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.
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, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
A CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The C stands for "closure-finite", and the W for "weak" topology.
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 geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group. These concepts are named after the mathematician J. H. C. Whitehead.
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, 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 EG → BG. 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. This means given a cohomology theory
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, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
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). 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 (1977) and Daniel Quillen (1969). This simplification of homotopy theory makes 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 the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension . The surgery structure set of a compact -dimensional manifold is a pointed set which classifies -dimensional manifolds within the homotopy type of .
In mathematics, specifically in surgery theory, the surgery obstructions define a map from the normal invariants to the L-groups which is in the first instance a set-theoretic map with the following property when :
In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension, one can construct an aspherical CW complex X of dimension n whose fundamental group is G. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics.
In algebra, Quillen's Q-construction associates to an exact category an algebraic K-theory. More precisely, given an exact category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. One puts
This is a glossary of properties and concepts in algebraic topology in mathematics.
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is mostly of interest for the study of infinite groups.
