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The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps |
Canad. Math. Bull. 55(2012), 523-536
Published:2011-06-29
Printed: Sep 2012
Printed: Sep 2012
- Norio Iwase,
- Mamoru Mimura,
- Nobuyuki Oda,
- Yeon Soo Yoon,
Abstract
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 |
| MSC Classifications: |
55P45 - $H$-spaces and duals 55P35 - Loop spaces |