The Whitehead product is a mathematical construction introduced in Whitehead (1941). It has been a useful tool in determining the properties of spaces. The mathematical notion of space includes every shape that exists in our 3-dimensional world such as curves, surfaces, and solid figures. Since spaces are often presented by formulas, it is usually not possible to visually determine their geometric properties. Some of these properties are connectedness (is the space in one or several pieces), the number of holes the space has, the knottedness of the space, and so on. Spaces are then studied by assigning algebraic constructions to them. This is similar to what is done in high school analytic geometry whereby to certain curves in the plane (geometric objects) are assigned equations (algebraic constructions). The most common algebraic constructions are groups. These are sets such that any two members of the set can be combined to yield a third member of the set (subject to certain restrictions). In homotopy theory, one assigns a group to each space X and positive integer p called the pth homotopy group of X. These groups have been studied extensively and give information about the properties of the space X. There are then operations among these groups (the Whitehead product) which provide additional information about the spaces. This has been very important in the study of homotopy groups.
Several generalisations of the Whitehead product appear in ( Blakers & Massey 1953 ) and elsewhere, but the most far-reaching one deals with homotopy sets, that is, homotopy classes of maps from one space to another. The generalised Whitehead product assigns to an element α in the homotopy set [ΣA, X] and an element β in the homotopy set [ΣB, X], an element [α, β] in the homotopy set [Σ(A ∧ B), X], where A, B, and X are spaces, Σ is the suspension (topology), and ∧ is the smash product. This was introduced by Cohen (1957) and Hilton (1965) and later studied in detail by Arkowitz (1962), (see also Baues (1989), p. 157). It is a generalization of the Whitehead product and provides a useful technique in the investigation of homotopy sets.
The relevant MSC code is: 55Q15, Whitehead products and generalizations.
Let and and consider elements and , where and are the homotopy classes of the projection maps. The commutator
in the group is trivial when restricted to , where denotes wedge sum. The generalised Whitehead product is then defined as the unique element
such that , where is the quotient map.
Naturality: f∗[α, β] = [f∗(α), f∗(β)], if is a map.
All [α, β] = 0, if X is an H-space.
E[α, β] = 0, where E : [Σ(A ∧ B), X] → [Σ2 (A ∧ B), ΣX] is the suspension homomorphism.
Bi-additivity, if A and B are suspensions.
A form of anti-commutativity.
An appropriate Jacobi identity for α and β as above and γ ∈ [ΣC, X], if A, B, and C are suspensions.
See Arkowitz (1962) for full statements of these results and proofs.
The product ΣA × ΣB has the homotopy type of the mapping cone of [ιΣA, ιΣB] ∈ [Σ(A ∧ B), ΣA ∨ ΣB] (Arkowitz (1962)).
Whitehead products for homotopy groups with coefficients are obtained by taking A and B to be Moore spaces (Hilton (1965), pp. 110–114)
There is a weak homotopy equivalence between a wedge of suspensions of finitely many spaces and an infinite product of suspensions of various smash products of the spaces according to the Hilton–Milnor theorem. The map is defined by generalised Whitehead products ( Baues & Quintero 2001 ).
If Y is a group-like H-space, then a product [A, Y] × [B, Y] → [A ∧ B, Y] is defined in analogy with the generalised Whitehead product. This is the generalised Samelson product denoted <σ, τ> for σ ∈ [A, Y] and τ ∈ [B, Y] ( Arkowitz 1963 ). If λU,V : [U, ΩV] → [ΣU, V] is the adjoint isomorphism, where Ω is the loop space functor, then λA∧B,X<σ, τ>= [λA,X (σ), λB,X (τ)] for Y = ΩX.
An Eckmann–Hilton dual of the generalised Whitehead product can be defined as follows. Let A♭B be the homotopy fiber of the inclusion j : A ∨ B → A × B, that is, the space of paths in A × B which begin in A ∨ B and end at the base point and let γ ∈ [X, ΩA] and δ ∈ [X, ΩB]. For (ΩιA)γ and (ΩιB)δ in [X, Ω(A ∨ B)], let d(γ, δ) ∈ [X, Ω(A ∨ B)] be their commutator. Since (Ωj) d(γ, δ) is trivial, there is a unique element {γ, δ} ∈ [X, Ω(A♭B)] such that (Ωp){γ, δ} = d(γ, δ), where p : A♭B → A ∨ B projects a path onto its initial point. For an application of this, let K(π, n) denote an Eilenberg–MacLane space and identify [X, K(π, n)] with the cohomology group Hn(X; π). If A = K(G, p) and B = K(G′, q), then there is a map θ : A♭B → K(G ⊗ G', p+q+1) such that (Ωθ){γ, δ} = γ ∪ δ, the cup product in Hp+q(X; G ⊗ G′). For details, see (( Arkowitz 1962 ), pp. 19–22) and ( Arkowitz 1964 ).
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold. It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n − p is called its codimension.
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.
In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.
The Duffing equation, named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.
The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.
In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.
A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space and those of the spaces and . The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.
In financial mathematics, tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.
In probability theory and statistics, the normal-inverse-gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.
The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function.
The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables.
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.
In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven Ricci scalars which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.
