• 研究課題をさがす
  • 研究者をさがす
  • KAKENの使い方
  1. 課題ページに戻る

2015 年度 実施状況報告書

コンパクト型可換群の構造及びMarkov稠密性を実現する群位相の導入の研究

研究課題

研究課題/領域番号 26400091
研究機関愛媛大学

研究代表者

D・B Shakhmatov  愛媛大学, 理工学研究科, 教授 (90253294)

研究期間 (年度) 2014-04-01 – 2018-03-31
キーワードdirect products / direct sums / topological group / topological vector space / dual group / reflexive group / multiplier convergence / compact
研究実績の概要

We introduce three new notions for a subset A of an abelian topological group G: that of an absolutely Cauchy summable subset, of an absolutely summable subset and of a topologically independent subset. With these three notions at hand, we prove that: (1) an abelian topological group contains a direct product (direct sum) of kappa-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality kappa; (2) a topological vector space contains the direct sum of countably many real lines as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains the direct product of countably many lines as its subspace if and only if it has a multiplier convergent series of non-zero elements. We answer a question of Husek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.

We prove that the group G of all homomorphisms from the Baer-Specker group to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a countable free closed non-reflexive subgroup of the direct product of continuum many copies of the integer group Z, thereby answering a problem of Galindo, Recorder-Nunez and Tkachenko.

現在までの達成度 (区分)
現在までの達成度 (区分)

3: やや遅れている

理由

While working on main topics of this research project, we discovered a new technique that, when combined with our own methods from previous publications, lead to a solution of the thirty years old problem of Comfort-Protasov-Remus on the existence of minimally almost periodic topologies on abelian groups, as well as a problem of Gabriyelyan about the realization of von Neuman kernel by some Hausdorff group topology. These unexpected new developments got us temporarily sidetracked from the original problems.

今後の研究の推進方策

We shall investigate the realization of the algebraic (Markov-Zariski closure) of a given countable subset in some pseudocompact group topology on this group. We shall also investigate the existence of Hausdorff group topologies on abelian group with a compactness-like property sandwiched between countable compactness and pseudocompactness.

次年度使用額が生じた理由

I was not able to attend an international conference that I was planning to attend due to unexpected work during the period of the conference.

次年度使用額の使用計画

We plan to invite Professors M. J. Chasco (University of Navarra, Spain) and M. Tkachenko (UAM, Mexico) to Ehime University for a joint research in topological groups. The Incurring Amount is to be used for covering their lodging and travel expenses.

  • 研究成果

    (8件)

すべて 2016 2015 その他

すべて 国際共同研究 (1件) 雑誌論文 (1件) (うち国際共著 1件、 査読あり 1件、 謝辞記載あり 1件) 学会発表 (5件) (うち国際学会 4件、 招待講演 2件) 備考 (1件)

  • [国際共同研究] Udine University(Italy)

    • 国名
      イタリア
    • 外国機関名
      Udine University
  • [雑誌論文] Direct sums and products in topological groups and vector spaces2016

    • 著者名/発表者名
      D. Dikranjan, D. Shakhmatov, J. Spevak
    • 雑誌名

      Journal of Mathematical Analysis and its Applications

      巻: 437 ページ: 1257-1282

    • DOI

      10.1016/j.jmaa.2016.01.037

    • 査読あり / 国際共著 / 謝辞記載あり
  • [学会発表] Minimally alsmost periodic and connected group topologies on abelian groups2015

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      1st Pan Pacific International Conference on Topology and Applications (PPICTA)
    • 発表場所
      Minnan Normal University, Zhangzhou (中国)
    • 年月日
      2015-11-25 – 2015-11-30
    • 国際学会 / 招待講演
  • [学会発表] 任意の可換群上面白い群位相の定めかたについて2015

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      RIMS研究集会「集合論的位相幾何学および幾何学的トポロジーの最近の動向と展望」
    • 発表場所
      京都大学数理解析研究所(京都府京都市)
    • 年月日
      2015-11-16 – 2015-11-18
  • [学会発表] Markov's problem on the existence of connected Hausdorff group topologies on abelian groups2015

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      Alexandroff Topology Seminar at Moscow State University
    • 発表場所
      Moscow (Russia)
    • 年月日
      2015-09-24 – 2015-09-24
    • 国際学会 / 招待講演
  • [学会発表] Complete solution of Markov's problem on the existence of connected Hausdorff group topologies2015

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      International Conference on Topology, Messina 2015 (ICTM2015), On the occasion of Filippo Cammaroto's 65th birthday
    • 発表場所
      Messina (Italy)
    • 年月日
      2015-09-07 – 2015-09-11
    • 国際学会
  • [学会発表] Topological groups with many small subgroups2015

    • 著者名/発表者名
      D. Shakhmatov
    • 学会等名
      International Conference on Set-Theoretic Topology and its Applications (Joint with the 50th Symposium of General Topology and 2015 General Topology Symposium)
    • 発表場所
      神奈川大学(神奈川県横浜市)
    • 年月日
      2015-08-24 – 2015-08-26
    • 国際学会
  • [備考] Homepage

    • URL

      http://www.math.sci.ehime-u.ac.jp/~dima/

URL: 

公開日: 2017-01-06  

サービス概要 検索マニュアル よくある質問 お知らせ 利用規程 科研費による研究の帰属

Powered by NII kakenhi