Simple Homotopy

	Simple Homotopy In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...es are simple-homotopy equivalent if they are related by a sequence of collapses and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map. The obstruction to a homotopy equivalence being a simple homotopy equivalence is the Whitehead torsion, \tau(f). A homotopy theory that studies simple-homotopy types is called simple homotopy theory. See also * Discrete Morse theory References * Homotopy theory Equivalence (mathematics) {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Homotopy Equivalence In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continuous de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k-ball B^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Collapse (topology) In topology, a branch of mathematics, a collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented by J. H. C. Whitehead. Collapses find applications in computational homology. Definition Let K be an abstract simplicial complex. Suppose that \tau, \sigma are two simplices of K such that the following two conditions are satisfied: # \tau \subsetneq \sigma, in particular \dim \tau < \dim \sigma; # $\sigma$ is a maximal face of $K$ and no other maximal face of $K$ contains $\tau,$ then $\tau$ is called a free face. A simplicial collapse of $K$ is the removal of all simplices $\gamma$ such that $\tau \subseteq \gamma \subseteq \sigma,$ where $\tau$ is a free face. If additionally we have $\dim \tau = \dim \sigma - 1,$ then this is called an elementary collap ... [...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]
	Simple Homotopy Theory 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". See also 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 \ope ... References * * Further reading A lecture by J. Lurie Homotopy theory Equivalence (mathematics) {{topology-stub ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Discrete Morse Theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Notation regarding CW complexes Let X be a CW complex and denote by \mathcal its set of cells. Define the ''incidence function'' \kappa\colon\mathcal \times \mathcal \to \mathbb in the following way: given two cells \sigma and \tau in \mathcal, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator is the endomorphism \partial of the free abelian group generated by \mathcal defined by :\partial(\sigma) = \sum_\kappa(\sigma,\tau)\tau. It is a defining property of boundary operators that \partial\circ\partial \equiv 0. In more axiomatic definitions one can find the requirement that \forall \sigma,\tau^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Homotopy Theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]