Homotopy fibrations with a section after looping (Q6605406)

In [Doc. Math. 27, 183--211 (2022; Zbl 1491.55010)], \textit{P. Beben} and \textit{S. Theriault} considered a certain cofibration related to the cell attachment problem and Ganea-type results, and developed new techniques that relate the action of a principal fibration to relative Whitehead products. In [Ann. Math. Stud. 113, 35--71 (1987; Zbl 0708.55011)], \textit{J. A. Neisendorfer} proved that, for an odd prime \(p\), \(p^{r+1}\) annihilates the homotopy groups of a simply connected mod \(p^r\) Moore space \(P(p^r)\).\N\NLet \(ad^k (i_1)(i_2)\) be the iterated Whitehead product \([i_1, ad^{k-1}(i_1)(i_2)]\) that is given by induction, and let \N\[\Sigma X^{\wedge k} \wedge \Sigma Y \xrightarrow{ad^k (i_1)(i_2)} \Sigma X \vee \Sigma Y \longrightarrow M_k\] \Nbe a homotopy cofibration, where \(i_1 : \Sigma X \rightarrow \Sigma X \vee \Sigma Y\), and \(i_2 : \Sigma Y \rightarrow \Sigma X \vee \Sigma Y\) are the first and second inclusions, respectively. In the paper under review, the author analyzes a general family of fibrations which, after looping, have sections. The methods are driven by applications to cofibers, Poincaré Duality complexes, the connected sum operation, and polyhedral products. More precisely, the author gives an alternative proof of the Neisendorfer's result on the homotopy equivalence between the homotopy fiber \(E\) of the pinch map \(\Sigma X \vee \Sigma Y \rightarrow \Sigma X\) to the first wedge summand and the certain type of smash products; that is, \(E \simeq \bigvee_{k=0}^{\infty} X^{\wedge k} \wedge \Sigma Y\). The author constructs homotopy equivalences:\N\begin{itemize}\N\item \(\Omega M_k \simeq \Omega \Sigma X \times \Omega (\bigvee_{t=0}^{k-1} X^{\wedge t} \wedge \Sigma Y)\),\N\item \(\Omega (M \sharp N) \simeq \Omega M \times \Omega (\Omega M \ltimes Y)\), and\N\item \(\Omega Q \simeq \Omega P^{n+1}(p^r) \times \Omega \left( (\Omega P^{n+1}(p^r) \ltimes \bar C ) \vee ( \bigvee_{i=2}^{m} P^{n+1} (p^r))\right)\).\N\end{itemize}\NHere, \(M\) and \(N\) are \(1\)-connected Poincaré Duality complexes, \(\ltimes\) is the left half-smash product, \(Q\) is a homotopy cofiber of \(S^{2n} \rightarrow \bigvee_{i=1}^m P^{n+1} (p^r)\), and \(\bar C \simeq \left( P^n(p^r) \wedge (\bigvee_{i=2, i \neq t}^{m}P^{n+1} (p^r))\right) \vee \left(S^{2n+1} \vee P^{2n}(p^r)\right)\). The author also investigates a homotopy cofibration satisfying a certain homotopy commutative diagram based on polyhedral products. The proofs in the paper are elegant and interesting.