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Integration of holomorphic differential forms along cycles embeds the first homology group as a full Euclidean lattice in $H^0(C,K_C{)}^{\vee }$. Fixing a base point, sum the path integrals to the points of an effective complex analytic divisor. Different path choices differ by that lattice. The Abel theorem for divisors identifies the fibres with linear equivalence of divisors.

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.

An element of a chain group is a cycle when its boundary operator kills it. A cycle need not itself be a chain boundary. The homology group $H_n(C)$ identifies cycles whose difference is a chain boundary.

Chromatic homotopy theory organizes stable homotopy theory by a prime number $p$ and a hierarchy of heights $n$. Its main probes include Morava K-theory, whose standard coefficient ring for $n\ge 1$ isThe index is the height of the associated formal group law. Rational homology plays the height-zero role. These different probes detect different parts of a spectrum, rather than replacing all stable homotopy information by one ordinary homology group.^[ref]

A basic chromatic question asks which heights detect a given finite spectrum: its type is the least $n$ with $K(n{)}_{\ast }(X)\ne 0$, when such an $n$ exists. Rational information already detects the sphere spectrum at height zero, whereas purely torsion phenomena require other probes. The chromatic viewpoint makes periodic patterns and localization of spectra into organizing tools for difficult stable computations.

Computational topology develops algorithms for topological questions about finite representations of spaces, such as simplicial complexes. It connects discrete data structures to invariants that ignore irrelevant geometric distortion. connected components and homology groups are examples of computable structural information.^[ref]

For a small simplicial homology calculation over a field, take three vertices joined by three edges in a cycle, without a filled face. The matrix of the first boundary operator has matrix rank two, so this linear map has a kernel of a linear map of dimension one; with no two-dimensional face, that cycle is not a boundary. Thus the first Betti number is one. Adding the filled triangular face makes the same cycle a boundary and reduces the first Betti number to zero. The vertex coordinates need not change: the encoded incidence structure determines whether the hole is present.

The Euler characteristic compresses a finite CW complex's homology into one integer: count its topological cells with alternating signs, or count its Betti numbers with the same signs. A filled triangle has Euler characteristic one; its cycle graph boundary has Euler characteristic zero. The missing face changes the answer even though the vertices and edges are identical.

For a finite CW complex $X$, including the geometric realization of a finite simplicial complex,Here $b_i={\dim }_{\mathbb{Q}}{H}_i(X;\mathbb{Q})$ is the rational Betti number. More generally this ordinary homology definition makes sense whenever every rational homology group is finite-dimensional and only finitely many are nonzero. An arbitrary infinite CW complex need not satisfy these assumptions; its formal infinite alternating cell count is not a definition of this invariant.

The usefulness is that a finite description can reveal something about a space without computing every homology group. By the cell-count formula for Euler characteristic, count pieces rather than solve all the boundary operators. If two spaces give different answers, topological invariance of Euler characteristic rules out a homeomorphism, and even a homotopy equivalence. Equal answers are less decisive: the Euler characteristic of an annulus and the Euler characteristic of a torus are both zero, while their homology groups differ. This is a compact diagnostic, with a clearly defined limit on what it can tell us.

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 concrete example of cellular homology of real projective space, the real projective plane ${\mathbb{RP}}^2$ has one topological cell in each of dimensions zero, one and two. Its one-dimensional topological cell is a loop, and the face attaches around that loop twice. Choose orientations so that the cellular chain complex over the integers iswith nonzero chain groups in degrees two, one and zero. The twice-around attachment is why the first boundary operator shown is multiplication by two.^[ref]

Over the rational numbers, multiplication by two is invertible, so $H_2$ and $H_1$ vanish, while $H_0\cong \mathbb{Q}$. Over the finite field ${\mathbb{F}}_2$, two equals zero. Both boundary operators then vanish, giving one-dimensional homology groups in all three degrees. Their respective Betti numbers areNevertheless,The direct cell count is also $1-1+1=1$.

This calculation makes the strength and loss of information visible. The Euler characteristic survives a change of the field used for coefficients, while individual homology groups can change. Over the integers, the same chain complex has $H_1\cong \mathbb{Z}/2\mathbb{Z}$ and $H_2=0$: a torsion subgroup that rational Betti numbers do not record. The rank cancellation in the Euler-Poincare formula explains why the alternating total still agrees with the cell count.

