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A cohomology theory assigns abelian groups $H^q(X;R)$ to a topological space $X$. For singular cohomology with coefficients in a commutative ring $R$, form the cochain complex $C^q(X;R)$ with coboundary $\delta$ and takeA cocycle represents a cohomology class; adding a coboundary leaves that class unchanged.

A cohomology class evaluates on a homology class by applying its cochain to a representing cycle. Replacing either representative changes the result by zero, explaining the relation to homology. A continuous map $f:X\to Y$ induces cochain pullback and hence $f^{\ast }:H^q(Y;R)\to H^q(X;R)$: the direction reverses.

The cup product combines degree-$p$ and degree-$q$ classes into degree $p+q$, giving a graded cohomology ring. The circle has one integral generator in degrees zero and one. A two-dimensional torus has two independent degree-one classes whose cup product generates degree two. That multiplication distinguishes it from $S^1\vee S^1\vee S^2$, which has the same additive cohomology groups but zero products between its positive-degree generators.

A simple closed curve in a closed orientable surface has an annular collar neighbourhood, so its collapse is governed by the collapsing a pair theorem. In the long exact sequence in relative homology, the decisive map is $H_1(A;\mathbb{Z})\to H_1({\Sigma }_g;\mathbb{Z})$, sending the circle generator to its homology class. That class is zero for a separating curve and a primitive homology class for a nonseparating curve. The two possibilities lead to homology after collapsing a separating surface curve and homology after collapsing a nonseparating surface curve.

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.

The core disk of a $k$-handle attachment $D^k\times D^{n-k}$ is $D^k\times \{0\}$. Together with its cap in the preceding stage, it often represents a homology class; for a surgery trace, capped two-handle cores give the basis of its surgery linking matrix.

For a finite CW complex, $b_0$ counts connected components: its connected components are its path components, as recorded by the zeroth homology group. For a finite graph regarded as a one-dimensional CW complex, no higher homology groups occur, so $b_1=E-V+b_0$ counts independent graph cycles, and $\chi =b_0-b_1$. For a connected planar surface obtained by removing $h$ disjoint open disks from the interior of a closed disk, $(b_0,b_1,b_2)=(1,h,0)$ and $\chi =1-h$; the annulus is $h=1$.

The word "hole" needs its dimension. A two-sphere encloses a cavity but has $b_1=0$ and $b_2=1$. A closed orientable surface of genus $g$ instead has $(b_0,b_1,b_2)=(1,2g,1)$ and $\chi =2-2g$: increasing the genus by one adds two independent one-dimensional homology classes. In higher dimensions, further homology groups enter with alternating signs; there is no universal "components minus handles" rule.

Identify opposite sides of a square by translations to obtain a torus. All four corners become one vertex; the horizontal sides give one topological cell $a$ and the vertical sides one topological cell $b$. The open square supplies one two-dimensional topological cell. Thus $(c_0,c_1,c_2)=(1,2,1)$ and $\chi =0$. The cellular boundary of each one-dimensional topological cell is zero because its endpoints agree. Traversing the face boundary gives the group commutator $ab{a}^{-1}{b}^{-1}$, whose signed counts of both $a$ and $b$ vanish. Hence its cellular boundary is also zero, and $(b_0,b_1,b_2)=(1,2,1)$.^[ref]

The boundary of a tetrahedron is a two-sphere, with four vertices, six edges and four triangular topological cells: $\chi =4-6+4=2$. Its rational Betti numbers are $(1,0,1)$ by homology of a sphere. The torus's two independent one-dimensional homology classes cancel its degree-zero and degree-two contributions. The torus and annulus both have Euler characteristic zero, yet different homology groups; one integer cannot recover all of homology.

For a compact symplectic manifold, smooth convergence of tame almost complex structures and a uniform area bound give a subsequence of J-holomorphic spheres converging, after reparametrization, to a finite tree of J-holomorphic spheres, with convergence smooth away from finitely many bubbling points. The total homology class and symplectic area are preserved. Constant components may be added for stability. In a spherical class which cannot split into two positive-area spherical classes, there is only one nonconstant component.

For a chain complex $(C_{\ast },\partial )$, its $n$th homology is the abelian group or moduleThe equation ${\partial }_n{\partial }_{n+1}=0$ says that every chain boundary is a chain cycle. Homology identifies two chain cycles when their difference is a chain boundary; a homology class records exactly what survives this identification.

