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

• Arkowitz Martin

  Dartmouth College

• Oshima Hideaki

  Ibaraki University

• Strom Jeffrey

  Western Michigan University

Abstract

For a based space X, we consider the group $¥mathscr{E}$_#n(X) of all self homotopy classes α of X such that α_# = id:π_i(X) → π_i(X), for all i≤n, where n≤∞, and the group $¥mathscr{E}$_Ω(X) of all α such that Ωα = id. Analogously, we study the semigroups $¥mathscr{Z}$_#n(X) and $¥mathscr{Z}$_Ω(X) 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 X which ensure that the $¥mathscr{E}$-groups and the $¥mathscr{Z}$-semigroups are equal. When X 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.

Journal

• Journal of the Mathematical Society of Japan

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

  The Mathematical Society of Japan

Keywords

• homotopy equivalences
• homotopy groups
• group-like spaces
• Lie groups

Details

• CRID: 1390282680093236224
• NII Article ID: 10018381284
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/1149166782
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 2228566
• NDL BIB ID: 7892114
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I7892114
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed