2012-01-20

I belabored some subtleties about handles last time and will continue to do so a little bit more here. The upshot of the story should be that, when $X$ is an $n$-dimensional cobordism from $M_0$ to $M_1$ with a Morse function $f : M \to [0,1]$ with a single critical point of index $k$, then $X$ is diffeomorphic to $[0,1] \times M_0$ with an $n$-dimensional $k$-handle attached to $\{1\} \times M_0$. Here are 3 different approaches to defining a handle and what it means to attach a handle, and hence making sense of the preceding sentence:

  1. The most standard thing is to say that an $n$-dimensional $k$-handle is $H = H^n_k = B^k \times B^{n-k}$. This is glued to the top $M$ of a cobordism $X$ via an embedding $\phi : S^{k-1} \times B^{n-k} \hookrightarrow M$. The boundary of $H$ is divided into two parts: the attaching region $(\partial B^k) \times B^{n-k} = S^{k-1} \times B^{n-k}$ and the free region $B^k \times \partial B^{n-k} = B^K \times S^{n-k-1}$. Note first that $H$ is not a smooth manifold, but a manifold with corners, and that, after attaching such a handle, we get a manifold with corners, which need to be smoothed. All this is illustrated in the figure below. There are subtleties one could discuss about what it means precisely to smooth corners and about the fact that any reasonable way of smoothing the corners produces the same smooth manifold. Some of these details are dealt with in Kosinski, Differential Manifolds.HandleAsProduct
  2. The second approach is the approach discussed first in the preceding lecture, in which the handle is itself a cobordism, but a cobordism with corners between manifolds with boundary. More precisely, $H = H^n_k$ is a subset of $\mathbb{R}^n$ defined as follows: Let $f = -x_1^2 - \ldots - x_k^2 + x_{k+1}^2 + \ldots + x_n^2$. Let $\partial_- H = f^{-1}(-1) \cap \{x_{k+1}^2 + \ldots x_n^2 \leq 1\}$. Then let $H$ be the closure of the intersection of $f^{-1}[-1,1]$ with the union of all flow lines for $\nabla f$ starting on $\partial_- H$. Note that $\partial_- H \cong S^{k-1} \times B^{n-k}$ and that $H$ is a cobordism from $\partial_- H$ to $\partial_+ H = f^{-1}(1) \cap \{x_1^2 + \ldots + x_k^2 \leq 1\} \cong B^k \times S^{n-k-1}$. Furthermore, $H$ has "product sides" $\partial_0 H \cong [0,1] \times S^{k-1} \times S^{n-k-1}$, the part of $\partial H$ consisting of flow lines starting at $\partial (partial_- H)$. In this case, what it means to attach $H$ is to choose an embedding $\phi : S^{k-1} \times B^k \hookrightarrow M$, glue $H$ using this attaching map, and then attach a product $[0,1] \times (M \setminus \phi(S^{k-1} \times B^k))$, glueing the sides of this product to the sides of $H$, resulting in a smooth manifold. This is illustrated below. HandleAsCobordism2
  3. The third approach is to describe the handle as something that immediately produces a smooth manifold after being attached; in this case it should have "flanges" instead of corners. Milnor, in his Morse Theory, does this by comparing the Morse function $f$ and a small perturbation of $f$ supported in a neighborhood of the critical point. Here is another way: $H^n_k$ is a subset of $\mathbb{R}^n$ described as follows: Let $f: \mathbb{R}^n \to \mathbb{R}$ and $\partial_- H$ be as in the preceding construction. Now let $\tau: \partial_- H \to (0,\infty]$ be the time it takes to flow from a point on $\partial_- H \subset f^{-1}(-1)$ to $f^{-1}(1)$. Thus $\tau = \infty$ along $S^{k-1} \times \{0\} \subset S^{k-1} \times B^{n-k} = \partial_- H$. Choose a bump function $\mu : B^{n-k} \to [0,1]$ and consider the function $\mu \tau : S^{k-1} \times B^{n-k} = \partial_- H \to (0,\infty]$. Then let $H$ be the closure of the union of flow lines for $\nabla f$ starting at $\partial_- H$ and flowing forward for time $\mu \tau$. This is illustrated below and, once again, is attached via an embedding of $S^{k-1} \times B^{n-k}$. The resulting manifold is immediately smooth and the new boundary is obtained from the old boundary by "replacing" the image of the embedding of $S^{k-1} \times B^{n-k}$ with the other part of the boundary of $H$, which is diffeomorphic to $B^k \times S^{n-k-1}$.HandleWithFlanges

Henceforth we will use the simplest $B^k \times B^{n-k}$ model, but I wanted to discuss these subtleties because, in some contexts, in can become important. For example, if one wants to build manifolds with certain additional structures (symplectic or metric structures, for example), one would like to be very careful about extending such structures across handles, and then one may need to be quite careful with the rounded corners.

In the rest of the lecture I went through examples of handles in dimensions one and two, and discussed the cases $k=0$ and $k=n$. I'll save that writeup for the next blog post. Here is the video: