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Steenrod Problem
In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds. Formulation Let M be a closed, oriented manifold of dimension n, and let \in H_n(M) be its orientation class. Here H_n(M) denotes the integral, n-dimensional homology group of M. Any continuous map f\colon M\to X defines an induced homomorphism f_*\colon H_n(M)\to H_n(X). A homology class of H_n(X) is called realisable if it is of the form f_* /math> where \in H_n(M). The Steenrod problem is concerned with describing the realisable homology classes of H_n(X). Results All elements of H_k(X) are realisable by smooth manifolds provided k\le 6. Moreover, any cycle can be realized by the mapping of a pseudo-manifold. The assumption that ''M'' be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of H_n(X,\Z_2), where \Z_2 denotes the integers modulo 2, ... [...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] |
Bordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network. It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 Septe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dimension are ''cobordant'' if their disjoint union is the ''boundary'' of a compact manifold one dimension higher. The boundary of an (''n'' + 1)-dimensional manifold ''W'' is an ''n''-dimensional manifold ∂''W'' that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A ''cobordism'' between manifolds ''M'' and ''N'' is a compact manifold ''W'' whose boundary is the disjoint union of ''M'' and ''N'', \partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin-Thom Construction
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the standard ''n''-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each ''n''-dimensional simplex to its (''n''−1)-dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. The resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. If n > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called K(G,n). The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Thom
René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as founder of catastrophe theory (later developed by Erik Christopher Zeeman). Life and career René Thom grow up in a modest family in Montbéliard, Doubs and obtained a Baccalauréat in 1940. After German invasion of France, his family took refuge in Switzerland and then in Lyon. In 1941 he moved to Paris to attend Lycée Saint-Louis and in 1943 he began studying mathematics at École Normale Supérieure, becoming agrégé in 1946. He received his PhD in 1951 from the University of Paris. His thesis, titled ''Espaces fibrés en sphères et carrés de Steenrod'' (''Sphere bundles and Steenrod squares''), was w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commentarii Mathematici Helvetici
The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society still owns and operates the journal; the publishing is currently handled on its behalf by the European Mathematical Society. The scope of the journal includes research articles in all aspects in mathematics. The editors-in-chief have been Rudolf Fueter (1929–1949), J.J. Burckhardt (1950–1981), P. Gabriel (1982–1989), H. Kraft (1990–2005), and Eva Bayer-Fluckiger (2006–present). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2019 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Space
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Notes
''Mathematical Notes'' is a peer-reviewed mathematical journal published by Springer Science+Business Media on behalf of the Russian Academy of Sciences that covers all aspects of mathematics. It is an English language translation of the Russian-language journal ''Matematicheskie Zametki'' (russian: Математические заметки) and is published simultaneously with the Russian version. The journal was established in 1967 as ''Mathematical Notes of the Academy of Sciences of the USSR'' and obtained its current title in 1991. The current editor-in-chief is Victor P. Maslov. According to the ''Journal Citation Reports'', the journal has a 2011 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 0.295. References External links * {{Official website, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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