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Relevant bibliographies by topics / Algebraic topology – Homology and cohomology theories – Other homology theories

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Academic literature on the topic 'Algebraic topology – Homology and cohomology theories – Other homology theories'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Algebraic topology – Homology and cohomology theories – Other homology theories"

1

Isaksen, Daniel C., and Armira Shkembi. "Motivic connective K-theories and the cohomology of A(1)." Journal of K-theory 7, no. 3 (May 24, 2011): 619–61. http://dx.doi.org/10.1017/is011004009jkt154.

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AbstractWe make some computations in stable motivic homotopy theory over Spec ℂ, completed at 2. Using homotopy fixed points and the algebraic K-theory spectrum, we construct over ℂ a motivic analogue of the real K-theory spectrum KO. We also establish a theory of motivic connective covers over ℂ to obtain a motivic version of ko. We establish an Adams spectral sequence for computing motivic ko-homology. The E2-term of this spectral sequence involves Ext groups over the subalgebra A(1) of the motivic Steenrod algebra. We make several explicit computations of these E2-terms in interesting special cases.

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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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NEHANIV, CHRYSTOPHER LEV. "ALGEBRAIC CONNECTIVITY." International Journal of Algebra and Computation 01, no. 04 (December 1991): 445–71. http://dx.doi.org/10.1142/s0218196791000316.

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Let [Formula: see text] be a type of algebra in the sense of universal algebra. By defining singular simplices in algebras and emulating singular [co] homology, we introduce for each variety, pseudo-variety, and divisional class V of type [Formula: see text], a homology and cohomology theory which measure the V-connectivity of type-[Formula: see text] algebras. Intuitively, if we were to think of an algebra as a space and subalgebras which lie in V as simplices, then V-connectivity describes the failure of subalgebras to lie in V, i.e., it describes the "holes" in this space. These [co]homologies are functorial on the class of type-[Formula: see text] algebras and are characterized by a natural topological interpretation. All these notions extend to subsets of algebras. One obtains for this algebraic connectivity, the long exact sequences, relative [co]homologies, and the analogues of the usual [co]homological notions of the algebraic topologists. In fact, we show that the [co]homologies are actually the same as the simplicial [co]homology of simplicial complexes that depend functorially on the algebras. Thus the connectivities in question have a natural geometric meaning. This allows the wholesale import into algebra of the concepts, results, and techniques of algebraic topology. In particular, functoriality implies that the [co]homology of a pair of algebras A ⊆ B is an invariant of the position of A in B. When one V contains another, we obtain relationships between the [co] homology theories in the form of long exact sequences. Furthermore for finite algebras, V-[co]homology is effectively computable if membership in V is. We obtain an analogue of the Poincaré lemma (stating that subsets of an algebra in V are V-homologically trivial), extremely general guarantees of the existence of subsets with non-trivial V-homology for algebras not in V, long exact V-homotopy sequences, as well as analogues of the powerful Eilenberg-Zilber theorems and Kunneth theorems in the setting of V-connectivity for V a variety or pseudo-variety. Also in the more general case of any divisionally closed V, we construct the long exact Mayer-Vietoris sequences for V-homology. Results for homomorphisms include an algebraic version of contiguity for homomorphisms (which implies they are V-homotopic) and a proof that V-surmorphisms are V-homotopy equivalences. If we allow the divisional classes to vary, then algebraic connectivity may be viewed as a functor from the category of pairs W ⊆ V of divisional classes of [Formula: see text]-algebras with inclusions as morphisms' to the category of functors from pairs of [Formula: see text]-algebras to pairs of simplicial complexes. Examples show the non-triviality of this theory (e.g. "associativity tori"), and two preliminary applications to semigroups are given: 1) a proof that the group connectivity of a torsion semigroup S is homotopy equivalent to a space whose points are the maximal subgroups of S, and 2) an aperiodic connectivity analogue of the fundamental lemma of complexity.

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4

Toën, Bertrand, and Gabriele Vezzosi. "Algèbres simplicialesS1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs." Compositio Mathematica 147, no. 6 (July 29, 2011): 1979–2000. http://dx.doi.org/10.1112/s0010437x11005501.

