In mathematics especially algebraic topology and abstract algebra , homology in Greek language Greek ... mathematics modules with a given mathematical object such as a topological space or a group mathematics group . See homology theory for more background, or singular homology for a concrete version ... space, the homology groups are generally much easier to compute than the homotopy group s, and consequently one usually will have an easier time working with homology to aid in the classification of spaces. The original motivation for defining homology groups is the commonplace observation that one ... obvious how to define a hole, or how to distinguish between different kinds of holes. Homology ... subtle kinds of holes that homology cannot see in which case homotopy groups may be what is needed. Construction of homology groups The construction begins with an object such as a topological ... th homology group of X to be the factor group or quotient module math H n X ker partial n mathrm .... The simplicial homology groups math H n X math of a simplicial complex math X math are defined ... by the math n math simplices of math X math . The singular homology groups math H n X math are defined for any topological space math X math , and agree with the simplicial homology groups for a simplicial ... 1 th map is always equal to the kernel of the n th map. The homology groups of math X math therefore ... from algebraic topology the simplicial homology of a simplicial complex math X math . Here math A n ... a field, then the dimension of the n th homology of math X math turns out to be the number of holes in math X math at dimension n . Using this example as a model, one can define a singular homology ... partial n math arise from the boundary maps of simplices. In abstract algebra , one uses homology to define ..., one obtains a chain complex the homology math H n math of this complex depends only on math F math ... math . Homology functors Chain complexes form a category theory category A morphism from the chain complex ... more details
Wiktionary homologyHomology may refer to Homology anthropology , analogy between human beliefs, practices or artifacts owing to genetic or historical connections Homology biology , any characteristic of biological organisms that is derived from a common ancestor. Homology chemistry , the relationship between compounds in a homologous series Homologymathematics , a procedure to associate a sequence of abelian groups or modules with a given mathematical object Homology modeling , a method of protein structure prediction Homology sociology , a structural resonance between the different elements making up a socio cultural whole Homologous may refer to Homologous chromosomes , chromosomes in a biological cell that pair up synapse during meiosis Homologous series chemistry , a series of organic compounds having different quantities of a repeated unit Homologous temperature , the temperature of a material as a fraction of its absolute melting point Homological may refer to Homological word Homological algebra disambig de Homologie el fa fr Homologie he nl Homologie ja no Homologi pl Homologia pt Homologia ru sr ... more details
In mathematics , a homology manifold or generalized manifold is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory . Definition A homology G manifold without boundary of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G cohomological dimension such that for any x &isin X , the homology groups math H p X,X x, G math are trivial unless p n , in which case they are isomorphic to G . Here H is some homology theory, usually singular homology. Homology manifolds are the same as homology Z manifolds. More generally, one can define homology manifolds with boundary, by allowing the local homology groups to vanish at some points, which are of course called the boundary of the homology manifold. The boundary of an n dimensional first countable homology manifold is an n &minus 1 dimensional homology manifold without boundary . Examples Any topological manifold is a homology manifold. An example of a homology manifold that is not a manifold is the suspension of a homology sphere that is not a sphere. If X × Y is a topological manifolds, then X and Y are homology manifolds. References springer id H h047800 title Homology manifold author E. G. Sklyarenko W. J .R. Mitchell, http links.jstor.org sici?sici 0002 9939 28199010 29110 3A2 3C509 3ADTBOAH 3E2.0.CO 3B2 R Defining the boundary of a homology manifold , Proceedings of the American Mathematical Society , Vol. 110, No. 2. Oct., 1990 , pp. 509 513. topology stub Category Algebraic topology Category Generalized manifolds ... more details