In singular homology, the chains are finite sums of maps from simplexes into a topological space. A continuous map induces homomorphisms on all homology groups, and homotopic maps induce the same homomorphisms. This makes homology a computable way to distinguish topological spaces that cannot be homotopy equivalent.^[ref]

For a circle, $H_0(S^1;\mathbb{Z})=\mathbb{Z}$ records its one path component, while $H_1(S^1;\mathbb{Z})=\mathbb{Z}$ records signed winding around it. A closed disk has the homology of a point: its boundary circle becomes a chain boundary when filled by the disk. Simplicial homology and cellular homology replace singular chains by smaller chain complexes suited to triangulated or cellular spaces.

For genus $g\ge 1$, collapse a nonseparating simple closed curve on a closed orientable surface. Its homology class is primitive, as a curve meeting it transversely once detects by the intersection pairing on an oriented surface. The long exact sequence in relative homology therefore gives $H_0=\mathbb{Z}$, $H_1={\mathbb{Z}}^{2g-1}$, $H_2=\mathbb{Z}$, and zero higher groups for the quotient. Geometrically it has the homotopy type ${\Sigma }_{g-1}\vee S^1$.

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

A homotopy equivalence of a closed surface cannot carry a simple closed curve of nonzero integral homology class onto a contractible simple closed curve. Any map into the latter factors through a circle whose inclusion induces zero on first homology; the former class would be killed by an isomorphism. Such a carrying map can nevertheless exist: project a handle generator onto a circle and compose with the contractible embedded circle.

Let $X$ be path-connected, locally path-connected and semilocally simply connected, and let $\alpha :{\pi }_1(X,x_0)\twoheadrightarrow \mathbb{Z}$ be a surjective group homomorphism. The classification of connected covering spaces supplies a connected covering space corresponding to the normal subgroup $\ker \alpha$. Its deck transformation group isChoose the deck transformation $T$ corresponding to $1\in \mathbb{Z}$. On each integral homology group, define $t\cdot z=T_{\ast }z$. Since $T_{\ast }$ is an automorphism, negative powers act too, making the homology group a module over the Laurent polynomial ring $\mathbb{Z}[t,t^{-1}]$. This records how a homology class changes when moved one sheet along the covering space.

For $X=S^1$ and the identity homomorphism ${\pi }_1(S^1)\cong \mathbb{Z}\to \mathbb{Z}$, the covering space is $\mathbb{R}\to S^1$, $u\mapsto e^{2\pi iu}$, with $T(u)=u+1$. Its only nonzero integral homology group is $H_0\cong \mathbb{Z}$, on which $t$ acts as the identity; as a Laurent polynomial ring module, it is $\mathbb{Z}[t,t^{-1}]/(t-1)$.

A compactly supported top-degree differential form on an oriented smooth manifold is integrated using orientation-preserving coordinates and a partition of unity. The coordinate change of variables formula makes the result independent of the choices. More generally, a form is integrated over a lower-dimensional oriented submanifold by its pullback. The Generalized Stokes theorem states ${\int }_{M}d\beta ={\int }_{\partial M}\beta$ with the induced boundary orientation. Integrating a closed differential form over a closed cycle depends only on its homology class, since its integral over a boundary vanishes.

For a closed manifold $M$ with an orientation and dimension $4k$, its integral intersection form is the symmetric bilinear form onobtained by pairing complementary middle-dimensional homology classes. Under Poincare duality, it is the cup product pairingon degree-$2k$ integral cohomology modulo its torsion subgroup. The even degree makes the pairing symmetric; integral duality makes it a unimodular intersection pairing. When representatives are transverse oriented submanifolds, the same value is the signed count of their intersection points.

The intersection form of a 4-manifold is the case $k=1$. On the Complex projective plane its matrix is $(1)$, whereas on $S^2\times S^2$ the factor spheres give $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

For a compact manifold with boundary, the perfect intersection pairing pairs absolute and relative homology, $H_{2k}(M)$ with $H_{2k}(M,\partial M)$; the induced absolute self-pairing can be degenerate. For instance the core sphere in $S^2\times D^2$ generates $H_2$ but has self-intersection zero.

For a closed manifold $M$ with an orientation and dimension four, its intersection form is the symmetric bilinear formon $H^2(M;\mathbb{Z})$ modulo its torsion subgroup. By Poincare duality it is the signed-intersection pairing on the free part of $H_2(M;\mathbb{Z})$, and it is a unimodular intersection pairing. Its real extension has positive and negative indices whose difference is its signature.

This is the four-dimensional case of the general intersection form. On a compact oriented four-dimensional manifold with boundary, the perfect intersection pairing uses one absolute and one relative homology class. An absolute self-pairing alone can be degenerate, so it should not be identified with the closed-manifold unimodular form without a boundary qualification.

