Homology

Homology assigns abelian groups $H_i(X)$ to a space, measuring cycles modulo boundaries in each dimension.

A positive-degree class whose reduced homology diagonal vanishes. In a torsion-free homology Hopf algebra, powers of an even primitive class have binomial diagonals. Dualizing that diagonal produces the multiplication of a divided power algebra.

A nonempty space is acyclic with the chosen coefficients if its reduced homology vanishes in every degree. Equivalently it has the homology of a point. It need not be contractible.

Suppose $X={\bigcup }_a{X}_a$ is directed by inclusion and every compact subset of $X$ lies in some $X_a$. Every finite singular chain then lies in one $X_a$, so $C_{\ast }(X)={\underrightarrow{\lim }}_a{C}_{\ast }(X_a)$. Exactness of filtered colimits gives

Every open subset $U\subseteq {\mathbb{R}}^N$ is the directed union of finite unions of closed rational cubes contained in $U$. Every compact subset of $U$ lies in one such finite polyhedron, whose homology is finitely generated. The homology of a directed union therefore expresses each $H_i(U;\mathbb{Z})$ as a direct limit over a countable family of countable groups, so it is countable.

Reduced homology modifies degree zero so that a one-point space has zero homology in every degree. For a nonempty space, ${\tilde{H}}_k(X)=H_k(X)$ when $k>0$, while $H_0(X)\cong {\tilde{H}}_0(X)\oplus \mathbb{Z}$.

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.

The zeroth homology group is the free abelian group on the path components of $X$. In particular, if $X$ has finitely many path components, then

A homology class is an equivalence class of chain cycles modulo chain boundaries in a chain complex.

A continuous map $f:X\to Y$ induces homomorphisms $f_{\ast }:H_n(X)\to H_n(Y)$ satisfying $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{id}}_X{)}_{\ast }={\mathrm{id}}_{H_n(X)}$.

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

The first homology group $H_1(X)$ is the group of one-dimensional cycles modulo boundaries. For a path-connected space it is the abelianization of the fundamental group.

Singular homology forms a chain complex from formal sums of continuous maps from standard simplices into a topological space. Its homology groups are invariant under homotopy.

A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.

A singular $n$-chain is a finite formal integer linear combination of singular simplices. Its support is contained in a compact subset of the ambient space.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.

A singular $n$-simplex in a space $X$ is a continuous map $\sigma :{\Delta }^n\to X$ from the standard $n$-simplex. Its alternating restrictions to the faces form its boundary.

For an oriented compact manifold of dimension $4k$, the intersection form is the bilinear pairing on middle-dimensional homology obtained by representing classes by transverse cycles and counting their signed intersection points. On an oriented four-manifold it is equivalently the cup-product pairing on $H^2$ evaluated on the fundamental class.

An integral intersection form on a finite-rank free abelian group $L$ is unimodular if its adjoint $x\mapsto \lambda (x,-)$ is an isomorphism to the integral dual. In an integral basis, this is equivalent to the pairing matrix having determinant $1$ or $-1$, not merely nonzero determinant. For example, projective lines generate a unimodular intersection form on the Complex projective plane, with matrix $(1)$ in its complex orientation.

When compact oriented four-manifolds are glued along an entire closed boundary three-manifold, the signature of the result is the sum of their intersection form signatures. It applies to $X_0{\cup }_{\partial X_0}(-X_1)$ with the difference of the two signatures. Gluing along only part of a boundary can require a correction; that is a different situation.

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.

For a compact three-manifold $W$, let $L=\ker (H_1(\partial W;{\mathbb{F}}_2)\to H_1(W;{\mathbb{F}}_2))$. The boundary intersection pairing satisfies $L^{\perp }=L$: the long exact sequence in relative homology identifies $L$ with boundaries of relative surfaces, and the boundary-interior adjoint identity identifies its orthogonal complement with the same kernel. Hence $2\dim L=\dim H_1(\partial W;{\mathbb{F}}_2)$. This also implies that a closed surface with odd first mod-two Betti number, such as ${\mathbb{RP}}^2$, cannot be the entire boundary of a compact three-manifold.

For a map $f:M\to N$ of connected closed oriented four-manifolds, naturality of the cup product givesWhen $\mathrm{deg}f\ne 0$, nondegeneracy makes $f^{\ast }:H^2(N;\mathbb{Q})\to H^2(M;\mathbb{Q})$ injective. Consequently the target intersection form, multiplied by the mapping degree, must embed into the source form. For equal second Betti numbers, a matrix $P$ for pullback satisfies $P^{\mathsf{T}}{Q}_{M}P=(\mathrm{deg}f)Q_N$; taking determinants often rules out nonzero degrees. In particular, a two-dimensional indefinite source form cannot realize a nonzero multiple of a two-dimensional definite target form.

On a compact oriented smooth manifold $N$ of dimension four without boundary, set $Q([\alpha ],[\beta ])={\int }_N\alpha \wedge \beta$ for closed real two-forms. The Stokes theorem makes the expression independent of representatives, and the even degrees make it symmetric. It is the real de Rham cohomology version of the intersection form. The Hodge decomposition theorem and Hodge star operator express it as $Q(h,k)=(h,\ast k{)}_{L^2}$ on harmonic representatives.

On a closed oriented Riemannian manifold of dimension four, the Hodge star operator preserves harmonic two-forms and splits them into orthogonal spaces ${\mathcal{H}}^{+}$ and ${\mathcal{H}}^{-}$ with eigenvalues $+1$ and $-1$. The real de Rham intersection form in dimension four equals the positive $L^2$ inner product on ${\mathcal{H}}^{+}$ and its negative on ${\mathcal{H}}^{-}$. It is nondegenerate, since $Q(h,\ast h)=\Vert h{\Vert }_2^2$. Its signature pair is $(\dim {\mathcal{H}}^{+},\dim {\mathcal{H}}^{-})$.

Integral homology is homology with coefficients in $\mathbb{Z}$. It retains both the free and torsion parts of each homology group.

For $n>0$,Writing $S^n$ as two contractible hemispheres whose intersection deformation retracts to $S^{n-1}$ gives the calculation inductively through the Mayer–Vietoris theorem.

For a map $f:S^m\to S^m$, the degree is the integer by which $f_{\ast }$ multiplies a chosen generator of $H_m(S^m;\mathbb{Z})$. It is invariant under homotopy.

Define mapping degree for $S^0$ using its rank-one reduced homology group ${\tilde{H}}_0(S^0;\mathbb{Z})$. Its generator is the difference of its two points. The four maps of that two-point set consist of the identity, the interchange, and two constant maps; their degree multipliers are $1,-1,0,0$. Thus sphere maps of arbitrary integer degree require positive sphere dimension.

For a map of oriented disk pairs, the multiplier on top relative homology is its relative mapping degree. Naturality of the connecting homomorphism identifies it with the mapping degree of the boundary map.

For a smooth map of disk pairs whose graph of a function is transverse to the zero slice, orient the graph by its domain. With the tangent space of the zero slice ordered before that of the graph, the local smooth intersection number at a zero is $\mathrm{sgn}\det DF$. Summing these signs gives the relative mapping degree and hence the boundary mapping degree. Reversing the order of these two $m$-dimensional tangent spaces changes the sign by $(-1{)}^m$.

Collapsing the boundary identifies the top relative homology of a disk pair with the top reduced homology of its quotient sphere. Together with the boundary connecting homomorphism, this shows that the induced quotient map and the boundary map have the same mapping degree, using the corresponding orientations.

Let $f:M\to N$ be a continuous map between oriented $d$-dimensional manifolds, with $d\ge 1$. Suppose $x$ is an isolated point of the fiber over $y=f(x)$. Choose an open neighborhood $U$ with $U\cap f^{-1}(y)=\{x\}$. The map of pairsinduces a map between rank-one degree-$d$ local homology groups. Relative to the local orientation of a manifold, this map is multiplication by an integer, the local degree. Excision makes it independent of the neighborhood. A smooth map between manifolds with invertible derivative at $x$ has local degree $\mathrm{sign}\det D{f}_x$, by the inverse function theorem and the chosen orientations.

For a continuous map $f:M\to N$ of closed connected oriented $d$-dimensional manifolds, suppose $f^{-1}(y)=\{x_1,\dots ,x_s\}$ is finite. ThenBy excision, the degree-$d$ relative homology group $H_d(M,M\setminus f^{-1}(y);\mathbb{Z})$ is the direct sum of the local groups at $x_j$. The fundamental class maps to the tuple of local orientation generators. Mapping this tuple to $H_d(N,N\setminus \{y\};\mathbb{Z})$ adds their local degrees. Naturality identifies the result with the image of $(\mathrm{deg}f)[N]$. If the fiber is empty, the same argument gives degree zero.

If $f:S^m\to S^m$, then its suspension $\Sigma f:S^{m+1}\to S^{m+1}$ has the same degree. This follows from naturality of the suspension isomorphism in reduced homology.

Every integer $m$ is the mapping degree of a continuous map $S^n\to S^n$ in every dimension $n\ge 1$. Start with the circle map $z\mapsto z^m$, whose argument winds $m$ times, and use degree under suspension repeatedly. Negative $m$ reverses the winding, and $m=0$ is constant before suspending. The degrees of maps of the zero-sphere are a genuine exceptional case.

For a map $f:M\to N$ between closed connected oriented $d$-manifolds, its degree is the integer determined by $f_{\ast }[M]=(\mathrm{deg}f)[N]$ in top-dimensional homology.

For maps of oriented connected closed manifolds of the same dimension, apply functoriality of homology to $f_{\ast }[M]=(\mathrm{deg}f)[N]$ and $g_{\ast }[N]=(\mathrm{deg}g)[L]$. The composite coefficient is their product. This recovers the multiplication of signed sheet counts for compositions of finite covering maps.

For maps between oriented connected closed manifolds of the same dimension, a homotopy induces equal maps on top-dimensional homology, hence equal coefficients on the fundamental class. For smooth maps and a smooth homotopy, the degree by integration of a pullback volume form gives another proof: the Generalized Stokes theorem on the homotopy cylinder makes the difference of endpoint pullback integrals zero because the target top form is closed. This is an obstruction to extending a degree-one sphere identity over a ball.

Although topological degree classifies maps $S^n\to S^n$ up to homotopy, it is not a complete invariant for other manifolds. On the torus, the identity and the integer shear $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ both have degree one, but induce distinct maps on $H_1(T^2;\mathbb{Z})$ and cannot be homotopic.

For a smooth map $f:M\to N$ between oriented connected closed manifolds of the same dimension, a normalized volume form on the target gives $\mathrm{deg}f={\int }_M{f}^{\ast }\omega$. The formula agrees with the degree as a sum of local degrees. A change of normalized top-form adds an exact form by top-dimensional de Rham cohomology, whose pullback has zero integral by Stokes theorem.

For a smooth map between oriented unit spheres, integrate the pullback of the target area form. Localizing a normalized top-degree form near a regular value, and using Stokes theorem to discard exact-form differences, identifies this integral with the degree as a sum of local degrees.

A degree of a map between oriented manifolds equal to one makes the pullback on degree-one cohomology injective by the Poincare duality pairing. Thus a degree-one map of closed oriented surfaces requires $2h\le 2g$. Conversely, collapse the extra handles in ${\Sigma }_h\#{\Sigma }_{g-h}$ onto a point. The retained oriented disc shows that this map has degree one.

If $S^n\to N$ has prime degree $p$ and $N$ is a closed connected oriented $n$-manifold, every integral homology group of $N$ in degrees strictly between zero and $n$ is a finite $p$-primary abelian group. Consequently one power of $p$ annihilates all these groups. The proof combines Poincare duality over each field ${\mathbb{F}}_q$, $q\ne p$, with the universal coefficient theorem for homology and finite generation.

