Cohomology ($H^{\ast }(X;R)$)

Cohomology is the contravariant counterpart of homology, formed from cochains and coboundaries. A coefficient ring equips it with the cup product and hence with a graded ring structure.

For an oriented proper smooth map $f:M^m\to N^n$ of oriented manifolds, the Gysin homomorphism is a degree-shifting pushforward $f_{!}:H^k(M)\to H^{k+n-m}(N)$, defined using Poincare duality and the appropriate pushforward on homology. For an oriented sphere bundle it is integration along the fiber and participates in the Gysin sequence of a sphere bundle. Oriented embeddings give the corresponding Thom-class construction.

A generalized cohomology theory assigns graded groups contravariantly to suitable topological pairs, with homotopy invariance, excision and a natural long exact sequence for each pair. It omits the dimension axiom of ordinary cohomology, so its coefficient groups at a point can occur in degrees other than zero. A multiplicative theory also has compatible external and relative cup products.

A topological spectrum $E$ represents the reduced generalized cohomology theory ${\tilde{E}}^q(X)=[{\Sigma }^{\infty }X,{\Sigma }^{q}E{]}_{\mathrm{st}}$. For an Omega-spectrum this equals $[X,E_q{]}_{\ast }$, taking $E_q={\Omega }^{-q}{E}_0$ for negative $q$. It is contravariant in $X$.

A generalized cohomology theory with natural unital cup products compatible with relative groups, suspension, and the cohomology axioms. Its coefficient groups form a graded ring. Thom orientations make the cohomology of a bundle's Thom space a shifted module over the cohomology of the base.

// Target: geometry-and-topology.bigb

The skeletal filtration of a CW complex gives this spectral sequence, with differential of bidegree $(r,1-r)$ in cohomological grading. For a finite complex the filtration is finite and convergence is bounded. In a multiplicative theory the associated graded product is the ordinary cohomology product with coefficients in the theory's coefficient ring.

The skeletal filtration of a CW complex and the cofiber sequences of spectra of its cell attachments produceFor a connective spectrum it is first quadrant and converges in each fixed total degree. Its reduced version has no contribution from the basepoint summand.

The torus and the wedge sum $S^1\vee S^1\vee S^2$ both have integral cohomology $\mathbb{Z},{\mathbb{Z}}^2,\mathbb{Z}$ in degrees zero, one and two. Their fundamental groups differ: the torus has ${\mathbb{Z}}^2$, while the wedge has the nonabelian free group on two generators. Thus even compact finite cell complexes can have the same additive cohomology without being homotopy equivalent.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.

A singular $n$-cochain with coefficients in an abelian group $A$ is a homomorphism from the singular chain group $C_n(X;\mathbb{Z})$ to $A$. It is specified by its values on all singular simplices, without a finite-support restriction. Its coboundary evaluates on the alternating sum of the faces. The resulting cochain complex computes singular cohomology.

For a continuous map $f:X\to Y$, precomposition with the induced map on singular chains sends a singular cochain on $Y$ to one on $X$. It commutes with the coboundary and induces the induced map on cohomology.

A cohomology operation is a natural transformation between cohomology functors. Thus for every continuous map $f$, its pullback commutes with the operation: $\theta (f^{\ast }a)=f^{\ast }\theta (a)$. This makes operations additional invariants of homotopy equivalence, beyond the additive groups and their cup products.

A cohomology operation is stable if it commutes with the suspension isomorphism on reduced cohomology. Ordinary cup products become zero on a suspension, but stable operations can still connect its nonzero classes. The Steenrod squares and Steenrod reduced powers are examples.

For cellular Omega-spectra $D,E$, natural degree-preserving additive operations $D^{\ast }\to E^{\ast }$ commuting with suspension are compatible families $[D_j,E_j{]}_{\ast }$ by the Yoneda lemma. Since $D\simeq {\mathrm{hocolim}}_j{\Sigma }^{-j}{\Sigma }^{\infty }{D}_j$, the Milnor exact sequence for maps of spectra realizes every such family. Its kernel consists precisely of hyperphantom maps of spectra. Thus operations determine stable maps only modulo that kernel.

