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Journal articles on the topic 'Homology theory'

To see the other types of publications on this topic, follow the link: Homology theory.

Author: Grafiati

Published: 4 June 2021

Last updated: 31 July 2024

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1

Kauffman, Louis H. "Simplicial homotopy theory, link homology and Khovanov homology." Journal of Knot Theory and Its Ramifications 27, no. 07 (June 2018): 1841002. http://dx.doi.org/10.1142/s021821651841002x.

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This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

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2

Špakula, Ján. "Uniform K-homology theory." Journal of Functional Analysis 257, no. 1 (July 2009): 88–121. http://dx.doi.org/10.1016/j.jfa.2009.02.008.

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3

Talbi, Mohamed Elamine, and Djilali Benayat. "Homology Theory of Graphs." Mediterranean Journal of Mathematics 11, no. 2 (November 12, 2013): 813–28. http://dx.doi.org/10.1007/s00009-013-0358-x.

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4

Crabb, M. C. "LOOP HOMOLOGY AS FIBREWISE HOMOLOGY." Proceedings of the Edinburgh Mathematical Society 51, no. 1 (February 2008): 27–44. http://dx.doi.org/10.1017/s0013091505001483.

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AbstractThe loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds.

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5

Zastrow, Andreas. "On the (non)-coincidence of Milnor–Thurston homology theory with singular homology theory." Pacific Journal of Mathematics 186, no. 2 (December 1, 1998): 369–96. http://dx.doi.org/10.2140/pjm.1998.186.369.

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6

Anand, Swadha, Bhusan K. Kuntal, Anwesha Mohapatra, Vineet Bhatt, and Sharmila S. Mande. "FunGeCo: a web-based tool for estimation of functional potential of bacterial genomes and microbiomes using gene context information." Bioinformatics 36, no. 8 (December 27, 2019): 2575–77. http://dx.doi.org/10.1093/bioinformatics/btz957.

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Abstract Motivation Functional potential of genomes and metagenomes which are inferred using homology-based methods are often subjected to certain limitations, especially for proteins with homologs which function in multiple pathways. Augmenting the homology information with genomic location of the constituent genes can significantly improve the accuracy of estimated functions. This can help in distinguishing cognate homolog belonging to a candidate pathway from its other homologs functional in different pathways. Results In this article, we present a web-based analysis platform ‘FunGeCo’ to enable gene-context-based functional inference for microbial genomes and metagenomes. It is expected to be a valuable resource and complement the existing tools for understanding the functional potential of microbes which reside in an environment. Availability and implementation https://web.rniapps.net/fungeco [Freely available for academic use]. Supplementary information Supplementary data are available at Bioinformatics online.

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7

Agüero-Chapin, Guillermin, Deborah Galpert, Reinaldo Molina-Ruiz, Evys Ancede-Gallardo, Gisselle Pérez-Machado, Gustavo A. De la Riva, and Agostinho Antunes. "Graph Theory-Based Sequence Descriptors as Remote Homology Predictors." Biomolecules 10, no. 1 (December 23, 2019): 26. http://dx.doi.org/10.3390/biom10010026.

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Alignment-free (AF) methodologies have increased in popularity in the last decades as alternative tools to alignment-based (AB) algorithms for performing comparative sequence analyses. They have been especially useful to detect remote homologs within the twilight zone of highly diverse gene/protein families and superfamilies. The most popular alignment-free methodologies, as well as their applications to classification problems, have been described in previous reviews. Despite a new set of graph theory-derived sequence/structural descriptors that have been gaining relevance in the detection of remote homology, they have been omitted as AF predictors when the topic is addressed. Here, we first go over the most popular AF approaches used for detecting homology signals within the twilight zone and then bring out the state-of-the-art tools encoding graph theory-derived sequence/structure descriptors and their success for identifying remote homologs. We also highlight the tendency of integrating AF features/measures with the AB ones, either into the same prediction model or by assembling the predictions from different algorithms using voting/weighting strategies, for improving the detection of remote signals. Lastly, we briefly discuss the efforts made to scale up AB and AF features/measures for the comparison of multiple genomes and proteomes. Alongside the achieved experiences in remote homology detection by both the most popular AF tools and other less known ones, we provide our own using the graphical–numerical methodologies, MARCH-INSIDE, TI2BioP, and ProtDCal. We also present a new Python-based tool (SeqDivA) with a friendly graphical user interface (GUI) for delimiting the twilight zone by using several similar criteria.

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8

MIYATA, TAKAHISA. "Homology decompositions in shape theory." Publicationes Mathematicae Debrecen 78, no. 1 (January 1, 2011): 15–35. http://dx.doi.org/10.5486/pmd.2011.4325.

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9

Cortinas, Guillermo. "$L$-theory and dihedral homology." MATHEMATICA SCANDINAVICA 73 (December 1, 1993): 21. http://dx.doi.org/10.7146/math.scand.a-12453.

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10

Kotelskiy, Artem. "Bordered theory for pillowcase homology." Mathematical Research Letters 26, no. 5 (2019): 1467–516. http://dx.doi.org/10.4310/mrl.2019.v26.n5.a11.

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11

LÜCK, WOLFGANG, and HOLGER REICH. "DETECTING $K$-THEORY BY CYCLIC HOMOLOGY." Proceedings of the London Mathematical Society 93, no. 3 (October 13, 2006): 593–634. http://dx.doi.org/10.1017/s0024611506015954.

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12

Poudel, Prayat. "Lescop's invariant and gauge theory." Journal of Knot Theory and Its Ramifications 24, no. 09 (August 2015): 1550050. http://dx.doi.org/10.1142/s0218216515500509.

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Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

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13

Niebrzydowski, M., and J. H. Przytycki. "Homology operations on homology of quandles." Journal of Algebra 324, no. 7 (October 2010): 1529–48. http://dx.doi.org/10.1016/j.jalgebra.2010.04.033.

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14

Noreldeen, Alaa Hassan. "On the Homology Theory of Operator Algebras." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/368527.

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We investigate the cyclic homology and free resolution effect of a commutative unital Banach algebra. Using the free resolution operator, we define the relative cyclic homology of commutative Banach algebras. Lemmas and theorems of this investigation are studied and proved. Finally, the relation between cyclic homology and relative cyclic homology of Banach algebra is deduced.

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15

Mukherjee, Sujoy. "A homology theory for a special family of semi-groups." Journal of Knot Theory and Its Ramifications 27, no. 03 (March 2018): 1840005. http://dx.doi.org/10.1142/s0218216518400059.

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In this paper, we construct a new homology theory for semi-groups satisfying the right self-distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one-term and two-term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley–Lieb algebras.

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16

Bullock, Doug, Charles Frohman, and Joanna Kania-Bartoszyńska. "Skein Homology." Canadian Mathematical Bulletin 41, no. 2 (June 1, 1998): 140–44. http://dx.doi.org/10.4153/cmb-1998-022-1.

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AbstractA new class of homology groups associated to a 3-manifold is defined. The theories measure the syzygies between skein relations in a skein module. We investigate some of the properties of the homology theory associated to the Kauffman bracket.

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17

Grigor’yan, Alexander, Yuri Muranov, Vladimir Vershinin, and Shing-Tung Yau. "Path homology theory of multigraphs and quivers." Forum Mathematicum 30, no. 5 (September 1, 2018): 1319–37. http://dx.doi.org/10.1515/forum-2018-0015.

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AbstractWe construct a new homology theory for the categories of quivers and multigraphs and describe the basic properties of introduced homology groups. We introduce a conception of homotopy in the category of quivers and we prove the homotopy invariance of homology groups.

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18

ROBINSON, ALAN, and SARAH WHITEHOUSE. "Operads and Γ-homology of commutative rings." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 2 (March 2002): 197–234. http://dx.doi.org/10.1017/s0305004102005534.

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We introduce Γ-homology, the natural homology theory for E∞-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from André–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

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19

Kaushal Rana. "Homological Algebra and Its Application: A Descriptive Study." Integrated Journal for Research in Arts and Humanities 2, no. 1 (January 31, 2022): 29–35. http://dx.doi.org/10.55544/ijrah.2.1.47.

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Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

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20

Poudel, Prayat, and Nikolai Saveliev. "Link homology and equivariant gauge theory." Algebraic & Geometric Topology 17, no. 5 (September 19, 2017): 2635–85. http://dx.doi.org/10.2140/agt.2017.17.2635.

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21

Zivaljevic, Rade T. "Infinitesimals, Microsimplexes and Elementary Homology Theory." American Mathematical Monthly 93, no. 7 (August 1986): 540. http://dx.doi.org/10.2307/2323028.

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22

Noreldeen, Alaa Hassan, and S. A. Abo-Quota. "Operations on the dihedral homology theory." Applied Mathematical Sciences 13, no. 20 (2019): 983–90. http://dx.doi.org/10.12988/ams.2019.98121.