For $n\ge 2$, a self-map of $S^n$ factoring through Real projective space ${\mathbb{RP}}^n$ has degree zero when $n$ is even, and can have exactly the even degrees when $n$ is odd. In the odd case the first map lifts through the double covering and the covering has degree two. The dimension-one exception allows every integer, since ${\mathbb{RP}}^1\cong S^1$.

The antipodal map $x\mapsto -x$ on $S^m$ has degree $(-1{)}^{m+1}$.

An odd map between spheres is a continuous map $f:S^n\to S^m$ satisfying $f(-x)=-f(x)$. Thus it is equivariant for the two antipodal maps. It descends to a continuous map of Real projective spaces.

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 an odd map between spheres $f:S^n\to S^m$ and its quotient $\bar{f}$, there is an isomorphism ${\gamma }_n\cong {\bar{f}}^{\ast }{\gamma }_m$ of real tautological line bundles. With $x$ a unit vector, send $tx$ to $tf(x)$; replacing $x$ by $-x$ and $t$ by $-t$ gives the same vector. This proves both linearity on fibers and well-definedness. Therefore ${\bar{f}}^{\ast }{w}_1({\gamma }_m)=w_1({\gamma }_n)$. The mod-two cohomology ring of real projective space then implies $n\le m$, since a vanishing $(m+1)$st power must pull back to a vanishing power.

Let a finite group $G$ act smoothly on a smooth manifold, and suppose that a $G$-invariant differential form $\omega$ is an exact differential form, say $\omega =d\eta$. Averaging gives the invariant primitiveIf the action is free, invariant forms descend uniquely through the resulting covering map.

For the double covering $\pi :S^{2n}\to {\mathbb{RP}}^{2n}$ and the antipodal map $a$, every pulled-back top form satisfies $a^{\ast }{\pi }^{\ast }\omega ={\pi }^{\ast }\omega$. Since $\mathrm{deg}a=-1$, its integral over the sphere is its own negative and hence vanishes. It is therefore exact on the sphere. Averaging a primitive under $a$ and descending it proves that $\omega$ is exact on real projective space.

A Euclidean homeomorphism is a proper map and extends to its one-point compactification $S^n$. Its degree is the degree of a continuous mapping of this extension, and belongs to $\{1,-1\}$. A linear isomorphism has degree equal to the sign of its determinant.

The factor swap ${\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^n\times {\mathbb{R}}^m$ has degree $(-1{)}^{mn}$. Moving an oriented basis of the first factor past one of the second requires $mn$ transpositions.

The relative chain complex is $C_{\ast }(X,A)=C_{\ast }(X)/C_{\ast }(A)$, and its homology is the relative homology $H_{\ast }(X,A)$. A pair $A\subseteq U\subseteq X$ produces a long exact sequence for the triple.

For a subspace $A\subseteq X$, the relative chain complex is the quotient group in each degree,with boundary operator $[c]\mapsto [\partial c]$. This is well defined because $C_{\ast }(A)$ is a chain subcomplex of $C_{\ast }(X)$.

If $L$ is a simplicial subcomplex of $K$, thenIts homology is the simplicial relative homology $H_k(K,L)$.

For a pair $A\subseteq X$, the short exact sequence of chain complexes $0\to C_{\ast }(A)\to C_{\ast }(X)\to C_{\ast }(X,A)\to 0$ induces

For $n\ge 1$,In the relative simplicial chain complex, every proper face vanishes, leaving one generator in degree $n$ and zero chain groups in all other degrees.

A pair $(X,A)$ is good when $A$ is closed and is a deformation retract of some neighborhood in $X$. Such a pair satisfies the hypotheses needed to compare relative homology with the reduced homology of a quotient.

For a good pair $(X,A)$, the quotient map induces natural isomorphisms

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.

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 ${\Sigma }_{g-1}\vee S^1$.

Collapse a separating simple closed curve on a closed orientable surface of genus $g$. If its two sides have genera $g_1,g_2$, the quotient is homeomorphic to ${\Sigma }_{g_1}\vee {\Sigma }_{g_2}$. Its integral homology is $\mathbb{Z}$ in degree zero, ${\mathbb{Z}}^{2g}$ in degree one, ${\mathbb{Z}}^2$ in degree two, and zero above degree two. In relative homology, the circle maps to zero in $H_1({\Sigma }_g)$, and the second relative group fits into a split sequence $0\to \mathbb{Z}\to H_2({\Sigma }_g,A)\to \mathbb{Z}\to 0$. The two collapsed sides retain independent fundamental classes.

Excision identifies relative homology after removing a subspace whose closure lies in the interior of the subspace being quotiented. It makes local homology computable in an arbitrarily small neighborhood.

An exact sequence has the image of each map equal to the kernel of the next.

A long exact sequence is a sequence of abelian groups or modules with arbitrarily many consecutive maps, for which the image of every map equals the kernel of the next. A short exact sequence of cochain complexes yields a long exact sequence of their cohomology, with connecting homomorphisms increasing degree by one. The exact sequences of relative cohomology and the Bockstein homomorphism are examples.

A commutative diagram is a diagram of objects and morphisms in which every two directed paths with the same endpoints define the same morphism.

A chain complex is a sequence of abelian groups or modules and homomorphisms $d_i:C_i\to C_{i-1}$ satisfying $d_{i-1}{d}_i=0$. Its homology is $H_i(C)=\ker d_i/\mathrm{im}{d}_{i+1}$.

The reversed dual module of a chain complex becomes a chain complex with the unsigned precomposition differential above. A functional on $C_{-i}$ is sent to a functional on $C_{1-i}$, lowering the reversed degree by one.

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

For finite free total chain complexes, evaluation identifies the underlying graded tensor product $X\otimes C^{\prime }$ with the graded Hom complex of chain complexes, where $X$ is the reversed dual chain complex. Under the precomposition-first Hom differential, conjugation by the sign $(-1{)}^{j(j-1)/2}$ makes this a chain isomorphism. The identity $\rho (j)-\rho (j-1)=j-1$ works for every integer degree. If the grading is unbounded and only degreewise finite, products on the Hom side need not equal direct sums on the tensor side.

The tensor product of homologically graded chain complexes has differential $D(x_p\otimes y)=d{x}_p\otimes y+(-1{)}^p{x}_p\otimes dy$. The sign makes the two mixed terms in $D^2$ cancel. Total degree is the sum of the two degrees.

Interchanging homogeneous objects of degrees $p$ and $q$ introduces $(-1{)}^{pq}$. In particular, moving a degree-minus-one differential past a degree-$p$ factor introduces $(-1{)}^p$. This convention produces the differential on the tensor product of chain complexes.

For $n\ge 1$, the chain complex with copies of $R$ in degrees $n,n-1$ and identity differential. It is acyclic; a map from it chooses an arbitrary degree-$n$ element and its boundary.

The chain complex with $R$ in a single degree $n$ and zero differential. A map from it into a complex chooses a cycle in degree $n$.

A double complex has modules $C_{p,q}$ and differentials lowering either index, each squaring to zero and anticommuting with the other. The total chain complex has term ${⨁}_{p+q=i}{C}_{p,q}$ and differential the sum of the two differentials. For the tensor product of two chain complexes, a sign $(-1{)}^p$ on the second differential ensures anticommutation. If the double complex is in the first quadrant and one direction has homology only in degree zero, its total homology is computed by the surviving degree-zero complex. Applying this twice to two free resolutions proves the balanced calculation of the Tor functor.

Two degree-raising differentials square to zero and anticommute. If an initial convention uses commuting differentials, introduce a sign in one of them before forming the total cochain complex.

Filtering the total cochain complex by each index yields two spectral sequences. One first takes vertical cohomology and the other horizontal cohomology; boundedness ensures both converge to total cohomology.

The total differential is $d_h+d_v$ for anticommuting differentials. For a double cochain complex bounded in both indices, both index filtrations are finite and have the same total cohomology as abutment.

