# 2012-01-25

Exercise: Let $\Sigma$ be a torus with two boundary components embedded in $\mathbb{R}^3$ as in the top picture at right, and let $f : \Sigma \to [0,1]$ be projection onto the vertical direction, a Morse function with two critical points of index $1$ and with $a \cup b$ being a level set. Identify $\Sigma$ with the square-with-opposite-sides glued minus 2 disks shown below the embedded picture, so that the indicated curves $a$, $b$ and $c$ match up. On the square picture, we have an automorphism $\phi : \Sigma \to \Sigma$ obtained by rotating the square $90^\circ$. Let $f_0 = f$ and let $f_1 = f \circ \phi$. Find a generic homotopy $f_t$ from $f_0$ to $f_1$, constant on $f^{-1}(0)$ and $f^{-1}(1)$. Generic means that $f_t$ is Morse for all but finitely many times $t$, and when not Morse, we have a simple birth or death of a pair of cancelling critical points of successive index. Also, if you don't consider distinct critical points with the same critical value as being Morse, then you should also allow finitely many times when two critical points cross. Draw the Cerf graphic for this homotopy, i.e. the graph in $[0,1] \times [0,1]$ of the critical values with their indices for all $t \in [0,1]$.

Heegaard splittings and Heegaard diagrams: We need three facts here that will be proved later:

1. For any closed connected $n$-manifold $X$ there is a Morse function on $X$ with exactly one index $0$ critical point (minimum) and one index $n$ critical point (maximum).
2. Given any Morse function $f : X \to \mathbb{R}$, there is a homotopy of maps $f_t: M \to \mathbb{R}$, with $f_0 = f$, and $f_t$ Morse for all $t$, such that the critical values of $f_1$ are ordered by index. In other words, if the indices of critical point $p$ and $q$ are $i$ and $j$, resp., and if $i < j$, then $f(p) < f(q)$. (Note that there if $f$ does not satisfy this property then there will necessarily be times $t$ where $f_t$ has two critical points with the same critical value, and these times should be isolated. We call these times "critical value crossing times".)
3. Given a Morse function $f : X \to \mathbb{R}$ with critical values ordered by index, a generic choice of an adapted metric allows us to assume that all the $k$-handles are attached simultaneously to the boundary of the union of the handles of index $< k$. In other words, if $p$ and $q$ are critical points of index $k$ with $f(p) < f(q)$ and no critical values in $(f(p),f(q))$, we can use the gradient flow to compare the free region of the boundary of the handle for $p$ and the attaching region of the boundary of the handle for $q$ inside an intermediate level set $f^{-1}(y)$ for $y \in (f(p),f(q))$. When the metric is chosen generically these will be disjoint, and thus we can flow from the attaching region of the handle for $q$ down along the gradient field past the handle for $p$ and see them as both attached to a level set below $f(p)$.
(As a consequence of these results, one sometimes works with self-indexing Morse functions, functions with the property that, for a critical point $p$, the index of $p$ equals $f(p)$. Note that these are not quite generic because distinct critical points do not have distinct critical values, so some might consider these not to be Morse functions, but that might be being too nitpicky. I will not use that terminology much, but you see it a lot in $3$-manifold topology.)

Now consider a closed connected oriented $3$-manifold $X^3$ with a Morse function $f : X \to \mathbb{R}$ and corresponding handle decomposition as in 3 above. Let $y$ be a regular value between the index $1$ and index $2$ critical values, and let $A = f^{-1}(-\infty,y]$, $B = f^{-1}[y,\infty)$ and $\Sigma = f^{-1}(y)$. Then $A$ is the result of attaching some number $g$ of $1$-handles to a ball ($0$-handle) and $\Sigma = \partial A$ is a genus $g$ surface. On $B$, consider the Morse function $-f$; the index $2$ critical points of $f$ become index $1$ criitical points for $-f$ and thus $B$ is the result of attaching $g'$ $1$-handles to a ball and $\Sigma = \partial B$ is a genus $g'$ surface.  Therefore $g = g'$, so $f$ has the same number of index $1$ and $2$ critical points, and both $A$ and $B$ are diffeomorphic to the standard genus $g$ handlebody (the solid object in $\mathbb{R}^3$ bounded by the standard embedding of a genus $g$ surface in $\mathbb{R}^3$). This decomposition of $X$ into two solid handlebodies is called a Heegaard splitting of $X$.

