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Journal articles on the topic 'Adams spectral sequences. Homotopy groups'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

Kato, Ryo, and Katsumi Shimomura. "The first line of the Bockstein spectral sequence on a monochromatic spectrum at an odd prime." Nagoya Mathematical Journal 207 (September 2012): 139–57. http://dx.doi.org/10.1017/s0027763000022339.

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AbstractThe chromatic spectral sequence was introduced by Miller, Ravenel, and Wilson to compute the E2-term of the Adams-Novikov spectral sequence for computing the stable homotopy groups of spheres. The E1-term of the spectral sequence is an Ext group of BP*BP-comodules. There is a sequence of Ext groups for nonnegative integers n with and there are Bockstein spectral sequences computing a module (n – s) from So far, a small number of the E1-terms are determined. Here, we determine the for p > 2 and n > 3 by computing the Bockstein spectral sequence with E1-term for s = 1, 2. As an app

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2

Baues, Hans-Joachim, and Mamuka Jibladze. "Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence." Journal of K-Theory 7, no. 2 (2011): 203–347. http://dx.doi.org/10.1017/is010010029jkt133.

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AbstractWe describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

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3

Smirnov, V. A. "Bott’s periodicity theorem and differentials of the Adams spectral sequence of homotopy groups of spheres." Mathematical Notes 84, no. 5-6 (2008): 710–17. http://dx.doi.org/10.1134/s0001434608110126.

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4

Bendersky, Martin, and John R. Hunton. "ON THE COALGEBRAIC RING AND BOUSFIELD–KAN SPECTRAL SEQUENCE FOR A LANDWEBER EXACT SPECTRUM." Proceedings of the Edinburgh Mathematical Society 47, no. 3 (2004): 513–32. http://dx.doi.org/10.1017/s0013091503000518.

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AbstractWe construct a Bousfield–Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum $E$ with unit and which is related to the homotopy groups of a certain unstable $E$ completion $X_E^{\wedge}$ of a space $X$. For $E$ an $\mathbb{S}$-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) $E_*(\underline{E}_*)$ for an arbitrary Landweber exact spectrum $E$, extending work of the second author with Hopkins and wi

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5

Hu, Po, Igor Kriz, and Kyle Ormsby. "Remarks on motivic homotopy theory over algebraically closed fields." Journal of K-Theory 7, no. 1 (2010): 55–89. http://dx.doi.org/10.1017/is010001012jkt098.

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AbstractWe discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

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6

Ray, Nigel. "BORDISM, STABLE HOMOTOPY AND ADAMS SPECTRAL SEQUENCES (Fields Institute Monographs 7)." Bulletin of the London Mathematical Society 30, no. 1 (1998): 109–11. http://dx.doi.org/10.1112/s0024609397263459.

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7

Isaksen, Daniel C., Guozhen Wang, and Zhouli Xu. "Stable homotopy groups of spheres." Proceedings of the National Academy of Sciences 117, no. 40 (2020): 24757–63. http://dx.doi.org/10.1073/pnas.2012335117.

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We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is the

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8

Greenlees, J. P. C. "Homotopy equivariance, strict equivariance and induction theory." Proceedings of the Edinburgh Mathematical Society 35, no. 3 (1992): 473–92. http://dx.doi.org/10.1017/s0013091500005757.

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An obvious question occurs at the very start of equivariant homotopy theory. What is the relationship between maps equivariant up to homotopy and strictly equivariant maps? This question has been studied by various people, usually away from the group order ([8, 11, 22, 25, 26]). We consider the problem stably and answer it by giving a spectral sequence proceeding from homotopy equivariant to strictly equivariant information. The form of the spectral sequence is not surprising, but there are three distinctive features of our approach: (1) we show that the spectral sequence may be viewed as an A

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9

Lin, Wen-Hsiung. "Some infinite families in the stable homotopy of spheres." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 3 (1987): 477–85. http://dx.doi.org/10.1017/s0305004100066858.

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10

Díaz, Antonio, Albert Ruiz, and Antonio Viruel. "Cohomological uniqueness of some p-groups." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (2012): 449–68. http://dx.doi.org/10.1017/s0013091512000247.

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AbstractWe consider classifying spaces of a family of p-groups and prove that mod p cohomology enriched with Bockstein spectral sequences determines their homotopy type among p-completed CW-complexes.

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11

Isaksen, Daniel C., and Armira Shkembi. "Motivic connective K-theories and the cohomology of A(1)." Journal of K-theory 7, no. 3 (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 speci

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12

Xiugui, Liu, and Jin Yinglong. "Detection of a new nontrivial family in the stable homotopy of spheres $ \mbf{\pi_{\ast}S} $." Tamkang Journal of Mathematics 39, no. 1 (2008): 75–83. http://dx.doi.org/10.5556/j.tkjm.39.2008.47.

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To determine the stable homotopy groups of spheres is one of the central problems in homotopy theory. Let $ A $ be the mod $ p $ Steenrod algebra and $S$ the sphere spectrum localized at an odd prime $ p $. In this article, it is proved that for $ p\geqslant 7 $, $ n\geqslant 4 $ and $ 3\leqslant s $, $ b_0 h_1 h_n \tilde{\gamma}_{s} \in Ext_A^{s+4,\ast}(\mathbb{Z}_p,\mathbb{Z}_p) $ is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element of order $ p $ in the stable homotopy groups of spheres $ \pi_{p^nq+sp^{2}q+(s+1)pq+(s-2)q-7}S $, where $ q=2(p-1 ) $.

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13

Greenlees, J. P. C. "Generalized Eilenberg–Moore spectral sequences for elementary abelian groups and tori." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (1992): 77–89. http://dx.doi.org/10.1017/s0305004100070778.

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AbstractIn this note we prove universal coefficient theorems for Borel cohomology and related theories. Whatever other merit this may have the comment of Borel [5] applies ‘ …elle a au moms l'utilité de bien mettre en évidence le rôle fondamental joué dans cette question par la cohomologie des groupes’.Indeed the purpose of the enterprise is to use homological properties of the group cohomology ring H*(BG+) to study properties of G-spaces. Because of the relative simplicity of ordinary cohomology much attention in the proofs and applications is concentrated on change of groups, and on changes

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14

Cuevas-Rozo, Julián, Jose Divasón, Miguel Marco-Buzunáriz, and Ana Romero. "Integration of the Kenzo System within SageMath for New Algebraic Topology Computations." Mathematics 9, no. 7 (2021): 722. http://dx.doi.org/10.3390/math9070722.

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This work integrates the Kenzo system within Sagemath as an interface and an optional package. Our work makes it possible to communicate both computer algebra programs and it enhances the SageMath system with new capabilities in algebraic topology, such as the computation of homotopy groups and some kind of spectral sequences, dealing in particular with simplicial objects of an infinite nature. The new interface allows computing homotopy groups that were not known before.

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15

Muranov, Yuri V., and Anna Szczepkowska. "Path homology theory of edge-colored graphs." Open Mathematics 19, no. 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 corresp

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