2022/08/17 更新

写真a

エダ カツヤ
江田 勝哉
所属
理工学術院
職名
名誉教授

学歴

  •  
    -
    1971年

    東京教育大学   理学研究科   応用数理学  

学位

  • 筑波大学   理学博士

所属学協会

  •  
     
     

    日本数学会

 

研究分野

  • 数学基礎

  • 応用数学、統計数学

  • 幾何学

研究キーワード

  • 応用集合論

論文

  • Existence and uniqueness of group structures on covering spaces over groups

    Katsuya Eda, Vlasta Matijevic

    FUNDAMENTA MATHEMATICAE   238 ( 3 ) 241 - 267  2017年  [査読有り]

     概要を見る

    Let f : X -> Y be a covering map from a connected space X onto a topological group Y and let x(0) is an element of X be a point such that f(x(0)) is the identity of Y. We examine if there exists a group operation on X which makes X a topological group with identity x(0) and f a homomorphism of groups. We prove that the answer is positive in two cases: if f is an overlay map over a locally compact group Y, and if Y is locally compactly connected. In this way we generalize previous results for overlay maps over compact groups and covering maps over locally path-connected groups. Furthermore, we prove that in both cases the group structure on X is unique.

    DOI

  • Cotorsion-free groups from a topological viewpoint

    Katsuya Eda, Hanspeter Fischer

    TOPOLOGY AND ITS APPLICATIONS   214   21 - 34  2016年12月  [査読有り]

     概要を見る

    We present a characterization of cotorsion-free abelian groups in terms of homomorphisms from fundamental groups of Peano continua, which aligns naturally with the generalization of slenderness to non-abelian groups. In the process, we calculate the first homology group of the Griffiths twin cone. (C) 2016 Elsevier B.V. All rights reserved.

    DOI

  • Singular homology groups of one-dimensional Peano continua

    K. Eda

    FUNDAMENTA MATHEMATICAE   232 ( 2 ) 99 - 115  2016年  [査読有り]

     概要を見る

    Let X be a one-dimensional Peano continuum. Then the singular homology group H-1(X) is isomorphic to a free abelian group of finite rank or the singular homology group of the Hawaiian earring.

    DOI

  • Maps from the minimal grope to an arbitrary grope

    Matija Cencelj, Katsuya Eda, Aleš Vavpetič

    International Journal of Algebra and Computation   23 ( 3 ) 503 - 519  2013年05月  [査読有り]

     概要を見る

    We consider open infinite gropes and prove that every continuous map from the minimal grope to another grope is nulhomotopic unless the other grope has a «branch» which is a copy of the minimal grope. Since every grope is the classifying space of its fundamental group, the problem is translated to group theory and a suitable block cancellation of words is used to obtain the result. © 2013 World Scientific Publishing Company.

    DOI

  • On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    MEDITERRANEAN JOURNAL OF MATHEMATICS   10 ( 1 ) 519 - 528  2013年02月  [査読有り]

     概要を見る

    We construct a functor AC(-, -) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)-dimensional; and (iii) For a path connected space X, pi (1)(X, x) is trivial if and only if pi (2)(AC(X, x)) is trivial. As a corollary, AC(S (1), x) is a 2-dimensional nonaspherical cell-like Peano continuum.

    DOI

  • Covering maps over solenoids which are not covering homomorphisms

    Katsuya Eda, Vlasta Matijevic

    FUNDAMENTA MATHEMATICAE   221 ( 1 ) 69 - 82  2013年  [査読有り]

     概要を見る

    Let Y be a connected group and let f : X -> Y be a covering map with the total space X being connected: We consider the following question: Is it possible to define a topological group structure on X in such a way that f becomes a homomorphism of topological groups. This holds in some particular cases: if Y is a pathwise connected and locally pathwise connected group or if f is a finite-sheeted covering map over a compact connected group Y. However, using shape-theoretic techniques and Fox's notion of an overlay map, we answer the question in the negative. We consider infinite-sheeted covering maps over solenoids, i.e. compact connected 1-dimensional abelian groups. First we show that an infinite-sheeted covering map f : X -> Sigma with a total space being connected over a solenoid Sigma does not admit a topological group structure on X such that f becomes a homomorphism. Then, for an arbitrary solenoid Sigma, we construct a connected space X and an infinite-sheeted covering map f : X -> Sigma, which provides a negative answer to the question.