Thinking now of constructions of manifolds, rather than decompositions of manifolds, we get the related notion of a Heegaard diagram. In the above figure, noting that $\Sigma$ is diffeomorphic to the standard genus $g$ surface $\Sigma_g$, we can now instead consider the Morse functions $-f$ on $A$ and $f$ on $B$, in which case we see both $A$ and $B$ as built by attaching $g$ $2$-handles and a $3$-handle to $\Sigma_g$. In other words, $X$ is built (or, rather, a $3$-manifold diffeomorphic to $X$ is built) by starting with $[-1,1] \times \Sigma_g$ and attaching $g$ $2$-handles and a $3$-handle to $1 \times \Sigma_g$ (producing $B$) and then turning things upside down and attaching $g$ more $2$-handles and a $3$-handle to $-1 \times \Sigma_g$ (producing $A$). This construction is completely determined by the $2g$ attaching circles (simple closed curves), often labelled $\alpha_1, \ldots, \alpha_g$ for $A$ and $\beta_1, \ldots, \beta_g$ for $B$. The $\alpha$ curves must be mutually disjoint and their complement in $\Sigma_g$ must be a $2g$-punctured sphere; ditto for the $\beta$ curves. Any such collection of simple closed curves in $\Sigma_g$ determines a closed $3$-manifold; this is a Heegaard diagram. The example below is $S^1 \times S^2$, again:

This is the data that is used to compute the famous Heegaard-Floer invariants of $3$-manifolds, but that story is beyond the scope of this course.

Surgery: The comment above that, in a Heegaard diagram, the complement of the $\alpha$ curves (resp. $\beta$ curves) should be a $2g$-punctured sphere merits further discussion. This is a condition that guarantees that, after attaching the $2$-handles along these curves, the new boundary is $S^2$ and so we can cap off with a $3$-handle. Surgery is the process by which the boundary of a manifold changes when one attaches a handle along that boundary. If $X^n$ is a cobordism from $M_0$ to $M_1$ and we attach an $n$-dimensional $k$-handle along an embedding $\phi : S^{k-1} \times B^{n-k} \hookrightarrow M_1$, we get a new cobordism from $M_0$ to $M_2$, and $M_2$ is diffeomorphic to $(M_1 \setminus \phi(S^{k-1} \times B^{n-k})) \cup_\phi B^k \times S^{n-k-1}$, because we "cover up" the image of the attaching region of the handle and "expose" the free region of the handle. The free region $B^k \times S^{n-k-1}$ is glued via $\phi : S^{k-1} \times S^{n-k-1} \hookrightarrow M_1$. This is $(n-1)$-dimensional $(k-1)$-surgery.

In the Heegaard diagram case, we have $n=3$ and $k=2$, so that we are doing $2$-dimensional $1$-surgery by removing a $S^1 \times B^1$ and replacing with $B^2 \times S^0$. In other words, we cut open $\Sigma$ along the attaching curve, leaving two new boundary components, and then we cap off each component with a disk.

