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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.
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, 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, specifically algebraic topology, a covering map is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood evenly covered by . In this case, is called a covering space and the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949, who discovered a solution to the equations of general relativity (GR) allowing CTCs known as the Gödel metric; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of quantum gravity which would replace GR, an idea which Stephen Hawking has labeled the chronology protection conjecture. Others note that if every closed timelike curve in a given space-time passes through an event horizon, a property which can be called chronological censorship, then that space-time with event horizons excised would still be causally well behaved and an observer might not be able to detect the causal violation.
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 topological soliton or "toron" occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise nor counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example.
In mathematics, an H-space, or a topological unital magma, is a topological space X together with a continuous map μ : X × X → X with an identity element e such that μ(e, x) = μ(x, e) = x for all x in X. Alternatively, the maps μ(e, x) and μ(x, e) are sometimes only required to be homotopic to the identity, sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes. Every topological group is an H-space; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.
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 mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to X
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to . J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead where he incorrectly claimed that no such manifold exists.
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is an immersion if
In mathematics, specifically geometric topology, the Borel conjecture asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, demanding that a weak, algebraic notion of equivalence imply a stronger, topological notion.
Suppose a Lorentzian manifold contains a closed timelike curve (CTC). No CTC can be continuously deformed as a CTC to a point, as that point would not be causally well behaved. Therefore, any Lorentzian manifold containing a CTC is said to be timelike multiply connected. A Lorentzian manifold that does not contain a CTC is said to be timelike simply connected.
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold. It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.
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 the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.
References
- J. Wolfgang Smith (1960). "Fundamental groups on a Lorentz manifold". Amer. J. Math. The Johns Hopkins University Press. 82 (4): 873–890. doi:10.2307/2372946. hdl: 2027/mdp.39015095257625 . JSTOR 2372946.
- André Avez (1963). "Essais de géométrie riemannienne hyperbolique globale. Applications à la relativité générale". Annales de l'Institut Fourier . 13 (2): 105–190. doi: 10.5802/aif.144 .
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