From: Robin Chapman
Subject: Re: Brouwer Fixed Point Theorem
Date: Tue, 30 Nov 1999 08:14:50 GMT
Newsgroups: sci.math
Keywords: Proving curves across a square must intersect
In article <81vkds$rvi$1@nntp9.atl.mindspring.net>,
"Daniel Giaimo" wrote:
> In the introductory chapter to Rotman's book "An Introduction to
> Algebraic Geometry", problem 0.5 reads:
^^^^^^^^^^^^^^^^^^^^^^]
Algebraic Topology.
> Let f,g:I->IxI be continuous, let f(0) = (a,0), f(1) = (b,1), g(0)=
> (0,c), and g(1) = (1,d) for some a,b,c,d in I. Show that f(s) = g(t) for
> some s,t in I.
>
> In the hint to this problem he claims that you can use Brouwer's Fixed
> Point Theorem to prove this. I have tried without success for a while now
> to figure out how, but I just can't seem to get it. I even asked my
> Topology professor and he said that he had never seen a proof of this using
> Brouwer's Fixed Point Theorem. Could someone please either help me see how
> Brouwer's Theorem applies? In fact, I really only need a proof in the case
> a=b=c=d=1/2.
It's quite a cunning argument. I couldn't find a proof myself, so
I resorted to looking up the cited paper by Maehara in the
Monthly (vol. 91, pp. 641-643, 1984).
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
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