On the image of the unstable Boardman map

TL;DR

This paper investigates the image of the unstable Boardman map at prime 2, identifying cases where the map's image is trivial or non-trivial, especially relating to H-space structures and Hopf invariant one elements.

Contribution

It determines the image of the unstable Boardman map for specific cases at prime 2, highlighting when the image is trivial or non-trivial based on space structures.

Findings

Image is trivial in most cases.

Non-trivial image occurs with H-space structures or Hopf invariant one elements.

Results depend on the relationships between m, n, and l.

Abstract

We consider the `unstable Boardman map' (homomorphism if $k>0$) $$b:\pi^{m+k}\Sigma^k\Omega^lS^{n+l}\simeq[\Omega^lS^{n+l},\Omega^kS^{m+k}]\longrightarrow \mathrm{Hom}(H_*\Omega^lS^{n+l},H_*\Omega^kS^{m+k})$$ defined by $h(f)=f_*$. 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.