In mathematics , K homology is a homologymathematicshomology theory on the Category mathematics category of compact space compact Hausdorff space s. It classifies the elliptic pseudo differential operator s acting on the vector bundle s over a space. In terms of math C math algebras, it classifies the Fredholm module s over an algebra . An operator homotopy between two Fredholm modules math mathcal H ,F 0, Gamma math and math mathcal H ,F 1, Gamma math is a norm mathematics norm Continuous function continuous Path topology path of Fredholm modules, math t mapsto mathcal H ,F t, Gamma math , math t in 0,1 . math Two Fredholm modules are then equivalent if they are related by unitary transformation s or operator homotopies. The math K 0 A math group mathematics group is the abelian group of equivalence relation equivalence classes of even Fredholm modules over A. The math K 1 A math group is the abelian group of equivalence classes of odd Fredholm modules over A. Addition is given by Direct sum of modules direct summation of Fredholm modules, and the Inverse mathematics inverse of math mathcal H , F, Gamma math is math mathcal H , F, Gamma . math References N. Higson and J. Roe, Analytic K homology . Oxford University Press, 2000. planetmath id 3330 title K homology Category K theory ... more details
Unreferenced date December 2009 In mathematics , reduced homology is a minor modification made to homology theory in algebraic topology , designed to make a point have all its homology group s zero. This change is required to make statements without some number of exceptional cases Alexander duality being an example . If P is a single point space, then with the usual definitions the integral homology group H sub 0 sub P is an infinite cyclic group , while for i 1 we have H sub i sub P 0 . More generally if X is a simplicial complex or finite CW complex , then the group H sub 0 sub X is the free abelian group on generators the connected space connected component s of X . The reduced homology should replace this group, of rank r say, by one of rank r &minus 1. Otherwise the homology groups should remain unchanged. An ad hoc way to do this is to think of a 0 th homology class not as a formal sum of connected components, but as such a formal sum where the coefficients add up to zero. A more fundamental way to do the same thing is to go back to the chain complex defining homology, and tweak the C sub 0 sub term in it. Namely, define the augmentation from C sub 0 sub to the integers, which expresses the sum of coefficients. Replace C sub 0 sub by the kernel of . Then calculate homology groups as usual, with the modified chain complex. Armed with this modified complex, the standard ways to obtain homology with coefficients by applying the tensor product , or reduced cohomology group s from the cochain complex made by using a Hom functor , can be applied. DEFAULTSORT Reduced Homology Category Homology theory ... more details
. SFH is the homology of the chain complex generated by the Fixed point mathematics fixed points ... inside Floer homology groups in Yang Mills theory . Cambridge tracts in mathematics 147 Cambridge ...Floer homology is a mathematical tool used in the study of symplectic geometry and low dimensional topology ..., Floer homology is a novel homology theory arising as an infinite dimensional analog of finite dimensional Morse homology . A similar construction, also introduced by Floer, provides a homology theory ... insights into the structure of three and four dimensional differentiable manifold s. Floer homology ... manifold version, it is the space of SU 2 connection mathematics connections on a three dimensional manifold. Loosely speaking, Floer homology is the Morse homology computed from a natural function ... space or the Chern Simons function on the space of connections. A homology theory is formed from the vector space spanned by the critical point mathematics critical points of this function. A linear ... two critical points. Floer homology is then the quotient space quotient vector space formed by identifying the image of this endomorphism inside its kernel mathematics kernel . Symplectic Floer homology Symplectic Floer Homology SFH is a homology theory associated to a symplectic manifold and a nondegenerate ... , the homology arises from studying the symplectic action functional on the universal cover ... with relative index 1. The symplectic Floer homology of a Hamiltonian symplectomorphism is isomorphic to the singular homology of the underlying manifold. Thus, the sum of the Betti numbers of that manifold ... M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the free ... Schwarz, and Cohen . The symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture. Floer homology of three manifolds There are several ... more details
In algebraic topology , Steenrod homology is a homology theory for compact metric space s introduced by harvs txt last Steenrod authorlink Norman Steenrod year1 1940 year2 1941 , based on regular cycles. It is similar to the homology theory introduced rather sketchily by Kolmogorov in 1936. References Citation last1 Milnor first1 John Willard author1 link John Milnor title Novikov conjectures, index theorems and rigidity, Vol. 1 Oberwolfach, 1993 origyear 1961 publisher Cambridge University Press series London Math. Soc. Lecture Note Ser. doi 10.1017 CBO9780511662676.005 id MR 1388297 year 1995 volume 226 chapter On the Steenrod homology theory pages 79 96 Citation last1 Steenrod first1 N. E. title Regular cycles of compact metric spaces url http www.jstor.org stable 1968863 id MR 0002544 year 1940 journal Annals of Mathematics Annals of Mathematics. Second Series issn 0003 486X volume 41 pages 833 851 Citation last1 Steenrod first1 N. E. title Lectures in Topology publisher University of Michigan Press location Ann Arbor id MR 0005298 year 1941 chapter Regular cycles of compact metric spaces pages 43 55 Category Homology theory ... more details
