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Assembly Map
In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data. Assembly maps for algebraic K-theory and L-theory play a central role in the topology of high-dimensional manifolds, since their homotopy fibers have a direct geometric Homotopy-theoretical viewpoint It is a classical result that for any generalized homology theory h_* on the category of topological spaces (assumed to be homotopy equivalent to CW-complexes), there is a spectrum E such that :h_*(X)\cong \pi_*(X_+\wedge E), where X_+:=X\sqcup \{*\}. The functor X\mapsto X_+ \wedge E from spaces to spectra has the following properties: * It is homotopy-invariant (preserves homotopy equivalences). This reflects the fact ... [...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] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M \hookrightarrow W \quad\mbox\quad N \hookrightarrow W are homotopy equivalences. The ''h''-cobordism theorem gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder ''M'' × , 1 Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Ranicki
Andrew Alexander Ranicki (born Andrzej Aleksander Ranicki; 30 December 1948 – 21 February 2018) was a British mathematician who worked on algebraic topology. He was a professor of mathematics at the University of Edinburgh. Life Ranicki was the only child of the well-known literary critic Marcel Reich-Ranicki and the artist Teofila Reich-Ranicki; he spoke Polish in his family. Born in London, he lived in Warsaw, in Frankfurt am Main and Hamburg, and attended school in England at the King's School, Canterbury from the age of sixteen.'Cambridge Tripos: English; Medical Sciences; Mathematics', ''Times'', 20 June 1969. Ranicki studied Mathematics at Trinity College, Cambridge, and graduated with a BA in 1969. At Cambridge he was a student of topologists Andrew Casson and John Frank Adams. He earned his doctoral degree in 1973 with a thesis on algebraic L-theory. Ranicki received numerous awards and honors for his scientific achievements during his studies. From 1972 to 1977 he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Quinn (mathematician)
Frank Stringfellow Quinn, III (born 1946) is an American mathematician and professor of mathematics at Virginia Polytechnic Institute and State University, specializing in geometric topology. Contributions He contributed to the mathematical field of 4-manifolds, including a proof of the 4-dimensional annulus theorem. In surgery theory, he made several important contributions: the invention of the assembly map, that enables a functorial description of surgery in the topological category, with his thesis advisor, William Browder, the development of an early surgery theory for stratified spaces, and perhaps most importantly, he pioneered the use of controlled methods in geometric topology and in algebra. Among his important applications of "control" are his aforementioned proof of the 4-dimensional annulus theorem, his development of a flexible category of stratified spaces, and, in combination with work of Robert D. Edwards, a useful characterization of high-dimensional manifold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dennis Sullivan
Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University. Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022. Early life and education Sullivan was born in Port Huron, Michigan, on February 12, 1941.. His family moved to Houston soon afterwards. He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words: He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, ''Triangulating h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Novikov (mathematician)
Sergei Petrovich Novikov (also Serguei) (Russian: Серге́й Петро́вич Но́виков) (born 20 March 1938) is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. In 1970, he won the Fields Medal. Early life Novikov was born on 20 March 1938 in Gorky, Soviet Union (now Nizhny Novgorod, Russia). He grew up in a family of talented mathematicians. His father was Pyotr Sergeyevich Novikov, who gave a negative solution to the word problem for groups. His mother, Lyudmila Vsevolodovna Keldysh, and maternal uncle, Mstislav Vsevolodovich Keldysh, were also important mathematicians. In 1955 Novikov entered Moscow State University, from which he graduated in 1960. Four years later he received the Moscow Mathematical Society Award for young mathematicians. In the same year he defended a dissertation for the ''Candidate of Science in Physics and Mathematics'' degree (equivalent to the PhD) at Moscow State University. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Browder (mathematician)
William Browder (born January 6, 1934)Curriculum vitae from Browder's web site, retrieved 2010-10-06. is an American , specializing in , and . Browder was one of the pioneers with [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surgery Exact Sequence
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension >4. The surgery structure set \mathcal (X) of a compact n-dimensional manifold X is a pointed set which classifies n-dimensional manifolds within the homotopy type of X. The basic idea is that in order to calculate \mathcal (X) it is enough to understand the other terms in the sequence, which are usually easier to determine. These are on one hand the normal invariants which form Cohomology#Cohomology theories, generalized cohomology groups, and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand, there are the L-theory, L-groups which are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the surgery obstruction maps from normal inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Controlled Topology
Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlling interest, a percentage of voting stock shares sufficient to prevent opposition * Foreign exchange controls, regulations on trade * Internal control, a process to help achieve specific goals typically related to managing risk Mathematics and science * Control (optimal control theory), a variable for steering a controllable system of state variables toward a desired goal * Controlling for a variable in statistics * Scientific control, an experiment in which "confounding variables" are minimised to reduce error * Control variables, variables which are kept constant during an experiment * Biological pest control, a natural method of controlling pests * Control network in geodesy and surveying, a set of reference points of known geospatial coo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
