Subject: Re: two postings From: "Vidhyanath Rao" Date: Wed, 26 Jan 2005 22:16:01 -0500 >> Subject: Re: Jeff Smith's email >> From: Nils Barth >> 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 continuous 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. For the weak equivalence claim, even this doesn't seem enough: Let $f: S^{n-1} \to C^{\infty}(M, N)$ and $g: R^{n} \to C0(M, N)$ be given such that $g$ restricted to $S^{n-1}$ agrees with $f$. We need to find $g_1: R^{n} \to C^{\infty}(M, N)$ that agrees with $f$ on $S^{n-1}$. The trouble is that the induced map $S^{n-1} \times M \to N$ need not be smooth. We can approximate $\tilde{g}: R^{n} \times M \to N$ by a smooth map and get $h: R^{n} \to C^{\infty}(M, N)$ whose restriction to $S^{n-1}$ is arbitrarily close to $f$ in the $C0$-topology. But now we need to modify $h$ to agree exactly with $f$. Of course, a standard argument using embeddings in $R^m$, partitions of unity and tubular neighborhoods will do this. The confusion seems to be that the gaps in Jeff's arguments cannot be filled in by "general abstract nonsense", but only by appealing to special features of smooth manifolds. ______________________________________________________ Subject: not so smooth From: Izak Grguric Date: Fri, 28 Jan 2005 14:43:31 -0800 (PST) This is a correction to the last e-mail I made: I dropped the word "smooth" at two key points, making the whole thing really confusing: 1: "...Either you do this (that's done in the Kosinski reference I quoted), or show that if F_0 and G_0 are within epsilon of each other, then they are homotopic..." - The last statement should read "..then they are SMOOTHLY homotopic..". 2: "I mean the whole point (the way I understood Boccellari's question) is that the smooth G should be a homotopy between something in [F_0] and something in [F_1] and not just any smooth map epsilon-close to F." - Here the change is the following: "between something in [F_0]" should be "between something in [F_0]_{SMOOTH}" i.e. [F_0] should stands for smooth homotopy classes (and not just any homotopy classes). The same goes for [F_1]. -------------------- Sorry about that, I promise no more e-mails on this topic. --------------------