Properties of Selick's filtration of the double suspension \(E^{2}\) (Q2874675)

We assume that all spaces are localized at an odd prime \(p\). It is known that there is a Hopf algebra isomorphism NEWLINE\[NEWLINEH_*(\Omega^2S^{2n+1})\cong \left(\bigotimes_{i=0}^\infty \Lambda(a_{2np^i-1})\right)\otimes \left(\bigotimes_{j=1}^\infty \mathbb Z/p\mathbb Z[b_{2np^j-2}]\right).NEWLINE\]NEWLINE Thus, we have a filtration of \(H_*(\Omega^2S^{2n+1})\) by Hopf algebras \(\{A_k(n)\}_{k\geq 0}\), where NEWLINE\[NEWLINEA_{2k-1}(n) = \left(\bigotimes_{i=0}^{k-1} \Lambda(a_{2np^i-1})\right)\otimes \left(\bigotimes_{j=1}^k \mathbb Z/p\mathbb Z[b_{2np^j-2}]\right)NEWLINE\]NEWLINE for \(k\geq 1\) and NEWLINE\[NEWLINEA_{2k}(n) = \left(\bigotimes_{i=0}^{k} \Lambda(a_{2np^i-1})\right)\otimes \left(\bigotimes_{j=1}^k \mathbb Z/p\mathbb Z[b_{2np^j-2}]\right)NEWLINE\]NEWLINE for \(k\geq 0\). Let \(E^2: S^{2n-1} \to \Omega^2 S^{2n+1}\) be the double suspension. It has been proved by Gray that the homotopy fiber \(W_n\) of \(E^2\) is a loop space of a space \(BW_n\), and there is a homotopy fibration \(S^{2n-1} @>{E^2}>> \Omega^2 S^{2n+1} @>{\nu}>> BW_n\) for any \(n\geq1\). It has also been proved by Gray that the fibration induces a Hopf algebra isomorphism \(H_*(\Omega^2 S^{2n+1}) \cong H_*(S^{2n-1}) \otimes H_*(BW_n)\). Now, it has been shown that \(BW_n\) is a homotopy associative, homotopy commutative \(H\)-space. Therefore, there is a Hopf algebra isomorphism NEWLINE\[NEWLINEH_*(BW_n)\cong \left(\bigotimes_{i=1}^\infty \Lambda(a_{2np^i-1})\right)\otimes \left(\bigotimes_{j=1}^\infty \mathbb Z/p\mathbb Z[b_{2np^j-2}]\right),NEWLINE\]NEWLINE and \(H_*(BW_n)\) is naturally filtered by Hopf algebras \(\{B_k(n)\}_{k\geq1}\), where \(B_k(n) \cong A_k(n)\). The aim of the paper under review is to geometrically realize the homological filtrations \(\{A_k(n)\}_{k\geq 0}\) and \(\{B_k(n)\}_{k\geq1}\). It is known that \(A_{2k-1}(n)\) is realized as \(H_*(\Omega J_{p^k-1}(S^{2n}))\), where \(J_{p^k-1}(S^{2n})\) is the \((p^k-1)^{st}\) stage of the James construction on \(S^{2n}\). Selick geometrically realised the Hopf algebra \(A_{2k}(n)\) as \(H_*(F_{2k}(n))\) by an \(H\)-space \(F_{2k}(n)\). By Selick, if we put \(F_0(n)= S^{2n-1}\) and \(F_{2k-1}(n)=\Omega J_{p^k-1}(S^{2n})\) for \(k\geq1\), we obtain a filtration \(F_0(n)= S^{2n-1}\to F_1(n) \to\cdots \to F_k(n) \to \cdots \to \Omega^2 S^{2n+1}\) where each \(F_k(n)\) is an \(H\)-space and each map \(F_k(n) \to \Omega^2 S^{2n+1}\) is an \(H\)-map. In this paper, the authors construct a filtration \(\{M_k(n)\}_{k=1}^\infty\) of \(BW_n\) which geometrically realizes the Hopf algebra filtration \(\{B_k(n)\}_{k=1}^\infty\). Then it is shown that each \(M_k(n)\) is an \(H\)-space, each map \(M_k(n) \to BW_n\) is an \(H\)-map, and \(H_*(M_k(n))\cong B_k(n)\). Moreover, it is proved that the filtration is compatible with Selick's filtration of \(\Omega^2S^{2n+1}\) and the homotopy fibration \(S^{2n-1} @>{E^2}>> \Omega^2 S^{2n+1} @>{\nu}>> BW_n\) : there is an \(H\)-fibration \(S^{2n-1} \to F_k(n) \to M_k(n)\) which induces a Hopf algebra isomorphism \(H_*(F_k(n)) \cong H_*(S^{2n-1}) \otimes H_*(M_k(n))\). Furthermore, the authors prove that \(F_{2k}(n)\) is a homotopy associative \(H\)-space for \(p>3\), and \(M_k(n)\) is a homotopy associative \(H\)-space either for \(p\geq 3\) when \(k\) is odd or for \(p>3\) when \(k\) is even. The authors also study the homotopy exponent of \(F_k(n)\).