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The cross product takes classes on two factors to a class on their product. Its multiplication rule has the graded tensor product sign: $(a\times b)(c\times d)=(-1{)}^{|b||c|}(ac)\times (bd)$. The Künneth theorem makes these products an additive basis when the integral cohomology of the finite cell complexes is torsion-free.

Suppose a continuous map $g:S^n\times S^n\to S^n$ changes sign on negating either input. Its quotient $\bar{g}:{\mathbb{RP}}^n\times {\mathbb{RP}}^n\to {\mathbb{RP}}^n$ pulls back the real tautological line bundle to the tensor product of vector bundles $p_1^{\ast }{\gamma }_n\otimes p_2^{\ast }{\gamma }_n$. The fiber map sends $(sx)\otimes (ty)$ to $stg(x,y)$. The First Stiefel–Whitney class of a tensor product of real line bundles consequently gives ${\bar{g}}^{\ast }a=a_1+a_2$. The Künneth theorem and the mod-two cohomology ring of real projective space implyEvery interior binomial coefficient in row $n+1$ must be even. By binomial coefficients with even interior terms, $n=2^k-1$ for some $k\ge 0$. This is a necessary condition, not an assertion of existence in every such dimension.

For odd $r\ge 1$ and $W_g={\#}_{i=1}^g(S^r\times S^r)$, the integral cohomology has one copy of $\mathbb{Z}$ in degrees zero and $2r$, ${\mathbb{Z}}^{2g}$ in degree $r$, and zero elsewhere. Choose the top orientation class $\omega$ and degree-$r$ classes ${\alpha }_i,{\beta }_i$ from the two factors of summand $i$. Their nonzero positive-degree basis products areAll ${\alpha }_i{\alpha }_j$ and ${\beta }_i{\beta }_j$ vanish, and $\omega$ annihilates positive-degree classes. The Künneth theorem computes each sphere product; the Mayer–Vietoris sequence computes the connected sum groups, and its degree-one pinch maps determine the products. Thus its middle-degree Poincare duality pairing is a symplectic vector space after changing coefficients to a field of characteristic different from two. The case $r=1$ is the usual cohomology ring of a closed oriented surface.

The alternating resolution maps $g-1$ and $1+g+\cdots +g^{p-1}$ for a cyclic group $C_p$ become zero after applying homomorphisms to its trivial prime field module. Thus there is one class in every degree. A two-step shift of the resolution represents a degree-two periodicity class $u$. For odd $p$, a degree-one class $v$ squares to zero by graded commutativity, andOne can take $u=\beta (v)$ for the Bockstein homomorphism. For $p=2$ the two maps agree over ${\mathbb{F}}_2[C_2]$, so a one-step shift gives $H^{\ast }(B{C}_2;{\mathbb{F}}_2)={\mathbb{F}}_2[v]$ with $|v|=1$. The Künneth theorem then gives dimension $i+1$ in degree $i$ for $B(C_p\times C_p)$, obstructing periodic group cohomology and hence such a free sphere action.

For positive unequal dimensions $p,q$, let $u,v$ be the pullbacks of the integral orientation classes of the factors. The Künneth theorem gives the free abelian basis $1,u,v,uv$, with degrees $0,p,q,p+q$. The products satisfy $u^2=v^2=0$ and $vu=(-1{)}^{pq}uv$, by naturality and graded commutativity of the cup product. In particular, for $p=2,q=4$, the cohomology ring is $\mathbb{Z}[u,v]/(u^2,v^2)$, with degrees two and four.

Since $T^r$ is a product of circles, the Künneth theorem identifies its cohomology ring with the exterior algebra on the coordinate degree-one classes. Inversion negates each integral degree-one class, so it acts as $(-1{)}^q$ in degree $q$. Even in characteristic two, each degree-one class has square zero.

In the product $(S^{2d}{)}^{2\ell }$, for $d,\ell \ge 1$, vary the factors indexed by an $\ell$-element subset $I$ and fix points in the other factors. Orient the resulting submanifold $S_I$ in increasing factor order. These homology classes form an integral middle-dimensional basis by the Künneth theorem. Complementary subsets give a transverse intersection at one point, with positive sign since all factor dimensions are even. For noncomplementary subsets some factor is fixed in both representatives; moving that point makes them disjoint. In particular every basis class has zero self-intersection number. Complementary pairs give hyperbolic blocks in the intersection form.

For $d\ge 1$ and $n>1$, no factor-permutation-invariant map $f:(S^d{)}^n\to S^d$ restricts to the identity on the diagonal. The Künneth theorem makes the degree-$d$ cohomology a direct sum of $n$ copies of $\mathbb{Z}$. Symmetry forces $f^{\ast }\alpha =a{\sum }_i{u}_i$ for one integer $a$. Pulling back to the diagonal multiplies this coefficient by $n$, so the identity restriction would give $na=1$. The contradiction uses integral coefficients and applies in both even and odd sphere dimensions.