In mathematics , Hochschild homology and cohomology is a homology theory for associative algebra ring theory algebras over ring mathematics rings . There is also a theory for Hochschild homology of certain functor s. Hochschild cohomology was introduced by harvs txt authorlink Gerhard Hochschild first Gerhard last Hochschild year 1945 . Definition of Hochschild homology of algebras Let k be a ring ... fold tensor product of A over k . The chain complex that gives rise to Hochschild homology is given ... sub n sub A , M , b is a chain complex called the Hochschild complex , and its homology is the Hochschild homology of A with coefficients in M . Remark The maps d sub i sub are face map s making the family of module mathematics modules C sub n sub A , M a Face map Simplicial objects simplicial object in the category mathematics category of k modules, ie. a functor sup o sup k mod, where is the simplicial ... a sub 0 sub a sub i sub 1 a sub i 1 sub a sub n sub . Hochschild homology is the homology of this simplicial module. Hochschild homology of functors The simplicial circle S sup 1 sup is a simplicial ... The homology of this simplicial module is the Hochschild homology of the functor F . The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor ... of Hochschild homology of algebras The Hochschild homology of a commutative algebra A with coefficients in a symmetric A bimodule M is the homology associated to the composition math Delta ... id 0011076 year 1945 journal Annals of Mathematics Annals of Mathematics. Second Series issn 0003 486X volume 46 pages 58 67 Jean Louis Loday, Cyclic Homology , Grundlehren der mathematischen ... Texts in Mathematics 88 , Springer, 1982. Teimuraz Pirashvili, http www.numdam.org item?id ASENS 2000 4 33 2 151 0 Hodge decomposition for higher order Hochschild homology See also Cyclic homology ... geometry which uses Hochschild homology to generalize differential forms . Category Ring theory ... more details
In mathematics , specifically in the field of differential topology , Morse homology is a homology theory ... homology . Morse homology also serves as a model for the various infinite dimensional generalizations known as Floer homology theories. Formal definition Given any compact smooth manifold, let ... homology depends on neither. The pair f , g gives us a gradient vector field. We say that f ... point mathematics critical points of f intersect each other transversality transversely . For any such f , g , it can be shown that the difference in index mathematics index between any two critical ... of orientation mathematics oriented points representing unparametrized flow lines. A chain ... mathematics module generated by the critical points. The differential d of the complex sends a critical ... of how the moduli spaces of gradient flows compactification mathematics compactify . Namely, in d ... of a compact one manifold is always zero proves that d sup 2 sup p 0. Invariance of Morse homology It can be shown that the homology of this complex is independent of the Morse Smale pair f , g used .... Another approach to proving the invariance of Morse homology is to relate it directly to singular homology. One can define a map to singular homology by sending a critical point to the singular .... The cleanest way to do this rigorously is to use the theory of Current mathematics currents . The isomorphism with singular homology can also be proved by demonstrating an isomorphism with cellular homology , by viewing an unstable manifold associated to a critical point of index i as an i cell, and showing ... implicit in John Milnor s book on the h cobordism theorem. From the fact that the Morse homology is isomorphic to the singular homology, the Morse inequalities follow by considering the number of generators &mdash that is, critical points &mdash necessary to generate the homology groups of the appropriate ... . The existence of Morse homology explains , in the sense of categorification , the Morse inequalities ... more details
In algebraic topology , a homology sphere is an n manifold X having the homology group s of an n sphere ... perfect see Hurewicz theorem . A rational homology sphere is defined similarly but using homology with rational coefficients. Poincar homology sphere Henri Poincar links here The Henri Poincar Poincar homology sphere also known as Poincar dodecahedral space is a particular example of a homology sphere. Being a spherical 3 manifold , it is the only homology 3 sphere besides the 3 sphere itself ... order 120. This shows the Poincar conjecture cannot be stated in homology terms alone. Construction ... 3 manifold . Alternatively, the Poincar homology sphere can be constructed as the quotient ... intuitively, this means that the Poincar homology sphere is the space of all geometrically distinguishable ... s and is homeomorphic to the 3 sphere. In this case, the Poincar homology sphere is isomorphic to S sup ... of I Embedding embed ded in S sup 3 sup . Another approach is by Dehn surgery . The Poincar homology ... a homology sphere. More generally, surgery on a link gives a homology sphere whenever the matrix ... 