Rational Homotopy Theory - The Sullivan Minimal Model of A Topological Space

The Sullivan Minimal Model of A Topological Space

For any topological space X Sullivan defined a commutative differential graded algebra A_PL(X), called the algebra of polynomial differential forms on X with rational coefficients. An element of this algebra consists of (roughly) a polynomial form on each singular simplex of X, compatible with face and degeneracy maps. This algebra is usually very large (uncountable dimension) but can be replaced by a much smaller algebra. More precisely, any differential graded algebra with the same Sullivan minimal model as A_PL(X) is called a model for the space X, and determines the rational homotopy type of X when X is simply connected.

To any simply connected CW complex X with all rational homology groups of finite dimension one can assign a minimal Sullivan algebra ΛV of A_PL(X), which has the property that V1 = 0 and all the Vk of finite dimension. This is called the Sullivan minimal model of X, and is unique up to isomorphism. This gives an equivalence between rational homotopy types of such spaces and such algebras, such that:

• The rational cohomology of the space is the cohomology of its Sullivan minimal model.
• The spaces of indecomposables in V are the duals of the rational homotopy groups of the space X.
• The Whitehead product on rational homotopy is the dual of the "quadratic part" of the differential d.
• Two spaces have the same rational homotopy type if and only if their minimal Sullivan algebras are isomorphic.
• There is a simply connected space X corresponding to each possible Sullivan algebra with V1 = 0 and all the Vk of finite dimension.

When X is a smooth manifold, the differential algebra of smooth differential forms on X (the de Rham complex) is almost a model for X; more precisely it is the tensor product of a model for X with the reals and therefore determines the real homotopy type. One can go further and define the p-adic homotopy type and the adelic homotopy type and compare them to the rational homotopy type.

The results above for simply connected spaces can easily be extended to nilpotent spaces (whose fundamental group is nilpotent and acts nilpotently on the higher homotopy groups). For more general fundamental groups things get more complicated; for example, the homotopy groups need not be finitely generated even if there are only a finite number of cells of the CW complex in each dimension.