Suppose two integral cohomology groups of a space are free of rank one. An integral isomorphism can send each chosen generator only to itself or its negative. After reduction modulo $p$, naturality of a Steenrod reduced power between those groups therefore permits its coefficient to change only by a sign, even though an abstract vector-space change of basis could rescale by any nonzero field element. At $p=5$, coefficients one and two cannot be related by signs, so they obstruct an integral homotopy equivalence. This constraint is stronger than comparing the isolated mod-five Steenrod modules with arbitrary bases.

The Steenrod algebra organizes stable mod-$p$ cohomology operations. At $p=2$ it is generated by the Steenrod squares; at an odd prime it is generated by the Steenrod reduced powers and Bockstein homomorphism. Cohomology is a module over this algebra, and every continuous map induces a homomorphism preserving its action.

An index sequence obeying the displayed inequalities represents an admissible product ${\mathrm{Sq}}^{i_1}\cdots {\mathrm{Sq}}^{i_r}$. Its degree increment is the sum of the indices. The Adem relations rewrite every square monomial in terms of admissible ones, whose Steenrod excess controls universal instability.

The excess measures the minimum degree on which an admissible Steenrod monomial can be nonzero. Strict excess below $n$ picks out polynomial generators in the mod-two cohomology of $K(\mathbb{Z}/2,n)$; equality can correspond to a square, rather than a new generator.

These mod-two identities rewrite nonadmissible products of Steenrod squares. For example, ${\mathrm{Sq}}^1{\mathrm{Sq}}^1=0$, ${\mathrm{Sq}}^1{\mathrm{Sq}}^2={\mathrm{Sq}}^3$, and ${\mathrm{Sq}}^1{\mathrm{Sq}}^3=0$. In general the sum ranges over $0\le t\le \lfloor a/2\rfloor$ and coefficients are reduced modulo two.

The Steenrod squares are natural operationsFor a degree-one class $x$, ${\mathrm{Sq}}^{1}x=x^2$, and ${\mathrm{Sq}}^1$ agrees with the mod-two Bockstein.

The Cartan product formula says that total Steenrod squares and total Steenrod reduced powers preserve the cup product:It is named after Henri Cartan and is distinct from Cartan's magic formula for the Lie derivative of a differential form, named after Élie Cartan.

For odd prime $p$, the reduced powers are natural stable cohomology operations. They satisfy $P^0=1$, the Cartan formula, and the instability conditions $P^{i}a=0$ for $2i>|a|$ and $P^{i}a=a^p$ for $|a|=2i$. In particular, a degree-two class has $P^{1}x=x^p$ and no higher nonzero reduced power. Stability lets these operations detect distinctions between suspensions whose additive cohomology and cup products agree.

Choose the degree-four generator with pullback $u\mapsto x^2$ under the complex inclusion into quaternionic projective space. On infinite complex projective space, the Cartan formula gives $P^i(x^{2k})=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)x^{2k+i(p-1)}$. Injectivity of the infinite-space pullback proves the displayed formula for quaternionic projective space. Restriction to ${\mathbb{HP}}^n$ sets powers above $n$ to zero. Coefficients are reduced modulo the odd prime $p$; the exponent is an integer because $p-1$ is even.

For a based space, reduced cohomology removes the coefficient-ring summand coming from the basepoint. In nonnegative degrees it is the kernel of restriction $H^{\ast }(X;R)\to H^{\ast }(\{x_0\};R)$. It agrees with ordinary cohomology in positive degrees and admits the suspension isomorphism and exact sequences for based cofibrations.

The Borel version of equivariant cohomology is the ordinary cohomology of the Borel construction. It retains information about both the space and its group action. For a point it is the cohomology of the classifying space; for a free action with the usual bundle hypotheses it is the cohomology of the orbit space.

