{\text{PL}}(X)$ is called the Sullivan minimal model of $X$. It satisfies $H^*(\wedge V, d) \cong H^*(X; \QQ)$ and $V^n \cong \Hom(\pi_n(X), \QQ)$. More generally, for each continuous mapping $f: X \to Y$, there is a commutative diagram

\begin{array}{ccccc} A_{\text{PL}}(Y) & \xrightarrow{A_\text{PL}(f)} & A_{\text{PL}}(X) \\ \big\uparrow \varphi & & \big\uparrow \psi \\ (\bigwedge V, d) & \xrightarrow{\ \ \ i\ \ \ } & (\bigwedge V \otimes \bigwedge W, D) & \xrightarrow{\ \ \ p\ \ \ } & (\bigwedge W, \overline D) \end{array}

where $\psi$ and $\varphi$ are quasi-isomorphisms, $d(V) \subset \wedge^{\ge 2} V$, $\overline D(W) \subset \wedge^{\ge 2} W$, and where $i$ and $p$ are the canonical injection and projection. In this case, the Grivel–Halperin–Thomas theorem asserts that $(\wedge W, \overline D)$ is a Sullivan minimal model for the homotopy fibre of $f$ [Ha2].

A key result in the theory is the so-called mapping theorem [FéHa]. Recall that the Lyusternik–Shnirel'man category of $X$ is the least integer $n$ such that $X$ can be covered by $n+1$ open sets each contractible in $X$ (cf. also Category (in the sense of Lyusternik–Shnirel'man)). If $f: X \to Y$ is a mapping between simply-connected spaces and if $\pi^*(f) \otimes \QQ$ is injective, then $\operatorname{cat}(X_0) \le \operatorname{cat}(Y_0)$. The Lyusternik–Shnirel'man category of $X_0$ can be computed directly from its Sullivan minimal model $(\wedge V, d)$. Indeed, consider the following commutative diagram:

\begin{array}{ccc} (\bigwedge V, d) & \xrightarrow{\ \ p \ \ } & (\bigwedge V / \bigwedge^{> n} V, d)\\ \big\| & & \big \uparrow \varphi\\ (\bigwedge V, d) & \xrightarrow{\ \ i \ \ } & (\bigwedge V \otimes \bigwedge W, D) \end{array}

where $p$ and $i$ denote the canonical projection and injection and $\varphi$ is a quasi-isomorphism. The category of $X_0$ is then the least integer $n$ such that $i$ admits a retraction [FéHa].

To obtain properties of simply-connected spaces with finite category, it is therefore sufficient to consider Sullivan minimal models $(\wedge V, d)$ with finite category. Using this procedure, Y. Félix, S. Halperin and J.-C. Thomas have obtained the following dichotomy theorem: Either $\pi^*(X) \otimes \QQ$ is finite-dimensional (the space is called elliptic), or else the sequence $\sum_{i=1}^N \dim \pi_i(X) \otimes \QQ$ has exponential growth (the space is thus called hyperbolic) [FéHaTh].

When $X$ is elliptic, the dimension of $H^*(X; \QQ)$ is finite, the Euler characteristic is non-negative and the rational cohomology algebra satisfies Poincaré duality [Ha].

The minimal model of $X$ contains all the rational homotopy invariants of $X$. For instance, the cochain algebra $(\wedge V^{\le m}, d)$ is a model for the $m$th Postnikov tower $X_0(m)$ of $X_0$ (cf. also Postnikov system), and the mapping $\widetilde{d\ } : V^{m+1} \to H^{m+1}(\wedge V^{\le m}, d)$ induced by $d$ is the dual of the $(m+1)$st $k$-invariant

$$k_{m+1} \in H^{m+1}(X_0(m), \pi_{m+1}(X_0)) = \Hom(H_{m+1}(X_0(m)), \pi_{m+1}(X_0)).$$

The quadratic part of the differential $d_1 : V \to \wedge^2 V$ is dual to the Whitehead product in $(\wedge V, d)$. More precisely, $(d_1 v; x, y) = (-1)^{k+n-1}(v, [x, y])$, $v \in V$, $x \in \pi(k(X)$, $y \in \pi_n(X)$.

References

[FéHa]	Y. Félix, S. Halperin, "Rational LS category and its applications" Trans. Amer. Math. Soc., 273 (1982) pp. 1–37 MR0664027
[FéHaTh]	Y. Félix, S. Halperin, J.C. Thomas, "Rational homotopy theory" (in preparation) MR1802847 Zbl 0961.55002 Zbl 0691.55001
[Ha]	S. Halperin, "Finiteness in the minimal models of Sullivan" Trans. Amer. Math. Soc., 230 (1977) pp. 173–199 MR0461508 Zbl 0364.55014
[Ha2]	S. Halperin, "Lectures on minimal models" Mémoire de la SMF, 9/10 (1983) MR0736299 MR0637558 Zbl 0536.55003 Zbl 0505.55014
[Qu]	D. Quillen, "Rational homotopy theory" Ann. of Math., 90 (1969) pp. 205–295 MR0258031 Zbl 0191.53702
[Su]	D. Sullivan, "Infinitesimal computations in topology" Publ. IHES, 47 (1977) pp. 269–331 MR0646078 Zbl 0374.57002