0 with this complex surface is a homology 3 sphere, called a Brieskorn 3 sphere p , q , r . It is homeomorphic ... sphere. The connected sum of two oriented homology 3 spheres is a homology 3 sphere. A homology 3 sphere that cannot be written as a connected sum of two homology 3 spheres is called irreducible or prime , and every homology 3 sphere can be written as a connected sum of prime homology 3 spheres in an essentially ... sub , ..., a sub r sub is a homology sphere, where the b s are chosen so that math b b 1 a 1 cdots b r a r 1 a 1 cdots a r . math There is always a way to chose the b &prime s, and the homology sphere ... 3 sphere otherwise they are distinct non trivial homology spheres. If the a &prime s are 2, 3 ... homology 3 sphere with infinite fundamental group that has a Thurston geometry modeled on the universal ... of homology 3 spheres. The Casson invariant is an integer valued invariant of homology 3 spheres ... more details
viewdoc summary?doi 10.1.1.29.3618 ref See also Hochschild homology Noncommutative geometry HomologymathematicsHomologyHomology theory References references Alain Connes , Noncommutative differential ...In homological algebra , cyclic homology and cyclic cohomology are co homology theories for associative algebra s introduced by Alain Connes around 1980, which play an important role in his noncommutative geometry . They were independently discovered by Boris Tsygan and studied by Connes, Karoubi, Feigin Tsygan, Loday, Quillen , and others. Hints about definition The first definition of the cyclic homology of a ring A over a field of characteristic algebra characteristic zero, denoted HC sub n sub ... homology Hochschild homology complex of A . Connes later found a more categorical approach to cyclic homology using a notion of cyclic object in an abelian category , which is analogous to the notion of simplicial object . In this way, cyclic homology and cohomology may be interpreted ... of the striking features of cyclic homology is the existence of a long exact sequence connecting Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence. Case ... was extensively developed by Connes. Variants of cyclic homology One motivation of cyclic homology was the need for an approximation of K theory that be defined, unlike K theory, as the homology of a chain ... than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates ... homology due to Alain Connes , analytic cyclic homology due to Ralf Meyer or asymptotic and local cyclic homology due to Michael Puschnigg. The last one is very near to K theory as it is endowed with a bivariant Chern character from KK theory . Applications One of the applications of cyclic homology ... Homology , Grundlehren der mathematischen Wissenschaften Vol. 301, Springer 1998 ISBN 3 540 63074 ... and Cyclic homology DEFAULTSORT Cyclic Homology Category Homological algebra fr Cohomologie cyclique ... more details
In mathematics , Khovanov homology is an invariant of oriented knot mathematics knots and links that arises as the homologymathematicshomology of a chain complex . It may be regarded as a categorification ... a knot mathematics link L , we assign the Khovanov bracket nowiki nowiki D nowiki nowiki , a chain ... complex C D . The homology of this chain complex turns out to be an invariant of L , and its graded ... th vector space or Module mathematics module in the complex is shifted along to the r s th place, with all the differential mathematics differential maps being shifted accordingly. Let V be a graded vector .... The Khovanov homology of L is then defined as the homology H L of this complex C D . It turns out that the Khovanov homology is indeed an invariant of L , and does not depend on the choice of diagram ... the Khovanov homology or category for any knot. ref New Scientist 18 Oct 2008 ref Related theories One of the most interesting aspects of Khovanov s homology is that its exact sequences are formally similar to those arising in the Floer homology of 3 manifolds . Moreover, it has been used to reprove ... see below . Conjecturally, there is a spectral sequence relating Khovanov homology with the Floer homology knot Floer homology of Peter Ozsv th and Zolt n Szab Dunfield et al. 2005 . Another spectral sequence Ozsv th Szab 2005 relates a variant of Khovanov homology with the Heegard Floer homology ... to a variant of the monopole Floer homology of the branched double cover. Khovanov homology is related ... in 2003 Khovanov homology to an invariant of tangles a categorified version of Reshetikhin Turaev ... exhibited a singly graded piece of the sl sub 2 sub Khovanov homology as a certain Lagrangian intersection Floer homology Ciprian Manolescu has since simplified their construction and shown how to recover ... of Khovanov homology was provided by Jacob Rasmussen, who defined the s invariant using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genus , and is sufficient ... more details
