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The Milnor–Stasheff Filtration on Spaces and Generalized Cyclic Maps
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 3 / 01 September 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 523-536
- Print publication:
- 01 September 2012
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- Article
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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.
