Homotopy classes of self-maps and induced homomorphisms of homotopy groups

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For a based space <i>X</i>, we consider the group $¥mathscr{E}$<sub>#<i>n</i></sub>(<i>X</i>) of all self homotopy classes α of <i>X</i> such that α<sub>#</sub> = id:π<sub><i>i</i></sub>(<i>X</i>) → π<sub><i>i</i></sub>(<i>X</i>), for all <i>i</i>≤<i>n</i>, where <i>n</i>≤∞, and the group $¥mathscr{E}$<sub>Ω</sub>(<i>X</i>) of all α such that Ωα = id. Analogously, we study the semigroups $¥mathscr{Z}$<sub>#<i>n</i></sub>(<i>X</i>) and $¥mathscr{Z}$<sub>Ω</sub>(<i>X</i>) defined by replacing 'id' by '0' above. There is a chain of containments of the $¥mathscr{E}$-groups and the $¥mathscr{Z}$-semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on <i>X</i> which ensure that the $¥mathscr{E}$-groups and the $¥mathscr{Z}$-semigroups are equal. When <i>X</i> is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the $¥mathscr{E}$-groups and $¥mathscr{Z}$-groups of certain low dimensional Lie groups.

収録刊行物

  • Journal of the Mathematical Society of Japan  

    Journal of the Mathematical Society of Japan 58(2), 401-418, 2006-04-01 

    The Mathematical Society of Japan

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各種コード

  • NII論文ID(NAID)
    10018381284
  • NII書誌ID(NCID)
    AA0070177X
  • 本文言語コード
    ENG
  • 資料種別
    ART
  • ISSN
    00255645
  • NDL 記事登録ID
    7892114
  • NDL 雑誌分類
    ZM31(科学技術--数学)
  • NDL 請求記号
    Z53-A209
  • データ提供元
    CJP書誌  NDL  J-STAGE 
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