For a group action on $X$ and a contractible free $G$-space $EG$, the diagonal orbit space $EG{\times }_{G}X$ maps to the classifying space $BG$ with fibre $X$. It defines equivariant cohomology. When $X\to X/G$ is a principal $G$-bundle, the map $EG{\times }_{G}X\to X/G$ has contractible fibre $EG$ and gives a homotopy equivalence for spaces of CW type. Freeness without the bundle hypotheses should not be substituted for this assertion for arbitrary topological groups.

A degree-$q$ cohomology group is the quotient of degree-$q$ cocycles by coboundaries in a cochain complex. Topological cohomology groups arise from singular or cellular cochains. Their direct sum carries the cohomology ring structure when equipped with the cup product.

The equivalence class of a cocycle in a cochain complex, where adding a coboundary leaves its class unchanged.

Integral cohomology is cohomology with integer coefficients, $H^{\ast }(X;\mathbb{Z})$. Its cup product makes it a graded cohomology ring, retaining torsion and integral information that can disappear after changing coefficients to a field.

Relative cohomology is the cohomology of cochains on $X$ vanishing on chains in $A$. A pair has a natural long exact sequence in cohomology, and excision identifies relative groups for suitable neighborhood pairs.

For an inclusion $A\subseteq X$, the restriction cochain map fits into an exact triangle with the relative cochain complex. Its cohomology gives $\cdots \to H^i(X,A)\to H^i(X)\to H^i(A)\stackrel{\delta }{\to }{H}^{i+1}(X,A)\to \cdots$. The same construction with derived sections of a sheaf complex gives relative hypercohomology and is natural under restrictions of pairs.

A continuous map $f:X\to Y$ induces a pullback $f^{\ast }:H^q(Y;R)\to H^q(X;R)$. It preserves sums and cup products, and $(g\circ f{)}^{\ast }=f^{\ast }\circ g^{\ast }$.

Homotopic continuous maps induce the same map on cohomology. In particular, a homotopy equivalence induces an isomorphism of cohomology rings.

If $X$ is a CW complex of dimension at most $n$, every class $a\in H^n(X;\mathbb{Z})$ has the formfor some map $f:X\to S^n$ and generator $u\in H^n(S^n;\mathbb{Z})$. Cellular obstruction theory constructs $f$ on the top cells; there are no cells of larger dimension that could obstruct the extension.

This is the cohomological form of the universal coefficient theorem, with Ext and Hom terms rather than the homological tensor and Tor terms.

For an abelian coefficient group $G$, cohomology fits into a split short exact sequence

The connected open set $X={\mathbb{R}}^2\setminus \{(n,0):n\ge 1\}$ retracts onto a locally finite graph having one independent loop around each puncture. Hence $H_1(X;\mathbb{Z})\cong {⨁}_{n\ge 1}\mathbb{Z}$, while the universal coefficient theorem for cohomology givesan uncountable group.

For a compact locally contractible subspace $A\subseteq S^n$, Alexander duality gives

For a closed connected $k$-manifold smoothly embedded in $S^n$, with $k<n$ and $n\ge 1$, its complement hasand zero reduced homology for $i\ge n-1$, where negative-index groups are zero. The Thom isomorphism theorem for the normal bundle and the pair's exact sequence give this formula. The top ambient fundamental class maps isomorphically to the fundamental class of $M$, canceling the exceptional top term.

A compact smooth $n$-manifold with nonempty boundary embedded in $S^n$ is a proper compact locally contractible subset. Alexander duality therefore expresses the reduced homology of its complement by the reduced cohomology of the abstract manifold. Thus these groups do not depend on its particular smooth embedding.

Compactly supported cohomology is the cohomology of cochains vanishing outside a compact subset. Equivalently,

For a connected n-manifold without boundary, a finite good cover presents its top compactly supported cohomology by one generator per disc and relations $u_i=\pm u_j$ on overlaps. This follows from the compact-support Mayer–Vietoris sequence and vanishing above dimension n. The overlap graph is connected, so the group is cyclic. Coherent local orientations give $\mathbb{Z}$; an orientation-reversing loop gives $\mathbb{Z}/2$.

