Homotopy monomorphisms and \(H\)-splitting in loop space fibrations
Let \(X,\) \(Y\) and \(Z\) be \(H\)-spaces and \(f:Y\to X\) and \(g:Z\to X\) be \(H\)-maps. The composite \(Y\times Z\to X\times X\to X\) is called an \(H\)-splitting if it is a homotopy equivalence and an \(H\)-map. The author considers here the special case of \(\Omega B\times\Omega F\to\Omega E,\) where \(p:E\to B\) is a fibration with fibre \(i:F\to E\) and \(\Gamma\;:\Omega B\to\Omega E\) is a, possibly naturally ocurring, homotopy right inverse of \(\Omega p.\) With these conditions \(\Omega B\times\Omega F\to\Omega E\) is known to be a homotopy equivalence, but is not in general an \(H\)-map. The author gives examples from the literature where it is, or is not, an \(H\)-map. For example, in the case of the extended Hopf fibration \(S^1\to S^3\to \mathbb CP^1\to K({\mathbb Z} ,2),\) the relevant homotopy equivalence \(\Omega S^2\to \Omega K({\mathbb Z},2)\times \Omega S^3\) is not an \(H\)-map, using the loop structure of \(\Omega S^2\), since \(\Omega S^2\) is not homotopy abelian and the other two spaces are. The author considers the case where \(F\) has an \(H\)-structure such that the holonomy action of the fibration \(F\times \Omega B\to F\) is homotopic to \(F\times \Omega B\to F\times F\to F,\) where \( \Omega B\to F\) is the connecting map; this condition is satisfied for the various definitions of principal fibration for example. His main result is that, if also the homotopy right inverse \(\Gamma\;:\Omega B\to\Omega E\) is an \(H\)-map, then \(\Omega B\times\Omega F\to\Omega E\) is an \(H\)-splitting. He also shows, without the assumption that \(\Gamma \) is an \(H\)-map, that \(F\to E\) is a homotopy monomorphism if, and only if, it is a weak homotopy monomorphism. Numerous special cases and examples are given.
scientific article; zbMATH DE number 5526281
scientific article