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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. Although algebraic topology primarily uses algebra to study topological ...
He was known for his book on non-Euclidean geometry (1st edition, 1974; 4th edition, 2008) [3] [4] and his book on algebraic topology (1st edition, 1967, published with the title Lectures on Algebraic Topology; revised edition published, with John R. Harper as co-author, in 1981 with the title Algebraic Topology: A First Course). [5] [6] [7]
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
Chain (algebraic topology) Betti number; Euler characteristic. Genus; Riemann–Hurwitz formula; Singular homology; Cellular homology; Relative homology; Mayer–Vietoris sequence; Excision theorem; Universal coefficient theorem; Cohomology. List of cohomology theories; Cocycle class; Cup product; Cohomology ring; De Rham cohomology; Čech ...
May, J. Peter (1999), A Concise Course in Algebraic Topology (PDF), University of Chicago Press, ISBN 0-226-51182-0, MR 1702278 Milnor, John (1959), "On spaces having the homotopy type of a CW-complex", Transactions of the American Mathematical Society , 90 (2): 272– 280, doi : 10.2307/1993204 , JSTOR 1993204 , MR 0100267
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
Category theory is the language of modern algebra, and has been widely used in the study of algebraic geometry and topology. It has been noted that "the key observation of [ 10 ] is that the persistence diagram produced by [ 8 ] depends only on the algebraic structure carried by this diagram."
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces Subcategories. This category has the following ...