The Milnor–Stasheff Filtration on Spaces and Generalized Cyclic Maps

Iwase, Norio ; Mimura, Mamoru ; Oda, Nobuyuki ; Yoon, Yeon Soo

Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 523-536

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-130-8

Mots-clés : 55P45, 55P35, Gottlieb sets for maps, L-S category, T-spaces