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Let $n>1$ and let $\mathfrak{X}(D_n)$ denote the set of continuous vector fields on the closed disc $D_n\subseteq \mathbb{R}^n$. Let $\operatorname{Homeo}_0(D_n)$ be the set of homeomorphism of the disc $D_n$ fixing $0$ which are isotopic to the identity. Equip both these with the compact-open topology.

Is there an explicit surjective continuous map from $\mathfrak{X}(D_n)$ onto $\operatorname{Homeo}_0(D_n)$?

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  • $\begingroup$ You're asking for a continuous map with no further assumption? it might exist for some "bad" reason (bad meaning quite unrelated to the context, working with $\mathrm{Homeo}_0(D_n)$ replaced with quite general nonempty connected Polish spaces). (This might be explicit too!) $\endgroup$
    – YCor
    Commented May 22, 2020 at 11:48
  • $\begingroup$ I added the word explicit. Otherwise, it goes well with me if it's for some unrelated reason just as long as it can be expressed explicitly. $\endgroup$
    – ABIM
    Commented May 22, 2020 at 11:50
  • $\begingroup$ But usually these kind of "general type reasons" can also be made explicit. For instance, a continuous surjection $[0,1]\to [0,1]^2$ can be made explicit, and one could imagine finding an explicit continuous surjection $X(D_n)\to \mathcal{H}$ ($\mathcal{H}$ separable infinite-dim Hilbert) as well as an explicit continuous surjection $\mathcal{H}\to \mathrm{Homeo}_0(D_n)$. $\endgroup$
    – YCor
    Commented May 22, 2020 at 11:52
  • $\begingroup$ The following "one" is straight-forward: $f \to \pi'\circ f\circ \iota$, where $\iota:D_n\to \mathbb{R}^n$ is the inclusion and ${\pi}$ is the identity on $D_n$ and on $\mathbb{R}^n-D_n$ it sends $x$ to $\frac{x}{\|x\|}$. However, when I try to incorporate the conditions that the image is a homeomorphism then such a map breaks (likewise with the constraint that the origin must be fixed). $\endgroup$
    – ABIM
    Commented May 22, 2020 at 12:00
  • $\begingroup$ By the way your title doesn't match your question (homeo vs continuous surjective) $\endgroup$
    – YCor
    Commented May 22, 2020 at 12:02

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