nLab formality

A rational homotopy type is “formal’‘ if its minimal model has differential zero. A dg-algebra, or, more generally, an A-infinity algebra is formal if it is quasiisomorphic to its own (co)homology.

The formality for simply-connected compact Kähler manifolds implies that their real homotopy type is determined by their de Rham cohomology ring as shown using Hodge theory in the seminal work

• Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274, MR382702 doi

This is later improved to rational homotopy type in

• Dennis Sullivan, Infinitesimal computations in topology, Publ. math. de l’ IHÉS. 47 (1977), p. 269-331, numdam,

A major formality result is the Kontsevich formality (see references there) and the Tamarkin formality of the little disks operad which implies it. For both see Kontsevich formality.

There are many recent cases of formality in geometry, topology and algebra

• Vasiliy Dolgushev, Dmitry Tamarkin, Boris Tsygan, Formality of the homotopy calculus algebra of Hochschild (co)chains, arxiv/0807.5117

category: algebra