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In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape
Cohomology_operation
In mathematics, a secondary cohomology operation is a functorial correspondence between cohomology groups. More precisely, it is a natural transformation
Secondary cohomology operation
Secondary_cohomology_operation
In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime
Pontryagin cohomology operation
Pontryagin_cohomology_operation
Algebra in algebraic topology
Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {\displaystyle p} cohomology. For a given prime number p {\displaystyle p}
Steenrod_algebra
Topological space with only one nontrivial homotopy group
homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology. A generalised Eilenberg–MacLane space
Eilenberg–MacLane_space
mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory
Adams_operation
Operation in cohomology theory
associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X {\displaystyle X} into a graded ring,
Cup_product
Operation in algebraic topology
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in (Massey 1958), which generalizes the cup product. The
Massey_product
Algebraic structure used in topology
mathematics, specifically in homology theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other
Cohomology
generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but
Complex_cobordism
Soviet mathematician (1908–1988)
vanish on a manifold that is a boundary. In 1942 he introduced the cohomology operations now called Pontryagin squares. Moreover, in operator theory there
Lev_Pontryagin
topology, a Postnikov square is a certain cohomology operation from a first cohomology group H1 to a third cohomology group H3, introduced by Postnikov (1949)
Postnikov_square
Tools for studying groups based on techniques from algebraic topology
specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic
Group_cohomology
Describes the Spanier–Whitehead dual of a secondary cohomology operation
Spanier–Whitehead dual of a secondary cohomology operation. Peterson, F. P.; Stein, N. (1960), "The dual of a secondary cohomology operation", Illinois Journal of Mathematics
Peterson–Stein_formula
American mathematician (1910–1971)
cup product structure of cohomology was understood by the early 1940s. Steenrod was able to define operations from one cohomology group to another (the so-called
Norman_Steenrod
Mathematical formulae
precisely, these five axioms define cohomology operations, which are natural transformations between cohomology functors, which in turn define Steenrod
Cartan_formula
Theorem relating Milnor K-theory and Galois cohomology
isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time
Norm residue isomorphism theorem
Norm_residue_isomorphism_theorem
Topics referred to by the same term
Wigan, purchased by Northern Counties in 1967 Massey product, a cohomology operation of higher order generalizing the cup product Massey Ferguson, an
Massey
British mathematician (1930–1989)
secondary cohomology operations. The Adams–Novikov spectral sequence is an analogue of the Adams spectral sequence using an extraordinary cohomology theory
Frank_Adams
Topological space associated to a vector bundle
rings of, say, topological manifolds from Thom spectra. Cobordism Cohomology operation Steenrod problem Hattori–Stong theorem Proof of the isomorphism.
Thom_space
Invariant of algebraic varieties and of more general schemes
algebraic geometry, motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and
Motivic_cohomology
Spectral sequence
The representability of the cohomology functor makes H*(X) a module over the algebra of its stable cohomology operations, the Steenrod algebra A. Thinking
Adams_spectral_sequence
Concept in algebraic topology
additional cohomology operations, and the cohomology algebra has addition structure mod p (as before, the mod p cohomology is the cohomology of the mod
Singular_homology
British mathematician and musicologist (1937–2018)
differentials can be better described. The family of higher cohomology operations on mod-2 cohomology that he constructed has been discussed by several authors
Richard_Maunder
Concept in algebraic geometry
algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many
Coherent_sheaf_cohomology
British-Lebanese mathematician (1929–2019)
was very long and complicated, using secondary cohomology operations. Atiyah showed how primary operations in K-theory could be used to give a short solution
Michael_Atiyah
