---
tag: V9079C40AZ
kind: definition
name: Sullivan minimal model
msc: [55P62]
defines: 3QQ2PBGY3N
uses: [08HK38GCSM, YZCBSRCZC7, 6N7BC70CX0]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_homotopy_theory
    locator: Sullivan algebras
---

## Definition

Work over $\Q$ with cochain differential graded algebras (dgas) that are graded-commutative: $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a$ and $b$.
For a graded vector space $V = \bigoplus_{n \ge 1} V^n$, let $\Lambda V$ be the free graded-commutative algebra on $V$.
It is the tensor product of the [[symmetric algebra|YZCBSRCZC7]] on the even part and the [[exterior algebra|08HK38GCSM]] on the odd part.
Let $\Lambda^{\ge 2} V$ be its ideal of decomposable elements, the span of products of at least two elements of $V$.

1. A *Sullivan algebra* is a dga $(\Lambda V, d)$ with $V = V^{\ge 1}$ and an increasing chain of graded subspaces $V(0) \subseteq V(1) \subseteq \cdots$ with union $V$.
   The chain must satisfy $d(V(0)) = 0$ and $d(V(k)) \subseteq \Lambda V(k-1)$ for $k \ge 1$.
2. A Sullivan algebra is *minimal* if $d(V) \subseteq \Lambda^{\ge 2} V$.
3. Let $(A, d_A)$ be a graded-commutative dga with $H^0(A) = \Q$.
   A *Sullivan minimal model* of it is a minimal Sullivan algebra $(\Lambda V, d)$ with a dga map $m \colon (\Lambda V, d) \to (A, d_A)$ that is an isomorphism in cohomology.

The minimal model of a path-connected space $X$ is the minimal model of Sullivan's algebra $A_{PL}(X)$ of rational polynomial differential forms on $X$.
The cohomology of $A_{PL}(X)$ is the rational [[singular cohomology|6N7BC70CX0]] $H^*(X; \Q)$.

## Remarks

If $X$ is simply connected and each $H^n(X; \Q)$ is finite-dimensional, then $V^n \cong \Hom(\pi_n(X), \Q)$ for all $n \ge 2$, and $V^1 = 0$.

Example: for $n$ odd, $(\Lambda(x), 0)$ with $\abs{x} = n$ is the minimal model of $S^n$.
Here $\Lambda(x) = \Q \oplus \Q x$ since $x^2 = 0$ by graded commutativity, which is $H^*(S^n; \Q)$.

Example: for $n$ even, the minimal model of $S^n$ is $(\Lambda(x, y), d)$ with $\abs{x} = n$, $\abs{y} = 2n - 1$, $dx = 0$ and $dy = x^2$.
Its cohomology has basis $1, x$: each $x^k$ with $k \ge 2$ equals $d(x^{k-2} y)$, and $d(x^k y) = x^{k+2} \ne 0$.

Non-example: $(\Lambda(x, y), d)$ with $\abs{y} = 1$, $\abs{x} = 2$ and $dy = x$ is a Sullivan algebra but not minimal, since $dy = x$ is not decomposable.
It is acyclic in positive degrees, and its minimal model is $\Q$.