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The Milnor–Stasheff Filtration on Spaces and Generalized Cyclic Maps

Published online by Cambridge University Press:  20 November 2018

Norio Iwase ,
Mamoru Mimura ,
Nobuyuki Oda  and
Yeon Soo Yoon
Norio Iwase
Affiliation:
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japane-mail: [email protected]
Mamoru Mimura
Affiliation:
Department of Mathematics, Okayama University, Okayama 700-8530, Japane-mail: [email protected]
Nobuyuki Oda
Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japane-mail: [email protected]
Yeon Soo Yoon
Affiliation:
Department of Mathematics Education, Hannam University, Daejeon 306-791, Koreae-mail: [email protected]
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Abstract

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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 cat $Z\,\le \,k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\sum B,\,X)\,=\,[\sum B,\,X]$ for any space $B$. Some results on the ${{C}_{k}}$-space are generalized to the $C_{k}^{f}$-space for a map $f\,:\,A\,\to \,X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of ${{C}_{k}}$-spaces, and non-${{C}_{k}}$-spaces.

Keywords

55P4555P35Gottlieb sets for mapsL-S categoryT-spaces
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 3 , 01 September 2012 , pp. 523 - 536
Copyright
Copyright © Canadian Mathematical Society 2012

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