Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?
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2$\begingroup$ @DanieleZuddas: Is that really true? How to see that? The number of connected components of the space of all diffeomorphisms fixing the boundary (but not the origin) is pretty complicated. $\endgroup$– Thomas RotCommented Apr 22, 2020 at 11:54
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4$\begingroup$ And I would guess the map $\mathrm{Diff}_\partial (D^n)\rightarrow \mathrm{int}(D^n)$ that sends a diffeomorphism that preserves the boundary to its evaluation at $0$ is a fibration with fiber the space in the OP. That would mean that the homotopy groups are the same. $\endgroup$– Thomas RotCommented Apr 22, 2020 at 12:04
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3$\begingroup$ people.math.harvard.edu/~kupers/teaching/272x/book.pdf $\endgroup$– Thomas RotCommented Apr 22, 2020 at 12:04
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2$\begingroup$ They are not all isotopic to the identity. Roughly speaking in high dimensions there is a bijection between these isotopy classes and the group of exotic $(n+1)$-spheres. Getting back to the original question, what kind of characterization do you want? You could view the homotopy-type of these diffeomorphism groups as the "generators" of the difference between the categories of topological and smooth manifolds. $\endgroup$– Ryan BudneyCommented Apr 22, 2020 at 14:02
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1$\begingroup$ As others say the diffeomorphism group of $D^n$ rel boundary is complicated, see e.g. pi.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf. $\endgroup$– Igor BelegradekCommented Apr 22, 2020 at 16:15
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