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Let $f$ be a locally flat embedding from $S^2 \times \mathbb R^2$ to $S^2 \times \mathbb R^2$ such that $f_*(\pi_k(S^2 \times \mathbb R^2))=0$ for any $k \ge 2$.

Can we find a domain $U$ that contains the image of $f$ such that $U$ is homeomorphic to $\mathbb R^4$?

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    $\begingroup$ In this situation, if $f_*=0$ on $\pi_2$ then $f$ is nullhomotopic (and therefore $f_*=0$ on $\pi_k$ for all $k$). $\endgroup$
    – Neil Strickland
    Commented Jul 13, 2021 at 8:12

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