    DOI

  • On snake cones, alternating cones and related constructions

    Katsuya Eda, Umed H. Karimov, Dušan Repovš, Andreas Zastrow

    Glasnik Matematicki   48 ( 1 ) 115 - 135  2013年  [査読有り]

     概要を見る

    We show that the Snake on a square SC(S1) is homotopy equivalent to the space AC(S1) which was investigated in the previous work by Eda, Karimov and Repovš. We also introduce related constructions CSC(-) and CAC(-) and investigate homotopical differences between these four constructions. Finally, we explicitly describe the second homology group of the Hawaiian tori wedge.

    DOI

  • The classification of the inverse limits of sequences of free groups of finite rank

    Katsuya Eda, Jun Nakamura

    Bulletin of the London Mathematical Society   45 ( 4 ) 671 - 676  2013年  [査読有り]

     概要を見る

    We classify the groups that are the inverse limits of sequences of free groups of finite rank. Except free groups of finite rank, there exist four groups that are the inverse limits of sequences of free groups of finite rank. © 2013 London Mathematical Society.

    DOI

  • Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces

    Katsuya Eda

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   63 ( 3 ) 769 - 787  2011年07月  [査読有り]

     概要を見る

    We strengthen previous results on the fundamental groups of the Hawaiian earring and wild Peano continua. Let X be a path-connected, locally path-connected, first countable space which is not locally semi-simply connected at any point. If the fundamental group pi(1)(X) is a subgroup of a free product *H-j is an element of J(j), then it is contained in a conjugate subgroup to some H-j.

    DOI

  • On the Singular Homology of One Class of Simply-connected Cell-like Spaces

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    MEDITERRANEAN JOURNAL OF MATHEMATICS   8 ( 2 ) 153 - 160  2011年06月  [査読有り]

     概要を見る

    In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called Snake space. In the sequel we introduced the functor SC(-,-) defined on the category of all spaces with base points and continuous mappings. For the circle S (1), the space SC (S-1, *) is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC (Z, *) for any path-connected compact spaces Z.

    DOI

  • Atomic property of the fundamental groups of the Hawaiian earring and wild Peano continua

      ( 63 ) 769 - 787  2011年  [査読有り]

    DOI

  • ON THE ASPHERICITY OF ONE-POINT UNIONS OF CONES

    Katsuya Eda, Kazuhiro Kawamura

    TOPOLOGY PROCEEDINGS, VOL 36   36   63 - +  2010年

     概要を見る

    We prove that the one-point union of two copies of the cone over the Hawaiian earring is aspherical.

  • Homotopy types of one-dimensional Peano continua

    Katsuya Eda

    FUNDAMENTA MATHEMATICAE   209 ( 1 ) 27 - 42  2010年

     概要を見る

    Let X and Y be one-dimensional Peano continua. If the fundamental groups of X and Y are isomorphic, then X and Y are homotopy equivalent. Every homomorphism from the fundamental group of X to that of Y is a composition of a homomorphism induced from a continuous map and a base point change isomorphism.

    DOI

  • ON THE SECOND HOMOTOPY GROUP OF SC(Z)

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    GLASNIK MATEMATICKI   44 ( 2 ) 493 - 498  2009年  [査読有り]

     概要を見る

    In our earlier paper we introduced a cone-like space SC(Z). In the present note we establish some new algebraic properties of SC(Z).

  • A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions

    Katsuya Eda, Umed H. Karimov, Dusan Repovs

    TOPOLOGY AND ITS APPLICATIONS   156 ( 3 ) 515 - 521  2009年01月

     概要を見る

    We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950s. (C) 2008 Elsevier B.V. All rights reserved.

