A version of the Borsuk--Ulam theorem states that a continuous antipodal map from the M-sphere into euclidean N-space has a zero provided that M is at least N. Clearly the general case follows from the case when M = N. But is the case when M >> N any easier to prove than the equidimensional case?
$\begingroup$
$\endgroup$
6
-
$\begingroup$ Are you asking if there is a direct and easier proof when $M >> N$? $\endgroup$– Somnath BasuCommented Jan 5, 2011 at 14:38
-
$\begingroup$ (Hi Tony!) Please explain: why do you expect the proof to be easier when M>>N? $\endgroup$– Piero D'AnconaCommented Jan 5, 2011 at 14:44
-
1$\begingroup$ It's a shame about that answer, as the question was a very nice one. I suppose there's still just about room for the optimist to wonder whether there is a short proof of the Borsuk--Ulam theorem that starts by proving it in very high dimensions. $\endgroup$– gowersCommented Jan 5, 2011 at 17:53
-
1$\begingroup$ Hi Tony! Could you specify what you mean by "easier", ie are you looking for a proof that doesn't appeal to the fact that an antipode-preserving map $f\colon S^n\to S^n$ is of odd degree, therefore homotopically non-trivial? $\endgroup$– Mark GrantCommented Jan 5, 2011 at 22:39
-
1$\begingroup$ Tim, there is a relatively short argument via dimensional reduction which is a hybrid of one of Shchepin's together with a low-dimensional argument reminiscent of the Stokes' theorem proof of the Brouwer fixed point theorem. See maths.ed.ac.uk/~carbery/analysis/links.html and click on Notes on the Borsuk--Ulam theorem. $\endgroup$– Tony CarberyCommented Jan 6, 2011 at 15:05
|
Show 1 more comment
1 Answer
$\begingroup$
$\endgroup$
1
I don't think so, since any antipodal (non-existent) map $S^n\to S^{n-1}$ would easily be "suspended" to an antipodal map $S^{n+1}\to S^n$. Iterating and composing would then yield antipodal maps $S^m\to S^{n-1}$ with arbitrarily large $m$. The $n=2$ case is somewhat easier though.
-
$\begingroup$ ..and the case $n=1$ and is even easier. $\endgroup$ Commented Oct 3, 2013 at 10:18