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In mathematics, specifically in algebraic topology, the **Eilenberg–Steenrod axioms** are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

In topology and related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.^{ [1] }

In mathematics, a **sequence** is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members. The number of elements is called the *length* of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers or the set of the first *n* natural numbers. The position of an element in a sequence is its *rank* or *index*; it is the natural number from which the element is the image. It depends on the context or a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the *n*th element of the sequence is denoted by this symbol with *n* as subscript; for example, the *n*th element of the Fibonacci sequence is generally denoted *F*_{n}.

In mathematics, a **functor** is a map between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism.

In mathematics, **K-theory** is, roughly speaking, the study of a ring generated by vector 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, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices.

The Eilenberg–Steenrod axioms apply to a sequence of functors from the category of pairs of topological spaces to the category of abelian groups, together with a natural transformation called the **boundary map** (here is a shorthand for . The axioms are:

In mathematics, a **category** is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces . Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and such that .

In mathematics, a **group** is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

**Homotopy**: Homotopic maps induce the same map in homology. That is, if is homotopic to , then their induced homomorphisms are the same.**Excision**: If is a pair and*U*is a subset of*X*such that the closure of*U*is contained in the interior of*A*, then the inclusion map induces an isomorphism in homology.**Dimension**: Let*P*be the one-point space; then for all .**Additivity**: If , the disjoint union of a family of topological spaces , then**Exactness**: Each pair*(X, A)*induces a long exact sequence in homology, via the inclusions and :

If *P* is the one point space, then is called the **coefficient group**. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.

Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.

The homology of some relatively simple spaces, such as *n*-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (*n* − 1)-sphere is not a retract of the *n*-disk. This is used in a proof of the Brouwer fixed point theorem.

A **sphere** is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace. A **deformation retraction** is a mapping which captures the idea of *continuously shrinking* a space into a subspace.

A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an ** extraordinary homology theory ** (dually, ** extraordinary cohomology theory **). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory, which are extraordinary *co*homology theories, and come with homology theories dual to them.

In mathematics, **topological K-theory** is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.

In mathematics, a **chain complex** is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, **cobordism** is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds of the same dimension are *cobordant* if their disjoint union is the *boundary* of a compact manifold one dimension higher.

In algebraic topology, a branch of mathematics, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In mathematics, specifically in algebraic topology, the **cup product** is a method of adjoining two cocycles of degree *p* and *q* to form a composite cocycle of degree *p* + *q*. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space *X* into a graded ring, *H*^{∗}(*X*), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944.

In mathematics, especially in homological algebra and algebraic topology, a **Künneth theorem**, also called a **Künneth formula**, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces *X* and *Y* and their product space . In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer.

In mathematics, and algebraic topology in particular, an **Eilenberg–MacLane space** is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

In mathematics, **Brown's representability theorem** in homotopy theory gives necessary and sufficient conditions for a contravariant functor *F* on the homotopy category *Hotc* of pointed connected CW complexes, to the category of sets **Set**, to be a representable functor.

In mathematics, the **Thom space,****Thom complex,** or **Pontryagin–Thom construction** of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.

In mathematics, in the field of algebraic topology, the **Eilenberg–Moore spectral sequence** addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper addresses this for singular homology.

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

In mathematics, a **normal map** is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex *X*, a normal map on *X* endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, *X* has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold *M* to *X* matching the fundamental classes and preserving normal bundle information. If the dimension of *X* is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to *X* actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.

In mathematics, **assembly maps** are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

In the mathematical disciplines of algebraic topology and homotopy theory, **Eckmann–Hilton duality** in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. A significantly deeper form argues that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. It is named after Beno Eckmann and Peter Hilton.

In mathematics, and particularly homology theory, **Steenrod's Problem** is a problem concerning the realisation of homology classes by singular manifolds.

This is a glossary of properties and concepts in algebraic topology in mathematics.

- Eilenberg, Samuel; Steenrod, Norman E. (1945). "Axiomatic approach to homology theory".
*Proceedings of the National Academy of Sciences of the United States of America*.**31**: 117–120. doi:10.1073/pnas.31.4.117. MR 0012228. PMC 1078770 . PMID 16578143. - Eilenberg, Samuel; Steenrod, Norman E. (1952).
*Foundations of algebraic topology*. Princeton, New Jersey: Princeton University Press. MR 0050886. - Bredon, Glen (1993).
*Topology and Geometry*. Graduate Texts in Mathematics.**139**. New York: Springer-Verlag. doi:10.1007/978-1-4757-6848-0. ISBN 0-387-97926-3. MR 1224675.

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