Subject: Re: two postings From: Jeffrey H Smith Date: Wed, 26 Jan 2005 21:14:15 -0500 (EST) To: Don Davis The mistake is assuming that I was trying to give a proof of anything. I was giving a one line statement of the essential point for standard results relating different function spaces. If you want proofs go read Milnor's notes. This is my final post on the matter. I will be using /dev/null from now on. Jeff On Wed, 26 Jan 2005, Don Davis wrote: >> Two postings: A question about K-12 education and a followup >> to the discussion of the past few days..........DMD >> _______________________________________________________ >> >> Subject: for the list >> From: "W. Stephen Wilson" >> Date: Wed, 26 Jan 2005 07:48:55 -0500 (EST) >> >> Subject: arithmetic advocacy >> >> Many colleges and universities run a local elementary school. >> Sometimes this is to make nice with the neighborhood, sometimes >> to make nice with the faculty and staff, and sometimes it is >> so the School of Education can experiment on the children. >> There are probably other motivations as well. >> >> Johns Hopkins University is going to open an elementary school >> in the next few years with the make-nice-with-the-neighborhood >> approach. I have the ear of the director and am trying to >> get him to take seriously the idea of teaching arithmetic. >> >> I know of some very high profile university run elementary >> schools which have gutted the math curriculum of arithmetic >> but the director asked me for an example of a university run >> school that is done right and I didn't have one. >> >> I have tried my usual math education experts and they have also >> come up blank and so I thought I'd toss this out to a wider >> group to see if I could find a college/university run elementary >> school that takes arithmetic seriously. Anyone know one? >> >> Steve >> >> >> W. Stephen Wilson (410) 516-7413 >> Department of Mathematics FAX (410) 516-5549 >> Johns Hopkins University >> Baltimore, MD 21218 wsw@math.jhu.edu >> >> http://www.math.jhu.edu/~wsw/ >> _________________________________________________________ >> >> Subject: Re: Jeff Smith's email >> From: Nils Barth >> Date: Wed, 26 Jan 2005 15:04:28 -0600 >> >> Dear Jeff (and mailing list readers), >> >> I'm not disputing that these results are true, standard and easy; >> however, the argument you outline confused me and others. >> You state two "therefores" that are either false or require elaboration. >> >> Let me explain my confusion, and a possible resolution. >> >> Your claim is: >> Smooth(M,N) < Cont(M,N) >> is a weak homotopy equivalence. >> (where Smooth, Cont mean the spaces of smooth/continuous >> maps, respectively, and < means subset) >> >> Here's how I read your argument: >> - Smooth(M,N) < Cont(M,N) is dense >> - and so every path in Cont(M,N) can be approximated by a path >> in Smooth(M,N) >> >> It's the "and so" that I object to: >> just because X < Y is dense does not mean that every path in >> Y can be approximated by a path in X. >> >> Now perhaps what you meant is: >> - a continuous path in Cont(M,N) is the same as an element >> of Cont(I x M, N), via the adjunction >> Hom(X,Hom(M,N)) = Hom(X x M, N) >> ...and likewise for smooth paths >> - I x M is also a manifold! >> - Smooth(I x M, N) < Cont(I x M, N) is dense >> - thus every continous path in Cont(M,N) can be approximated >> by a smooth path in Smooth(M,N) >> >> This is still not quite enough: >> we've replaced a continuous homotopy between two smooth functions f,g >> by a nearby smooth homotopy between two NEARBY functions. >> >> We need a relative statement: >> let X, Y be manifolds, K < X a closed subset. >> Given a continuous map H: X -> Y that is smooth on K, >> there is a nearby smooth function H~: X -> Y that agrees >> with H on K. >> >> [This is Kosinski's 2.5(b); >> note that by "nearby" he doesn't just use one epsilon, >> but a functional on Y (since he doesn't assume compact).] >> >> The result we want follows by letting >> X = I x M >> Y = N >> K = {0} x M union {1} x M >> H restricted to K = two smooth functions f, g >> H = the continuous homotopy between f, g >> >> and then H~ is the nearby smooth homotopy between f and g. >> [This is Kosinski's Corollary 2.6; >> note that he doesn't give a proof of the Corollary, and so >> it was filling in the details that I understood (I presume) >> our confusion.] >> >> * * * * * * * >> >> So perhaps the confusion was that you meant: >> smooth maps are dense in continuous maps >> in particular, Smooth(I x M, N) is dense in Cont(I x M, N) >> >> ...and I didn't fill this in in my head, >> and you didn't think it bore elaboration. >> >> * * * * * * * >> >> On the other statement, that density (and density of paths) >> implies homotopy equivalence, is wrong without >> added conditions: >> Consider X = {1/n : n in N} and Y = X union {0} >> Then >> - X is dense in Y >> - for all n, >> Hom(I^n, X) = X, Hom(I^n, Y) = Y >> [as X and Y are both totally disconnected] >> so Hom(I^n, X) is dense in Hom(I^n, Y) >> - but X -> Y is not a weak homotopy equivalence: >> it's not even onto pi_0 >> >> Admittedly this is a gross point-set example, >> but I don't know enough about the point-set topology of >> function spaces to reject this out of hand. >> >> * * * * * * * >> >> Hopefully this explains my concerns; if I'm missing >> something and you've the time to elaborate, I'd be >> appreciative. >> >> best, >> nils >> >> >> >> Subject: Re: 3 postings >> >> From: Jeffrey H Smith >> >> Date: Tue, 25 Jan 2005 11:34:47 -0500 (EST) >> >> >> >> JFC >> >> >> >> These are standard results from the fifties and sixties. The space of >> >> smooth maps is dense in the space of continuous functions and so the >> every >> >> continuous family of maps can be approximated within epsilon by a smooth >> >> family of maps. From which it follows that the space of smooth maps is >> >> weak equivalent to the space of continuous maps. >>