On the homotopy type of the loops on a 2-cell complex (Q2752870)

Let \(\theta: S^{2n-1}\to S^{2m-1}\). Write \(P = S^{2m-1}\cup_\theta D^{2n}\), and \(P^r = \Sigma^{r-2n} P\). If \(n= m\), \(P\) could be a Moore space for the group \({\mathbb Z}/p^a\); in this special case, \textit{F. R. Cohen, J. C. Moore} and \textit{J. A. Neisendorfer} showed that the loop space \(\Omega P\) decomposes into a product of spaces of the form \(T^{2k+1}(p^a)\) and \(S^{2m+1} \{ p^a \} \), each of which sits in a fibration sequence involving only spheres, their loop spaces, and the fibers of maps between spheres [Lect. Notes Math. 763, 1-22 (1979; Zbl 0436.55013)]. The present paper is a largely successful attempt to generalize this loop space product decomposition to the loop space of the mapping cone of any \(\theta: S^{2n-1}\to S^{2m-1}\). The paper is written \(p\)-locally, where \(p>2\) is prime. The main results are the following.NEWLINENEWLINENEWLINEIf the \((n+3)\)-fold suspension \(S^{n+3}\theta\) of \(\theta\) is nonzero, then NEWLINE\[NEWLINE \Omega P^{2n+2} \simeq F_{ S^2\theta} \times \Omega( P^{4n+3}\wedge (\Omega S^{2m+1})^+). NEWLINE\]NEWLINE If \(\theta = p\varphi\) is stably essential and \(p>3\), then \(\Omega p^{2n+1} \simeq T \times \Omega W\), where \(W = \bigvee_\alpha P^{n_\alpha}\) and \(T\) sits in an explicit fibration sequence involving only spheres, their loop spaces, and the fibers of maps between spheres . On the other hand, if \(S^{2n(p-3)+4m -2} \theta\sim *\), then \(\Omega P^{2n+1} \simeq T_\infty\times \Omega W \times \prod_{i\geq 1} (S^{2np^i-\sigma}\times \Omega S^{2np^i-2\sigma + 1})\), where \(\sigma = 2n-2m +1\). Here \(W\) is as above, and \(T_\infty\) also is placed in an explicit fibration sequence involving only spheres, their loop spaces, and the fibers of maps between spheres. The proofs of these results roughly follow the same pattern as the proof of the Cohen, Moore and Neisendorfer result. The proofs also depend on some more general and more technical clutching results. These results are applied to give a decomposition of the loop space of a Smith-Toda complex \(V(1)\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00013].