Tool to track locally defined data attached to the open sets of a topological space
for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic
Sheaf_(mathematics)
Algebraic topology uses abstract algebra to study topological spaces
theorem Cohomology List of cohomology theories Cocycle class Cup product Cohomology ring De Rham cohomology Čech cohomology Alexander–Spanier cohomology Intersection
List of algebraic topology topics
List_of_algebraic_topology_topics
American mathematician
Charles (2006). "The units of a ring spectrum and a logarithmic cohomology operation". Journal of the American Mathematical Society. 19 (4): 969–1015
Charles_Rezk
Characteristic class of oriented, real vector bundles
Euler class e ( E ) {\displaystyle e(E)} is an element of the integral cohomology group H r ( X ; Z ) , {\displaystyle H^{r}(X;\mathbf {Z} ),} constructed
Euler_class
Mathematician (born 1937)
he spent one year attending the lectures of Norman Steenrod on cohomology operations, making notes and revisions to them, later published as a book by
David_B._A._Epstein
Soviet and Russian mathematician (1938–2024)
(at that time) cohomology theory typified by cobordism and K-theory. This required the development of the idea of cohomology operations in the general
Sergei Novikov (mathematician)
Sergei_Novikov_(mathematician)
French mathematician (1928–2014)
categories, "six operations") Yoga of the Grothendieck–Riemann–Roch theorem K-theory relation with intersection theory Schemes Topoi Étale cohomology and l-adic
Alexander_Grothendieck
American mathematician
supervision of Norman Steenrod. His dissertation was titled Secondary cohomology operations induced by the diagonal mapping. In 1963 he became a member of the
Paul_A._Schweitzer
Mathematics glossary
{\displaystyle [X,S^{n}]} is called the n-th cohomotopy group of X. cohomology operation collapse An informal phrase but usually means taking a quotient;
Glossary of algebraic topology
Glossary_of_algebraic_topology
Tool in homological algebra
found that the cohomology groups of the pushforward formed a natural chain complex, so that he could take the cohomology of the cohomology. This was still
Spectral_sequence
proof. Piecewise algebraic space Hess, Kathryn. Deligne’s Hochschild Cohomology Conjecture (PDF). Tamarkin, Dmitry E. (1998). "Another proof of M. Kontsevich
Deligne's conjecture on Hochschild cohomology
Deligne's_conjecture_on_Hochschild_cohomology
Pontryagin, 79, Soviet mathematician (Pontryagin duality, Pontryagin cohomology operation). Abraham Seidenberg, 71, American mathematician (Tarski–Seidenberg
Deaths_in_May_1988
Algebraic structure associated with a topological space
notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological
Homology_(mathematics)
French mathematician (1904–2008)
algebraic topology. Among his major contributions, he worked on cohomology operations and homology of the Eilenberg–MacLane spaces, he introduced the
Henri_Cartan
American mathematician (1930–2000)
Peterson, Franklin P.; Stein, Norman (1960), "The dual of a secondary cohomology operation", Illinois Journal of Mathematics, 4 (3): 397–404, doi:10.1215/ijm/1255456056
Franklin_P._Peterson
Formalism in homological algebra
étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight was that many of the elementary facts relating cohomology on X and
Six_operations
Association of cohomology classes to principal bundles
associating to each principal bundle of a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted"
Characteristic_class
Homological construction in category theory
this is in general not an exact sequence anymore. But we can compute its cohomology at the i-th spot (the kernel of the map from F(Ii) modulo the image of
Derived_functor
Since the cohomology operation kills the homotopy information, and not every differential graded algebra is quasi-isomorphic to its cohomology algebra,
Homotopy_associative_algebra
Theorem describing the Milnor K-theory (mod 2) by means of the Galois cohomology
characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996
Milnor_conjecture_(K-theory)
Technique in mathematical group theory
linear representations of finite groups of Lie type using ℓ-adic cohomology (Étale cohomology) with compact support, introduced by Pierre Deligne and George
Deligne–Lusztig_theory
Algebraic structure in homological algebra
Rham cohomology of the manifold. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover
Differential_graded_algebra
Cohomology theory
theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished
Morava_K-theory
Construction in homological algebra
algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in
Ext_functor
Graduate-level textbooks in mathematics
Homotopy Groups of Spheres. Hirosi Toda 1963 193 9780691095868 50 Cohomology Operations: Lectures by N. E. Steenrod David B. A. Epstein 1962-10-21 138 9780691079240
Annals_of_Mathematics_Studies
Relates the homology of two objects to the homology of their product
Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic. These many results are named
Künneth_theorem
Formulation to quantize gauge field theories in physics
pure operators are graded by integral ghost numbers and we have a BRST cohomology. From a practical perspective, a quantum field theory consists of an action
BRST_quantization
Cohomology class
allow for operations in the underlying cohomology theory, analogous to (and generalizing) the well-known Steenrod operations in ordinary cohomology. As not
Highly structured ring spectrum
Highly_structured_ring_spectrum
Topological spaces whose union is a boundary
algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological
Cobordism
Annual session of lectures
analysis and probability. 1957 Norman Steenrod (Princeton University): Cohomology operations. 1959 Joseph L. Doob (University of Illinois, Urbana-Champaign):
Colloquium_Lectures_(AMS)
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group H 1 ( X , O X ∗ ) . {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})
Picard_group
Group whose operation is a composition of braids
x_{i}=x_{j}{\text{ for some }}i\neq j\}.} The cohomology of a group G {\displaystyle G} is defined as the cohomology of the corresponding Eilenberg–MacLane classifying
Braid_group
Algebraic geometry analog of a principal bundle in algebraic topology
1-coboundaries in cohomology; that is, the g i j {\displaystyle g_{ij}} define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf
Torsor_(algebraic_geometry)
Branch of mathematics
words, to give a generalized cohomology theory is to give a spectrum. A K-theory is an example of a generalized cohomology theory. A basic example of a
Homotopy_theory
Set with associative invertible operation
Kampen theorem for an example. An example is group cohomology of a group which equals the singular cohomology of its classifying space, see Weibel 1994, §8
Group_(mathematics)
General concept and operation in mathematics
cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent to group cohomology
Duality_(mathematics)
Generalized cohomology theory
In algebraic topology, Johnson–Wilson theory E(n) is a generalized cohomology theory introduced by David Copeland Johnson and W. Stephen Wilson (1975)
Johnson–Wilson_theory
History of maths
iteratively approximating cohomology groups by previous approximate cohomology groups. In the limiting case it gives the sought cohomology groups. 1948 Cartan
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Duality for sheaves of k-modules over a locally compact space
spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said
Verdier_duality
Algebraic structure with addition and multiplication
algebras in functional analysis, rings of differential operators, and cohomology rings in topology. The conceptualization of rings spanned the 1870s to
Ring_(mathematics)
Relate the direct image and the pull-back of sheaves
closely related to the cohomology of the fiber of the point under f, this statement is paraphrased by saying that "cohomology commutes with base extension"
Base_change_theorems
Mathematical theory
differentiate between different links using multiplicative operations on these cohomology spaces, like cup-products and the Massey products. For example
Alexander_duality
Invariant of mathematical knots
mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the
Khovanov_homology
Graduate-level textbooks in mathematics
Y. Eliashberg, N. Mishachev 2002 978-0-8218-3227-1 49 Secondary Cohomology Operations John R. Harper 2002 978-0-8218-3270-7 50 An Invitation to Operator
Graduate Studies in Mathematics
Graduate_Studies_in_Mathematics
Type of topological space
statements remain true. Cellular approximation theorem Singular homology and cohomology of CW complexes is readily computable via cellular homology. Moreover
CW_complex
Concept in ring theory
there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined as
Azumaya_algebra
Generalization of vector bundles
and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a
Coherent_sheaf
Sheaf consisting of modules on a ringed space; generalizing vector bundles
