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Problem: Suppose that $f:S^n\to S^n$ is a mapping from the n-dimensional sphere ($n\geq 3$) into itself which maps circles into (instead of onto) circles. Can we say that f maps (n-1)-dimensional spheres into (n-1)-dimensional spheres?

Here, we make no any other assumption on f, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.

Note that neither of two "into"s in assumptions means "onto".

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In your questions are you assuming continuity? –  alvarezpaiva Dec 28 '12 at 20:14
    
I don't see why this is tagged soft-question. –  HJRW Dec 28 '12 at 20:15
    
@Woodbass: do you have an example of a map that sends circles into circles, but which is not a Moebius transformation? –  alvarezpaiva Dec 28 '12 at 20:36
    
@alvarezpavia: Take any map which sends sphere to the circle. –  Misha Dec 29 '12 at 4:53
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What is a "circle" for your purposes -- great circle, round circle, or perhaps something else? –  Ryan Budney Dec 29 '12 at 6:09
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1 Answer

Take two circles $C_1, C_2$ intersecting in two points $p_1, p_2$. These are contained in a unique 2-sphere $S$. One way to see this is to assume $p_1=\infty$, $S^n-p_1=\mathbb{R}^n$ (considering $S^n$ as the one-point compactification of $\mathbb{R}^n$). Then the circles $C_1, C_2$ correspond to two lines intersecting in a point $p_2$, which span a unique 2-plane in $\mathbb{R}^n$ corresponding to the sphere $S$.

The images $f(C_i)\subset D_i$, where $D_i$ are circles, and $D_1\cap D_2 = \{f(p_1),f(p_2)\}$. Let $T$ be the unique 2-sphere containing $D_1\cup D_2$. Then I claim $f(S)\subset T$. Normalize it by post-composition with a Moebius transformation so that $f(p_1)=\infty$, then any line intersecting $(C_1\cup C_2)-\infty$ in two points distinct from $p_2$ will map to a line intersecting the lines $D_1\cup D_2-\infty$ in two points, and thus is contained in $T$. So $f(S)\subset T$ since every point of $S$ is contained in a line intersecting $D_1, D_2$ in points distinct from $f(p_2)$.

One can perform a similar inductive argument to show that higher-dimensional spheres are preserved. Take $k$ circles intersecting two points, say $0, \infty$, which span a $k$-plane $K$ in $\mathbb{R}^n$. Then any point on $K$ is contained in a $k-1$-plane $J$ intersecting the $k$ lines in points different from the origin. Then the image is contained in a similar configuration by induction, so the $k$-spheres are taken to $k$-spheres.

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Your argument requires that the inverse image of $p_1$ (or say $\infty$) consists of a single point. So, you can not reduce the question to $\mathbb{R}^n.$ –  woodbass Dec 31 '12 at 2:29
    
@woodbass: Actually, Agol's arguments (with minor corrections) suffice. Also, a much more general result is Theorem 4.8 in rupertmccallum.com/thesis11.pdf –  Misha Jan 1 '13 at 0:31
    
I do not think so. Please think of: Why f cannot map $S^2$ exactly to 5 points which is not in a 2-sphere and no four of which lie in a circle? –  woodbass Jan 1 '13 at 8:04
    
Of course, if assume injectivity, the situation reduced to R^n. There are essential gaps in MaCcallum's thesis as I noted in another comment (see: mathoverflow.net/questions/117433/…). I do not know how to add a simple super-link to that comment. –  woodbass Jan 1 '13 at 8:26
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