For a nonnegative integer $d$ and an abelian group $A$,The one-point compactification is the sphere $S^d$, and it is locally contractible at the added point. Apply the compact-support comparison with a one-point compactification and the reduced cohomology of a sphere. In dimension zero the compactification is $S^0$, giving the same formula.

If the one-point compactification $X^{+}$ is Hausdorff and has a basis of contractible neighbourhoods at the added point, thenThe Excision theorem identifies $H^{\ast }(X,X\setminus K)$ with $H^{\ast }(X^{+},X^{+}\setminus K)$. The contractible neighbourhoods form a cofinal family of the complements of compact $K$, and their pair long exact sequences identify each relative group naturally with reduced cohomology. Passing to the direct limit gives the comparison. Local contractibility matters for singular cohomology; the Hawaiian earring illustrates its failure.

A continuous map is proper when inverse images of compact sets are compact. Proper maps induce contravariant maps on compactly supported cohomology.

The cap product pairs a homology class and a cohomology class to lower degree:

For a singular simplex $\sigma$ of dimension $n$ and a degree-$p$ cochain, use $\sigma ⌢\alpha =\alpha (\sigma |[v_0,\dots ,v_p])\sigma |[v_p,\dots ,v_n]$, setting the result to zero if $p>n$. A continuous map $f$ commutes with both face restrictions, and the pullback cochain evaluates by composing with $f$. Therefore $f_{\ast }(\sigma ⌢f^{\ast }\alpha )=(f\circ \sigma )⌢\alpha$ already on chains. Passing to homology and cohomology proves the displayed cap product identity.

For an oriented closed connected $d$-manifold, the fundamental class is the unique generator of $H_d(M;\mathbb{Z})$ that restricts to the orientation generator in every local homology group.

A local $R$-orientation of a $d$-manifold $M$ at $x$ is a generator of the rank-one $R$-module $H_d(M,M\setminus \{x\};R)$. An $R$-orientation is a locally coherent choice of such a generator at every point.

An $R$-fundamental class of a closed $d$-manifold is a class in $H_d(M;R)$ whose image in every local homology group $H_d(M,M\setminus \{x\};R)$ is a generator. These local images form an $R$-orientation.

For a closed oriented $d$-manifold, cap product with the fundamental class gives isomorphisms

Cap product with the locally finite fundamental class of an $n$-manifold without boundary gives the displayed duality, including a possibly nontrivial orientation local system ${\mathcal{O}}_M$. It is the noncompact version of Poincare duality; for a compact manifold Borel-Moore homology is ordinary homology. Locally it is the relative fundamental-class calculation in an oriented coordinate ball; compatibility on overlaps and the Mayer–Vietoris sequence give the global statement. Since the chain complex of locally finite homology is zero in negative degrees, this duality implies $H^k(M;\mathbb{Z})=0$ for $k>n$.

For a continuous map $f:M\to N$ of nonzero mapping degree between closed connected oriented manifolds of the same dimension, $f^{\ast }:H^q(N;\mathbb{Q})\to H^q(M;\mathbb{Q})$ is injective in every degree. Pair a nonzero class $a$ with a complementary class $b$ using Poincare duality. Naturality gives $⟨f^{\ast }a⌣f^{\ast }b,[M]⟩=(\mathrm{deg}f)⟨a⌣b,[N]⟩\ne 0$, so $f^{\ast }a\ne 0$. Over coefficients of positive characteristic the argument requires that the degree remain nonzero in that field.

For a connected closed oriented four-manifold, Poincare duality identifies $H_3$ with $H^1$. The universal coefficient theorem for cohomology identifies $H^1$ with $\mathrm{Hom}(H_1,\mathbb{Z})$ because $H_0=\mathbb{Z}$ has zero Ext. Consequently $H_3$ is torsion-free and its rank equals that of $H_1$.