    DOI

  • Finite index supergroups and subgroups of torsion free abelian groups of rank two

    Katsuya Eda, Vlasta Matijevic

    JOURNAL OF ALGEBRA   319 ( 9 ) 3567 - 3587  2008年05月

     概要を見る

    Every torsionfree abelian group A of rank two is a subgroup of Q circle plus Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 x 2-matrices as the bonding maps. Using these direct systems we classify all subgroups of Q circle plus Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A <= Q circle plus Q and all supergroups are subgroups of Q circle plus Q:
    (1) for each natural numbers there are Sigma(q vertical bar s),(gcd(p,q)=1) q s-index supergroups and also Sigma(q vertical bar s),(gcd(p,q)=1) s-index subgroups;
    (2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index, subgroups are non-isonnorphic. (c) 2008 Elsevier Inc. All rights reserved.

    DOI

  • On the fundamental group of ℝ3 modulo the case-Chamberlin continuum

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Glasnik Matematicki   42 ( 1 ) 89 - 94  2007年06月

     概要を見る

    It has been known for a long time that the fundamental group of the quotient of ℝ3 by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.

    DOI

  • A construction of noncontractible simply connected cell-like two-dimensional Peano continua

    Katsuya Eda, Urned H. Karimov, Dusan Repovs

    FUNDAMENTA MATHEMATICAE   195 ( 3 ) 193 - 203  2007年

     概要を見る

    Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts from a noncontractible n-dimensional Peano continuum for any n > 0, then our construction yields a simply connected noncontractible (n + l)-dimensional cell-like Peano continuum. In particular, starting from the circle S-1, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.

    DOI

  • Finite sheeted covering maps over 2-dimensional connected, compact Abelian groups

    K Eda, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 7 ) 1033 - 1045  2006年01月

     概要を見る

    Let G be a 2-dimensional connected, compact Abelian group and s be a positive integer. We prove that a classification of s-sheeted covering maps over G is reduced to a classification of s-index torsionfree supergroups of the Pontrjagin dual (G) over cap. Using group theoretic results from earlier paper we demonstrate its consequences. We also prove that for a connected compact group Y:
    (1) Every finite-sheeted covering map from a connected space over Y is equivalent to a covering homomorphism from a compact, connected group.
    (2) If two finite-sheeted covering homomorphisms over Y are equivalent, then they are equivalent as topological homomorphisms. (C) 2005 Elsevier B.V. All rights reserved.

    DOI

  • Torus-like continua which are not self-covering spaces

    K Eda, J Mandic, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 359 - 369  2005年09月

     概要を見る

    For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T-2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f(0): X-0 -> Y, f(1) : X-1 -> Y, f(2): X-2 -> Y such that the total spaces X-0 = Y, X-1 and X-2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-shected covering map f(3): X-3 -> Y such that X-3 and Y are nonhomeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps. (c) 2004 Elsevier B.V. All rights reserved.

    DOI

  • Torus-like continua which are not self-covering spaces

    K Eda, J Mandic, Matijevic, V

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 359 - 369  2005年09月

     概要を見る

    For each non-quadratic p-adic integer, p > 2, we give an example of a torus-like continuum Y (i.e., inverse limit of an inverse sequence, where each term is the 2-torus T-2 and each bonding map is a surjective homomorphism), which admits three 4-sheeted covering maps f(0): X-0 -> Y, f(1) : X-1 -> Y, f(2): X-2 -> Y such that the total spaces X-0 = Y, X-1 and X-2 are pair-wise non-homeomorphic. Furthermore, Y admits a 2p-shected covering map f(3): X-3 -> Y such that X-3 and Y are nonhomeomorphic. In particular, Y is not a self-covering space. This example shows that the class of self-covering spaces is not closed under the operation of forming inverse limits with open surjective bonding maps. (c) 2004 Elsevier B.V. All rights reserved.

