nLab formal dg-algebra

This entry is about quasi-isomorphism to cochain cohomology. For infinitesimal thickening of smooth manifolds see at formal smooth manifold.

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Definition

A dg-algebra $A$ is a formal dg-algebra if in the homotopy category of the model structure on dg-algebras it is isomorphic

$$A \stackrel{\simeq}{\to} H^\bullet(A)$$

to its chain (co)homology (regarded as a dg-algebra with trivial differential). Since all dg-algebras are fibrant in the standard model, this is equivalent to the existence of a span

$$A \stackrel{\simeq_w}{\leftarrow} Q A \stackrel{\simeq}{\to} H^\bullet(A)$$

of quasi-isomorphisms of dg-algebras.

Examples

In rational homotopy theory rational topological spaces are encoded in their dg-algebras of Sullivan forms. A simply connected topological space $X$ whose dg-algebra of Sullivan forms $\Omega^\bullet(X)$ is formal is called a formal topological space. (One can also say a formal rational space, to distinguish from the unrelated formal spaces in formal topology.) Such a space represents a formal homotopy type.

Examples are

• compact Kähler manifolds (e.g. smooth projective varieties);

• Lie groups;

• classifying spaces$B G$ of Lie groups $G$;

• some homogeneous spaces $G/H$

• the unstable Thom spaces $M U_n$ and $M S O_n$

• the Eilenberg-MacLane space $K(\mathbb{Z},2)$,

  whose Sullivan minimal model is the dg-algebra on a single degree-2 generator with trivial differential.

• the little n-disk operad is formal

Related concepts

• H-cohomology

References

For an early discussion of formal dg-algebras in the context of rational homotopy theory see section 12 of

• Dennis Sullivan, Infinitesimal computations in topology , Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam: djvu, pdf)

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

A survey is around definition 2.1 of

• Kathryn Hess, Rational homotopy theory: a brief introduction (arXiv)

Review with an eye towards the dd^c-lemma and H-cohomology is in

• Gil Cavalcanti, p. 12 of New aspects of the $d d^c$-lemma (arXiv:math/0501406)

The case of Sullivan algebras of sphere bundles:

• Jiawei Zhou, Formality of Sphere Bundles [arXiv:2304.09594]