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Topology of Infinite-Dimensional Manifolds and Universal Spaces

Research Project

Project/Area Number 14540059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionUniversity of Tsukuba

Principal Investigator

SAKAI Katsuro  University of Tsukuba, Garduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (50036084)

Co-Investigator(Kenkyū-buntansha) KAWAMURA Kazuhiro  University of Tsukuba, Garduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (40204771)
YAMAZAKI Kaori  University of Tsukuba, Garduate School of Pure and Applied Sciences, Instructor, 大学院・数理物質科学研究科, 助手 (80301076)
YAGASAKI Tasuhiko  Kyoto Institute of Technology, Department of Mathematics, Associate Professor, 工芸学部, 助教授 (40191077)
IWAMOTO Yutaka  Yuge National College of Maritime Technology, Associate Professor, 助教授 (10300641)
UEHARA Shigenori  Takamatsu National College of Technology, Assistant Professor, 講師 (80321496)
加藤 久男  筑波大学, 数学系, 教授 (70152733)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2002: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsinfinite-dimensional manifold / universal space / topology / hyperspace / mapping space / AR, ANR / absolute Borel class / Nobeling space / 下半連続実数値関数 / Hillert立方体 / 有限部分集合 / 非可分 / Nobeling空間 / Hausdorff距離位相 / Fell位相 / 閉凸集合 / Hilbert立方体 / Hilbert空間 / Hyperspace / closed convex set / normed linear space / Hausdorff metric / Attouch-Wets topology / Wijsman topology / uniform AR / AR
Research Abstract

Throughout three years, making our efforts to achieve the following three intensions in this subject, we have many results relating to each item.
1.Characterizing universal spaces for various classes of non-separable spaces which had not been investigated ;
2.In order to enrich the theory of infinite-dimensional manifolds, finding natural examples of non-separable infinite-dimensional manifolds and investigating their topological and geometrical structures ;
3.Studying other subjects related to this subject and applying each others.
For the first item, by the joint work of Sakai and his student Yaguchi and the work of the other student Mine, Bestvina-Mogilski's theory of absorbing sets have been extended to non-separable absolute Borel classes. By Iwamoto's studies, it have been clear that Ageev's proof of the characterization of Nobeling spaces contains gaps. The complete proof is excepted.
For the second item, we have many results concering hyperspaces and mapping spaces. The hyperspace of non-empty closed sets in a non-compact metric space is studied by introducing various topologies instead of the classical Vietoris topology, that is, the Hausdorff metric topology, the Fell topology, the Attouch-Wets topology, the Wijsman topology, etc. Sakai has studied with Banakh, Yang, Kubis and his students, Kurihara, Yaguchi and Mine, and obtained results that such hyperspaces are homeomorphic to infinite-dimensional universal spaces for some classes and that they are AR's even if they are not known about the above. On the other hand, Yagasaki have been working on the spaces of embeddings and homeomorphisms and he have many results. Moreover, Uehara had a result concerning the spaces of upper semi-ccontinuous set-valued functions.
Concering the third item, there are results by Sakai and Sakai-Iwamoto and Kawamura and Yamazaki have many interesting results.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (60 results)

All 2005 2004 2003 2002 Other

All Journal Article (48 results) Publications (12 results)

  • [Journal Article] Hyperspaces of separable Banach spaces with the Wijsman topology2005

    • Author(s)
      W.Kubis, 酒井 克郎, 矢口 雅人
    • Journal Title

      Topology and its Applications 148

      Pages: 7-32

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On a conjecture of Wood2005

    • Author(s)
      川村 一宏
    • Journal Title

      Glasgow Mathematical Journal 47

      Pages: 1-5

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of separable Banach spaces with the Wijsman topology2005

    • Author(s)
      W.Kubis, K.Sakai, M.Yaguchi
    • Journal Title

      Topology and its Applications vol.148

      Pages: 7-32

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On a conjecture of Wood2005

    • Author(s)
      K.Kawamura
    • Journal Title

      Glasgow Mathematical Journal vol.47

      Pages: 1-5

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of separable Banach spaces with the Wijsman topology2005

    • Author(s)
      W.Kulis, K.Sakai, M.Yaguchi
    • Journal Title

      Topology and its Application 148

      Pages: 7-32

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hyperspaces of Banach spaces with the Attouch-Wets topolog2004

    • Author(s)
      酒井 克郎, 矢口 雅人
    • Journal Title

      Set-Valued Analysis 12

      Pages: 329-344

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of Banach spaces with the Attouch-Wets topology2004

    • Author(s)
      K.Sakai, M.Yaguchi
    • Journal Title

      Set-Valued Analysis vol.12

      Pages: 329-344

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of Banach spaces with the Attouch-Wets topology2004

    • Author(s)
      K.Sakai, M.Yaguchi
    • Journal Title

      Set-Valued Analysis 12

      Pages: 329-344

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Characterizing of manifold modeled on certain dense subspaces of non-separable Hilbert spaces2003

    • Author(s)
      酒井 克郎, 矢口 雅人
    • Journal Title

      Tsukuba Journal of Mathematics 27

      Pages: 143-159

    • NAID

      120001139458

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The space of limits of continua in the Fell topology2003

    • Author(s)
      Z.Yang, 酒井 克郎
    • Journal Title

      Houston Journal of Mathematics 29

      Pages: 327-337

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of normed linear spaces with the Attouch-Wets topology2003

    • Author(s)
      T.Banakh, 栗原 正幸, 酒井 克郎
    • Journal Title

      Set-Valued Analysis 11

      Pages: 21-36

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The groups of PL and Lipschitz homeomorphisms of noncompact 2-manifolds2003

    • Author(s)
      矢ヶ崎 達彦
    • Journal Title

      Bulletin of Polish Academy of Sciences, Mathematics 53

      Pages: 445-466

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Spaces of upper semi-continuous multi-valued functions which are absolute retracts2003

    • Author(s)
      上原 成功
    • Journal Title

      Bulletin of Polish Academy of Sciences, Mathematics 51

      Pages: 319-328

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extending point-finite partitions of unity2003

    • Author(s)
      山崎 薫里
    • Journal Title

      Houston Journal of Mathematics 29

      Pages: 353-359

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Selections and sandwich-like properties via semi-continuous Banach-valued functions2003

    • Author(s)
      V.Gutev, 大田 春外, 山崎 薫里
    • Journal Title

      Journal of the Mathematical Society of Japan 55

      Pages: 499-521

    • NAID

      10011796682

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Characterizing of manifold modeled on certain dense subspaces of non-separable Hilbert spaces2003

    • Author(s)
      K.Sakai, M.Yaguchi
    • Journal Title

      Tsukuba Journal of Mathematics vol.27

      Pages: 143-159

    • NAID

      120001139458

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The space of limits of continua in the Fell topology2003

    • Author(s)
      Z.Yang, K.Sakai
    • Journal Title

      Houston Journal of Mathematics vol.29

      Pages: 327-337

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of normed linear spaces with the Attouch-Wets topology2003

    • Author(s)
      T.Banakh, M.Kurihara, K.Sakai
    • Journal Title

      Set-Valued Analysis vol.11

      Pages: 21-36

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The groups of PL and Lipschitz homeomorphisms of noncompact 2 -manifolds2003

    • Author(s)
      T.Yagasaki
    • Journal Title

      Bulletin of Polish Academy of Sci., Math. vol.51

      Pages: 445-466

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Spaces of upper semi-continuous multi-valued functions which are absolute retracts2003

    • Author(s)
      S.Uehara
    • Journal Title

      Bulletin of Polish Academy of Sci., Math. vol.51

      Pages: 319-328

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extending point-finite partitions of unity2003

    • Author(s)
      K.Yamazaki
    • Journal Title

      Houston Journal of Mathematics vol.29

      Pages: 353-359

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Selections and sandwich-like properties via semi-continuous Banach-valued functions2003

    • Author(s)
      V.Gutev, H.Ohta, K.Yamazaki
    • Journal Title

      Journal of the Math.Soc.Japan vol.55

      Pages: 499-521

    • NAID

      10011796682

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube2002

    • Author(s)
      酒井 克郎, Z.Yang
    • Journal Title

      Topology and its Applications 127

      Pages: 331-342

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Strong n-shape theory2002

    • Author(s)
      酒井 克郎, 岩本 豊
    • Journal Title

      Topology and its Applications 122

      Pages: 253-267

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An alternative proof of Cauty-Geoghegan's characterization of ANR's2002

    • Author(s)
      酒井 克郎
    • Journal Title

      Topology and its Applications 122

      Pages: 407-414

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Actions of finitely generated groups on the universal Menger curve2002

    • Author(s)
      川村 一宏
    • Journal Title

      Proceeding of the American Mathematical Society 130

      Pages: 609-611

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A note on singular homology groups of infinite product of compacta2002

    • Author(s)
      川村 一宏
    • Journal Title

      Fundamenta Mathematicae 175

      Pages: 285-289

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Embedding spaces and hyperspaces of polyhedra in 2-manifolds2002

    • Author(s)
      矢ヶ崎 達彦
    • Journal Title

      Topology and its Applications 121

      Pages: 247-254

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube2002

    • Author(s)
      K.Sakai, Z.Yang
    • Journal Title

      Topology and its Applications vol.127

      Pages: 331-342

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Strong n-shape theory2002

    • Author(s)
      Y.Iwamoto, K.Sakai
    • Journal Title

      Topology and its Applications vol.122

      Pages: 253-267

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] An alternative proof of Cauty-Geoghegan's characterization of ANR's2002

    • Author(s)
      K.Sakai
    • Journal Title

      Topology and its Applications vol.122

      Pages: 407-414

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Actions of finitely generated groups on the universal Menger curve2002