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AbstractThis work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi,Chern character, loop spaces and derived algebraic geometry, inAlgebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. IfAis a smooth commutativek-algebra andkhas characteristic 0, we show that two objects,S1⊗Aand ϵ(A), determine one another, functorially inA. The objectS1⊗Ais theS1-equivariant simplicialk-algebra obtained by tensoringAby the simplicial groupS1:=Bℤ, while the object ϵ(A) is the de Rham algebra ofA, endowed with the de Rham differential, and viewed as aϵ-dg-algebra(see the main text). We define an equivalence φ between the homotopy theory of simplicial commutativeS1-equivariantk-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying theS1-equivariant functions on the derived loop spaceLXof a smoothk-schemeXwith the algebraic de Rham cohomology of X/k. As corollaries, we obtainfunctorialandmultiplicativeversions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separatedk-schemes. By construction, these decompositions aremoreovercompatible with theS1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

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Shi, Dinghua, Linyuan Lü, and Guanrong Chen. "Totally homogeneous networks." National Science Review 6, no. 5 (April 9, 2019): 962–69. http://dx.doi.org/10.1093/nsr/nwz050.

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Abstract In network science, the non-homogeneity of node degrees has been a concerning issue for study. Yet, with today's modern web technologies, the traditional social communication topologies have evolved from node-central structures into online cycle-based communities, urgently requiring new network theories and tools. Switching the focus from node degrees to network cycles could reveal many interesting properties from the perspective of totally homogenous networks or sub-networks in a complex network, especially basic simplexes (cliques) such as links and triangles. Clearly, compared with node degrees, it is much more challenging to deal with network cycles. For studying the latter, a new clique vector-space framework is introduced in this paper, where the vector space with a basis consisting of links has a dimension equal to the number of links, that with a basis consisting of triangles has the dimension equal to the number of triangles and so on. These two vector spaces are related through a boundary operator, for example mapping the boundary of a triangle in one space to the sum of three links in the other space. Under the new framework, some important concepts and methodologies from algebraic topology, such as characteristic number, homology group and Betti number, will play a part in network science leading to foreseeable new research directions. As immediate applications, the paper illustrates some important characteristics affecting the collective behaviors of complex networks, some new cycle-dependent importance indexes of nodes and implications for network synchronization and brain-network analysis.

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Dissertations / Theses on the topic "Algebraic topology – Homology and cohomology theories – Other homology theories"

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Cho, Karina Elle. "Enhancing the Quandle Coloring Invariant for Knots and Links." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/228.

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Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

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Krempasky, Seyide Denise. "Symmetric Squaring in Homology and Bordism." Doctoral thesis, 2011. http://hdl.handle.net/11858/00-1735-0000-0006-B3F0-3.

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Betrachtet man das kartesische Produkt X × X eines topologischen Raumes X mit sich selbst, so kann auf diesem Objekt insbesondere die Involution betrachtet werden, die die Koordinaten vertauscht, die also (x,y) auf (y,x) abbildet. Das sogenannte 'Symmetrische Quadrieren' in Čech-Homologie mit Z/2-coefficients wurde von Schick et al. 2007 als Abbildung von der k-ten Čech-Homologiegruppe eines Raumes X in die 2k-te Čech-Homologiegruppe von X × X modulu der oben genannten Involution definiert. Es stellt sich heraus, dass diese Konstruktion entscheidend ist für den Beweis eines parametrisierten Borsuk-Ulam-Theorems.Das Symmetrische Quadrieren kann zu einer Abbildung in Bordismus verallgemeinert werden, was der Hauptgegenstand dieser Dissertation ist. Genauer gesagt werden wir zeigen, dass es eine wohldefinierte, natürliche Abbildung von der k-ten singulären Bordismusgruppe von X in die 2k-te Bordismusgruppe von X × X modulu der obigen Involution gibt.Insbesondere ist dieses Quadrieren wirklich eine Verallgemeinerung der Konstruktion in Čech-Homologie, denn es ist vertauschbar mit dem Übergang von Bordismus zu Homologie via dem Fundamentalklassenhomomorphismus. Auf dem Weg zu diesem Resultat wird das Konzept des Čech-Bordismus als Kombination aus Bordismus und Čech-Homologie zunächst definiert und dann mit Čech-Homologie verglichen.