    DOI

  • Algebraic topology of Peano continua

    K Eda

    TOPOLOGY AND ITS APPLICATIONS   153 ( 2-3 ) 213 - 226  2005年09月

     概要を見る

    Let X be a Peano continuum. Then the following hold: (1) The singular cohomology group H I (X) is isomorphic to the tech cohomology group h I (X). (2) For each homomorphism h : pi(1)(X) -> *(i is an element of l)G(i) there exists a finite subset F of I such that Im(h) subset of *(i is an element of F)G(i). (3) For each injective homomorphism h : pi(1) (X) -> G(0) * G(1) there exists a finitely generated subgroup F-0 of G(0) or a finitely generated subgroup F-1 of G(1) such that Im(h) subset of F-0 * G(1) or Im(h) subset of G(0) * F-1. (c) 2004 Published by Elsevier B.V.

    DOI

  • Fundamental groups having the whole information of spaces

    G Conner, K Eda

    TOPOLOGY AND ITS APPLICATIONS   146   317 - 328  2005年01月

     概要を見る

    We introduce a new construction of spaces from groups using homomorphic images of the fundamental group of the Hawaiian earring H. According to this construction the Menger sponge, Sierpinski gasket, Sierpinski carpet and the direct product of their countable many copies are recovered from their fundamental groups. (C) 2004 Published by Elsevier B.V

    DOI

  • The fundamental groups of one-dimensional spaces and spatial homomorphisms

    Katsuya Eda

    Topology and its Applications   123 ( 3 ) 479 - 505  2002年09月

     概要を見る

    Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π1(ℍ) to π1(X) is induced from a continuous map up to the base-point-change isomorphism on π1(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π1(X) is isomorphic to π1(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π1(X) and π1(Y) are isomorphic. Moreover, each isomorphism from π1(X) to π1(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. © 2001 Elsevier Science B.V. All rights reserved.

    DOI

  • Fundamental groups of one-dimensional spaces and spatial homomorphisms

    K Eda

    TOPOLOGY AND ITS APPLICATIONS   123 ( 3 ) 479 - 505  2002年09月

     概要を見る

    Let X be a one-dimensional metric space and H be the Hawaiian earring.
    (1) Each homomorphism from pi(1) (H) to pi(1) (X) is induced from a continuous map up to the base-point-change isomorphism on pi(1) (X).
    (2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of H if and only if pi(1) (X) is isomorphic to pi(1) (H), if and only if X has a unique point at which X is not semi-locally simply connected.
    (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if pi(1) (X) and pi(1) (Y) are isomorphic. Moreover, each isomorphism from pi(1) (X) to pi(1) (Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism. (C) 2001 Elsevier Science B.V. All rights reserved.

    DOI

  • The non-commutative Specker phenomenon in the uncountable case

    K Eda, S Shelah

    JOURNAL OF ALGEBRA   252 ( 1 ) 22 - 26  2002年06月

    DOI

  • The non-commutative Specker phenomenon in the uncountable case

    K Eda, S Shelah

    JOURNAL OF ALGEBRA   252 ( 1 ) 22 - 26  2002年06月

    DOI

  • On (co)homology locally connected spaces

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Topology and its Applications   120 ( 3 ) 397 - 401  2002年05月

     概要を見る

    We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon. © 2002 Elsevier Science B.V. All rights reserved.

    DOI

  • On (co)homology locally connected spaces

    Katsuya Eda, Umed H. Karimov, Dušan Repovš

    Topology and its Applications   120 ( 3 ) 397 - 401  2002年05月

     概要を見る

    We prove that there exists a cohomology locally connected compact metrizable space which is not homology locally connected. In the category of compact Hausdorff spaces a similar result was proved earlier by G.E. Bredon. © 2002 Elsevier Science B.V. All rights reserved.

    DOI

  • Free subgroups of free complete products

    G Conner, K Eda

    JOURNAL OF ALGEBRA   250 ( 2 ) 696 - 708  2002年04月

    DOI

  • Free subgroups of free complete products

    G Conner, K Eda

    JOURNAL OF ALGEBRA   250 ( 2 ) 696 - 708  2002年04月

    DOI

  • The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

    Katsuya Eda

    Proceedings of the American Mathematical Society   130 ( 5 ) 1515 - 1522  2002年

     概要を見る

    Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C. Then the fundamental group of X cannot be embeddable into a free σ-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

    DOI

  • The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

    K Eda

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   130 ( 5 ) 1515 - 1522  2002年

     概要を見る

    Let X be a one-dimensional space which contains a copy C of a circle and let it not be semi-locally simply connected at any point on C: Then the fundamental group of X cannot be embeddable into a free sigma-product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.

