|
J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, New Jersey, in 1960. Life J. H. C. (Henry) Whitehead was the son of the Right Rev. Henry Whitehead, Bishop of Madras, who had studied mathematics at Oxford, and was the nephew of Alfred North Whitehead and Isobel Duncan. He was brought up in Oxford, went to Eton and read mathematics at Balliol College, Oxford. After a year working as a stockbroker, at Buckmaster & Moore, he started a PhD in 1929 at Princeton University. His thesis, titled ''The representation of projective spaces'', was written under the direction of Oswald Veblen in 1930. While in Princeton, he also worked with Solomon Lefschetz. He became a fellow of Balliol in 1933. In 1934 he married the concert pianist Barbara Smyth, great-great-granddaughter of Elizabeth Fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fellow Of The Royal Society
Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, including mathematics, engineering science, and medical science". Overview Fellowship of the Society, the oldest known scientific academy in continuous existence, is a significant honour. It has been awarded to :Fellows of the Royal Society, around 8,000 fellows, including eminent scientists Isaac Newton (1672), Benjamin Franklin (1756), Charles Babbage (1816), Michael Faraday (1824), Charles Darwin (1839), Ernest Rutherford (1903), Srinivasa Ramanujan (1918), Jagadish Chandra Bose (1920), Albert Einstein (1921), Paul Dirac (1930), Subrahmanyan Chandrasekhar (1944), Prasanta Chandra Mahalanobis (1945), Dorothy Hodgkin (1947), Alan Turing (1951), Lise Meitner (1955), Satyendra Nath Bose (1958), and Francis Crick (1959). More recently, fellow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Conjecture
The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical. A group presentation G=(S\mid R) is called ''aspherical'' if the two-dimensional CW complex K(S\mid R) associated with this presentation is aspherical or, equivalently, if \pi_2(K(S\mid R))=0. The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group ''G'' so that either ''G'' is a counterexample to the Eilenberg–Ganea conjecture The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group ''G'' has cohomological dimension 2, then it h ..., or there mus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biographical Memoirs Of Fellows Of The Royal Society
The ''Biographical Memoirs of Fellows of the Royal Society'' is an academic journal on the history of science published annually by the Royal Society. It publishes obituaries of Fellows of the Royal Society. It was established in 1932 as ''Obituary Notices of Fellows of the Royal Society'' and obtained its current title in 1955, with volume numbering restarting at 1. Prior to 1932, obituaries were published in the '' Proceedings of the Royal Society''. The memoirs are a significant historical record and most include a full bibliography of works by the subjects. The memoirs are often written by a scientist of the next generation, often one of the subject's own former students, or a close colleague. In many cases the author is also a Fellow. Notable biographies published in this journal include Albert Einstein, Alan Turing, Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Senior Berwick Prize
The Berwick Prize and Senior Berwick Prize are two prizes of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ... awarded in alternating years in memory of William Edward Hodgson Berwick, a previous Vice-President of the LMS. Berwick left some money to be given to the society to establish two prizes. His widow Daisy May Berwick gave the society the money and the society established the prizes, with the first Senior Berwick Prize being presented in 1946 and the first Junior Berwick Prize the following year. The prizes are awarded "in recognition of an outstanding piece of mathematical research ... published by the Society" in the eight years before the year of the award. The Berwick Prize was known as the Junior Berwick Prize up to 1999, and was given its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanier–Whitehead Duality
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space ''X'' may be considered as dual to its complement in the ''n''-sphere, where ''n'' is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds. The theory is also referred to as ''S-duality'', but this can now cause possible confusion with the S-duality of string theory. It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory. Statement Let ''X'' be a compact neighborhood retract in \R^n. Then X^+ and \Sigma^\Sigma'(\R^n \setminus X) are dual object In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Lemma (Lie Algebras)
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. One usually makes the distinction between Whitehead's first and second lemma for the corresponding statements about first and second order cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead. The first Whitehead lemma is an important step toward the proof of Weyl's theorem on complete reducibility. Statements Without mentioning cohomology groups, one can state Whitehead's first lemma as follows: Let \mathfrak be a finite-dimensional, semisimple Lie algebra over a field of characteristic zero, ''V'' a finite-dimensional module over it, and f\colon \mathfrak \to V a linear map such tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead's Algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group ''Fn''. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead.J. H. C. Whitehead, ''On equivalent sets of elements in a free group'', Ann. of Math. (2) 37:4 (1936), 782–800. It is still unknown (except for the case ''n'' = 2) if Whitehead's algorithm has polynomial time complexity. Statement of the problem Let F_n=F(x_1,\dots, x_n) be a free group of rank n\ge 2 with a free basis X=\. The automorphism problem, or the automorphic equivalence problem for F_n asks, given two freely reduced words w, w'\in F_n whether there exists an automorphism \varphi\in \operatorname(F_n) such that \varphi(w)=w'. Thus the automorphism problem asks, for w, w'\in F_n whether \operatorname(F_n)w=\operatorname(F_n)w'. For w, w'\in F_n one has \operatorname(F_n)w=\operatorname(F_n)w' if and only if \operatorname(F_n) \operatorn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Postnikov System
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space X there is a list of spaces \_ where\pi_k(X_n) = \begin \pi_k(X) & \text k \leq n \\ 0 & \text k > n \endand there is a series of maps \phi_n: X_n \to X_ that are fibrations with fibers Eilenberg-MacLane spaces K(\pi_n(X),n). In short, we are decomposing the homotopy type of X using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X_n with the homotopy type of X for degrees \leq n, these spaces have null homotopy groups \pi_(X_n) =0 for 1 is classified by a homotopy class : _n\in _,K(\pi_n(X), n+1)\cong H^(X_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Torsion
In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \operatorname(\pi_1(Y)). 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. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the early 1960s by Stephen Smale, for differentiable manifolds. The development of handlebody theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Theorem
In 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. Statement In more detail, let ''X'' and ''Y'' be topological spaces. Given a continuous mapping :f\colon X \to Y and a point ''x'' in ''X'', consider for any ''n'' ≥ 0 the induced homomorphism :f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), where π''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π0(''X'') just means the set of path components of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Product
In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in . The relevant MSC code is: 55Q15, Whitehead products and generalizations. Definition Given elements f \in \pi_k(X), g \in \pi_l(X), the Whitehead bracket : ,g\in \pi_(X) is defined as follows: The product S^k \times S^l can be obtained by attaching a (k+l)-cell to the wedge sum :S^k \vee S^l; the attaching map is a map :S^ \stackrel S^k \vee S^l. Represent f and g by maps :f\colon S^k \to X and :g\colon S^l \to X, then compose their wedge with the attaching map, as :S^ \stackrel S^k \vee S^l \stackrel X . The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of :\pi_(X). Grading Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so \pi_k(X) has degree (k-1); equivalently, L_k = \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