In algebraic topology , a branch of mathematics , singular homology refers to the study of a certain set of algebraic invariant s of a topological space X , the so called homology groups math H n X math . Intuitively spoken, singular homology counts, for each dimension n , the n dimensional holes of a space. Singular homology is a particular example of a homology theory , which has now grown to be a rather ... a singular chain complex . The singular homology is then the homologymathematicshomology of the chain complex. The resulting homology groups are the same for all Homotopy Homotopy equivalence and null ... to understand, being built on fairly concrete constructions. In brief, singular homology is constructed ... can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory , where the homology group becomes a functor from the category of topological spaces ... chain complex The usual construction of singular homology proceeds by defining formal sums of simplices ... can define a certain group, the homology group of the topological space, involving the boundary operator ... circ partial n 1 0 math . The math n math th homology group of math X math is then defined as the factor group math H n X Z n X B n X math . The elements of math H n X math are called homology classes ... H n X H n Y , math for all n &ge 0. This means homology groups are topological invariants. In particular, if X is a connected contractible space , then all its homology groups are 0, except math H 0 X mathbb Z math . A proof for the homotopy invariance of singular homology groups can be sketched as follows ... is a chain complex Chain maps chain map , which descends to homomorphisms on homology math f H n X rightarrow ... homology theory can be recast in the language of category theory . In particular, the homology group ... axiom, one has that math H n math is also a functor, called the homology functor , acting on hTop , the quotient homotopy category math H n bold hTop to bold Ab . math This distinguishes singular homology ... more details
In mathematics , in the area of algebraic topology , simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory . Simplicial homology concerns ... gives singular homology . The simplicial homology of a simplicial complex is naturally isomorphic to the singular homology of its geometric realization. This implies, in particular, that the simplicial homology of a space does not depend on the triangulation chosen for the space. It has been shown ... possible to resolve the simplicial homology of a simplicial complex automatically and efficiently ... homology.jpg thumb 100 px A simplicial complex with 2 1 holes The k sup th sup homology group H sub k sub of S is defined to be the quotient math H k S Z k B k , . math A homology group H sub k sub .... The rank of an abelian group rank of the homology groups, the numbers math beta k rm rank H ... In order to compute the homology groups of the triangle, one should compute the different groups ... mathbb Z mathbb Z oplus mathbb Z cong mathbb Z math As for the other homology groups, computations ..., dark pixels in a bit map, etc. in which one wishes to find a topological feature. Homology can ... of persistent homology http graphics.stanford.edu projects lgl paper.php?id elz tps 02 Edelsbrunner et al.2002 http at.yorku.ca b a a k 28.htm Robins, 1999 involves analysis of homology at different resolutions, registering homology classes holes that persist as the resolution is changed. Such features ... in complex data. A MATLAB toolbox for computing persistent homology, Plex Vin de Silva , Gunnar Carlsson , is available at http math.stanford.edu comptop programs this site . See also Homology theory Singular homology Cellular homology References Lee, J.M., Introduction to Topological Manifolds , Springer Verlag , Graduate Texts in Mathematics, Vol. 202 2000 ISBN 0 387 98759 2 Allen Hatcher Hatcher ... Press 2002 ISBN 0 521 79540 0. Detailed discussion of homology theories for simplicial complexes ... more details
In topology , a branch of mathematics , intersection homology is an analogue of singular homology especially ... MacPherson approach The homology group s of a compact, oriented, n dimensional manifold X have ... intersection yields a 0 dimensional cycle. One may prove that the homology class of this cycle depends only on the homology classes of the original i and n &minus i dimensional cycles ... this equivalence relation intersection homology . They furthermore showed that the intersection ... homology class is well defined. Stratifications Intersection homology was originally defined on suitable ... spaces. A convenient one for intersection homology is an n dimensional topological pseudomanifold ... Middle perversity Intersection homology groups I sup p sup H sub i sub X depend on a choice of perversity ... invariance of intersection homology groups under change of stratification. The complementary perversity q of p is the one with math p k q k k 2 math Intersection homology groups of complementary dimension ... variety then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent. Singular intersection homology ... singular simplexes. The singular intersection homology groups with perversity p math I pH i X math are the homology groups of this complex. If X has a triangulation compatible with the stratification, then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups. The intersection homology groups are independent of the choice of stratification of X . If X is a topological manifold, then the intersection homology groups for any perversity are the same as the usual homology groups. Small resolutions A resolution ... induces an isomorphism from the intersection homology of X to the intersection homology of Y with the middle ... co homology. Sheaf theory Deligne s formula for intersection cohomology states that math ... more details
