Note on homotopy normality and the \(n\)-connected fiber space (Q2731413)

An \(H\)-map \(i:Y\to X\) of simply connected homotopy associative \(H\)-spaces is called mod \(p\) homotopy normal, if \(c_X(X_{(p)} \times i(Y_{(p)}))\) can be deformed into \(i(Y_{(p)})\). Here \(c_X:X \times X\to X\) is the commutator map \((x,y)\mapsto (x\cdot y)\cdot (\sigma(x) \cdot\sigma (y))\) with \(\sigma:X\to X\) a homotopy inversion. The authors study \(H\)-spaces satisfying \(H^*(X; \mathbb{F}_p) \cong \mathbb{F}_p [y]/(y^p)\otimes \Lambda\), \(|y|= 2p+2,\) where \(p\) is an odd prime and \(\Lambda\) is an exterior algebra generated by odd degree elements. They show that the \(n\)-connective covering \(X\langle n\rangle\to X\) is mod \(p\) homotopy normal for \(n\geq 3\). They use this fact to obtain information about the ordinary mod \(p\) cohomology and the Morava \(K\)-theory of \(X\langle n \rangle\).