The singular chain complex of a product is naturally chain-homotopy equivalent to the tensor product of the factors' chain complexes. The shuffle map and the front-face/back-face map implement the equivalence. It is the chain-level input to the Künneth theorem.

For smooth projective varieties $X,Y$, pull back Chow classes from the two factors and take their intersection product on $X\times Y$. This gives the external product and a graded map ${\mathrm{CH}}^{\ast }(X)\otimes {\mathrm{CH}}^{\ast }(Y)\to {\mathrm{CH}}^{\ast }(X\times Y)$. Unlike the cohomological Künneth theorem, it need not generate every class of the product.

Suppose a finite group $G$ acts freely on $S^n$, $n>1$. The quotient has fundamental group $G$ and higher homotopy groups equal to those of $S^n$. Attach one $(n+1)$-cell killing a generator of ${\pi }_n\cong \mathbb{Z}$, then attach higher cells successively to kill all remaining higher homotopy. This constructs a Eilenberg–MacLane space $K(G,1)$ with only one $(n+1)$-cell. The one-cell bound on next-degree cohomology gives dimension at most one in degree $n+1$. But if $G=C_p\times C_p$, the mod-p cohomology ring of a cyclic group and Künneth theorem give dimension $n+2$ there. This contradiction excludes that group, and also excludes such a subgroup of any group acting freely on a sphere.

The tensor product of graded algebras uses the displayed multiplication, incorporating the sign for interchanging homogeneous factors. This is the algebra structure in the cohomological Künneth theorem, and explains the signs of mixed intersection products on product manifolds.

On the torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$, an integer matrix $A$ defines an endomorphism. The coordinate circles give a basis of the first homology group, and the induced map on that group is $A$. The Künneth theorem identifies higher integral homology with the exterior powers of ${\mathbb{Z}}^d$, so the degree-$q$ map is ${\bigwedge }^{q}A$. In top degree it is multiplication by $\det A$. The map is a homeomorphism precisely when $A$ is a unimodular matrix; a merely nonsingular integer matrix need not have an integer inverse.

The cellular cochain complex of finite Real projective space consists of alternating maps zero and two. Its groups can be tensored using the Künneth theorem; the integral Künneth torsion polynomial efficiently counts all tensor and Tor functor summands. A product of factors of dimensions two, three and four has free-rank polynomial $1+t^3$ and torsion polynomial $3t^2+3t^3+5t^4+6t^5+5t^6+4t^7+2t^8+t^9$.

Each real projective plane has cells $e_0,e_1,e_2$ with $d{e}_2=2e_1$ and $d{e}_1=0$. Their product cellular chain complex has ranks $1,2,3,2,1$. Its degree-two cycle $e_1\times e_1$ has order two in homology. The degree-three cycle $e_2\times e_1+e_1\times e_2$ also has order two, because its double is $d(e_2\times e_2)$. Consequently $H_0=\mathbb{Z}$, $H_1=(\mathbb{Z}/2{)}^2$, $H_2=H_3=\mathbb{Z}/2$ and all other groups vanish. The degree-three torsion is also the Tor functor term in the Künneth theorem.

For spaces whose integral cohomology has only free and $\mathbb{Z}/2$ summands, let $F$ encode free ranks and $T$ count order-two summands by degree. An additively split integral Künneth theorem gives $F_{A\times B}=F_A{F}_B$ and the displayed rule for $T$. The $t^{-1}$ term is the cohomological degree shift of the Tor functor contribution. The formula requires this torsion type and the usual finite-type hypotheses.

For smooth projective curves $C$ and $D$, the Künneth theorem gives

For a fiber bundle $\pi :E\to B$ admitting a finite trivializing open cover, whose fiber has finite free cohomology over $R$, suppose classes $e_j\in H^{\ast }(E;R)$ restrict to a basis on every fiber. Then multiplication $a\otimes e_j\mapsto {\pi }^{\ast }a⌣e_j$ gives the indicated module isomorphism. A finite trivializing open cover suffices, by the local Künneth theorem and the Mayer–Vietoris sequence. This determines an additive module structure; multiplicative relations must still be computed. Powers of the hyperplane class on a complex projective bundle provide an important application.

On a trivialized fiber bundle over $U$, assume the fiber has finite-dimensional total cohomology over a field. The Künneth theorem identifies the total-space cohomology with a finite free graded module over $H^{\ast }(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an isomorphism over $U$.

For a closed connected orientable surface of genus $g$, the Künneth theorem gives the Betti numbers of $S^1\times {\Sigma }_g$ as $(1,2g+1,2g+1,1)$. The weak Morse inequalities give the displayed lower bound. A surface Morse function with one minimum, $2g$ saddles and one maximum exists from its standard handle decomposition. Adding a positive multiple of the cosine Morse function on the circle makes a product of Morse functions, whose critical counts are precisely these four Betti numbers. The lower bound is attained.