    DOI

  • SLENDER MODULES, ENDO-SLENDER ABELIAN-GROUPS AND LARGE CARDINALS

    K EDA

    FUNDAMENTA MATHEMATICAE   135 ( 1 ) 5 - 24  1990年  [査読有り]

  • Algebraic Topology of Peano Continua

    K.Eda

    Topology Appl.   153   317 - 328

▼全件表示

共同研究・競争的資金等の研究課題

  • 無限生成の対象の研究(野性的空間の基本群)

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • Coarse幾何学におけるコホモロジー次元論

    科学研究費助成事業(静岡大学)  科学研究費助成事業(基盤研究(C))

  • 無限生成の対象の研究(1ー2次元の野性的空間と基本群)

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 無限生成の対象の研究(野性的空間の群論的研究)

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • 数理論理学の総合的研究

    科学研究費助成事業(九州大学)  科学研究費助成事業(総合研究(A))

  • 幾何学と情報数学

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • モデルの理論とその関連分野の総合的研究

    科学研究費助成事業(筑波大学)  科学研究費助成事業(総合研究(A))

  • 代数及び位相幾何における無限生成の対象の研究(数理論理学及び集合論の応用)

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • 情報伝達における数学の役割について

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • 曲面の写像類群と3次元多様体の研究

    科学研究費助成事業(筑波大学)  科学研究費助成事業(一般研究(C))

  • 高次無限公理とそれに関連する数学諸分野における命題についての研究

    科学研究費助成事業(神戸大学)  科学研究費助成事業(基盤研究(B))

  • 連続体の基数不変量とその応用

    科学研究費助成事業(神戸大学)  科学研究費助成事業(基盤研究(C))

  • 超準モデルとその計算量理論への応用

    科学研究費助成事業(名古屋大学)  科学研究費助成事業(基盤研究(C))

  • 無限生成の対象についての研究

    科学研究費助成事業(早稲田大学)  科学研究費助成事業(基盤研究(C))

  • プログラム自動生成システム

    文部科学省 

▼全件表示

特定課題研究

  • フラクタルの基本群(非可算非可換群)

    2001年  

     概要を見る

    1.フラクタルの多くはペアノ空間(局所連結、連結なコンパクト距離空間)である。成果発表予定の1番目の論文の内容はペアノ空間の基本群、ホモロジー群に関する結果である。定理1:局所的 n-連結なペアノ空間 X とスレンダ-群 S、特異ホモロジ-群 H_{n+1}(X) に対して、Hom (H_{n+1}(X), S) は有限被覆のなす多面体のホモロジー群の帰納極限と同型となる。とくに、ペアノ空間 X の特異1次コホモロジー群はチェックコホモロジー群と同型である。また定理1の非可換化として、ペアノ空間 X と非可換スレンダ-群 S、基本群に対して同様の結果が成立する。定理2:ペアノ空間 X の基本群から無限個の群の自由積への準同型像は有限個の群の自由積に含まれる。準同型写像が単射であるときはそれより強い次の定理が成立する。定理3:ペアノ空間 X の基本群が2つの群の自由積 G*H の部分群と同型ならば G の有限生成部分群 G' が存在して G'*H の部分群となるか、H の有限生成部分群 H' が存在して G*H' の部分群となる。2.一般に群 G に対して、空間 X_G を定義し、次が成立することを示した。定理4:空間が1次元局所連結、連結な距離空間またはそれらの直積であるとき、G をその空間の基本群とすれば、X_G はもとの空間と同相である。

海外研究活動

  • 無限生成の対象の研究(野性的空間の代数的位相幾何)

    2013年04月
    -
    2014年03月

    オーストリア   ウィーン工科大学

    アメリカ   ブリガムヤング大学