For a closed connected manifold, cap product with $[M]\in H_n(M;{\mathcal{O}}_M)$ gives the displayed isomorphism. The ordinary integral theorem results when the orientation system is trivial. Modulo two the sign representation is always trivial, so duality holds without an orientability assumption. In particular a nonorientable connected closed $n$-manifold has $H_n(M;\mathbb{Z})=0$ and $H^n(M;\mathbb{Z})=\mathbb{Z}/2$.

Zero Euler characteristic and $H_3=0$ give free rank $b_1-1$. The universal coefficient theorem for cohomology identifies $H^3(M;\mathbb{Z})=\mathbb{Z}/2$ with $\mathrm{Ext}(H_2(M;\mathbb{Z}),\mathbb{Z})$, so the only torsion in $H_2$ is one copy of $\mathbb{Z}/2$. In particular $b_1\ge 1$.

Represent one-dimensional homology classes on a closed oriented topological surface by transverse oriented curves and sum their signed crossing points. This gives a bilinear form that is skew-symmetric. A symplectic handle basis has matrix $\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right)$, which is unimodular. In particular, a class detected with intersection $1$ or $-1$ by another integral class is a primitive homology class: the detecting homomorphism splits its rank-one subgroup.

For a closed $n$-dimensional manifold, Poincare duality holds over ${\mathbb{F}}_2$ without an orientation assumption. The mod-two fundamental class exists because orientation signs disappear. The resulting complementary-degree intersection pairing is a perfect pairing.

For a closed oriented manifold of dimension $4r+2$, Poincare duality pairs all rational Betti numbers except the middle one, making their total contribution to the Euler characteristic even. The middle cohomology degree is odd, so its nondegenerate Poincare duality pairing is an alternating bilinear form and has even dimension. Hence the Euler characteristic is even. Nonorientable manifolds can have odd Euler characteristic, as Real projective space of even dimension demonstrates.

Take a connected sum of $r$ copies of Complex projective space ${\mathbb{CP}}^3$ and $s$ copies of $S^3\times S^3$, with the empty sum interpreted as $S^6$. Their Euler characteristics are four, zero and two, respectively. Removing a six-ball subtracts one from the Euler characteristic and gluing along its boundary $S^5$ subtracts zero, so each connected sum subtracts two. The result is $2+2r-2s$. For any even integer $n$, choose $r=\max (n/2-1,0)$ and $s=\max (1-n/2,0)$.

For a map between closed oriented manifolds of the same dimension, Poincare duality turns homological pushforward into a degree-preserving map on cohomology. It is adjoint to pullback under the duality pairing: $⟨f^{!}a⌣b,[N]⟩=⟨a⌣f^{\ast }b,[M]⟩$. If $f$ has degree $d$, nondegeneracy gives $f^{!}{f}^{\ast }=d\mathrm{id}$. Maps of different dimensions require the corresponding degree shift.

For a closed oriented $d$-dimensional manifold over a field $F$, the cup product pairingis a perfect pairing between complementary degrees. The cap product with the fundamental class is an isomorphism by Poincare duality, and the universal coefficient theorem for cohomology over $F$ identifies the complementary cohomology with the full dual space of the resulting homology. Evaluation is therefore a perfect pairing. Taking the top-degree part of the same formula defines a nondegenerate bilinear form on the full graded cohomology. A boundary requires relative groups and Poincare-Lefschetz duality instead.

For a closed oriented manifold, the rational middle-dimensional cup-product pairing is nondegenerate by Poincare duality. Its degree is odd, so graded commutativity makes it skew-symmetric. A nondegenerate skew-symmetric form has even dimension. In particular a closed oriented six-manifold cannot have a rank-one third cohomology group.

