• Title, Summary, Keyword: cohomology groups

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THE KÜNNETH ISOMORPHISM IN BOUNDED COHOMOLOGY PRESERVING THE NORMS

  • Park, HeeSook
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.873-890
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    • 2020
  • In this paper, for discrete groups G and K, we show that the cohomology of the complex of projective tensor product B*(G)⨶B*(K) is isomorphic to the bounded cohomology Ĥ*(G × K) of G × K, which is the cohomology of B*(G × K) as topological vector spaces, where B*(G) is a complex of bounded cochains of G with real coefficients ℝ. In fact, we construct an isomorphism between these two cohomology groups that carries the canonical seminorm in Ĥ*(G × K) to the seminorm in the cohomology of B*(G)⨶B*(K).

SOME REMARKS ON BOUNDED COHOMOLOGY GROUP OF PRODUCT OF GROUPS

  • Park, HeeSook
    • Honam Mathematical Journal
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    • v.41 no.3
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    • pp.631-650
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    • 2019
  • In this paper, for discrete groups G and K, we show that the bounded cohomology group of $G{\times}K$ is isomorphic to the cohomology group of the complex of the projective tensor product $B^*(G){\hat{\otimes}}B^*(K)$, where $B^*(G)$ and $B^*(G)$ are the complexes of bounded cochains with real coefficients ${\mathbb{R}}$ of G and K, respectively.

Cohomology Groups of the Separated Spaces

  • Park, Boo Ja
    • Journal of the Chungcheong Mathematical Society
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    • v.3 no.1
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    • pp.57-62
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    • 1990
  • In this note we analyse establish the cohomology groups of the topological space which is separated by infinitely many open subspaces.

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SOME RESULTS ON THE SECOND BOUNDED COHOMOLOGY OF A PERFECT GROUP

  • Park, Hee-Sook
    • Honam Mathematical Journal
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    • v.32 no.2
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    • pp.227-237
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    • 2010
  • For a discrete group G, the kernel of a homomorphism from bounded cohomology $\hat{H}^*(G)$ of G to the ordinary cohomology $H^*(G)$ of G is called the singular part of $\hat{H}^*(G)$. We give some results on the space of the singular part of the second bounded cohomology of G. Also some results on the second bounded cohomology of a uniformly perfect group are given.

Cohomology of flat vector bundles

  • Kim, Hong-Jong
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.391-405
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    • 1996
  • In this article, we calculate the cohomology groups of flat vector bundles on some manifolds.

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SOME EXAMPLES OF RELATIONS BETWEEN NON-STABLE INTEGRAL COHOMOLOGY OPERATIONS

  • Percy, Andrew
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.275-286
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    • 2010
  • The algebraic structure of the natural integral cohomology operations is explored by means of examples. We decompose the generators of the groups $H^m(\mathbb{Z},\;n)$ with $2\;{\leq}\;n\;{\leq}\;7$ and $2\;{\leq}\;m\;{\leq}\;13$ into the operations of cup products, cross-cap products and compositions. Examination of these decompositions and comparison with other possible generators demonstrates the existence of relations between integral operations that have withheld formulation. The calculated groups and generators are collected in a table for practical reference.

GENERALIZED COHOMOLOGY GROUP OF TRIANGULAR BANACH ALGEBRAS OF ORDER THREE

  • Motlagh, Abolfazl Niazi;Bodaghi, Abasalt;Tanha, Somaye Grailoo
    • Honam Mathematical Journal
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    • v.42 no.1
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    • pp.105-121
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    • 2020
  • The main result of this article is to factorize the first (σ, τ)-cohomology group of triangular Banach algebra 𝓣 of order three with coefficients in 𝓣 -bimodule 𝓧 to the first (σ, τ)-cohomology groups of Banach algbras 𝓐, 𝓑 and 𝓒, where σ, τ are continuous homomorphisms on 𝓣. As a direct consequence, we find necessary and sufficient conditions for 𝓣 to be (σ, τ)-weakly amenable.

ON THE ACTIONS OF HIGMAN-THOMPSON GROUPS BY HOMEOMORPHISMS

  • Kim, Jin Hong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.449-457
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    • 2020
  • The aim of this short paper is to show some rigidity results for the actions of certain finitely presented groups by homeomorphisms. As an interesting and special case, we show that the actions of Higman-Thompson groups by homeomorphisms on a cohomology manifold with a non-zero Euler characteristic should be trivial. This is related to the wellknown Zimmer program and shows that the actions by homeomorphism could be very much different from those by diffeomorphisms.

STABLE SPLITTINGS OF BG FOR GROUPS WITH PERIODIC COHOMOLOGY AND UNIVERSAL STABLE ELEMENTS

  • Lim, Pyung-Ki
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.109-114
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    • 1989
  • This paper deals with the classifying spaces of finite groups. To any finite group G we associate a space BG with the property that .pi.$_{1}$(BG)=G, .pi.$_{i}$ (BG)=0 for i>1. BG is called the classifying space of G. Consider the problem of finding a stable splitting BG= $X_{1}$$^{V}$ $X_{1}$$^{V}$..$^{V}$ $X_{n}$ localized at pp. Ideally the $X_{i}$ 's are indecomposable, thus displaying the homotopy type of BG in the simplest terms. Such a decomposition naturally splits $H^{*}$(BG). The main purpose of this paper is to give the classification theorem in stable homotopy theory for groups with periodic cohomology i.e. cyclic Sylow p-subgroups for p an odd prime and to calculate some universal stable element. In this paper, all cohomology groups are with Z/p-coefficients and p is an odd prime.prime.

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