Let $X$ be simply connected, with finitely generated integral homology groups, and suppose $H_i(X;\mathbb{Z})=0$ for $i>n$, where $n\ge 2$. Then $X$ admits a weak homotopy equivalence $C\to X$ from a finite CW complex of dimension at most $n+1$. If $X$ already has CW complex homotopy type, this map is a homotopy equivalence.

Start with a finite $n$-dimensional CW complex $C_n$ and an $n$-connected map $f:C_n\to X$, constructed by the finite type CW approximation. This means that $f$ induces isomorphisms on homotopy groups below $n$ and a surjection in degree $n$; it does not mean that $C_n$ itself is $n$-connected.

The kernel of $H_n(C_n;\mathbb{Z})\to H_n(X;\mathbb{Z})$ is a finitely generated free abelian group, because $H_n(C_n;\mathbb{Z})$ is a subgroup of its finite free top cellular chain group. The Relative Hurewicz theorem represents a basis of this kernel by attaching $(n+1)$-cells whose maps extend into $X$. Their cellular boundary is injective, killing exactly that kernel without adding top homology. All integral homology groups now agree, so the homological Whitehead theorem proves the conclusion. The acyclic simply connected case is already weakly equivalent to a point.

Given a compact planar set $K$ and an open neighborhood $V$, a finite square-grid construction gives a compact polygonal neighborhood inside $V$ with finitely many complementary components. Remove thin polygonal corridors disjoint from $K$ joining holes that lie in the same component of $\mathbb{C}\setminus K$, and joining holes in its unbounded component to the exterior. Corridors can also include a prescribed point from each of finitely many specified bounded components. There remains a neighborhood whose bounded holes lie in distinct bounded components of $\mathbb{C}\setminus K$, each containing its chosen point. Standard small polygonal rounding gives a finite union of planar surfaces with boundary. The index map from its first integral homology group to one integer per bounded hole is an isomorphism. Triangulate a surrounding disc: a one-cycle bounds a two-chain whose coefficients are constant in each complementary component. Zero hole coefficients give a filling supported inside the neighborhood; the face chain of each hole supplies a dual cycle. The cycles can be sums of boundary curves in different components. This uses finite triangulation, not the general Jordan curve theorem.

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.

Identifying two distinct points of $S^2$ produces a space with homology groups $H_0=H_1=H_2=\mathbb{Z}$ and all higher groups zero. Cover the identified point by the image of two small disks, a contractible wedge of disks, and take as the other open set the complement of the point, a cylinder. Their intersection is two annuli. In the Mayer–Vietoris sequence, $H_1$ of this intersection maps onto $H_1$ of the cylinder with a rank-one kernel, producing $H_2=\mathbb{Z}$. The kernel at $H_0$ has rank one and produces $H_1=\mathbb{Z}$.

For the standard coordinate inclusion of spheres, write points as $(u,v)\in {\mathbb{R}}^{m+1}\times {\mathbb{R}}^{n-m}$, with $v\ne 0$ in the complement. The map $(u,v)\mapsto (u,v/\Vert v\Vert )$ is a homeomorphism to $B_{\mathrm{open}}^{m+1}\times S^{n-m-1}$, with inverse $(u,w)\mapsto (u,\sqrt{1-\Vert u{\Vert }^2}w)$. Contracting the ball factor proves the displayed homotopy equivalence. In codimension one there are two contractible components, so integral $H_0$ is ${\mathbb{Z}}^2$; in higher codimension the only nonzero homology groups are $H_0=\mathbb{Z}$ and $H_{n-m-1}=\mathbb{Z}$.