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23

Naot, Gad. "The universal Khovanov link homology theory." Algebraic & Geometric Topology 6, no. 4 (November 1, 2006): 1863–92. http://dx.doi.org/10.2140/agt.2006.6.1863.

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24

Everitt, Brent, and Paul Turner. "The homotopy theory of Khovanov homology." Algebraic & Geometric Topology 14, no. 5 (November 5, 2014): 2747–81. http://dx.doi.org/10.2140/agt.2014.14.2747.

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25

Barcelo, Hélène, Valerio Capraro, and Jacob A. White. "Discrete homology theory for metric spaces." Bulletin of the London Mathematical Society 46, no. 5 (June 17, 2014): 889–905. http://dx.doi.org/10.1112/blms/bdu043.

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26

Auslander, M., and Ø. Solberg. "Relative homology and representation theory 1." Communications in Algebra 21, no. 9 (January 1993): 2995–3031. http://dx.doi.org/10.1080/00927879308824717.

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27

Auslander, M., and Ø. Solberg. "Relative homology and representation theory II." Communications in Algebra 21, no. 9 (January 1993): 3033–79. http://dx.doi.org/10.1080/00927879308824718.

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28

Auslander, M., and Ø. Solberg. "Relative homology and representation theory III." Communications in Algebra 21, no. 9 (January 1993): 3081–97. http://dx.doi.org/10.1080/00927879308824719.

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29

Tang, Zhongming. "Local homology theory for artinian modules." Communications in Algebra 22, no. 5 (January 1994): 1675–84. http://dx.doi.org/10.1080/00927879408824928.

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30

Kuz’min, Yu V. "Homology theory of free abelianized extensions." Communications in Algebra 16, no. 12 (January 1, 1988): 2447–533. http://dx.doi.org/10.1080/00927879808823701.

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31

Loday, Jean-Louis. "Algebraic K-theory and cyclic homology." Journal of K-theory 11, no. 3 (April 30, 2013): 553–57. http://dx.doi.org/10.1017/is012011006jkt200.

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The following are personal reminiscences of my research years in algebraic K-theory and cyclic homology during which Dan Quillen was everyday present in my professional life.In the late sixties (of the twentieth century) the groups K0;K1;K2 were known and well-studied. The group K0 had been introduced by Alexander Grothendieck, then came K1 by Hyman Bass [2] (as a variation of the Whitehead group), permitting one to generalize the notion of determinant, and finally K2 by John Milnor [9] and Michel Kervaire. The big problem was: how about Kn? Having in mind topological K-theory and all the other generalized (co)homological theories, one was expecting higher K-groups which satisfy similar axioms, in particular the Mayer-Vietoris exact sequence. The discovery by Richard Swan of the existence of an obstruction for this property to hold shed some embarrassment. What kind of properties should we ask of Kn? There were various attempts, for instance by Max Karoubi and Orlando Villamayor [4]. And suddenly Dan Quillen came with a candidate sharing a lot of nice properties. He had even two different constructions of the same object: the “+” construction and the “Q” construction [14, 15]. Not only did he propose a candidate but he already got a computation: the higher K-theory of finite fields. This was a fantastic step forward and Hyman Bass organized a two week conference at the Battelle Institute in Seattle during the summer of 1972, which was attended by Bass, Borel, Husemoller, Karoubi, Priddy, Quillen, Segal, Stasheff, Tate, Waldhausen, Wall and sixty other mathematicians. The Proceedings appeared as Springer Lecture Notes 341, 342 and 343. I met Quillen for the first time on this occasion.

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32

Cavicchioli, Alberto, and Mauro Meschiari. "A homology theory for colored graphs." Discrete Mathematics 137, no. 1-3 (January 1995): 99–136. http://dx.doi.org/10.1016/0012-365x(93)e0146-u.

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33

Cortiñas, Guillermo. "L-theory and dihedral homology II." Topology and its Applications 51, no. 1 (June 1993): 53–69. http://dx.doi.org/10.1016/0166-8641(93)90014-5.

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34

Anderson, Douglas R., and Hans Jørgen Munkholm. "Geometric modules and Quinn homology theory." K-Theory 7, no. 5 (September 1993): 443–75. http://dx.doi.org/10.1007/bf00961537.

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35

Živaljević, Rade T. "Infinitesimals, Microsimplexes and Elementary Homology Theory." American Mathematical Monthly 93, no. 7 (August 1986): 540–44. http://dx.doi.org/10.1080/00029890.1986.11971878.

