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Let $X$ be a CW complex of dimension at most $n$, with $n\ge 1$. Every class $a\in H^n(X;\mathbb{Z})$ is $f^{\ast }u$ for a continuous map $f:X\to S^n$ and an orientation generator $u\in H^n(S^n;\mathbb{Z})$.

Choose a cellular cocycle representing $a$ and collapse the $(n-1)$-skeleton to a point. The quotient is a wedge sum of one $n$-dimensional sphere for every $n$-cell. map each sphere to $S^n$ with mapping degree equal to the chosen cellular coefficient. Composing with the quotient gives the required continuous map. There are no higher-dimensional cells whose attachments could obstruct it; the weak topology of a CW complex also permits an infinite wedge.

The restriction $n\ge 1$ is essential. On a three-point discrete space, an integral degree-zero cohomology class can take the three values $0,1,2$. pullback along a map to $S^0$ can take at most the two values of the chosen class on $S^0$, so it cannot realize that class.

A cohomology operation is stable when it commutes with the suspension isomorphism on reduced cohomology:Products of positive-degree reduced classes on a suspension vanish, but stable operations can still connect its nonzero cohomology classes. The Steenrod squares and, at odd primes, Steenrod reduced powers are examples.

Let $x\in H^2({\mathbb{CP}}^2;{\mathbb{F}}_2)$ be the hyperplane class modulo two. It satisfies ${\mathrm{Sq}}^{2}x=x^2\ne 0$. Stability gives ${\mathrm{Sq}}^2(\sigma x)=\sigma (x^2)\ne 0$ on $\Sigma {\mathbb{CP}}^2$. The suspension of $S^2\vee S^4$ has the same additive groups and vanishing positive-degree products, but its corresponding degree-three class has zero ${\mathrm{Sq}}^2$, by pullback from $S^3$. Thus these suspensions have different cohomology operations even though their additive groups and cup products agree.^[ref]

The Stiefel–Whitney classes of a real rank-$r$ vector bundle $E\to X$ are characteristic cohomology classesThey are natural under pullback vector bundles and obey the Whitney product formula for Stiefel–Whitney classes, $w(E\oplus F)=w(E)w(F)$. For a real line bundle $L$, the normalization is $w(L)=1+w_1(L)$, with $w_1$ the class classifying its orientation double cover. These properties determine the classes through the splitting principle for real vector bundles.

The first Stiefel–Whitney class vanishes exactly when the vector bundle is orientable. The top Stiefel–Whitney class is its Euler class with mod-two coefficients and obstructs a nowhere-zero section. A trivial vector bundle has total class one; the Möbius line bundle has nonzero $w_1$.

Vanishing of every positive-degree Stiefel–Whitney class need not imply triviality. The tangent bundle of the two-dimensional sphere has $w_1=0$ and $w_2=\chi (S^2)\mathrm{mod}2=0$, although its integral Euler class is nonzero. The classes detect useful obstructions while forgetting information lost by reduction modulo two.

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.

The Whitehead square of a class $\alpha \in {\pi }_{m}Y$ is $[\alpha ,\alpha ]\in {\pi }_{2m-1}Y$. For the identity class of an even-dimensional sphere, the Hopf invariant is two with the standard product-cell orientation: folding the two sphere summands makes the middle cohomology class of the mapping cone pull back to $a+b$ on $S^m\times S^m$, and $(a+b{)}^2=2ab$. For odd $m$, graded symmetry makes the square two-torsion, hence rationally zero.