List of Topological Homology and Cohomology

Introduction

This article lists the homology and cohomology theories of Parts I to III, each with the coefficient system it takes and the article that introduces it. The theories fall into four families: the singular, simplicial and cellular theories of a space, which take coefficients in a commutative ring or a module over it; the de Rham, Dolbeault and Hodge theories of a smooth or complex manifold, which take real or complex coefficients through differential forms; the sheaf and Grothendieck theories, which take coefficients in a sheaf and are defined for a space, a scheme or a site; and the algebraic theories of a group, a Galois extension, a Lie algebra or an algebra, which take coefficients in a module over the group ring, the Galois group, the enveloping algebra or the enveloping algebra of the algebra. The generalised cohomology theories and the duality theorems are listed with them.

Every entry points to the article that introduces the theory or the coefficient system. The article introduces nothing and proves nothing: it records the coefficients the introducing article allows and the degree in which the theory begins, and it neither restates a definition nor gives a proof.

The article records examples and non-examples side by side. Beside the theories that compute the same invariant it lists the pairs that do not agree — the de Rham cohomology, which takes real coefficients and is blind to torsion, so that it misses the $\mathbb{Z}/2$ in $H_1(\mathbb{RP}^2)$; the Čech cohomology, which agrees with sheaf cohomology over a good cover and disagrees with it in general; the Hochschild and cyclic theories, which are related by the Connes sequence and are not equal — each with the failure named and the article that records it.

The Singular, Simplicial and Cellular Theories

These theories attach a graded module to a space, a pair or a simplicial complex, and their coefficients are a commutative ring $R$ with identity $1 \neq 0$, or a module $G$ over it. The default is $R = \mathbb{Z}$ and the modules are abelian groups.

Theory	Its coefficients	Introduced in
simplicial homology $H_n^\Delta(K;R)$	an $R$-module, $R$ commutative with $1 \neq 0$; the chain groups are free of rank the number of $n$-simplices	Simplicial and Singular Homology
singular homology $H_n(X;R)$	an $R$-module; the $n$-chains are formal sums of continuous maps $\Delta_n \to X$	Simplicial and Singular Homology
reduced homology $\tilde H_n(X;R)$	an $R$-module; the augmentation removes the degree-zero class of a point	Simplicial and Singular Homology
relative homology $H_n(X,A;R)$	an $R$-module; the long exact sequence of a pair has connecting map $\partial$	Simplicial and Singular Homology
cellular homology $H_n^{\mathrm{CW}}(X;R)$	an $R$-module; agrees with the singular theory on a CW complex	CW Complexes and Cellular Approximation
singular cohomology $H^n(X;G)$	an $R$-module $G$; contravariant in $X$, with the universal coefficient sequence and the Bockstein	Cohomology and the Universal Coefficient Theorem
the cohomology ring $H^*(X;R)$	a commutative ring $R$; the cup product makes the direct sum a graded-commutative ring	Cup and Cap Products
homology and cohomology with local coefficients	a local system, that is a representation of $\pi_1(X)$	Poincaré Duality; The Leray–Serre Spectral Sequence
the Künneth and cross products	a commutative ring $R$, with a $\operatorname{Tor}$ correction over a general ring	Cup and Cap Products
the Bockstein homomorphism $\beta_m$	the coefficient sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/m \to 0$	Cohomology and the Universal Coefficient Theorem
the Euler characteristic $\chi(X)$	integral; the alternating sum of the ranks of $H_n(X;\mathbb{Z})$	Simplicial and Singular Homology
Alexander duality	integral coefficients; the duality of a compact subset of $S^n$ with its complement	Degree Theory and the Brouwer Fixed Point Theorem

The de Rham, Dolbeault and Hodge Theories

The theories of a manifold take their coefficients from the differential forms, so the coefficient field is $\mathbb{R}$ or $\mathbb{C}$ and the torsion of the integral theory is invisible.

