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The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps

Open Access article
 Printed: Sep 2012
  • Norio Iwase,
    Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
  • Mamoru Mimura,
    Department of Mathematics, Okayama University, Okayama 700-8530, Japan
  • Nobuyuki Oda,
    Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan
  • Yeon Soo Yoon,
    Department of Mathematics Education, Hannam University, Daejeon 306-791, Korea
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The concept of $C_{k}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The $C_{k}$-space corresponds to the $k$-th Milnor-Stasheff filtration on spaces. It is proved that a space $X$ is a $C_{k}$-space if and only if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\, Z\le k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a map $f\colon A \to X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of $C_{k}$-spaces, and non-$C_{k}$-spaces.
Keywords: Gottlieb sets for maps, L-S category, T-spaces Gottlieb sets for maps, L-S category, T-spaces
MSC Classifications: 55P45, 55P35 show english descriptions $H$-spaces and duals
Loop spaces
55P45 - $H$-spaces and duals
55P35 - Loop spaces

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