    • Author(s)
      K.Kawamura
    • Journal Title

      Proceedings of the Amer.Math.Soc. vol.130

      Pages: 609-611

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A note on singular homology groups of infinite product of compacta2002

    • Author(s)
      K.Kawamura
    • Journal Title

      Fundamenta Mathematicae vol.175

      Pages: 285-289

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Embedding spaces and hyperspaces of polyhedra in 2-manifolds2002

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its Applications vol.121

      Pages: 247-254

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces with the Hausdorff metric and uniform ANR's

    • Author(s)
      栗原 正幸, 酒井 克郎, 矢口 雅人
    • Journal Title

      Journal of the Mathematical Society of Japan (印刷中)

    • NAID

      10016420078

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The AR-property of the spaces of closed convex sets

    • Author(s)
      酒井 克郎, 矢口 雅人
    • Journal Title

      Colloquium Mathematicum (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds

    • Author(s)
      矢ヶ崎 達彦
    • Journal Title

      Topology and its Applications (印刷中)

    • NAID

      110001127618

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extending pointwise bounded equicontinuous collections of functions

    • Author(s)
      山崎 薫里
    • Journal Title

      Tsukuba Journal of Mathematics (掲載予定)

    • NAID

      120002598887

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Controlled extensions of products of continuous functions

    • Author(s)
      山崎 薫里
    • Journal Title

      Houston Journal of Mathematics (掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces with the Hausdorff metric and uniform ANR's

    • Author(s)
      M.Kurihara, K.Sakai, M.Yaguchi
    • Journal Title

      Journal of the Math.Soc.Japan (in press)

    • NAID

      10016420078

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The AR-property of the spaces of closed convex sets

    • Author(s)
      K.Sakai, M.Yaguchi
    • Journal Title

      Colloquium Mathematicum (to appear.)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its Applications (in press)

    • NAID

      110001127618

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Extending pointwise bounded equicontinuous collections of functions

    • Author(s)
      K.Yamazaki
    • Journal Title

      Tsukuba Journal of Mathematics (to appear.)

    • NAID

      120002598887

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Controlled extensions of products of continuous functions

    • Author(s)
      K.Yamazaki
    • Journal Title

      Houston Journal of Mathematics (to appear.)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Hyperspaces with the Hausdorff metric and uniform ANR's

    • Author(s)
      M.Kurihara, K.Sakai, M.Yaguchi
    • Journal Title

      Journal of Mathematical Society of Japan (印刷中)

    • NAID

      10016420078

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The AR-property of the space of closed convex sets

    • Author(s)
      K.Sakai, M.Yaguchi
    • Journal Title

      Colloquium Mathematicum (掲載決定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Homology types of the components of spaces of embeddings of compact polyhedra into 2-manifolds

    • Author(s)
      T.Yagasaki
    • Journal Title

      Topology and its Applications (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Controlled extensions of products of continuous functions

    • Author(s)
      K.Yamazaki
    • Journal Title

      Houston Journal of Mathematics (掲載決定)

    • Related Report
      2004 Annual Research Report
  • [Publications] K.Sakai, Z.Yang: "The space of limits of continua in the Fell topology"Houston Journal of Mathematics. 29(2). 325-335 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Sakai, M.Yaguchi: "Characterizing manifolds modeled on certain dense subspaces of nonseparable Hilbert spaces"Tsukuba Journal of Mathematics. 27(1). 143-159 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Uehara: "Spaces of upper semi-continuous multi-valued functions which are absolute retracts"Bulletin of Polish Academy of Sciences, Mathematics. 51(3). 319-328 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Yagasaki: "The groups of PL and Lipschitz homeomorphisms of noncompact 2-manifolds"Bulletin of Polish Academy of Sciences, Mathematics. 51(4). 445-466 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Kawamura: "Low dimensional homotopy groups of suspensions of the Hawaiian earring"Colloquium Mathematicum. 96(1). 27-39 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Yamazaki: "Extending point-finite partitions of unity"Houston Journal of Mathematics. 29(2). 353-359 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 酒井克郎: "An alternative proof of Cauty-Geoghegan's characterization of ANRs"Topology and its Applications. 122・1〜2. 407-414 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 酒井克郎, 岩本豊: "Strong n-Shape theory"Topology and its Applications. 122・1〜2. 253-267 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 酒井克郎, Z.Yang: "Hyperspaces of noncompact metrizable spaces which are homeomorphlc to the Hilbert cube"Topology and its Applications. 127・3. 331-342 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 酒井克郎, T.Banakh: "Hyperspaces of normed linearspaces with the Attouch-wets topology"Set-Valued Analysis. 11・1. 21-36 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 酒井克郎, Z.Yang: "The space of limits of continua in the Fell topology"Hauston Journal of Mathematics. (掲載予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 酒井克郎, 矢口雅人: "Characterizing manifolds modeled on certain deuse subspaces of non-separable Hilbert spaces"Tsukuba Journal of Mathematics. (掲載予定).

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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