In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. It can be broadly defined as the study of homology theories on topological spaces. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is
Contents |
Simple explanation
At the intuitive level homology is taken to be an equivalence relation, such that chains C and D are homologous on the space X if the chain C − D is a boundary of a chain of one dimension higher. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Algebraic topology, a simplicial k - chain is a formal linear combination of k - simplices. The simplest case is in graph theory, with C and D vertices and homology with a meaning coming from the oriented edge E from P to Q having boundary Q — P. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects A collection of edges from D to C, each one joining up to the one before, is a homology. In general, a k-chain is thought of as a formal combination
where the ai are integers and the di are k-dimensional simplices on X. In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle The boundary concept here is that taken over from the boundary of a simplex; it allows a high-dimensional concept which for k = 1 is the kind of telescopic cancellation seen in the graph theory case. In Mathematics, a telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels This explanation is in the style of 1900, and proved somewhat naive, technically speaking.
Example of a torus surface
For example if X is a 2-torus T, a one-dimensional cycle on T is in intuitive terms a linear combination of curves drawn on T, which closes up on itself (cycle condition, equivalent to having no net boundary). In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object If C and D are cycles each wrapping once round T in the same way, we can find explicitly an oriented area on T with boundary C − D. Topologists can prove that the homology classes of 1-cycles with integer coefficients form a free abelian group with two generators, one generator for each of the two different ways round the 'doughnut'. In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in
The nineteenth century
This level of understanding was common property in the mathematics of the nineteenth century, starting with the idea of Riemann surface. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional At the end of the century, the work of Poincaré had provided a much more general, though still intuitively-based, setting. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician
For example, it is considered that the general Stokes' theorem was first stated in 1899 by Poincaré: it involves necessarily both an integrand (we would now say, a differential form), and a region of integration (a p-chain), with two kinds of boundary operators, one of which in modern terms is the exterior derivative, and the other a geometric boundary operator on chains that includes orientation and can be used for homology theory. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms The two boundaries appear as adjoint operators, with respect to integration. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator.
Twentieth century beginnings
Rather loose, geometric arguments with homology were only gradually replaced at the beginning of the twentieth century by rigorous techniques. The twentieth century of the Common Era began on To begin with, the style of the era was to use combinatorial topology (the fore-runner of today's algebraic topology). In Mathematics, combinatorial topology was an older name for Algebraic topology, dating from the time when topological invariants of spaces (for example Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic That assumes that the spaces treated are simplicial complexes, while the most interesting spaces are usually manifolds, so that artificial triangulations have to be introduced to apply the tools. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Trigonometry and Geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either Pioneers such as Solomon Lefschetz and Marston Morse still preferred a geometric approach. Solomon Lefschetz ( 3 September 1884 – 5 October 1972) was an American Mathematician who did fundamental work on Marston Morse (born Harold Calvin Marston Morse; born 24 March, 1892 – 22 June, 1977) was an American Mathematician best The combinatorial stance did allow Brouwer to prove foundational results such as the simplicial approximation theorem, at the base of the idea that homology is a functor (as it would later be put). Brouwer is the last name of different people Adriaen Brouwer (1605–1638 was a Flemish painter Dirk Brouwer (1902–1966 was a Dutch-American In Mathematics, the simplicial approximation theorem is a foundational result for Algebraic topology, guaranteeing that Continuous mappings can be (by a In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Brouwer was able to prove the Jordan curve theorem, basic for complex analysis, and the invariance of domain, using the new tools; and remove the suspicion attached to topological arguments as handwaving. In Topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside" Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Invariance of domain is a theorem in Topology about Homeomorphic Subsets of Euclidean space R n. The term handwaving is an informal term that describes either the Debate technique of failing to Rigorously address an Argument in an attempt to bypass the
Towards algebraic topology
The transition to algebraic topology is usually attributed to the influence of Emmy Noether, who insisted that homology classes lay in quotient groups — a point of view now so fundamental that it is taken as a definition. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In fact Noether in the period from 1920 onwards was with her students elaborating the theory of modules for any ring, giving rise when the two ideas were combined to the concept of homology with coefficients in a ring. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Before that, coefficients (that is, the sense in which chains are linear combinations of the basic geometric chains traced on the space) had usually been integers, real or complex numbers, or sometimes residue classes mod 2. In the new setting, there would be no reason not to take residues mod 3, for example: to be a cycle is then a more complex geometric condition, exemplified in graph theory terms by having the number of incoming edges at every vertex a multiple of 3. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects But in algebraic terms, the definitions present no new problem. The universal coefficient theorem explains that homology with integer coefficients determines all other homology theories, by use of the tensor product; it is not anodyne, in that (as we would now put it) the tensor product has derived functors that enter into a general formulation. In Mathematics, the universal coefficient theorem in Algebraic topology establishes the relationship in Homology theory between the integral homology In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones
Cohomology, and singular homology
The 1930s were the decade of the development of cohomology theory, as several research directions grew together and the De Rham cohomology that was implicit in Poincaré's work cited earlier became the subject of definite theorems. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable Cohomology and homology are dual theories, in a sense that required detailed working out; at the same time it was realised that homology, that was, simplicial homology, was far from being at the end of its story. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is The definition of singular homology avoided the need for the apparatus of triangulations, at a cost of moving to infinitely-generated modules. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space
Axiomatics and extraordinary theories
The development of algebraic topology from 1940 to 1960 was very rapid, and the role of homology theory was often as 'baseline' theory, easy to compute and in terms of which topologists sought to calculate with other functors. The axiomatisation of homology theory by Eilenberg and Steenrod (the Eilenberg-Steenrod axioms) revealed that what various candidate homology theories had in common was, roughly speaking, some exact sequences and in particular the Mayer-Vietoris theorem, and the dimension axiom that calculated the homology of the point. Eilenberg is a surname and may refer to Samuel Eilenberg, Polish mathematician Richard Eilenberg, German composer see. Norman Earl Steenrod ( April 22, 1910 – October 14, 1971) was a preeminent Mathematical Topologist who most widely known for his contributions In Mathematics, specifically in Algebraic topology, the Eilenberg-Steenrod axioms are properties that homology theories of Topological spaces In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Algebraic topology and related branches of Mathematics, the Mayer–Vietoris sequence (named after Walther Mayer and Leopold Vietoris) is The dimension axiom was relaxed to admit the (co)homology derived from topological K-theory, and cobordism theory, in a vast extension to the extraordinary (co)homology theories that became standard in homotopy theory. In Mathematics, K-theory is a tool used in several disciplines In Mathematics, an n+1 cobordism is a Triple (WMN where W is an (n+1-dimensional Manifold, whose In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical These can be easily characterised for the category of CW complexes. In Topology, a CW complex is a type of Topological space introduced by J
Current state of homology theory
For more general (i. This is a list of some of the ordinary and generalized (or extraordinary homology and cohomology theories in Algebraic topology that are defined on the categories of CW e. less well-behaved) spaces, recourse to ideas from sheaf theory brought some extension of homology theories, particularly the Borel-Moore homology for locally compact spaces. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, Borel-Moore homology or homology with closed support is a Homology theory for Locally compact spaces For Compact spaces In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks
The basic chain complex apparatus of homology theory has long since become a separate piece of technique in homological algebra, and has been applied independently, for example to group cohomology. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper Therefore there is no longer one homology theory, but many homology and cohomology theories in mathematics.
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic