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Hopf Invariant One Problem
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. __TOC__ Motivation In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map'' :\eta\colon S^3 \to S^2, and proved that \eta is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles :\eta^(x),\eta^(y) \subset S^3 is equal to 1, for any x \neq y \in S^2. It was later shown that the homotopy group \pi_3(S^2) is the infinite cyclic group generated by \eta. In 1951, Jean-Pierre Serre proved that the rational homotopy groups :\pi_i(S^n) \otimes \mathbb for an odd-dimensional sphere (n odd) are zero unless i is equal to 0 or ''n''. However, for an even-dimensional sphere (''n'' even), there is one more bit of infinite cyclic homotopy in degree 2n-1. Definition Let \phi \colon S^ \to S^n be a continuous map (assume n>1). Then we can form the cell complex : C_\phi = S^n \cup_\phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Adams
John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his PhD from the University of Cambridge in 1956. His thesis, written under the direction of Shaun Wylie, was titled ''On spectral sequences and self-obstruction invariants''. He held the Fielden Chair of Pure Mathematics, Fielden Chair at the University of Manchester (1964–1970), and became Lowndean Professor of Astronomy and Geometry at the University of Cambridge (1970–1989). He was elected a Fellow of the Royal Society in 1964. His interests included mountaineering—he would demonstrate how to climb right round a table at parties (a Hassler Whitney, Whitney traverse)—and the game of go (game), Go. He died in a car crash in Brampton, Cambr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quarterly Journal Of Mathematics
The ''Quarterly Journal of Mathematics'' is a quarterly peer-reviewed mathematics journal established in 1930 from the merger of ''The Quarterly Journal of Pure and Applied Mathematics'' and the ''Messenger of Mathematics''. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.681. References External links * {{Official website, http://qjmath.oxfordjournals.org/ Mathematics journals Publications established in 1930 English-language journals Oxford University Press academic journals Quarterly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Suspension
In topology, a branch of mathematics, the suspension of a topological space ''X'' is intuitively obtained by stretching ''X'' into a cylinder and then collapsing both end faces to points. One views ''X'' as "suspended" between these end points. The suspension of ''X'' is denoted by ''SX'' or susp(''X''). There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by Σ''X''. The "usual" suspension ''SX'' is sometimes called the unreduced suspension, unbased suspension, or free suspension of ''X'', to distinguish it from Σ''X.'' Free suspension The (free) suspension SX of a topological space X can be defined in several ways. 1. SX is the quotient space (X \times ,1/(X\times \, X\times \). In other words, it can be constructed as follows: * Construct the cylinder X \times ,1/math>. * Consider the entire set X\times \ as a single point ("glue" all its points together). * Consider the entire set X\times \ as a single point (" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point At Infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point). In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-point Compactification
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let ''X'' be a topological space. Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an open embedding ''c'' : ''X'' → ''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞. The map ''c'' is a Hausdorff compactification if and only if ''X'' is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Differential Form
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another differential form ''β''. Thus, an ''exact'' form is in the '' image'' of ''d'', and a ''closed'' form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''. Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré's Lemma
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another differential form ''β''. Thus, an ''exact'' form is in the ''image'' of ''d'', and a ''closed'' form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''. Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Differential Form
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another differential form ''β''. Thus, an ''exact'' form is in the ''image'' of ''d'', and a ''closed'' form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''. Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback (differential Geometry)
Suppose that is a smooth map between smooth manifolds ''M'' and ''N''. Then there is an associated linear map from the space of 1-forms on ''N'' (the linear space of sections of the cotangent bundle) to the space of 1-forms on ''M''. This linear map is known as the pullback (by ''φ''), and is frequently denoted by ''φ''∗. More generally, any covariant tensor field – in particular any differential form – on ''N'' may be pulled back to ''M'' using ''φ''. When the map ''φ'' is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from ''N'' to ''M'' or vice versa. In particular, if ''φ'' is a diffeomorphism between open subsets of R''n'' and R''n'', viewed as a change of coordinates (perhaps between different charts on a manifold ''M''), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Form
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form'' or ''pseudo-volume form''. It als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
