This is a glossary of properties and concepts in algebraic topology in mathematics.
See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of manifolds.
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups. The fundamental group of a topological space is denoted by .
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with 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 topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber.
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
In mathematics, specifically in homotopy theory, a classifying spaceBG of a topological group G is the quotient of a weakly contractible space EG by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle . As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory
,
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group.
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling.
In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy.
In mathematics, the Serre spectral sequence is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor. Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan and Daniel Quillen. This simplification of homotopy theory makes certain calculations much easier.
In homotopy theory, a branch of algebraic topology, a Postnikov system is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.
In mathematics, especially in 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 topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'.
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site taking values in simplicial sets. Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).
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