On the iterated suspension
The author considers the question of whether the fibre \(W_{n,r}\) of the iterated suspension map i: \(S^ n\to \Omega^ rS^{n+r}\) is a loop space. In particular he constructs in many cases a fibration sequence (i.e. a sequence of spaces and maps homotopy equivalent to that defined by a fibration): \[ S^ n\to^{i}\Omega^ rS^{n+r}\to^{\nu_{n,r}}B_{n,r} \] so that \(W_{n,r}=B_{n,r}\). In such a case \(B_{n,r}\) is said to exist. Suppose all spaces are localized at a fixed prime p and let \(q=2(p-1).\) The author proves: (a) If \(p=2\) and \(r\leq n+1\) then \(B_{n,r}\) exists. (b) If \(p>2\) and \(r\leq nq+1\) then \(B_{2n-1,r}\) exists. Moreover in these cases \(B_{n,r}\) is the fibre of a map \(W_{n+1,r- 1}\to S^{(n+1)p-1}\); if \(p>3\) then \(B_{n,r}\) is an associative H space and \(\nu_{n,r}\) is an H map; \(S^ r(\Omega^ rS^{n,r})\simeq S^ r(S^ n\times B_{n,r})\). The author next pays particular attention to \(B_ n=B_{2n-1,2}\). He shows that there are fibrations localized at p: \[ S^{2n-1}\to^{i}\Omega^ 2S^{2n+1}\to^{\nu_ n}B_ n,\quad B_ n\to^{j}\Omega^ 2S^{2np+1}\to^{\phi}S^{2np- 1} \] where i is the suspension, \(j\cdot \nu_ n\) is the loops on the p'th James Hopf invariant and \(\phi\) has degree p in dimension 2np-1. Further: (c) If \(p>3\), \(B_ n\) is an associative H space and \(\nu_ n\) and j are H maps. (d) If \(p>2\) there is a map \(\xi\) : \(\Omega\) \(S^{2np-1}\to \Omega^ 3S^{2np+1}\) of degree 1 such that \(\xi \cdot \Omega \phi \simeq \Omega^ 3(p_{\iota}).\) (e) \(\Omega \phi =H'\cdot P: \Omega^ 3S^{2np+1}\to \Omega S^{2n}_{p-1}\to \Omega S^{2np-1}\) for \(p\geq 2\), where \(H'\) is a version of the Toda Hopf invariant map and P is the loops on the inclusion of the fibre of the inclusion map \(S^{2n}_{p-1}\to \Omega S^{2n+1}\).
scientific article; zbMATH DE number 4094375