# nLab formal dg-algebra

### Context

#### Rational homotopy theory

and

rational homotopy theory

# Contents

## 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.

## Applications

### In rational homotopy theory

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);

• classifying spaces of Lie groups;

• some homogeneous spaces $G/H$

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

• the space $K(\mathbb{Z},2)$,

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

## 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

Revised on September 11, 2010 01:08:43 by Toby Bartels (98.19.62.220)