In algebraic topology , a branch of mathematics , the singular homology of a topological space relative to a subspace is a construction in singular homology , for pairs of spaces . The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Definition Given a subspace math A subset X math , one may .... The corresponding homology is called relative homology math H n X,A H n C bullet X C bullet A . math One says that relative homology is given by the relative cycles , chains whose boundaries ... delta to H n 1 A to cdots . math The connecting map takes a relative cycle, representing a homology ... X , x sub 0 sub , where x sub 0 sub is a point in X , is the n th reduced homology group of X . In other ... x sub 0 sub becomes trivial in relative homology. The excision theorem says that removing a sufficiently nice subset Z A leaves the relative homology groups H sub n sub X , A unchanged. Using the long ... th reduced homology groups of the quotient space X A . The n th local homology group of a space ... homology of X close to x sub 0 sub . Relative homology readily extends to the triple X , Y , Z ... chain complexes of abelian groups . Examples One important use of relative homology is the computation of the homology groups of quotient spaces math X A math . In the case that math A math is a subspace ... use this fact to compute the homology of a sphere. We can realize math S n math as the quotient ... homology gives the following br math ...H n D n rightarrow H n D n,S n 1 rightarrow H n 1 S n 1 rightarrow H n 1 D n .... math Because the disk is contractible, we know its homology groups vanish ... homology Excision theorem Excision Theorem Mayer Vietoris sequence References planetmath reference id 3722 title Relative homology groups Joseph J. Rotman, An Introduction to Algebraic Topology , Springer Verlag, ISBN 0 387 96678 1 Category Homology theory ... more details
In complex geometry , a polar homology is a group which captures holomorphic invariants What s that? of a complex manifold in a similar way to usual Homologymathematicshomology of a manifold in differential topology . Polar homology was defined by B. Khesin and A. Rosly in 1999. Definition Let M be a complex projective manifold . The space math C k math of polar k chains is a vector space over math Bbb C math defined as a quotient math A k R k math , with math A k math and math R k math vector spaces defined below. Defining math A k math The space math A k math is freely generated by the triples math X, f, alpha math , where X is a smooth, k dimensional complex manifold, math f X mapsto M math a holomorphic map, and math alpha math is a rational k form on X , with first order poles on a normal crossing divisor divisor with normal crossing . Defining math R k math The space math R k math is generated by the following relations. math lambda X, f, alpha X, f, lambda alpha math math X,f, alpha 0 math if math dim f X k math . math sum i X i,f i, alpha i 0 math provided that math sum if i alpha i equiv 0, math where math dim f i X i k math for all math i math and the push forwards math f i alpha i math are considered on the smooth part of math cup i f i X i math . Defining the boundary operator The boundary operator math partial C k mapsto C k 1 math is defined by math partial X,f, alpha 2 pi sqrt 1 sum i V i, f i, res V i , alpha math , where math V i math are components of the polar divisor of math alpha math , res is the Poincare residue , and math f i f V i math are restrictions of the map f to each component of the divisor. Khesin and Rosly proved that this boundary operator is well defined, and satisfies math partial 2 0 math . They defined the polar cohomology as the quotient ... arxiv.org abs math 0102152 Polar Homology and Holomorphic Bundles Phil. Trans. Roy. Soc. Lond. A359 ... Homology theory ... more details
In mathematics , cellular homology in algebraic topology is a homology theory for CW complex es. It agrees with singular homology , and can provide an effective means of computing homology modules. Definition If X is a CW complex with n skeleton X sub n sub , the cellular homology modules are defined as the homology group s of the cellular chain complex math cdots to H n 1 X n 1 , X n to H n X n, X n 1 to H n 1 X n 1 , X n 2 to cdots . math The module math H n X n, X n 1 , math is Free module free , with generators which can be identified with the n cells of X . Let math e n alpha math be an n cell of X , let math chi n alpha partial e n alpha cong S n 1 to X n 1 math be the attaching map, and consider the composite maps math chi n alpha beta S n 1 to X n 1 to X n 1 X n 1 e n 1 beta cong S n 1 math where math e n 1 beta math is an math n 1 math cell of X and the second map is the quotient map identifying math X n 1 e n 1 beta math to a point. The boundary map math d n H n X n,X n 1 to H n 1 X n 1 ,X n 2 , math is then given by the formula math d n e n alpha sum beta deg chi n alpha beta e n 1 beta , math where math deg chi n alpha beta math is the Degree of a continuous mapping degree of math chi n alpha beta math and the sum is taken over all math n 1 math cells of X , considered as generators of math H n 1 X n 1 ,X n 2 , math . Other properties One sees from the cellular chain complex that the n skeleton determines all lower dimensional homology math H k X cong H k X n math ... in consecutive dimensions, all its homology modules are free. For example, complex projective space ... method of computing the co homology of a CW complex, for an arbitrary Homology theory extraordinary co homology theory . Euler characteristic For a cellular complex X , let X sub j sub be its j th skeleton ... sequence of relative homology for the triple X sub n sub , X sub n 1 sub , &empty math cdots to H ... Press ISBN 978 0 521 79540 1 Category Homology theory ... more details
about the homology of topological spaces the homology of other mathematical objects Homological algebra In mathematics , homology theory is the axiom atic study of the intuitive geometric idea of homology of cycles on topological space s. It can be broadly defined as the study of Homologymathematicshomology theories on topological spaces. The general idea Image torus cycles.png thumb right A torus ... homology classes. There is a well defined way to add and subtract homology classes, which makes math H k X math into an abelian group , called the math k math th homology group of math X math ... two holes, which in this context count as being one dimensional. The corresponding homology ... rigorous way this is a central purpose of homology theory. For a more intricate example, if math Y math ..., see singular homology . There is also a version called simplicial homology that works when ... H 1 T math is again isomorphic to math mathbb Z oplus mathbb Z math . Cohomology As well as the homology ... of cohomology over homology is that it has a natural ring structure there is a way to multiply ... an math i j math dimensional cohomology class. Applications Notable theorems proved using homology include ... n k M math , which we can use to transfer the ring structure from cohomology to homology. For any compact ... a homology class, or in other words an element of math H 1 X math . It is also important that in the case ... more general relationship between homology and integration, which is most efficiently formulated ... operators Divergence div , Gradient grad and Curl mathematics curl from vector calculus ... C alpha math depends only on the homology class of math C math , provided that math d alpha 0 math ... math H k X times H k dR X to R math given by integration. Axiomatics and generalised homology There are various ... connective and so on. Brown Peterson cohomology Brown Peterson homology , Morava K theory , Morava E ... cohomology elliptic homology These are called generalised homology theories they carry much richer ... more details
beta right math . Hence math f math induces a homomorphism between the homology groups math f H n left ... Main singular homology Homotopy invariance Two basic properties of the push forward are math left ... to the identity isomorphism of homology groups. br A main result about the push forward is the homotopy ... that the homology groups of homotopy equivalent spaces are isomorphic The maps math f H n left X right ... more details
Unreferenced stub auto yes date December 2009 In anthropology and archaeology , homology is a type of analogy whereby two human beliefs, practices or Cultural artifact artifact s are separated by time but share similarities due to genetics genetic or history historical connections. Specifically in anthropology, a homology is a structure that is shared through descent from a common ancestor. The concept was explored by the American archaeologist William Duncan Strong in his direct historical approach to Archaeology Archaeological theory archaeological theory . DEFAULTSORT Homology Anthropology Anthropology stub Category Anthropology ... more details
Protein homology is homology biology biological homology between proteins, meaning that the proteins are derived from a common ancestor . ref name pmid3621342 cite journal author Reeck GR, de Ha n C, Teller DC, et al. title Homology in proteins and nucleic acids a terminology muddle and a way out of it journal Cell volume 50 issue 5 pages 667 year 1987 month August pmid 3621342 doi url ref The proteins may be in different species, with the ancestral protein being the form of the protein that existed in the ancestral species orthology . Or the proteins may be in the same species, but have evolved from a single protein whose gene was gene duplication duplicated in the genome paralogy . See also homology biology References Reflist Biochem stub Category Molecular evolution Category Proteins pt Prote na hom loga ... more details
