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In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy types". [1]
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
John Willard Milnor is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the five mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize.
John Henry Constantine Whitehead FRS, known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai, in India, and died in Princeton, New Jersey, in 1960.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group. These concepts are named after the mathematician J. H. C. Whitehead.
In mathematics, and especially in homotopy theory, a crossed module consists of groups and , where acts on by automorphisms (which we will write on the left, , and a homomorphism of groups
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).
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.
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.
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety X embedded in projective space and a hyperplane section Y, the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if
In mathematics, Reidemeister torsion is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by Wolfgang Franz (1935) and Georges de Rham (1936). Analytic torsion is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer as an analytic analogue of Reidemeister torsion. Jeff Cheeger and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups,. They were introduced in to prove that every subgroup of a free group is free, but are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory. The textbook devotes all of chapter 3 to Nielsen transformations.
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions, and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map.
In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis,. Notably, the link group is not in general the fundamental group of the link complement.
In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum . More precisely, it states that for any ring spectrum , the kernel of the map consists of nilpotent elements. It was conjectured by Douglas Ravenel (1984) and proved by Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith (1988).
In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces.
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, 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 are assigned equations. 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. 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 which provide additional information about the spaces. This has been very important in the study of homotopy groups.
Dan Burghelea is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.
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