Attach two disks to a simply connected space $A$ along disjoint null-homotopic boundary circles, and additionally identify one interior point of each disk. Assume the attachments are CW complexes or admit compatible triangulations. The extra interior identification contributes a copy of $\mathbb{Z}$ to the first homology group, in addition to the two new second-homology classes. Indeed, after the first attachment $Y=A\cup D_1$, the intersection with $D_2$ has two components, $S^1\bigsqcup \{p\}$. In the Mayer–Vietoris sequence, the map $H_0(S^1\bigsqcup \{p\})\to H_0(Y)\oplus H_0(D_2)$ sends $(m,n)$ to $(m+n,-m-n)$, whose kernel is $\mathbb{Z}$. If $H_1(A)=0$, exactness gives $H_1(Y\cup D_2)=\mathbb{Z}$. The degree-two sequence is $0\to H_2(Y)\to H_2(Y\cup D_2)\to \mathbb{Z}\to 0$, which splits because $\mathbb{Z}$ is free. For $A=S^3$, the integral homology groups are $\mathbb{Z},\mathbb{Z},{\mathbb{Z}}^2,\mathbb{Z}$ in degrees zero through three.

Homotopic maps induce the same homomorphism on every homology group. Consequently, homotopy equivalent spaces have isomorphic homology groups.

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 $\mathbb{Z}[t,t^{-1}]/(t-1)$.

For a continuous self-map $f:X\to X$, suppose the rational homology groups $H_i(X;\mathbb{Q})$ are finite-dimensional and vanish in sufficiently high degrees. Its Lefschetz number is the finite alternating traceFor a self-map of a finite simplicial complex, it also equals the alternating trace on any finite simplicial chain complex representing the induced map. The traces on boundaries cancel in consecutive degrees, leaving precisely the traces on homology. Thus a potentially large chain computation reduces to the homology groups.

The identity map has $L(1_X)=\chi (X)$, the Euler characteristic. On a circle, a map of mapping degree $d$ has $L(f)=1-d$. A nonzero value forces a fixed point under the hypotheses of the Lefschetz fixed-point theorem; finiteness of the displayed sum alone is not that theorem's full geometric hypothesis.

Let $M$ be the given closed three-dimensional manifold. It is connected by the meaning of simply connected. An orientation chosen at one point can be transported along paths, and its sign change around loops defines a homomorphism ${\pi }_1(M)\to \{\pm 1\}$. Since the fundamental group is trivial, this monodromy vanishes; thus $M$ is orientable.

The first homology group is the abelianization of the fundamental group, giving $H_1(M;\mathbb{Z})=0$. Also $H_0(M;\mathbb{Z})=\mathbb{Z}$. The universal coefficient theorem for cohomology in degree one givesBoth outer groups vanish: the first because $\mathbb{Z}$ is free and the second because $H_1=0$. Hence $H^1=0$. Integral Poincare duality now gives $H_2(M;\mathbb{Z})\cong H^1(M;\mathbb{Z})=0$ and $H_3(M;\mathbb{Z})\cong H^0(M;\mathbb{Z})=\mathbb{Z}$. Homology in degrees greater than three vanishes by the same duality with negative-degree cohomology. ConsequentlyThese are precisely the homology of a sphere for $S^3$, proving that simply connected closed three-manifolds are homology spheres. No homeomorphism classification of three-manifolds is needed.

The group $G(K)$ consists of the nonvanishing continuous functions, and $E(K)$ is a subgroup because multiplication is commutative: $e^g{e}^h=e^{g+h}$. The objective is a logarithm decomposition with finitely many integer obstructions, one for each bounded complementary component. No local connectivity of $K$ is assumed.

We use two elementary finite-neighborhood facts, which make the outline independent of the general Jordan theorem. Given a neighborhood $V$ of $K$, take a finite grid neighborhood $N\subset V$ with $K\subset \mathrm{int}N$. It can be chosen to be a finite union of polygonal surfaces with boundary. Its complement has finitely many bounded holes. Holes that lie in the same component of $\mathbb{C}\setminus K$ can be joined by thin polygonal corridors disjoint from $K$, which are removed from $N$. A hole lying in the unbounded component is similarly joined to the exterior. For the remaining finitely many components, corridors may be continued to include their chosen representatives $a_i$. Small polygonal adjustments preserve $K\subset \mathrm{int}N$ and $N\subset V$. Thus every remaining bounded hole lies in a distinct bounded component of $\mathbb{C}\setminus K$ and contains its representative. One may also arrange this for any prescribed finite list of bounded components by first keeping their representative points out of the grid neighborhood. All paths used are compact and disjoint from $K$, so sufficiently thin corridors remain disjoint from $K$.