An intersection pairing counts signed intersections of transverse representatives of complementary homology classes, using one relative representative when the manifold has boundary. With ${\mathbb{F}}_2$ coefficients it is defined without orientation choices. Poincare-Lefschetz duality makes this a perfect pairing for compact manifolds.

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 an open cover $X=U\cup V$, integral singular homology has the long exact sequenceAll four maps other than $\delta$ are induced by inclusions. The minus sign makes consecutive maps compose to zero. Exactness identifies the homology classes created by gluing $U$ and $V$ with cycles in their overlap.^[ref]

To construct $\delta$, subdivide a representing cycle in singular homology until it is a sum $c_U+c_V$ of chains in the two open sets. Since $\partial c_U=-\partial c_V$, this common boundary lies in the overlap; its class is $\delta [c]$. This chain construction proves exactness and also gives the version for reduced homology.

Cover $S^n$, $n\ge 1$, by two contractible enlarged hemispheres. Their overlap has the homotopy type of $S^{n-1}$, giving ${\tilde{H}}_n(S^n)\cong {\tilde{H}}_{n-1}(S^{n-1})$. Iterating starts from ${\tilde{H}}_0(S^0)\cong \mathbb{Z}$ and computes the top homology of every sphere.

A chain cycle is null-homologous with the chosen coefficients when its homology class is zero. Equivalently it is a chain boundary: there is a chain $b$ one degree higher such that $\partial b=c$.

A geometric bounding argument over the integers needs compatible orientations. If a compact oriented submanifold $W$ has oriented boundary $C$, a chain representing its relative fundamental class has boundary representing $C$, so $[C]=0$ in the ambient integral homology. With mod-two coefficients, orientation signs disappear and an unoriented bounding submanifold suffices.

For example, the boundary circle of a Möbius band bounds that band as an unoriented surface, but represents twice the core generator in $H_1(\text{Mo¨bius band};\mathbb{Z})\cong \mathbb{Z}$. It is not integrally null-homologous inside the band. Its mod-two homology class is zero. This distinguishes bounding a geometric subset from bounding a chain with the required coefficients.

DefineEvery projective point with $z_3\ne 0$ has a unique unit representative whose third coordinate is positive real, so $F$ is a diffeomorphism. More explicitly, in the affine coordinate $u=(z_1/z_3,z_2/z_3)$ its inverse is $w=u/\sqrt{1+|u{|}^2}$. Lift $F$ to the unit sphere by $\tilde{F}(w)=(w,\sqrt{1-|w{|}^2})$. The last coordinate is real, so its contribution $d{x}_3\wedge d{y}_3$ pulls back to zero. The reduction identity therefore yieldsThis proves the symplectic ball chart in complex projective space directly, including the precise unit radius.

For the final unheaded request, embed the two disjoint balls in this chart. Fix arbitrary $0<{\rho }_i<r_i$, and choose intermediate radii ${\rho }_i<{\rho }_i^{\prime }<r_i$. On the images of the balls of radius ${\rho }_i^{\prime }$, transport the standard complex structure by the given symplectic embeddings. It is compatible with $\Omega$. Their smaller closed neighborhoods are compact and disjoint. Choose a global Riemannian metric agreeing there with the associated compatible metrics, using a partition of unity outside these neighborhoods, and apply the metric construction. The resulting global compatible almost complex structure $J$ agrees with the transported standard structure on each ball of radius ${\rho }_i$ and on a slightly larger neighborhood. This extends metrics, rather than averaging almost complex structures.

Let $p_1,p_2$ be the centres. They are distinct. The degree-one J-holomorphic curve supplied in the question has total area $\pi$, because its homology class is the projective line class. We use the sharp Euclidean version of the Monotonicity theorem for a J-holomorphic curve: a nonconstant holomorphic curve through the centre of a standard complex ball, with no boundary in the ball and counted with its parametrization multiplicities, has area in the radius-$\rho$ ball at least $\pi {\rho }^2$. This includes branched points: their density is a positive integer $m$, and the stronger lower bound is $m\pi {\rho }^2$. One way to see the constant is that complex curves are calibrated minimal surfaces; their stationary area ratio $\mathrm{area}(C\cap B(\rho ))/{\rho }^2$ is nondecreasing and its limit at the centre is $m\pi$.

The closed degree-one curve cannot stay entirely inside either chart ball: there the form is exact, so Stokes theorem would give zero total area, contradicting its positive area. Thus the branch through each centre crosses the surrounding spheres, and the stated monotonicity bound applies to each restriction. The two ball images are disjoint, so their area contributions add without overlap. It follows thatLetting both smaller radii increase to their original radii givesThis two-ball packing obstruction in the projective plane is sharper than volume alone: a volume comparison would only give $r_1^4+r_2^4\le 1$.