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36

Xia, Qinglan. "Intersection homology theory via rectifiable currents." Calculus of Variations and Partial Differential Equations 19, no. 4 (April 1, 2004): 421–43. http://dx.doi.org/10.1007/s00526-003-0222-0.

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37

Imamura, Takuma. "Nonstandard homology theory for uniform spaces." Topology and its Applications 209 (August 2016): 22–29. http://dx.doi.org/10.1016/j.topol.2016.05.016.

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38

Usher, Michael, and Jun Zhang. "Persistent homology and Floer–Novikov theory." Geometry & Topology 20, no. 6 (December 21, 2016): 3333–430. http://dx.doi.org/10.2140/gt.2016.20.3333.

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39

Kleiman, Steven L. "The Development of Intersection Homology Theory." Pure and Applied Mathematics Quarterly 3, no. 1 (2007): 225–82. http://dx.doi.org/10.4310/pamq.2007.v3.n1.a8.

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40

Jacobs, Philippe. "A sheaf homology theory with supports." Illinois Journal of Mathematics 44, no. 3 (September 2000): 644–66. http://dx.doi.org/10.1215/ijm/1256060422.

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41

Kashiwara, Masaki, and Pierre Schapira. "Persistent homology and microlocal sheaf theory." Journal of Applied and Computational Topology 2, no. 1-2 (September 18, 2018): 83–113. http://dx.doi.org/10.1007/s41468-018-0019-z.

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42

Anderson, Douglas R., and Hans Jørgen Munkholm. "Geometric modules and algebraicK-homology theory." K-Theory 3, no. 6 (November 1990): 561–602. http://dx.doi.org/10.1007/bf01054451.

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43

Lodder, Jerry M. "Dihedral homology and Hermitian K-theory." K-Theory 10, no. 2 (March 1996): 175–96. http://dx.doi.org/10.1007/bf00536611.

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44

Akbarov, S. S. "Absolute homology theory of stereotype algebras." Functional Analysis and Its Applications 34, no. 1 (January 2000): 60–63. http://dx.doi.org/10.1007/bf02467068.

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45

Wigderson, Yuval. "The Bar-Natan theory splits." Journal of Knot Theory and Its Ramifications 25, no. 04 (April 2016): 1650014. http://dx.doi.org/10.1142/s0218216516500140.

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We show that over the binary field [Formula: see text], the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch’s analogous result for ordinary Khovanov homology, without the perturbation.

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46

Muranov, Yuri V., and Anna Szczepkowska. "Path homology theory of edge-colored graphs." Open Mathematics 19, no. 1 (January 1, 2021): 706–23. http://dx.doi.org/10.1515/math-2021-0049.

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Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

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47

Kiem, Young-Hoon, and Jun Li. "Quantum singularity theory via cosection localization." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 766 (September 1, 2020): 73–107. http://dx.doi.org/10.1515/crelle-2019-0018.

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AbstractWe generalize the cosection localized Gysin map to intersection homology and Borel–Moore homology, which provides us with a purely topological construction of the Fan–Jarvis–Ruan–Witten invariants and some GLSM invariants.

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48

KHOVANOV, MIKHAIL. "TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES." International Journal of Mathematics 18, no. 08 (September 2007): 869–85. http://dx.doi.org/10.1142/s0129167x07004400.

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We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

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49

Bauer, F. W. "Strong Homology Theories as Localizations." gmj 11, no. 4 (December 2004): 635–43. http://dx.doi.org/10.1515/gmj.2004.635.

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Abstract Let 𝔎 be a category of pairs of spaces, 𝔏 ⊂ 𝔎 the category of pairs of ANRs or CW-spaces, 𝐴∗ a chain functor (e.g., one associated with a spectrum). Then the derived homology 𝑠ℎ∗ of the 𝔏-localization of 𝐴∗ is the strong homology theory on 𝔎 which is up to an isomorphism uniquely determined by the fact that 𝑠ℎ∗ | 𝔏 agrees with the derived homology of 𝐴∗ on 𝔏. This establishes a relationship between localization theory and strong homology theory (e.g., Steenrod–Sitnikov homology theory, whenever all pairs are compact metric).

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50

Staecker, P. Christopher. "Digital homotopy relations and digital homology theories." Applied General Topology 22, no. 2 (October 1, 2021): 223. http://dx.doi.org/10.4995/agt.2021.13154.

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In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.<br /><br />We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.<br /><br />We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.

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