The Milnor-Stasheff filtration on spaces and generalized cyclic maps (Q2909637)

Let \(k\) be an integer, \(1\leq k\leq \infty\). In [\textit{J. Aguadé}, Can. J. Math. 39, No. 4, 938--955 (1987; Zbl 0644.55008)], Aguadé uses the Milnor--Stasheff filtration of a space to define \(T_k\)-spaces as intermediate spaces between \(H\)-spaces and \(T\)-spaces in the sense that a \(T_{\infty}\)-space is an \(H\)-space and a \(T_1\)-space is a \(T\)-space.NEWLINENEWLINEInspired by the work of Aguadé, the authors define a \(0\)-connected space \(X\) to be a \(C_k\)-space if the inclusion \(P^k(\Omega X)\to P^{\infty}(\Omega X)\simeq X\) in the Milnor--Stasheff filtration is a cyclic map. Let \(G(Z,X)\) denote the set of all homotopy classes of cyclic maps from \(Z\) to \(X\), the so-called Gottlieb set. The authors show that \(X\) is \(C_k\) if and only if \(G(Z,X)=[Z,X]\) for any space \(Z\) with \(\mathrm{cat}(Z)\leq k\). Here \(\mathrm{cat}(Z)\) is the Lusternik--Schnirelmann category of \(Z\).NEWLINENEWLINESeveral results relating \(H, T_k,\) and \(C_k\) spaces are proven, as well as results relating \(C_k\) spaces with products and covering spaces. Some of these results are used to prove the following new characterization of \(H\)-spaces: Let \(\mathrm{cat}(X)=k\geq 1\). Then \(X\) is an \(H\)-space if and only if \(X\) is a \(C_n\)-space for some \(n\geq k\).NEWLINENEWLINEInteresting examples and non-examples of \(C_k\)-spaces are given by projective spaces and lens spaces. In particular, a computation shows that the lens space \(L^3(p)\) is a \(C_2\)-space if and only if \(p=2\).