Sullivan minimal model

The theory of minimal models began with the work of D. Quillen [a5]. A simply-connected topological space (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 a mapping , such that is rational and is an isomorphism. The interest of this construction is that the homotopy category of rational spaces has an algebraic nature. More precisely, in [a5], 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 [a6], D. Sullivan associated to each space a commutative differential graded algebra (CDGA), , which is linked to the cochain algebra 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 -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, , such that , , and for each .

The correspondence

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 be a commutative differential graded algebra such that , , and for each . There exists then a quasi-isomorphism of commutative differential graded algebras , where denotes the free commutative algebra on the graded vector space of finite type , and . The cochain algebra is called the Sullivan minimal model of ; it is unique up to isomorphism.

The Sullivan minimal model of is called the Sullivan minimal model of . It satisfies and . More generally, for each continuous mapping , there is a commutative diagram

where and are quasi-isomorphisms, , , and where and are the canonical injection and projection. In this case, the Grivel–Halperin–Thomas theorem asserts that is a Sullivan minimal model for the homotopy fibre of [a4].

A key result in the theory is the so-called mapping theorem [a1]. Recall that the Lyusternik–Shnirel'man category of is the least integer such that can be covered by open sets each contractible in (cf. also Category (in the sense of Lyusternik–Shnirel'man)). If is a mapping between simply-connected spaces and if is injective, then . The Lyusternik–Shnirel'man category of can be computed directly from its Sullivan minimal model . Indeed, consider the following commutative diagram:

where and denote the canonical projection and injection and is a quasi-isomorphism. The category of is then the least integer such that admits a retraction [a1].

To obtain properties of simply-connected spaces with finite category, it is therefore sufficient to consider Sullivan minimal models with finite category. Using this procedure, Y. Félix, S. Halperin and J.-C. Thomas have obtained the following dichotomy theorem: Either is finite-dimensional (the space is called elliptic), or else the sequence has exponential growth (the space is thus called hyperbolic) [a2].

When is elliptic, the dimension of is finite, the Euler characteristic is non-negative and the rational cohomology algebra satisfies Poincaré duality [a3].

The minimal model of contains all the rational homotopy invariants of . For instance, the cochain algebra is a model for the th Postnikov tower of (cf. also Postnikov system), and the mapping induced by is the dual of the st -invariant

The quadratic part of the differential is dual to the Whitehead product in . More precisely, , , , .

References

[a1]	Y. Félix, S. Halperin, "Rational LS category and its applications" Trans. Amer. Math. Soc. , 273 (1982) pp. 1–37
[a2]	Y. Félix, S. Halperin, J.C. Thomas, "Rational homotopy theory" (in preparation)
[a3]	S. Halperin, "Finiteness in the minimal models of Sullivan" Trans. Amer. Math. Soc. , 230 (1977) pp. 173–199
[a4]	S. Halperin, "Lectures on minimal models" Mémoire de la SMF , 9/10 (1983)
[a5]	D. Quillen, "Rational homotopy theory" Ann. of Math. , 90 (1969) pp. 205–295
[a6]	D. Sullivan, "Infinitesimal computations in topology" Publ. IHES , 47 (1977) pp. 269–331