In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .
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.
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. This means that, given a cohomology theory
,
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.
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.
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a -bundle over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
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 (1961). 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 topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.
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 topology, a branch of mathematics, a string group is an infinite-dimensional group introduced by Stolz (1996) as a -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups
In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map
This is a glossary of properties and concepts in algebraic topology in mathematics.
