Homotopy theory and generalized dimension subgroups

Ivanov, Sergei O.; Mikhailov, Roman; Wu, Jie

doi:10.1016/j.jalgebra.2017.04.012

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…Here X is the homotopy pushout described above, ΩX is the loop space, the lower horizontal map is the Hurewicz homomorphism, the upper one is the natural map induced by g → g − 1. For a connected space Y , the kernel of the second Hurewicz homomorphism π 2 (Y ) → H 2 (Y ) is a 2-torsion group [20]. As a consequence we get the following theorem from [20].…”

Section: Consider a Homotopy Pushoutmentioning

confidence: 90%

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Homotopy patterns in group theory

Mikhailov¹

2021

Preprint

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This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of non-additive functors, homotopy groups of spheres, group homology etc appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.

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Section: Consider a Homotopy Pushoutmentioning

confidence: 90%

“…For a connected space Y , the kernel of the second Hurewicz homomorphism π 2 (Y ) → H 2 (Y ) is a 2-torsion group [20]. As a consequence we get the following theorem from [20].…”

Section: Consider a Homotopy Pushoutmentioning

confidence: 90%

Homotopy patterns in group theory

Mikhailov¹

2021

Preprint

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“…For example, the socialist core values can guide a healthy online culture, carry out a variety of campus cultural activities, strengthen the humanities education for students, and enhance the humanity of students. An open and diverse campus network culture is shaped [5].…”

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confidence: 99%

College Network Culture Construction based on Socialist Core Values

Niu¹,

Zhang²

2019

Proceedings of the 2018 International Workshop on Education Reform and Social Sciences (ERSS 2018)

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In order to promote advanced culture to effectively lead the healthy and harmonious development of campus network culture, socialist core values were integrated into the construction of university network culture. Relevant theories and concepts were introduced. Based on this, the data obtained from the survey were analyzed and summarized through empirical analysis. The shortage of college network culture construction was summarized. Combined with the socialist core values, corresponding suggestions were put forward. The results showed that the socialist core values promoted the healthy and harmonious development of campus network culture. which created a warm spiritual home for teachers and students. Therefore, the core values of socialism can guide a healthy network culture, strengthen the humanities education for students, enhance the humanities quality of students, so as to create a more open and diverse campus network culture.

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On lengths of Hℤ-localization towers

Ivanov

Mikhailov

2018

Isr. J. Math.

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In this paper, the HZ-length of different groups is studied. By definition, this is the length of HZ-localization tower or the length of transfinite lower central series of HZ-localization. It is proved that, for a free noncyclic group, its HZ-length is ≥ ω + 2. For a large class of Z[C]-modules M, where C is an infinite cyclic group, it is proved that the HZ-length of the semi-direct product M ⋊ C is ≤ ω + 1 and its HZ-localization can be described as a central extension of its pro-nilpotent completion. In particular, this class covers modules M , such that M ⋊ C is finitely presented and H 2 (M ⋊ C) is finite.

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Homotopy theory and generalized dimension subgroups

Cited by 3 publications

References 17 publications

Homotopy patterns in group theory

Homotopy patterns in group theory

College Network Culture Construction based on Socialist Core Values

On lengths of Hℤ-localization towers

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