In chemistry , homology refers to the appearance of homologues. A homologue also spelled as homolog is a chemical compound compound belonging to a series of compounds differing from each other by a repeating unit, such as a methylene methylene group , a peptide residue, etcetera. ref name iupac http www.chem.qmul.ac.uk iupac medchem ah.html h5 Glossary of Terms Used in Medicinal Chemistry IUPAC Recommendations 1998 ref A homolog is a special case of an analog chemistry analog . Examples are alkane s and compounds with alkyl sidechain s of different length the repeating unit being a methylene group CH sub 2 sub . On the periodic table , homologous elements share many chemical properties and appear in the same group column of the table. For example, all noble gases are colorless, monatomic gases with very low reactivity. These similarities are due to similar structure in their Electron shell outer shells of valence electrons . Homology in medicine refers to being from the same family. For example, the gastrointestinal tract hormone gastrin is homologous to cholecystokinin CCK , meaning they are from the same family of hormones. See also Homologous series Analog chemistry Analog Congener Chemistry Congener Structure activity relationship References references Category Chemical nomenclature ja lv Homologs zh ... more details
In evolutionary developmental biology , the concept of deep homology is used to describe cases where Cell growth growth and Cellular differentiation differentiation processes are governed by genetic mechanisms that are Homology biology Homology of sequences in genetics homologous and deeply Conserved sequence conserved across a wide range of species . Textbook examples common to metazoa include the homeotic gene s that control differentiation along major body axis body axes , and pax genes especially PAX6 involved in the development of the eye and other sensory organ s. In early 2010, a team at The University of Texas at Austin led by Edward Marcotte developed an algorithm which identifies deeply homologous genetic modules in unicellular organisms, plants, and non human animals based on phenotype phenotypes such as traits and developmental defects . The technique aligns phenotypes across organisms based on orthology a type of homology of genes involved in the phenotypes. ref Zimmer, Carl. http www.nytimes.com 2010 04 27 science 27gene.html?hp &pagewanted all The Search for Genes Leads to Unexpected Places , The New York Times , New York, April 26, 2010. ref ref name McGary2010 cite journal author1 McGary KL authorlink1 Kriston McGary author2 Park TJ author3 Woods JO author4 Cha HJ author5 Wallingford JB author6 Marcotte EM authorlink6 Edward Marcotte author separator , title Systematic discovery of nonobvious human disease models through orthologous phenotypes journal Proceedings of the National Academy of Sciences volume 107 issue 14 pages 6544 9 year 2010 month April pmid 20308572 doi 10.1073 pnas.0910200107 url http www.marcottelab.org paper pdfs PNAS Phenologs 2010.pdf ref References cite journal author1 Shubin N authorlink1 Neil Shubin author2 Tabin C author3 Carroll S authorlink3 Sean B. Carroll author separator , title Fossils, genes and the evolution of animal limbs journal Nature volume 388 issue 6643 pages 639 48 year 1997 month August pmid 9262397 doi 10.1038 ... more details
Homology modeling , also known as comparative modeling of protein refers to constructing an atomic resolution ... three dimensional structure of a related homologous protein the template . Homology modeling relies ... 325. ref The quality of the homology model is dependent on the quality of the sequence alignment ... of protein sequence homology. Structure 4 1123 27. ref Taken together, these various atomic position errors are significant and impede the use of homology models for purposes that require atomic resolution ... of a protein may be difficult to predict from homology models of its subunit s . Nevertheless, homology models can be useful in reaching qualitative conclusions about the biochemistry of the query ... acid. Homology modeling can produce high quality structural models when the target and template are closely ... 953. ref The chief inaccuracies in homology modeling, which worsen with lower sequence identity , derive .... ref Like other methods of structure prediction, current practice in homology modeling is assessed ... Prediction, or CASP . Motive The method of homology modeling is based on the observation that protein ... and protein NMR for every protein of interest, homology modeling can provide useful structural ... acid sequence are required to take on a related function. Steps in model production The homology modeling ... and sequence alignment The critical first step in homology modeling is the identification of the best ... also be used as a search technique for identifying templates to be used in traditional homology modeling ... a reliable homology model. Other factors may tip the balance in marginal cases for example, the template ... homology models are produced for a single query sequence, with the most likely candidate chosen ... pmid 18436442 pmc 2680823 ref Fragment assembly The original method of homology modeling relied ... B, Elofsson A. 2005 . All are not equal A benchmark of different homology modeling programs. Protein ... of spatial restraints The most common current homology modeling method takes its inspiration from ... more details
Encyclopedia
top