// Target: algebra.bigb

If $f:M\to N$ is a map between closed connected oriented manifolds of the same dimension and its degree is nonzero in a coefficient field $F$, then $f^{\ast }:H^{\ast }(N;F)\to H^{\ast }(M;F)$ is injective. For a nonzero class $a$, Poincare duality supplies $b$ with $⟨a⌣b,[N]⟩\ne 0$, and naturality gives

For a compact oriented $n$-manifold $W$ with boundary, cap product with its relative fundamental class gives

Reverse a handle decomposition of a compact $n$-dimensional manifold. Absolute $k$-handles become relative $(n-k)$-handles based on the boundary. The absolute boundary matrices count attaching sphere-belt sphere intersections; the corresponding relative matrices are their transposes over ${\mathbb{F}}_2$. The absolute chain complex is thus the complementary-degree dual of the relative chain complex. The universal coefficient theorem for cohomology proves Poincare-Lefschetz duality without an orientation hypothesis.

For an oriented codimension-$r$ closed submanifold $Y\subset M$, its Poincare dual is the class in $H^r(M)$ whose cup-product evaluation against any complementary class equals intersection with $Y$.

For a compact oriented submanifold without boundary in an oriented manifold, the normal Thom class gives a compactly supported cohomology class dual to its pushed-forward fundamental class. It is independent of the tubular neighborhood and its ordinary-cohomology image vanishes on the complement of the submanifold. A noncompact submanifold closed only as a subset generally has its dual in ordinary cohomology, using locally finite homology instead.

If a closed oriented manifold is covered by two open sets and a positive-degree cohomology class restricts to zero on both, then its cap product with the fundamental class vanishes in every degree strictly below the top. A chain proof uses the small simplex theorem and replaces the two restricted cocycles by coboundaries.

A degree-one map $f:M\to N$ between closed connected oriented manifolds of the same dimension induces an injection $f^{\ast }:H^{\ast }(N;R)\to H^{\ast }(M;R)$. The wrong-way map defined using Poincare duality is a left inverse by the projection formula and $f_{\ast }[M]=[N]$.

An integral homology $d$-sphere is a space with the same integral homology as $S^d$. Removing one point from a closed oriented integral homology sphere produces an acyclic space by the long exact sequence of the pair and the local orientation isomorphism in degree $d$.

A simply connected closed three-dimensional manifold is orientable, since its orientation monodromy factors through its trivial fundamental group. Its first integral homology is the abelianization of that fundamental group and is zero. The universal coefficient theorem for cohomology gives $H^1(M;\mathbb{Z})=0$, and Poincare duality gives $H_2=0$, $H_3=\mathbb{Z}$. Connectedness gives $H_0=\mathbb{Z}$, and duality gives zero homology above dimension three. Thus it is an integral homology sphere, without needing any homeomorphism classification.

A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.

Every closed topological manifold with the homotopy type of $S^d$ is homeomorphic to $S^d$. This is a classification theorem about homeomorphisms, and does not assert a diffeomorphism between prescribed smooth structures. It is the precise final step needed after proving that a cone link is a simply connected integral homology sphere in the manifold criterion for a suspension.

The cup product is a natural bilinear operation $H^p(X;R)\times H^q(X;R)\to H^{p+q}(X;R)$. It is graded-commutative: $a⌣b=(-1{)}^{pq}b⌣a$.

For a cohomology class $a$, its rth cup power is its r-fold product in the cohomology ring. Naturality gives $f^{\ast }(a^r)=(f^{\ast }a{)}^r$. A nilpotent class has some positive cup power zero; the nilpotence order is preserved by ring isomorphisms and can obstruct a homotopy equivalence or a nonzero mapping degree.

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.

The largest number of positive-degree cohomology classes whose cup product is nonzero, with a specified coefficient ring $R$; it can be infinite. It is zero if there is no nonzero positive-degree class. The vanishing cup product from an open cover argument implies that Lusternik–Schnirelmann category is at least cup length plus one. For Complex projective space ${\mathbb{CP}}^r$, its integral cup length is $r$.

For open subsets $U,V$ of a topological space $Y$, the relative cup product givesCompatibility with the maps to absolute cohomology identifies its image with the ordinary cup product. More generally the construction requires an excisive pair of subsets. The open-set case can be proved using subdivision to compute on chains subordinate to the open cover.