Theory	Its coefficients	Introduced in
de Rham cohomology $H^\bullet_{dR}(M)$	real coefficients, through the differential forms; $H^\bullet_{dR}(M) \cong H^\bullet(M;\mathbb{R})$ for a smooth manifold	Sheaves and the de Rham Complex
the de Rham complex as a resolution	the constant sheaf $\underline{\mathbb{R}}$; the Poincaré lemma makes the complex a resolution, so sheaf cohomology computes de Rham cohomology	Sheaf Cohomology; Sheaves and the de Rham Complex
Stokes' theorem and the de Rham pairing	real coefficients; the integration of an $n$-form over an $n$-cycle	Differential Forms and Stokes' Theorem
Dolbeault cohomology $H^{p,q}_{\bar\partial}(M)$	complex coefficients; the $\bar\partial$-closed $(p,q)$-forms modulo the $\bar\partial$-exact ones	Sheaves and the de Rham Complex; Several Complex Variables
the Hodge decomposition	complex coefficients on a compact Kähler manifold; $H^k(M;\mathbb{C}) = \bigoplus_{p+q=k}H^{p,q}$	Kähler Geometry; Sheaves and the de Rham Complex
the Hodge theorem	real coefficients; each de Rham class has a unique harmonic representative	Differential Forms and Stokes' Theorem; Kähler Geometry
the de Rham theorem	the comparison of the smooth theory with the singular one, over $\mathbb{R}$	Sheaves and the de Rham Complex
the Dolbeault theorem	the identification of $H^{p,q}_{\bar\partial}$ with the sheaf cohomology $H^q(M,\Omega^p)$	Sheaves and the de Rham Complex; Coherent Sheaves

The Sheaf and Grothendieck Theories

A sheaf cohomology theory takes its coefficients in a sheaf of abelian groups on a space, a scheme or a site; the groups are the right derived functors of the global-section functor, and the coefficients range from the constant sheaf to the coherent, the étale and the constructible.

Theory	Its coefficients	Introduced in
sheaf cohomology $H^i(X,\mathcal{F})$	a sheaf of abelian groups $\mathcal{F}$; $H^0$ is the global sections, and flabby and soft resolutions compute	Sheaf Cohomology
the comparison with singular cohomology	the constant sheaf $\underline{A}$ on a locally contractible paracompact space; $H^i(X,\underline{A}) \cong H^i(X;A)$	Sheaf Cohomology
cohomology with support	a family $\Phi$ of supports, in particular the compact supports $H^i_c$	Sheaf Cohomology
higher direct images and the Leray spectral sequence	a sheaf $\mathcal{F}$ and a map $f$; $E_2^{p,q} = H^p(Y,R^qf_*\mathcal{F})$	Sheaf Cohomology; The Leray–Serre Spectral Sequence
Čech cohomology $\check H^p(X,\mathcal{F})$	a sheaf $\mathcal{F}$; agrees with sheaf cohomology over a good cover by the Leray theorem	Čech Cohomology
coherent sheaf cohomology $H^i(X,\mathcal{F})$	a coherent $\mathcal{O}_X$-module on a scheme or a complex manifold	Coherent Sheaves; Sheaves in Algebraic Geometry
Serre duality	a coherent sheaf on a smooth projective variety; $H^i(X,\mathcal{F}) \cong H^{n-i}(X,\mathcal{F}^\vee\otimes\omega_X)^\vee$	Coherent Sheaves; Sheaves in Algebraic Geometry
sheaf cohomology on a site, and étale cohomology	a sheaf on a Grothendieck site; the étale site of a scheme with coefficients in a sheaf of abelian groups	Sheaves on Sites; Schemes
the derived functors of a left exact functor	an abelian category with enough injectives; $R^iF$ with the long exact sequence	Derived Functors; Derived Categories
the Grothendieck spectral sequence	the composite of two derived functors; $E_2^{p,q} = R^pG(R^qF(A))$	Spectral Sequences
the Čech-to-derived spectral sequence	a sheaf and its presheaf of cohomology; converges to the sheaf cohomology	Čech Cohomology

The Algebraic Theories: Groups, Galois, Lie and Hochschild

These theories attach a graded module to an algebraic object, and the coefficients are modules over the group ring, the Galois group, the universal enveloping algebra, the algebra or the enveloping algebra of the algebra.

Theory	Its coefficients	Introduced in
group cohomology $H^n(G,M)$	a $G$-module $M$, equivalently a module over the group ring $\mathbb{Z}[G]$	Group Cohomology
group homology $H_n(G,M)$	a $G$-module; $H_1(G,\mathbb{Z}) = G^{\mathrm{ab}}$ and $M(G) = H_2(G,\mathbb{Z})$ is the Schur multiplier	Group Cohomology
Tate cohomology $\hat H^n(G,M)$	a $G$-module for a finite group $G$; defined in all degrees $n \in \mathbb{Z}$	Group Cohomology
the classification of group extensions	an abelian group $M$; $H^2(G,M)$ is the extension classes	Group Cohomology
group cohomology through a classifying space	a $G$-module, through $H^(G;M) \cong H^(BG;M)$	Classifying Spaces and Cohomology Operations
Galois cohomology $H^n(K,A)$	a Galois module $A$: $\mu_n$, $\overline K^\times$, $\mathbb{Z}/n$, with the action of $G_K$	Galois Cohomology
the Brauer group as a second cohomology	$H^2(K,\overline K^\times)$; the $n$-torsion ${}_n\operatorname{Br}(K)$	Galois Cohomology; Central Simple Algebras and the Brauer Group
Lie algebra cohomology $H^\bullet(\mathrm{G};M)$	a $\mathrm{G}$-module $M$, equivalently a module over $U(\mathrm{G})$; the Chevalley–Eilenberg complex	Lie Algebra Cohomology
Hochschild homology and cohomology $HH_\bullet(A,M)$	an $A$-bimodule $M$, equivalently a module over $A^{\mathrm{e}} = A\otimes_kA^{\mathrm{op}}$	Hochschild Homology
Hochschild–Kostant–Rosenberg	the identification of $HH_\bullet(A)$ with the Kähler differential forms for a smooth commutative algebra	Hochschild Homology
cyclic homology $HC_\bullet(A)$	an algebra $A$ over a commutative ring, with a cyclic structure on the Hochschild complex	Cyclic Homology
cyclic cohomology $HC^\bullet(A)$	an algebra over a field of characteristic zero; the dual complex, pairing with the $K$-theory	Cyclic Cohomology
the Dennis trace and the Chern character	the maps $K_n(R) \to HH_n(R)$ and to cyclic homology	Cyclic Homology
the $\operatorname{Ext}$ and $\operatorname{Tor}$ families	$R$-modules; the universal coefficient and Künneth corrections	Ext and Tor
algebraic $K$-theory	the $K$-theory of a ring; coefficients in the category of projective modules	K-Theory of Rings

The Generalised Cohomology Theories and the Dualities

A generalised cohomology theory is a graded functor on spaces that satisfies the Eilenberg–Steenrod axioms except the dimension axiom; the corpus's examples take their coefficients in vector bundles, in spectra or in analytic data, and the duality theorems are the pairings that organise the coefficients.

Theory	Its coefficients	Introduced in
topological $K$-theory $K^*(X)$	real or complex vector bundles, equivalently finitely generated projective modules over $C(X)$	Topological K-Theory
the Atiyah–Singer index theorem	the $K$-theory of the symbol and the index pairing; the topological index equals the analytic one	The Atiyah–Singer Index Theorem and K-Theory
$K$-theory of operator algebras	the projection classes of a $\mathrm{C}^*$-algebra in $K_0$ and the unitaries in $K_1$	K-Theory of Operator Algebras
$KK$-theory	the Kasparov bimodules; the bivariant theory whose $KK(A,B)$ classifies the extensions	KK-Theory
higher algebraic $K$-theory	the Quillen $K$-groups of an exact category; coefficients in the category of modules	Higher Algebraic K-Theory
cohomology operations and the Steenrod algebra	$\mathbb{Z}/2$ and $\mathbb{Z}/p$ coefficients; the algebra of stable operations	Classifying Spaces and Cohomology Operations
cobordism	the bordism classes of manifolds, with the Thom spectrum coefficients	Cobordism and Surgery Theory
Floer homology	analytic coefficients, through the moduli spaces of trajectories; the homology of an infinite-dimensional Morse theory	Floer Homology
Poincaré duality	a fundamental class over $\mathbb{Z}$ for a closed orientable manifold, and over $\mathbb{Z}/2$ in general	Poincaré Duality
Lefschetz and Poincaré–Lefschetz duality	a compact manifold with boundary and the cohomology with compact supports	Poincaré Duality
the homology of a Lie group and its homogeneous spaces	the classifying-space coefficients, through the Leray–Serre spectral sequence	Homology of Classical Groups and Homogeneous Spaces

The Coefficients

The coefficient system is the part of the theory that the article introducing the theory fixes, and the same letters appear in different theories with different meanings. The table collects the principal coefficients and the theory each serves.

Coefficient system	The theory it serves	Introduced in
a commutative ring $R$ with $1 \neq 0$	simplicial, singular, cellular homology and cohomology; the default is $\mathbb{Z}$	Simplicial and Singular Homology
an $R$-module $G$	singular cohomology, through $\operatorname{Hom}_R$; the universal coefficient theorem	Cohomology and the Universal Coefficient Theorem
$\mathbb{Z}$, $\mathbb{Z}/m$	the integral theories and the Bockstein; the torsion of the coefficient group	Cohomology and the Universal Coefficient Theorem
$\mathbb{Z}/2$	characteristic classes, Steenrod operations and the unoriented cobordism	Classifying Spaces and Cohomology Operations
$\mathbb{R}$	de Rham cohomology and the de Rham theorem; torsion is invisible	Sheaves and the de Rham Complex
$\mathbb{C}$	Dolbeault cohomology and the Hodge decomposition of a Kähler manifold	Several Complex Variables
a local system	homology and cohomology with local coefficients, through $\pi_1$	Poincaré Duality
a sheaf of abelian groups	sheaf cohomology, with the comma convention $H^i(X,\mathcal{F})$	Sheaf Cohomology
a coherent $\mathcal{O}_X$-module	coherent sheaf cohomology and Serre duality	Coherent Sheaves
a $G$-module, or a $\mathbb{Z}[G]$-module	group cohomology and homology	Group Cohomology
a Galois module, such as $\mu_n$ or $\overline K^\times$	Galois cohomology and the Brauer group	Galois Cohomology
a $\mathrm{G}$-module	Lie algebra cohomology, through the universal enveloping algebra	Lie Algebra Cohomology
an $A$-bimodule	Hochschild homology and cohomology	Hochschild Homology
a vector bundle	topological $K$-theory	Topological K-Theory

Non-examples and Warnings

The failures of agreement are recorded beside the theories, since the reader who expects one theorem of comparison must find where it fails.

Object	Why the expected statement fails	Introduced in
$H_1(\mathbb{RP}^2;\mathbb{Z}) = \mathbb{Z}/2$	the torsion is invisible to de Rham cohomology, which takes real coefficients; the de Rham theorem sees only $H^\bullet(M;\mathbb{R})$	Sheaves and the de Rham Complex
the non-abelian fundamental group $\pi_1(X,x)$	there is no homology or cohomology with coefficients in a non-abelian group; $H_1$ is the abelianisation, and the description of $\pi_1$ is of The Fundamental Group and Covering Spaces	Simplicial and Singular Homology
Čech cohomology in general	it agrees with sheaf cohomology over a good cover and over a paracompact space, and it is only an approximation in general; the Leray theorem names the hypothesis	Čech Cohomology
singular cohomology of the constant sheaf	the comparison $H^i(X,\underline{A}) \cong H^i(X;A)$ needs the space locally contractible and paracompact; the proof is by a flabby resolution	Sheaf Cohomology
group cohomology versus Lie algebra cohomology	the two are different theories; Van Est's comparison requires the exponential, a manifold and a limit, and belongs to Parts II and III	Lie Algebra Cohomology
Hochschild homology versus cyclic homology	the two are related by the Connes $SBI$ sequence and are not equal; the cyclic theory remembers the cyclic symmetry of the Hochschild complex	Cyclic Homology
Tate cohomology for an infinite group	it is defined only for a finite group; for an infinite group the norm map has no meaning	Group Cohomology
Floer homology	an infinite-dimensional homology whose definition uses the analytic moduli spaces of Part III; it is named here with its coefficients and not computed	Floer Homology

Objects that a reader may expect to find in a list of homology and cohomology theories, and does not.

Object	Why it is not listed	Introduced in
Borel–Moore homology	the corpus does not introduce it; the cohomology with compact supports $H^i_c$ is the form in which it meets the compactly supported theory	Sheaf Cohomology
Deligne cohomology	not introduced; the Hodge-theoretic combination of the de Rham and integral theories is not one of the corpus's theories	Sheaves and the de Rham Complex
intersection cohomology	not introduced as such; it appears only as a name in the Hecke-theoretic literature	Hecke Algebras
crystalline cohomology	recorded with the divided-power constructions of the algebraic geometry and not as a theory of this list	Divided Powers
étale cohomology as a topological theory	it is a sheaf cohomology on the étale site of a scheme and is listed with the sheaf theories	Schemes; Sheaves on Sites
Morse homology	the corpus reaches it through Floer homology and does not introduce the finite-dimensional Morse theory	Floer Homology
orbifold and equivariant cohomology	the corpus treats the equivariant theories through the classifying spaces and the group cohomology	Classifying Spaces and Cohomology Operations

Summary

This article has listed the homology and cohomology theories of the corpus with their coefficients: the simplicial, singular and cellular theories with coefficients in a ring or a module; the de Rham, Dolbeault and Hodge theories with real or complex coefficients; the sheaf, Čech, coherent, étale and derived-functor theories with coefficients in a sheaf on a space, a scheme or a site; the group, Galois, Lie algebra, Hochschild and cyclic theories with coefficients in a module over the appropriate algebra; and the generalised theories of $K$-theory, cobordism and Floer homology, together with the duality theorems of Poincaré, Lefschetz, Serre and Alexander. Beside the theories that agree stand the pairs that do not: de Rham cohomology blind to torsion, Čech cohomology agreeing with the derived theory only under the Leray hypothesis, and the Hochschild and cyclic theories separated by the Connes sequence. The list introduces and proves nothing; it is the index of the (co)homology theories of the corpus.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$H_n(X;R)$, $H^n(X;G)$	singular homology and cohomology; the semicolon separates the space from the coefficient
$H^i(X,\mathcal{F})$	sheaf cohomology; the comma is the convention of Sheaf Cohomology, deliberately distinct from the semicolon
$\check H^p$	Čech cohomology, with the check
$\tilde H_n$, $\tilde H^n$	reduced homology and cohomology
$\partial$, $\delta$	the boundary and the coboundary
$\smile$, $\frown$	the cup and cap products
$\chi(X)$	the Euler characteristic
$[M] \in H_n(M;\mathbb{Z})$	the fundamental class of a closed orientable manifold
$H^i_c$	cohomology with compact supports
$R$, $G$, $M$, $A$	a commutative coefficient ring with $1 \neq 0$; an $R$-module; a module of coefficients; a coefficient group or algebra
$\mathbb{Z}[G]$	the group ring of $G$
$A^{\mathrm{e}} = A\otimes_kA^{\mathrm{op}}$	the enveloping algebra of an algebra $A$
$U(\mathrm{G})$	the universal enveloping algebra of a Lie algebra
$H^n(G,M)$, $H^n(K,A)$, $H^\bullet(\mathrm{G};M)$, $HH_\bullet(A,M)$	group, Galois, Lie algebra and Hochschild theories
$HC_\bullet(A)$, $HC^\bullet(A)$	cyclic homology and cyclic cohomology
$K^(X)$, $K_(A)$	topological $K$-theory; the $K$-theory of an operator algebra or a ring
$\beta_m$	the Bockstein homomorphism of the coefficient sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/m \to 0$