first cohomology group H 1 ( X , O ∗ ) {\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})} (by the standard argument with Čech cohomology). If
Sheaf_of_modules
Lλ(Lμ(V)). Littlewood (1936, p. 52, 1944, p. 329) Weyman, Jerzy (2003). Cohomology of Vector Bundles and Syzygies. Cambridge University Press. doi:10.1017/CBO9780511546556
Plethysm
Branch of mathematics
Retrieved 2024-01-22. Carlson, Stephan C. (2024). "Topology – Homology, Cohomology, Manifolds". Encyclopædia Britannica. Retrieved 2024-10-02. Carstensen
Algebra
Mathematical structure with multiplication as its operation
varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory. Multiplicative
Multiplicative_group
Way to join two given mathematical manifolds together
{\displaystyle V} to the circle, which in turn equals the first integral cohomology group H 1 ( V ) {\displaystyle H^{1}(V)} . So the diffeomorphism type
Connected_sum
Algebraic structure with addition, multiplication, and division
The cohomological study of such representations is done using Galois cohomology. For example, the Brauer group, which is classically defined as the group
Field_(mathematics)
All points in the topological closure not belonging to the interior
the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for
Boundary_(topology)
crucial that the cohomology of a stack is with respect to the smooth topology (not etale). When X is a variety, the smooth cohomology is the same as etale
Behrend's_trace_formula
Mathematical theory
terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic
Chern–Weil_homomorphism
algebro-geometric methods. Voevodsky, Vladimir (2003). "Reduced power operations in motivic cohomology". Publications Mathématiques de l'IHÉS. 98: 1–57. arXiv:math/0107109
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
German mathematician (born 1958)
theory, is an arithmetic analogue of Poincaré duality, a duality for sheaf cohomology on a compact manifold. In this parallel, the (spectrum of the) ring of
Christopher_Deninger
Subject area in mathematics
development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic
Algebraic_K-theory
Result in algebraic geometry
Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
equivalence, homological equivalence for a fixed cohomology theory (such as singular cohomology or étale cohomology), numerical equivalence, as well as all of
Algebraic_cycle
Construction in algebra
functions on the group by evaluating the function on the summed elements. The cohomology algebra (over a field K {\displaystyle K} ) of a Lie group G {\displaystyle
Hopf_algebra
Branch of algebraic geometry
abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties. p-adic
Arithmetic_geometry
Set on which a group acts freely and transitively
It leads at once to questions of Galois cohomology, since the torsors represent classes in group cohomology H1. The concept of a principal homogeneous
Principal_homogeneous_space
Homological map
by the usual argument (snake lemma). A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have β : H i ( C
Bockstein_homomorphism
for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G). The Kantor–Koecher–Tits construction applied to an Albert
Albert_algebra
constitutes a realisation of the cocycles in the second cohomology group in group cohomology. Suppose G is a group and A is an abelian group. For a group
Factor_system
Mathematical space
cohomology of the Grassmannians is generated, as a ring, by the Chern classes of E {\displaystyle E} . In particular, all of the integral cohomology is
Grassmannian
Branch of mathematics
bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry
K-theory
Theory of ideals in commutative rings in mathematics
and is a closure operation (this notion is closely related to the study of local cohomology). See also tight closure. Local cohomology can sometimes be
Ideal_theory
Concept in topology
homology and cohomology groups. For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf
H-space
geometry, the Cartier isomorphism is a certain isomorphism between the cohomology sheaves of the de Rham complex of a smooth algebraic variety over a field
Cartier_isomorphism
Type of generalized function
The motivation can be concretely implemented using ideas from sheaf cohomology. Let O {\displaystyle {\mathcal {O}}} be the sheaf of holomorphic functions
Hyperfunction
Concept in mathematics
Toda brackets vanish. This parallels the theory of Massey products in cohomology. The direct sum π ∗ S = ⨁ k ≥ 0 π k S {\displaystyle \pi _{\ast }^{S}=\bigoplus
Toda_bracket
COHOMOLOGY OPERATION
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