For such a finite polygonal $N$, winding indices identify the homology group $H_1(N;\mathbb{Z})$ with the free abelian group on its bounded holes. Here is a finite-chain explanation. Triangulate a large disc with $N$ as a subcomplex. Every integral one-cycle bounds a two-chain in the disc. Its filling coefficient is constant on each component of the complement of $N$, is zero on the exterior, and is the winding index there. If all bounded-hole indices vanish, the filling is supported in $N$, so the cycle already bounds in $N$. Conversely the sum of the oriented faces in one hole has its chain boundary in $N$ and has index one in that hole and zero in the others. Thus cycles ${\gamma }_j$ can be chosen with $\mathrm{wind}({\gamma }_j,a_i)={\delta }_{ij}$. These can be integral sums of boundary curves, which matters when components of $N$ are nested. This proves the needed statement using finite triangulation, not separation of an arbitrary simple closed curve. Together these are finite-hole neighborhoods of a planar compact set.

Now extend $f\in G(K)$ to a continuous $F$ near $K$, using the Tietze extension theorem on its real and imaginary parts. Since ${\min }_K|f|>0$, shrink the neighborhood until $F$ has no zeros, and choose $N$ within it as above. The winding of $F$ on the cycles ${\gamma }_j$ defines integers $k_j$. Setwhere only the finitely many represented holes are used. The winding of $F/r$ on every cycle of $N$ is zero. On each component choose a local complex logarithm at a base point and continue it along paths. Zero winding number around every loop makes the continued value independent of the path; local logarithm charts make it continuous. The continuous logarithm lifting criterion therefore gives $F/r=e^g$ on $N$, and restriction gives $f=e^{g{|}_K}r{|}_K$. Thus $G(K)=E(K)G(\mathbf{a})$, even when $K$ has infinitely many holes: each individual function uses finitely many factors.

To prove uniqueness, suppose a finite product $r=\prod (z-a_i{)}^{k_i}$ equals $e^g$ on $K$. Extend $g$ continuously to a neighborhood. There $r{e}^{-g}$ is close to $1$ after shrinking, so it has its small-branch logarithm. Hence $r$ itself has a continuous function giving a complex logarithm on that neighborhood. Choose $N$ there detecting all the finitely many $a_i$ as separate holes. Its dual cycles give $\mathrm{wind}(r\circ {\gamma }_j,0)=k_j$, while a function with a logarithm has zero winding number. Therefore every $k_j=0$. This proves both freeness of the listed rational functions and $E(K)\cap G(\mathbf{a})=\{1\}$. All subgroups commute, soThis is planar logarithm factorization. If there are no bounded components the second factor is trivial.

For the Jordan curve deduction, let $J$ be homeomorphic to the circle. Parametrize it by $\gamma :[0,1]\to J$ with just the two endpoints identified. For $f\in G(J)$, lift a logarithm of $f\circ \gamma$ along the interval. Its endpoint increment is $2\pi in$. The integer $n$ is additive, every integer is attained by a function with $f(\gamma (t))=e^{2\pi int}$, and $n=0$ precisely when the lift descends to a logarithm on $J$. Hence $G(J)/E(J)\cong \mathbb{Z}$. The factorization says this is free abelian on the bounded complementary components; comparison of ranks, or tensoring with $\mathbb{Q}$, shows that exactly one bounded component exists. There is also the unique unbounded component, so the complement has exactly two components.

We still need the common-boundary assertion. A proper closed arc has trivial exponential quotient because every nonvanishing function on an interval admits a logarithm. Factorization consequently makes the complement of such an arc connected, and since it is open it is path-connected. Fix $p\in J$ and any neighborhood $V$ of $p$, and choose a small open arc $A\subset J\cap V$ containing $p$. The remainder $J\setminus A$ is a proper closed arc. Join a point in each of the two complementary components by a path avoiding $J\setminus A$. This path must meet $J$, and every such meeting lies in $A$. The segment before its first meeting lies in the first region, and the segment after its last meeting lies in the second; continuity makes both regions meet $V$. Thus every point of $J$ lies on both boundaries. Conversely either boundary is contained in the closed set $J$. There are exactly two regions, one bounded and one unbounded, with common boundary $J$, the Jordan curve theorem. This is Jordan separation from the exponential quotient; it does not assume the theorem in the factorization proof.