We just touched on $4$-manifolds in this lecture, but in the notes I'll start that in the next post. Here's the video (thanks Eddie)

# 2012-01-23

Here we focus on examples of handles. Recall that an $n$-dimensional $k$-handle is $H = H^n_k = B^k \times B^{n-k}$ with $\partial H$ divided into two regions, the attaching region $S^{k-1} \times B^{n-k}$ and the free region $B^k \times S^{n-k-1}$, and a handle is attached to a pre-existing cobordism $X$ from $M_0$ to $M_1$ via an embedding of the attaching region into $M_0$, producing a new cobordsim $X'$ from $M_0$ to $M_1'$, containing $X$. Note that we have not yet discussed carefully how $M_1'$ is obtained from $M_1$, but when we do discuss this, the general method will be known as surgery.

First we note that all of this even makes sense when $k=0$ or $k=n$, with the convention that $B^0$ is a point and $S^{-1} = \emptyset$. For example, $S^1$ is built with a $0$-handle $B^0 \times B^1$, attached along an embedding of $\emptyset$, i.e. not attached to anything, following by a $1$-handle $B^1 \times B^0$ attached along an embedding of $S^0 \times B^0$, as in the figure to the right. This picture obviously generalizes to $S^n$ built with a $0$-handles and a $n$-handle.

Our main results thus far concern Morse functions with zero or one critical points, but these immediately imply the following general result:

Corollary: Every cobordism decomposes into a sequence of products and handles. In particular, every closed $n$-manifold can be built starting with a $0$-handle, then attaching some number of other handles of index $0 \leq k \leq n$, and then capping off with a $n$-handle.

(In fact we can always arrange that, if the manifold is connected, we only need one $0$-handle and one $n$-handle, but this fact is not entirely trivial and we will try to prove it carefully later.)

So here is a sequence of examples:

Dimension $1$: $S^1$ again, but with more handles; notice the different possible ways you might cancel pairs of $0$- and $1$-handles:

Dimension $2$: Here is a standard picture of a torus decomposed into a $0$-handle, two $1$-handles and  a $2$-handle, with products in between. In terms of this decomposition into elementary cobordisms, the second $1$-handle is attached to the top of the product above the first $1$-handle. However, we can let the attaching map for the second $1$-handle flow down along the gradient vector field through the product and past the first $1$-handle (as long as we are not unlucky and don't get sucked into the first $1$-handle's critical point - this is again an issue to be discussed more carefully soon), and then see both $1$-handle attached simultaneously to the boundary of the $0$-handle.

Dimension $3$: Now it gets interesting. First, we must abandon hope of embedding the $3$-manifold in $\mathbb{R}^3$ and seeing the Morse function as the height function. So instead we will just draw some handle decompositions and, perhaps, some level sets of the Morse functions. First recall the three kinds of handles in dimension $3$:

We can put these together as follows, for a simple example:

Note that, after attaching the $2$-handle, we have a ball again, so we might as well not have attached the $1$- and the $2$-handle at all. I.e. these two handles can be cancelled in a way that will be made precise in due time. (This is the convertible roof, see end of Lecture 5 video for the hand gestures,) This picture is hard to look at so we can flatten it and draw only the images of the attaching maps in the boundary of the $0$-handle (identifying $S^2$ with $\mathbb{R}^2 \cup \infty$) as follows:

Note that we could have many $1$-handles, so some labelling of the feet is appropriate, and note that we only need to draw the core of the attaching map of each $2$-handle (the image of $S^1 \times \{0\} \subset S^1 \times B^1$) to specify the isotopy class of the attaching map. In fact, the same could be said for $1$-handles (assuming everything is oriented) but it is visually convenient to draw the whole disk. So here is another example:

So this has three $1$-handles, labelled $A$, $B$ and $C$, and three $2$-handles, labelled $a$, $b$ and $c$, and, of course, a $0$-handle that is the "background" to this picture and a $3$-handle that caps everything off. First, to see that you can cap it off with a $3$-handle you need to verify that the boundary is $S^2$.

Exercise: show that this manifold is $S^1 \times S^2$.

I'll end here, although in the lecture I then discussed Heegaard splittings. I'll write that up next time. Here's the video:

# 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.
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.
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 $B^k \times S^{n-k-1}$.

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: