Cospherical classes in some iterated loop spaces on spheres

Abstract

For a nice topological space X, working at the prime $p=2$, we consider the ‘unstable Boardman map’ (homomorphism if $k>0$)

$$\begin{aligned} b:\pi ^{m+k}\Sigma ^kX\simeq [X,\Omega ^kS^{m+k}]\longrightarrow \mathrm {Hom}_{\mathbb Z/2}(H^*\Omega ^kS^{m+k},H^*X) \end{aligned}$$

defined by $b(f)=f^*$ where $k\ge 0$ and $m\ge 0$. We use classic maps, such as the Kahn–Priddy map, to provide examples of X so that b is nonzero in many dimensions. We also consider the case of $X=\Omega ^lS^{n+l}$, with particular interest in the cases with $0\le k<l\le +\infty$, and consider the problem of computing the image of

$$\begin{aligned} b:\pi ^{m}\Omega ^lS^{n+l}\simeq [\Omega ^lS^{n+l},S^{m}]\longrightarrow \mathrm {Hom}_{\mathbb Z/2}(H^*S^{m},H^*\Omega ^lS^{n+l}). \end{aligned}$$

Our results concern with the extreme values of k given by $k=0,l$. For $k=l$, a simple interpretation of well known facts about James-Hopf maps shows that the image of b when $m=2n$ is always nontrivial; we have not completely determined the image of b in this case. For $k=0$ we completely determine the image of b in the following cases: (1) $m=n$ and $l>0$ arbitrary; (2) $m>n$ and $l=1$. We observe that in most of the cases the image is trivial with the exceptions corresponding to the cases when either there is a (commutative) H-space structure on $S^n$ or there is a Hopf invariant one element.