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Books on the topic "Algebraic topology – Homology and cohomology theories – Other homology theories"

1

Topological modular forms. Providence, Rhode Island: American Mathematical Society, 2014.

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1975-, Panov Taras E., ed. Toric topology. Providence, Rhode Island: American Mathematical Society, 2015.

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Conference Board of the Mathematical Sciences and NSF-CBMS Regional Conference in the Mathematical Sciences on Deformation Theory of Algebras and Modules (2011 : Raleigh, N.C.), eds. Deformation theory of algebras and their diagrams. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2012.

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Basterra, Maria, Kristine Bauer, Kathryn Hess, and Brenda Johnson. Women in topology: Collaborations in homotopy theory : WIT, Women in Topology Workshop, August 18-23, 2013, Banff International Research Station, Banff, Alberta, Canada. Providence, Rhode Island: American Mathematical Society, 2015.

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1974-, Zomorodian Afra J., ed. Advances in applied and computational topology: American Mathematical Society Short Course on Computational Topology, January 4-5, 2011, New Orleans, Louisiana. Providence, R.I: American Mathematical Society, 2012.

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1974-, Nelson Sam, ed. Quandles: An introduction to the algebra of knots. Providence, Rhode Island: American Mathematical Society, 2015.

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Modern classical homotopy theory. Providence, R.I: American Mathematical Society, 2011.

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Ausoni, Christian, 1968- editor of compilation, Hess, Kathryn, 1967- editor of compilation, Johnson Brenda 1963-, Lück, Wolfgang, 1957- editor of compilation, and Scherer, Jérôme, 1969- editor of compilation, eds. An Alpine expedition through algebraic topology: Fourth Arolla Conference, algebraic topology, August 20-25, 2012, Arolla, Switzerland. Providence, Rhode Island: American Mathematical Society, 2014.

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Stanford Symposium on Algebraic Topology: Applications and New Directions (2012 : Stanford, Calif.), ed. Algebraic topology: Applications and new directions : Stanford Symposium on Algebraic Topology: Applications and New Directions, July 23--27, 2012, Stanford University, Stanford, CA. Providence, Rhode Island: American Mathematical Society, 2014.

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Center for Mathematics at Notre Dame and American Mathematical Society, eds. Toplogy and field theories: Center for Mathematics at Notre Dame, Center for Mathematics at Notre Dame : summer school and conference, Topology and field theories, May 29-June 8, 2012, University of Notre Dame, Notre Dame, Indiana. Providence, Rhode Island: American Mathematical Society, 2014.

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Book chapters on the topic "Algebraic topology – Homology and cohomology theories – Other homology theories"

1

Switzer, Robert M. "Homology and Cohomology Theories." In Algebraic Topology — Homotopy and Homology, 99–132. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/978-3-642-61923-6_8.

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Dodson, C. T. J., and Phillip E. Parker. "Homology and Cohomology Theories." In A User’s Guide to Algebraic Topology, 105–42. Boston, MA: Springer US, 1997. http://dx.doi.org/10.1007/978-1-4615-6309-9_5.

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Adhikari, Mahima Ranjan. "Homology and Cohomology Theories." In Basic Algebraic Topology and its Applications, 347–406. New Delhi: Springer India, 2016. http://dx.doi.org/10.1007/978-81-322-2843-1_10.

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Adhikari, Mahima Ranjan. "Spectral Homology and Cohomology Theories." In Basic Algebraic Topology and its Applications, 475–509. New Delhi: Springer India, 2016. http://dx.doi.org/10.1007/978-81-322-2843-1_15.

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Dieudonné, Jean. "The Various Homology and Cohomology Theories." In A History of Algebraic and Differential Topology, 1900 - 1960, 67–157. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4907-4_5.

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Adhikari, Mahima Ranjan. "Eilenberg–Steenrod Axioms for Homology and Cohomology Theories." In Basic Algebraic Topology and its Applications, 419–31. New Delhi: Springer India, 2016. http://dx.doi.org/10.1007/978-81-322-2843-1_12.

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