$\phi$-Postnikov systems and extensions of $H$-spaces

Abstract:

Let $f:X \to Y$ be a map of CW complexes and let $\pi :{F_f} \to X$ be the fibration induced by $f$. The following theorems are proven: Theorem. Assume $F( = {F_f})$ and $\Omega Y$ are simply connected and that (a) ${f^ \ast }:{H^n}(Y;{\pi _n}(Y)) \to {H^n}(X;{\pi _n}(Y))$ is epic for all $n$, (b) ${(i \wedge i)^ \ast }:{H^n}(F \wedge F;\operatorname {co} \ker {f_{{n^ \ast }}}) \to {H^n}(\Omega Y \wedge \Omega Y;\operatorname {co} \ker {f_{{n^ \ast }}})$ is monic for all $n$ (where ${f_{{n^ \ast }}}:{\pi _n}(X) \to {\pi _n}(Y))$). If $X$ is an $H$-space then $F$ is an $H$-space such that $\pi :F \to X$ is an $H$-map. Theorem. Assume $Y$ is $(p - 1)$-connected, $F$ is $(q - 1)$-connected $(p - 1 \geq 2,q - 1 \geq 1)$ of dimension $< \min (2p - 1,p + q - 1)$, and ${H^ \ast }(Y)$ is free. If $X$ is an $H$-space and ${f^ \ast }:{H^n}(Y) \to {H^n}(X)$ is onto for all $n$ then $F$ is an $H$-space and the map $\pi :F \to X$ is an $H$-map. Analogous theorems are shown to hold for loop spaces.