Homotopy theoryIn 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 studied 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 (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
Homotopy groupIn 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. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
A¹ homotopy theoryIn algebraic geometry and algebraic topology, branches of mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A1, which is.
HomotopyIn topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.
Simplicial complexIn mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.
Homotopy groups of spheresIn 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.
Stable homotopy theoryIn mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example, In the two examples above all the maps between homotopy groups are applications of the suspension functor.
Homotopy categoryIn mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
Homotopy fiberIn mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups.
Cellular approximation theoremIn algebraic topology, the cellular approximation theorem states that a map between CW-complexes can always be taken to be of a specific type. Concretely, if X and Y are CW-complexes, and f : X → Y is a continuous map, then f is said to be cellular, if f takes the n-skeleton of X to the n-skeleton of Y for all n, i.e. if for all n. The content of the cellular approximation theorem is then that any continuous map f : X → Y between CW-complexes X and Y is homotopic to a cellular map, and if f is already cellular on a subcomplex A of X, then we can furthermore choose the homotopy to be stationary on A.
CW complexA CW complex (also called cellular complex or cell 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 properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.
Simplicial setIn mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
Abstract simplicial complexIn combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
Whitehead theoremIn homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.
Whitehead torsionIn 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. The Whitehead torsion is important in applying surgery theory to non-simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same.
Mathematical inductionMathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
Logical disjunctionIn logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction , is a logical connective typically notated as and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula , assuming that abbreviates "it is sunny" and abbreviates "it is warm". In classical logic, disjunction is given a truth functional semantics according to which a formula is true unless both and are false.
Hurewicz theoremIn 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. The Hurewicz theorems are a key link between homotopy groups and homology groups. For any path-connected space X and positive integer n there exists a group homomorphism called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients).
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Fundamental groupIn 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 (or the stronger case of homeomorphic) have isomorphic fundamental groups.
