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Dennis Sullivan developed algebraic methods for topology, particularly rational homotopy theory. His Sullivan minimal model replaces a space's differential-form information by a controlled differential graded algebra.^[ref]

Generators are added to represent missing cohomology classes and impose necessary relations. Under the theory's hypotheses, the model records rational homotopy information that ordinary cohomology alone can miss. This makes nonlinear questions about spaces accessible through algebra while retaining more structure than a list of Betti numbers.^[ref]

A Sullivan minimal model of a topological space $X$ is a quasi-isomorphismfrom a free graded-commutative differential graded algebra over the rational numbers to the rational polynomial differential forms of $X$. Here $d$ raises degree by one and $d^2=0$. The notation $\Lambda V$ means a polynomial ring on even-degree generators tensored with an exterior algebra on odd-degree generators.

There is a well-ordered homogeneous basis of $V$ such that the expression $d(v)$ for each generator uses only earlier generators. Minimality additionally requires each $d(v)$ to be decomposable, with no linear term. These are distinct requirements; freeness and absence of linear terms alone do not replace the earlier-generator condition.^[ref]

For odd spheres $S^{2r+1}$, $r\ge 1$, the model is $(\Lambda (a_{2r+1}),0)$. For even spheres $S^{2r}$, $r\ge 1$, it isThe equation $db=a^2$ kills the square while retaining the fundamental cohomology class. For simply connected finite-type CW complexes, $V^k$ is dual to ${\pi }_k(X)\otimes \mathbb{Q}$: the extra generator for an even sphere records a second nonzero rational homotopy group. A proposed model must therefore specify its actual quasi-isomorphism and cohomology, beyond listing plausible relations.