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, ...'" Greg Egan, _Teranesia_ Sent via Deja.com http://www.deja.com/ Before you buy.