Infinitely generated objects (1-2 dimensional wild spaces and fundamental groups)

• Project/Area Number: 23540110
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Waseda University
• Principal Investigator: Eda Katsuya 早稲田大学, 理工学術院, 教授 (90015826)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: fundamental group / wild spaces / one dimensional / two dimensional / Peano continua / singular homology group / topological group / grope group / singular homology / uncountable group / covering homomorphism / overlay / wild space / uncountable groups / one dimension / inverse limit / free groups / grope groups / Hawaiian Earring / n-slender / solenoid / cover

Outline of Final Research Achievements

We studied on the subjects (1) 2 dimensional nonaspherical cell-like continua [2,3]; (2) The inverse limits of finitely generated free groups [4]; (3) Grope groups [5] (4) Covering maps over topological groups [6]; (5) Singular homology groups of one-dimensional Peano continua [1].

(1) We propose four constructions of spaces which produce 2 dimensional nonaspherical cell-like continua. There exists a Peano continuum for each two of them which shows the difference of the two constructions. (2) The inverse limits of inverse sequences of free groups of finite rank are free groups of finite rank or four non-isomrphic groups. (3) If there exists a nontrivial homomorphism from the minimal grope group to another grope group, then the grope has a binary braching part as the minimal grope. (4) There exist infinite sheeted covering maps over any solenoids. (5) Singular homology groups of one-dimensional Peano continua are free abelian groups of finite rank or that of the Hawaiian earring.

Research Products

• [Journal Article] The classification of the inverse limits of free groups of finite rank (2013)
  • Author(s): K. Eda and J. Nakamura
  • Journal Title: Bull. London Math. Soc. Volume: 45 Issue: 4 Pages: 671-676
  • DOI: 10.1112/blms/bds123
  • Peer Reviewed
• [Journal Article] On Snake cones, alternating cones and related constructions (2013)
  • Author(s): K. Eda, U. Karimov, Dusan Repovs and A. Zastrow
  • Journal Title: Glasnik Mate. Volume: 48(68) Issue: 1 Pages: 155-135
  • DOI: 10.3336/gm.48.1.11
  • Peer Reviewed
• [Journal Article] Maps from the minimal grope to arbitrary gropes (2013)
  • Author(s): M. Cencelj, K. Eda, and Ales Vavpetic
  • Journal Title: Inter. Jour. Algebra Comp. Volume: 23 Issue: 03 Pages: 503-519
  • DOI: 10.1142/s0218196713500070
  • Peer Reviewed
• [Journal Article] On Snake cones, alternating cones and related constructions (2013)
  • Author(s): K. Eda, U. Karimov, D. Repovs and A. Zastrow
  • Journal Title: Glasnik Mate. Volume: -
  • Peer Reviewed
• [Journal Article] Maps from the minimal grope to arbitrary gropes (2013)
  • Author(s): M. Cencelj, K. Eda, and A. Vavpetic
  • Journal Title: Inter. Jour. Algebra Comp Volume: -
  • Peer Reviewed
• [Journal Article] Covering maps over solenoids which are not covering homomorphisms (2013)
  • Author(s): K. Eda and V. Matijevic
  • Journal Title: Fund. Math. Volume: -
  • Peer Reviewed
• [Journal Article] On 2-dimensional nonaspherical cell-like Peano continua: a simple approach (2013)
  • Author(s): K. Eda, U. Karimov and D. Repovs
  • Journal Title: Meditarranean J. Math. Volume: 10
  • Peer Reviewed
• [Journal Article] Atomic property of the fundamental group of the Hawaiian earring and wild Peano continua (2011)
  • Author(s): K. Eda
  • Journal Title: J. Math. Soc. Japan Volume: 63
  • Peer Reviewed
• [Journal Article] On the Singular Homology of One Class of Simply-connected Cell-like Spaces (2011)
  • Author(s): K. Eda, Umed H. Karimov and Dusan Repovs
  • Journal Title: Meditarranean J. Math. Volume: 8 Pages: 153-160
  • Peer Reviewed
• [Presentation] Covering maps over topological groups (2015)
  • Author(s): Katsuya Eda
  • Organizer: Dubrovnik VIII, Geometric Topology, Geometric Group Theory and Dynamical systems
  • Place of Presentation: Inter-University center Dubrovnik
  • Year and Date: 2015-06-22 – 2015-06-26
  • Invited
• [Presentation] Singular homology of Peano continua (2014)
  • Author(s): Katsuya Eda
  • Organizer: DMV-PTM 2014 Poznan
  • Place of Presentation: University of Poznan
  • Year and Date: 2014-09-18
• [Presentation] Atomic property of the fundamental group of the Hawaiian earring and wild Peano continua (2011)
  • Author(s): K. Eda
  • Organizer: Dubrovnik VII - Geometric Topology
  • Place of Presentation: Dubrovnik
• [Presentation] Group theoretic properties for wild algebraic topology (2011)
  • Author(s): K. Eda
  • Organizer: Workshop on Topology of Wild Spaces and Fractals(招待講演)
  • Place of Presentation: Strobl