Comments on: Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem

By: The closed graph theorem in various categories « What’s new

The closed graph theorem in various categories « What’s new — Wed, 21 Nov 2012 03:46:58 +0000

[...] theorem can be proven by applying invariance of domain (discussed in this previous post) to the projection of to , to show that it is open if has the same dimension as [...]

By: Jon Sjogren

Jon Sjogren — Mon, 25 Jun 2012 14:13:50 +0000

Greetings Terry Tao from af friends esp Robert Bonneau. Also greeting to A. Carberry from Prof. Wm Moran of Melbourne. I would not be hasty about attributing the proof of Brouwer FPT (using differential forms) to “E. Lima”. I know that reference comes from a book of do Carmo. On the other hand, this exact proof is found in Amer Math Monthly article April 1981, 264-268, by Yakar Kannai. He credits discussions with Prof. H.Scarf. First Dr. Kannai goes through the proof from the point of view of classical divergence theorem. However he also covers the proof again, in seven lines, using Stokes’ theorem in its formulation using exterior forms. A refinement that could be suggested to both “differential forms” proofs, is two consider two mappings g, h from the n-ball to (n-1) sphere both fixing every boundary point. In the integral of dg1 ^ dg2 ^ … ^ dgn over the ball, you want to replace each “g” by and “h”. You finally replace the retraction g by the identity h. First apply Stokes’ to equate the first integral with g1 dg2 ^ dg3 ^ … , integrated over the sphere, and note that now g1 may be replaced by h1 . Perform Stokes’ now to the new expression (“in reverse”) to obtain dh1 ^ dg2 ^ dg3 ^ … ^ dgn (over the ball), and now Stokes’ again to get
dh1 ^ (g2) ^ dg3 ^ … (over the sphere). Now replace g2 by h2 (equal on the domain of integration). Repeat this process to replace all gi by hi, resulting in the volume form for the n-ball, which computes to a non-zero value. As Yakar Kannai points out in the Monthly selection, the same proof also applies to any reasonable n-manifold with boundary, embedded in Euclidean space of the same dimension (attributed to M. Hirsch amongst others). J Sjogren

By: Terence Tao

Terence Tao — Sat, 08 Oct 2011 19:11:49 +0000

Dear Tony,

I think the connection with degree theory comes in the fact that the degree of a smooth map can be expressed as the integral (normalised by the surface area of the sphere). An application of Stokes’ theorem shows that this quantity is invariant under smooth deformations of g; since the constant map has zero degree and the identity map has non-zero degree, one therefore cannot continuously deform one to the other. If one expresses your map in polar coordinates one sees that this argument is basically equivalent to Lima’s argument.

By: Tony Carbery

Tony Carbery — Sun, 02 Oct 2011 13:49:21 +0000

There’s a really simple proof of the Brouwer fixed point theorem, along the lines of the one in Evans’ book, but even simpler. It seems to be due to E.Lima.

As usual it’s enough to show that there is no continuous map which restricts to the identity on the boundary. As usual it’s enough to prove there is no such map. Suppose there were such a map. Consider : on the one hand this is zero as has less than full rank, and on the other hand it equals
by Stokes’ theorem.
But as is the identity on the boundary, we can replace by here (pause for thought) and reverse the argument, giving the alternate answer , the volune of the unit ball in . Hence no such exists. Is there degree theory lurking here?

A similar argument, combined with some ideas of Shchepin can be used to give a quite easy proof of the Borsuk–Ulam theorem. See http://www.maths.ed.ac.uk/~carbery/analysis/notes/bu3_public.pdf

By: Anonymous

Anonymous — Tue, 30 Aug 2011 09:31:51 +0000

Typo, just after Lemma 6: “is a not an interior point”
Just before the Weierstrass approximation theorem is mentioned, in “{0 < \delta < 0.1}", I thought you were saying {\delta} was the infimum of {G} on {\Sigma_1} and necessarily was bounded above by 0.1 for some mysterious but important reason. Maybe changing it to "for some {\delta} small enough, {G} is bounded…" (as a flag that we might need to further decrease the naive bound) would prevent similar misreadings.

If you know, is it possible to go backwards in a sense and deduce the Brouwer fixed point theorem from the fact that {{\bf R}^n} and {{\bf R}^m} are not homeomorphic when the exponents differ, or from the invariance of domain theorem?

[Corrected, thanks. As for the last point, see Remark 1. -T.]

By: 254A, Notes 0 – Hilbert’s fifth problem and related topics « What’s new

Sat, 27 Aug 2011 19:35:24 +0000

[...] We will spend several lectures proving this theorem (due to Gleason and to Yamabe). As stated, may depend on , but one can in fact take the open subgroup to be uniform in the choice of ; we will show this in later notes. Theorem 6 can in fact be deduced from Theorem 7 and some topological arguments involving the invariance of domain theorem; we will see this later in this course (or see this previous blog post). [...]

By: Vadim Kulikov

Vadim Kulikov — Sat, 02 Jul 2011 00:34:54 +0000

My favourite proof of Brouwer fixed point theorem uses combinatorial game theory. It is funny, because Brouwer f.p.t is in turn used to show the existence of a Nash equilibrium. Also this approach, as is the Sperner’s lemma approach, very elementary and I even presented the proof to a group of high school students last fall. Here is the proof by David Gale:

http://www.cs.cmu.edu/afs/cs/academic/class/15859-f01/www/notes/brouwer-hex.pdf

By: Eighth Linkfest

Eighth Linkfest — Sat, 25 Jun 2011 20:07:53 +0000

[...] Tao: Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem, Hilbert’s Fifth Problem and Gleason metrics, The C^{1,1} Baker-Campbell-Hausdorff [...]

By: Terence Tao

Terence Tao — Mon, 20 Jun 2011 18:44:53 +0000

That is indeed a strange but lovely proof! (The idea is to deduce the fixed point theorem from the hairy ball theorem, and to prove the latter by observing that a smooth unit vector field on gives rise (for small t) to a diffeomorphism between the unit ball and the ball of radius whose Jacobian depends polynomially on t, implying that is a polynomial, which is absurd for n odd.) It may be that a degree-like invariant is somehow disguised inside the volume computation (it does faintly resemble the analytic representations of degree) but I don’t see exactly how. In any case, this is the first proof I’ve seen that doesn’t obviously involve some sort of degree (or disguised version thereof).

By: Noud

Noud — Sun, 19 Jun 2011 10:52:53 +0000

Dear Terry Tao,

Thank you for this post. There is a small typo: in remark 2, the second delta (on line 10) should be in uppercase. [Corrected, thanks - T.]