Answered Sullivan-Model Existence Question

Definitions and mathematical statement

The supplied question is: Does every connected manifold admit a free commutative differential graded algebra model with a morphism to its de Rham algebra inducing an isomorphism in cohomology?

A conventional interpretation uses a connected smooth manifold $M$ and real differential forms. This interpretation supplies smoothness and coefficients that the extracted question does not explicitly specify; its boundary convention is also unspecified. Write $(\Omega^*(M;{\mathbb{R}}),d_{\mathrm{dR}})$ for the algebra of smooth real differential forms with exterior derivative.

For a graded real vector space $V$, let $\wedge V$ denote the free unital graded-commutative algebra generated by $V$. Thus homogeneous elements satisfy $ab=(-1)^{|a||b|}ba$; the notation does not mean an ordinary exterior algebra on generators of every degree. A differential on this algebra is a linear map $d$ of degree $+1$ such that $d^2=0$ and \[ d(ab)=d(a)b+(-1)^{|a|}a\,d(b). \] A morphism $\phi$ of differential graded algebras preserves degrees, multiplication and the unit, and satisfies $\phi\circ d=d_{\mathrm{dR}}\circ\phi$. It is a quasi-isomorphism if its induced maps on cohomology are isomorphisms in every degree, where \[ H^k(A,d)=\frac{\ker(d:A^k\to A^{k+1})}{\operatorname{im}(d:A^{k-1}\to A^k)}. \]

Under the real smooth interpretation, the extracted question asks whether, for every connected $M$, there exist $V$, $d$, and a morphism \[ \phi:(\wedge V,d)\longrightarrow(\Omega^*(M;{\mathbb{R}}),d_{\mathrm{dR}}) \] such that \[ H^k(\phi):H^k(\wedge V,d)\xrightarrow{\;\cong\;}H^k(\Omega^*(M;{\mathbb{R}}),d_{\mathrm{dR}}) \quad\text{for every }k. \] The source does not specify the permitted generator degrees or additional conditions on the model. In particular, no minimality condition, Sullivan filtration, simple connectivity assumption, or finite-type hypothesis is imposed here.

Unresolved source qualification. The supplied registry labels the record open, and its dated literature assessment describes an inadequately specified open question. However, its supplied statement-verification field says that the report immediately answers this introductory question affirmatively for connected manifolds, before developing Sullivan minimal models, and does not present it as an open problem. The record marks its statement status as unrecoverable and supplies no separate clean statement. Consequently, the display above formalizes the retained existence question under an explicit interpretation; it does not establish an unresolved mathematical problem.

Short English statement

The retained question asks whether every connected manifold has a free graded-commutative differential algebra mapping to its de Rham algebra and computing the same cohomology. A real smooth interpretation is described explicitly, but some conventions remain unspecified. The supplied verification says the report answers the question affirmatively, conflicting with the registry's open label and dated assessment. No distinct open target is recoverable from this record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001753$, exact source problem code catalogue entry, title Answered Sullivan-Model Existence Question. The original statement is retained for provenance; no clean statement is supplied.
• Arbeitsgemeinschaft: Rational Homotopy Theory in Mathematics and Physics (2011), Oberwolfach Reports, pp. 993--1045, DOI 10.4171/owr/2011/18. Supplied report reference and literature-source link: https://doi.org/10.4171/owr/2011/18. The supplied collection describes these reports as documenting mathematical workshops and their open-problem discussions. No individual proposer is supplied; the proposed year is 2011.
• Supplied literature assessment, checked exactly 2026-08-21: the extraction allegedly omits the intended model and hypotheses, and no authoritative resolution of the exact claim was identified in that curation, which labels it open. The listed source checked is Rational Homotopy Theory in Mathematics and Physics, OWR 18/2011, at https://doi.org/10.4171/owr/2011/18. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field instead reports an immediate affirmative answer in the cited report and says the question is not presented as open. This source-specific qualification is retained as supplied, without independent verification. No precise page within the report or passage identifying that answer is supplied.