Nonvanishing of consecutive suspended Hopf composites by Adem relation

By [F1], $C_{{\eta }_2}={\mathbb{C}\mathbb{P}}^2$ and $S{q}^2$ sends its bottom degree-two generator to the nonzero degree-four class by [F2]. For reduced cones, suspension commutes with this cell cofiber: both $\mathrm{\Sigma }{C}_f$ and $C_{\mathrm{\Sigma }f}$ are the quotient of $I\times S^n\bigsqcup I\times S^{n+1}\times I$ by the same relations, collapsing suspension ends, cone tips and based tracks and identifying $(s,x,0)$ with $(s,f(x))$. Quotient descent therefore gives inverse continuous comparison maps. By [F5], ordinary cones are based homotopy equivalent to these reduced cones. Thus for every $n\ge 2$ there is a based equivalence $C_{{\eta }_n}\simeq {\mathrm{\Sigma }}^{n-2}{C}_h$, preserving the bottom and top cohomology classes. Suspension compatibility in [F2] therefore gives a nonzero $S{q}^2:H^n(C_{{\eta }_n};{\mathbb{F}}_2)\to H^{n+2}(C_{{\eta }_n};{\mathbb{F}}_2)$.

Suppose for contradiction that ${\eta }_n{\eta }_{n+1}$ is null. Put $f={\eta }_n:S^{n+1}\to S^n$ and $g={\eta }_{n+1}:S^{n+2}\to S^{n+1}$. Let $K:C{S}^{n+2}\to S^n$ be a based cone extension of $f\circ g$, and let $\chi :C{S}^{n+1}\to C_f$ be the characteristic map of the cell attached by $f$. Decompose $S^{n+3}$ into two cones on $S^{n+2}$. On the first define $k:S^{n+3}\to C_f$ by $\chi \circ Cg$; on the second define it by $K$ followed by $S^n\hookrightarrow C_f$. On the common equator both formulas equal $f\circ g$, and on the chosen basepoint both equal the target basepoint, so they paste to a based continuous map. Attach an $(n+4)$-cell along $k$, obtaining $X=C_f{\cup }_k{D}^{n+4}$. Its cells have dimensions $0,n,n+2,n+4$. Collapsing the bottom $S^n$ gives an attaching map $\bar{k}=(\mathrm{\Sigma }g)\circ c$, where $c:S^{n+3}\to S^{n+3}$ collapses the hemisphere carrying $K$ and expands the other hemisphere. In suspension coordinates $[x,s]$, $-1\le s\le 1$, take $c[x,s]=[x,\phi (s)]$, where $\phi (s)=-1$ for $s\le 0$ and $\phi (s)=2s-1$ for $s\ge 0$. The coordinate maps ${\phi }_t(s)=(1-t)s+t\phi (s)$ preserve the endpoints and give a based homotopy from the identity to $c$. Hence $\bar{k}$ is based homotopic to $\mathrm{\Sigma }g={\eta }_{n+2}$. An attaching homotopy $H$ induces a cone comparison equal to the identity on the bottom sphere, equal to $H(x,2t)$ on the first half of the cone, and equal to the target cone point $[x,2t-1]$ on the second half. The reverse homotopy gives an inverse up to homotopy by cancelling the traversed homotopy and its reverse. This comparison has degree $+1$ on the relative top cell. Consequently $X\mathrm{/}{S}^n\simeq C_{{\eta }_{n+2}}$ by a based homotopy equivalence preserving both upper cellular generators. No covariant homotopy exact sequence for cofibers is assumed.

The cells of $X$ have dimensions $0,n,n+2,n+4$, so its cellular cochain differential is zero and $H^{n+1}(X;{\mathbb{F}}_2)=0$. Let $x,y,z$ denote the nonzero classes in degrees $n,n+2,n+4$, respectively. Inclusion $C_{{\eta }_n}\hookrightarrow X$ induces isomorphisms in degrees $n$ and $n+2$, since these cellular cochain groups and their differentials are unchanged by the single top cell. By step 1.1 and naturality [F2], $S{q}^{2}x=y$. The based equivalence $X\mathrm{/}{S}^n\simeq C_{{\eta }_{n+2}}$ from step 1.2 preserves the upper generators, so the quotient has nonzero $S{q}^2$ between degrees $n+2$ and $n+4$ by step 1.1 at index $n+2$. Collapsing the bottom subcomplex removes precisely its $0$- and $n$-cells from the cellular cochains; the quotient map preserves the upper cellular generators and their differentials. Naturality [F2] therefore gives $S{q}^{2}y=z$. Thus $S{q}^{2}S{q}^{2}x=z\mathrm{\ne }0$. But $H^{n+1}(X;{\mathbb{F}}_2)=0$, so $S{q}^{1}x=0$.

On the other hand, [F3] gives $S{q}^{2}S{q}^{2}x=S{q}^{3}S{q}^{1}x=0$ by step 2.1. This contradicts $S{q}^{2}S{q}^{2}x=z\mathrm{\ne }0$, so the assumed null composite is impossible. Hence ${\eta }_n{\eta }_{n+1}\mathrm{\ne }0$ for every $n\ge 2$.