7

I am reading the book Introduction to Smooth Manifolds by John Lee. In his book he proves a theorem called the Whitney's Approximation theorem which essentially states that any continuous map can be approximated by a smooth map.

In the end he gives an application where he proves that any homotopy between smooth manifolds is an isotopy. I am wondering if it has more applications.Thank You.

happymath
  • 6,148
  • 1
    For example, you can calculate the homotopy group $\pi_k(S^n), k < n$ using Sard's theorem. – Qiaochu Yuan Oct 31 '15 at 08:25
  • @QiaochuYuan can you please give a reference since I have never heard of higher homotopy groups. – happymath Oct 31 '15 at 16:41
  • You can extend Proposition 6.34 in Lee's smooth manifold to "smoothly homotopic" by first find a smooth curve between two points $s_1,s_2 $ in connected manifold $S$. The details for finding the curve is given here – Kelvin Lois Mar 29 '18 at 14:00

1 Answers1

5

What you mean in the last paragraph that two homotopic smooth maps are smoothly homotopic. I find this a pretty pleasant fact, that "smooth homotopy theory" on manifolds is the same thing as continuous.

Smooth maps are well-behaved, much better than continuous maps. Here's a proof of the Brouwer fixed point theorem without any algebraic topology.

Suppose I have a map $f: D^n \to S^{n-1}$ that's the identity on the boundary. Then, using a version of Whitney's theorem, I may as well assume this map is smooth. Pick a regular value $x \in S^{n-1}$; these exist by Sard's theorem. Then by a version of the regular value theorem $f^{-1}(x)$ is a properly embedded (that is, $\partial f^{-1}(x) = f^{-1}(x) \cap \partial D^n$) compact submanifold of dimension 1. Thus it has an even number of endpoints. But the very fact that $f$ is the identity on the boundary means that $f^{-1}(x) \cap \partial D^n = \{x\}$ - which is one point. Contradiction. Hence Brouwer's.

You can imagine that other arguments can also benefit from the assumption that the map is smooth.

  • But I have heard that whitneys approximation theorem is not valid for manifolds with boundary so can you please tell how you are using it here? – happymath Oct 31 '15 at 16:47
  • 2
    You have heard incorrectly. Perhaps the proof you were given is not valid for manifolds with boundary. In fact, there's an even stronger version, which I'm using here: if $f: M \to N$ is continuous with the restriction to $\partial M$ smooth, then $f$ is homotopic to a smooth map, with the homotopy not changing $f$ on the boundary. –  Oct 31 '15 at 17:16
  • can you please give some reference? – happymath Nov 01 '15 at 08:43
  • @happymath: Hirsch, Differential Topology, the chapters on approximation and $\delta$-approximation. I warn that this is not an enitirely friendly book to read, but at least you'll be able to see the correct statements. –  Nov 01 '15 at 13:48