The Koszul complex on $f_1,\dots ,f_r$ has term $K_i={\bigwedge }^i{R}^r$ and differential $\partial (e_{j_1}\wedge \cdots \wedge e_{j_i})={\sum }_a(-1{)}^{a-1}{f}_{j_a}{e}_{j_1}\wedge \cdots \wedge \widehat{e_{j_a}}\wedge \cdots \wedge e_{j_i}$. Its degree-zero homology is $R/(f_1,\dots ,f_r)$. For a regular sequence it is exact in positive degrees, hence a Koszul resolution of that quotient. The variables of a polynomial ring form a regular sequence, so this resolves the coefficient field in length equal to the number of variables.

The perfect exterior pairing in complementary degrees identifies the dual of a Koszul complex with its degree reversal, with suitable differential signs. For a regular sequence, its Koszul resolution consequently gives ${\mathrm{Ext}}_R^i(R/(f),R)=0$ except in degree $r$, where it is $R/(f)$.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M$ be nonzero and finitely generated, and let every $f_i$ lie in $\mathfrak{m}$. Then positive Koszul homology vanishes exactly when $f$ is a regular sequence on a module $M$. The mapping cone exact sequence proves the forward implication by induction; surjectivity of the last generator on earlier homology and the Nakayama lemma prove the converse.

Tensor the finite free Koszul complex on $f_1,\dots ,f_r$ with the module $M$. For a regular sequence on a module, its positive Koszul homology vanishes. If the sequence is regular on $R$, this complex instead computes ${\mathrm{Tor}}^R(R/(f),M)$ for arbitrary $M$.

Koszul homology is the homology of the Koszul complex with module coefficients. Its degree-zero term is $M/(f)M$, and its positive terms record failures of regularity. Each generator acts trivially on homology by the Koszul homotopy for multiplication by a generator.

Exterior multiplication $h_j(u)=e_j\wedge u$ is a chain homotopy between multiplication by $f_j$ and zero on a Koszul complex. A linear combination of these homotopies is a contracting homotopy when the generators span the unit ideal.

For a possibly noncommutative ring and central $t_i$, the formal exterior basis gives free bimodules with differential $d(e_{i_1}\wedge \cdots \wedge e_{i_k})={\sum }_a(-1{)}^{a-1}{t}_{i_a}{e}_{i_1}\wedge \cdots \widehat{e_{i_a}}\cdots \wedge e_{i_k}$. Centrality makes this a bimodule differential. Pairwise cancellation proves $d^2=0$.

An augmented chain complex includes an augmentation $\epsilon :C_0\to \mathbb{Z}$ with $\epsilon d_1=0$. For a nonempty simplicial complex, the augmentation sends every vertex to $1$ and the augmented homology in degree zero is its reduced homology.

The chain group $C_i$ is the group in degree $i$ of a chain complex. Its elements are $i$-chains.

A chain subcomplex $D_{\bullet }\subseteq C_{\bullet }$ consists of subgroups $D_i\le C_i$ preserved by the boundary operator: $\partial (D_i)\subseteq D_{i-1}$.

The boundary operator of a chain complex is its degree-minus-one differential $\partial :C_i\to C_{i-1}$. The identity ${\partial }^2=0$ says that every chain boundary is a chain cycle.

For a CW complex, $C_n^{\mathrm{cell}}(X)=H_n(X^n,X^{n-1})$ is freely generated by the oriented $n$-cells. Connecting maps of skeleton pairs define its differential.

For a CW complex, the homology of its cellular chain complex naturally equals its singular homology. The relative groups of successive skeleton pairs, free on the cells in the corresponding dimension and zero in other dimensions, yield this identification through their connecting maps.

For a CW complex, applying $\mathrm{Hom}(-,A)$ to its integral cellular chain complex gives a cochain complex. The differential is precomposition with the cellular boundary. For finitely many cells it is represented by the transpose of the boundary matrix, reduced in the coefficient group when appropriate.

The cohomology of the cellular cochain complex naturally agrees with singular cohomology. A CW complex with no odd cells has zero cellular differentials and free integral cohomology on its even cells, although its cup product still requires a separate computation.

The coefficient of an $(n-1)$-cell in the cellular boundary of an $n$-cell is the degree of the attaching map after collapsing the complement of that $(n-1)$-cell to obtain a map $S^{n-1}\to S^{n-1}$.