Rational Homotopy Theory - Sullivan Algebras

Sullivan Algebras

A Sullivan algebra is a commutative differential graded algebra over the rationals Q, whose underlying algebra is the free commutative graded algebra Λ(V) on a graded vector space

satisfying the following "nilpotence condition on d ": V is the union of an increasing series of graded subspaces V(0)⊆V(1)⊆ where d = 0 on V(0) and d(V(k)) is contained in Λ(V(k − 1)). Here "commutative" means commutative in the graded sense, sometimes called supercommutative. Thus ab = (−1)deg(a)deg(b)ba.)

The Sullivan algebra is called minimal if the image of d is contained in Λ+(V)2, where Λ+(V) is the direct sum of the positive degree subspaces of Λ(V).

A Sullivan model for a commutative differential graded algebra A is an algebra homomorphism from a Sullivan algebra Λ(V) that is an isomorphism on cohomology. If A0 = Q then A has a minimal Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology as A need not be a minimal Sullivan model for A: it is also necessary that the isomorphism of cohomology be induced by an algebra homomorphism. There are examples of non-isomorphic minimal Sullivan models with the same cohomology algebra.)