Abstract : Given a topological space \(X\) we can compute the cohomology groups(modules) of it. Call them \(H^k(X;R)\), where $R$ is the ring of coefficients. We want to give $\bigoplus_k H^{k}(X,R)$ a graded ring structure. In order to do so we have defined cup product, $$\smile : H^{k}(X;R) \times H^{n}(X;R)\to H^{n+k}(X;R)$$
We define this graded ring as $H^*(X;R)$, this ring can be a polynomial ring over $R$. The natural question arose in our mind when can we write a graded commutative $R$-algebra can as a cup-product algebra $H^\ast(X;R)$ for some space $X$. The answer is affirmative for a large class of algebra over $\mathbb{Q}$. In order to construct/ get such a space we will introduce Rational homotopy theory (as it was done by mathematician D.Quillen and later Sullivan).
Introduction#
In the previous discussion, we asked when a graded commutative algebra can be realized as the cohomology ring of some topological space. Over $\mathbb{Q}$, this question is beautifully answered using rational homotopy theory.
The basic idea is to replace spaces by algebraic models that encode their rational homotopy type. Two major frameworks exist — one due to Quillen, who used differential graded Lie algebras, and one due to Sullivan, who used commutative differential graded algebras (CDGAs). We will focus on the Sullivan picture.
Rational Homotopy Theory and Minimal Models#
Sullivan showed that the category of simply connected CW complexes of finite type (up to rational homotopy) is equivalent to the opposite of the category of $1$-connected minimal CDGAs of finite type over $\mathbb{Q}$. Symbolically,
$$ \text{(Simply connected spaces of finite type)}_{\mathbb{Q}} \simeq$$
$$\text{(1-connected minimal CDGAs of finite type)}_{\mathbb{Q}}^{op}$$
For every simply connected space $X$, there exists a CDGA $A_{PL}(X)$ (the algebra of piecewise-linear differential forms on $X$). Its minimal model $(\Lambda V, d)$ is a free graded commutative algebra generated by a graded vector space $V$, equipped with a differential satisfying $$ d(V) \subseteq \Lambda^{\ge 2} V. $$ This minimal model encodes all rational homotopy information of $X$ — in fact, the rational homotopy groups of $X$ are dual to the graded pieces of $V$.
Realization of Algebras as Cohomology Rings#
Now suppose we start with a graded commutative $\mathbb{Q}$-algebra $A$ with $A^0 = \mathbb{Q}$. We may regard $A$ as a CDGA $(A, d=0)$.
Rational homotopy theory guarantees that every such $1$-connected CDGA of finite type is quasi-isomorphic to $A_{PL}(X)$ for some simply connected CW complex $X$. Hence,
$$H^{\ast}(X;\mathbb{Q}) \cong H^{\ast}(A,0) \cong A.$$
Thus, over $\mathbb{Q}$, every finite-type graded commutative algebra can indeed be realized as the cohomology ring of some space — not by an ad-hoc geometric cell-by-cell construction, but because there exists a minimal model corresponding to it, and that model determines a unique rational homotopy type of a space.
Sketch of the Algebraic Construction (Minimal Model Viewpoint)#
To make the above less mysterious, here is a succinct description of how one obtains a minimal model whose cohomology is a given algebra $A$.
Start with $A$ (connected, finite type) and view it as the CDGA $(A, d=0)$.
One constructs a minimal Sullivan algebra $(\Lambda V, d)$ together with a quasi-isomorphism $$ (\Lambda V, d) {\simeq} (A,0). $$ This is done inductively: introduce generators in $V$ to account for cohomology in each degree, and define differentials on new generators to kill unwanted cohomology classes or to realize relations present in $A$. The condition $d(V)\subset \Lambda^{\ge 2}V$ guarantees minimality (no redundant linear parts).
The resulting minimal model $(\Lambda V,d)$ is then the algebraic object corresponding to a unique rational homotopy type; by Sullivan’s realization functor there is a simply connected CW complex $X$ with
$$ A_{PL}(X)\simeq (\Lambda V,d), $$ hence $H^{\ast}(X;\mathbb{Q})\cong H^{\ast}(\Lambda V,d)\cong A$.
This algebraic extension procedure replaces geometric cell attachments by algebraic generators-and-differentials; it is canonical up to quasi-isomorphism and yields the desired realization when possible.
Conceptual Picture#
- Topological side: simply connected CW complexes of finite type.
- Algebraic side: minimal differential graded algebras $(\Lambda V, d)$.
- The bridge: the functor $A_{PL}$ (piecewise-linear forms) and its inverse realization functor.
In this correspondence:
- adding a generator in $V$ corresponds to introducing a new rational homotopy group (algebraically),
- the differential $d$ records relations/attachments (algebraically killing unwanted cohomology),
- quasi-isomorphism classes of minimal models correspond to rational homotopy types of spaces.
Example — Non-Formal Space via a Given CDGA#
Consider the CDGA $$(\Lambda(x_2, y_3, z_5), d), \quad |x_2|=2, |y_3|=3, |z_5|=5,$$ with differential $$dx_2 = 0, \quad dy_3 = 0, \quad dz_5 = x_2 y_3.$$
This algebra is minimal and 1-connected.
The cohomology ring is $$H^{\ast}(\Lambda(x_2, y_3, z_5), d) \cong \mathbb{Q}[x_2, y_3]/(x_2 y_3),$$ so $x_2 y_3 = 0$ in cohomology.
The differential $dz_5 = x_2 y_3$ encodes a nontrivial Massey product, which makes this space non-formal — its rational homotopy type is not determined by the cohomology ring alone.
By Sullivan’s realization theorem, there exists a simply connected CW complex $X$ such that $$A_{PL}(X) \simeq (\Lambda(x_2, y_3, z_5), d), \quad H^{\ast}(X;\mathbb{Q}) \cong \mathbb{Q}[x_2, y_3]/(x_2 y_3).$$
One way to construct $X$ explicitly:
- Start with $S^2 \vee S^3$ (two spheres corresponding to $x_2$ and $y_3$).
- Attach a 5-cell along a map representing the element $x_2 \smile y_3$ in cohomology. This corresponds to the differential $dz_5 = x_2 y_3$.
- The resulting CW complex $X$ has rational cohomology ring as above and a nontrivial rational homotopy type that is not formal, i.e., the minimal model is strictly richer than the cohomology algebra.
Final Remarks#
- The statement “every graded commutative $\mathbb{Q}$-algebra is realizable” is shorthand for: every connected, finite-type graded commutative $\mathbb{Q}$-algebra $(A,0)$ arises as the cohomology of some simply connected rational space because it is quasi-isomorphic (via a minimal model) to $A_{PL}(X)$ for some $X$.
- Over rings with torsion (e.g. $\mathbb{Z}$ or $\mathbb{Z}_p$), additional obstructions (Steenrod operations, Bocksteins, etc.) can prevent realization; the rational world is special because torsion vanishes and the Sullivan–Quillen machinery applies.