On the image of the unstable Boardman map

Abstract

We consider the `unstable Boardman map' (homomorphism if $k>0$) $$b:{\pi }^{m+k}{\Sigma }^k{\Omega }^l{S}^{n+l}\simeq [{\Omega }^l{S}^{n+l},{\Omega }^k{S}^{m+k}]⟶\mathrm{Hom}(H_{\ast }{\Omega }^l{S}^{n+l},H_{\ast }{\Omega }^k{S}^{m+k})$$ defined by $h(f)=f_{\ast }$. We work at the prime $2$, with $k=0$, and determine the image for various 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.

This is an update, with an updated title, to an older arxiv post with this identifier as a note on `cospherical' classes

References in corpus (1)

• Algebraic topology and modular forms