If a pointed CW complex is covered by $l$ contractible subcomplexes containing the basepoint, every reduced class lifts to a relative class for each appropriate subcomplex. Their relative cup product lies in $h^{\ast }(X,{\bigcup }_j{A}_j)=h^{\ast }(X,X)=0$, so every product of $l$ reduced classes vanishes. The conclusion also holds if $X$ is path connected. Without either condition a two-point space covered by its two points is a counterexample in ordinary degree-zero cohomology.

Let $U_1,\dots ,U_k$ be an open cover of $Y$, and suppose ${\alpha }_i\in H^{d_i}(Y;R)$ restricts to zero on $U_i$. The pair long exact sequence lifts each ${\alpha }_i$ to $H^{d_i}(Y,U_i;R)$. Their relative cup product lies in $H^{\sum d_i}(Y,{\bigcup }_i{U}_i;R)=H^{\sum d_i}(Y,Y;R)=0$, so ${\alpha }_1⌣\cdots ⌣{\alpha }_k=0$. This proves characteristic-class products vanish from local trivializations without cancelling possible zero divisors in the coefficient ring.

Homogeneous cohomology classes satisfyIn particular, an odd-degree class has square killed by two, and its square vanishes whenever two is invertible in the coefficient ring.

For projections $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$, the exterior product isThe same definition gives $H^p(X,A;R)\otimes H^q(Y;R)\to H^{p+q}(X\times Y,A\times Y;R)$.

The cohomology ring is the graded ring whose multiplication is the cup product. It records intersection information that the graded cohomology groups alone do not detect.

The property that every degree-one element has square zero is invariant under a graded ring isomorphism. Over ${\mathbb{F}}_2$ it distinguishes the cohomology ring of a torus from the mod-two cohomology ring of an inversion mapping torus of dimension at least two.

For positive $n$, let $x,y$ be the pullbacks of the orientation classes from the two factors. Then $x^2=y^2=0$, $xy$ generates degree $2n$, and $yx=(-1{)}^{n}xy$. When $n$ is even, every graded ring automorphism restricts on $H^n$ to a signed permutation matrix; when $n$ is odd the ring admits every matrix in $G{L}_2(\mathbb{Z})$.

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.

For even $n\ge 2$, attach an $(n+1)$-cell to $S^n\times S^n$ along the diagonal. The cellular chain complex has boundary $(1,1)$ for the new cell. The degree-$n$ cohomology generator restricts to $a-b$, and the top generator restricts to $ab$. Naturality of the cup product therefore gives $u^2=-2v$, since $(a-b{)}^2=-2ab$. Thus the resulting cohomology ring is $\mathbb{Z}[u,v]/(u^2+2v,uv,v^2)$, with $|u|=n$ and $|v|=2n$. This distinguishes the attachment from a wedge of spheres, whose positive-degree products vanish.

Two finite CW complexes can have isomorphic integral cohomology rings but different cohomology rings modulo $p$. For example, attaching a three-cell of degree $p$ to the projective line in ${\mathbb{CP}}^2$ preserves the nonzero square of the degree-two class modulo $p$, whereas the corresponding Moore space wedged with $S^4$ has zero square. The spaces therefore cannot be homotopy equivalent.

Choose degree-one classes ${\alpha }_i,{\beta }_i$ dual to a symplectic basis of one-cycles and let $\omega$ generate $H^2({\Sigma }_g;\mathbb{Z})$. The only nonzero products of positive-degree basis elements are

For chain complexes of free modules over a principal ideal domain $R$, the Künneth theorem gives a natural split short exact sequencealthough the splitting is not natural. Over a field the Tor term vanishes, and the cohomological cross product gives $H^{\ast }(X\times Y;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(Y;k)$ as graded rings under standard finiteness hypotheses.

For classes on $X$ and $Y$, apply the induced maps on cohomology of the coordinate projections and take their cup product on $X\times Y$. The result has degree $|a|+|b|$. On a tensor product of graded rings, the sign rule $(a\otimes b)(c\otimes d)=(-1{)}^{|b||c|}ac\otimes bd$ makes this a ring homomorphism.