Rational Homotopy Theory - Examples

Examples

• If X is a sphere of odd dimension 2n + 1 > 1, its minimal Sullivan model has 1 generator a of degree 2n + 1 with da = 0, and a basis of elements 1, a.
• If X is a sphere of even dimension 2n > 0, its minimal Sullivan model has 2 generators a and b of degrees 2n and 4n − 1, with db = a2, da = 0, and a basis of elements 1, a, b→ a2, ab→a3, a2b→a4, ... where the arrow indicated the action of d.
• Suppose that V has 4 elements a, b, x, y of degrees 2, 3, 3 and 4 with differentials da = 0, db = 0, dx = a2, dy = ab. Then this algebra is a minimal Sullivan algebra that is not formal. The cohomology algebra has nontrivial components only in dimension 2,3,6, generated respectively by a, b and xb-ay. Any homomorphism from V to its cohomology algebra would map y to 0, x to a multiple of b, so it would surely map xb-ay to 0. So V cannot be a model for its cohomology algebra. The corresponding topological spaces are two spaces with the same rational cohomology ring but different rational homotopy types. Notice that xb-ay is in the Massey product .