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 it is quasi-isomorphic (i.e.: isomorphic in the homotopy category of the model structure on dg-algebras)

$$A \xrightarrow{\;\;\; \simeq \;\;\;} H^\bullet(A)$$

to its chain (co)homology regarded as a dg-algebra with trivial differential.

Since all dg-algebras are fibrant in the standard model structure, this is equivalent to the existence of a span

(1)$$A \overset{\;\; \simeq_w \;\;}{\leftarrow} Q A \overset{\;\; \simeq \;\;}{\rightarrow} H^\bullet(A)$$

of quasi-isomorphisms of dg-algebras.

Examples

In rational homotopy theory rational topological spaces are encoded in their Sullivan dg-algebras (cf. The fundamental theorem of dg-algebraic rational homotopy theory). A simply connected topological space $X$ whose Sullivan dg-algebra $\Omega^\bullet(X)$ 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:

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

• Lie groups,

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

• some homogeneous spaces $G/H$ (see (Amann 12) for nonformal examples),

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

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

  whose Sullivan minimal model is the dg-algebra on a single degree$=n$ generator with trivial differential.,

• the little n-disk operad is formal.

Related concepts

• H-cohomology

References

Early discussion of formal Sullivan model dg-algebras in the context of rational homotopy theory:

• Dennis Sullivan, section 12 of: Infinitesimal computations in topology, Publications mathématiques de l’ I.H.É.S 47 (1977) 269-331 [numdam:PMIHES_1977__47__269_0, pdf]

• Pierre Deligne, Phillip Griffiths, John Morgan, Dennis Sullivan: Real homotopy theory of Kähler manifolds, Invent Math 29 (1975) 245-274 [doi:10.1007/BF01389853]

Since Sullivan models are also cofibrant, the above authors get away with considering a direct comparison map instead of a span as in (1). The general definition is reviewed in:

• Kathryn Hess, Def. 2.1 in: Rational homotopy theory: a brief introduction, contribution to Summer School on Interactions between Homotopy Theory and Algebra, University of Chicago, July 26-August 6, 2004, Chicago [arXiv:math.AT/0604626], chapter in Luchezar Lavramov, Dan Christensen, William Dwyer, Michael Mandell, Brooke Shipley (eds.), Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS (2007) [doi:10.1090/conm/436]

• Torgeir Aambø, Def. 1.15 & Theorem A (p.2): On formal DG-algebras, MSc thesis, NTNU (2021) [hndl:11250/2778399]

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]

A construction of nonformal homogeneous spaces:

• Manuel Amann. Non-formal Homogeneous Spaces (2012). (arXiv:1206.0786).