Brian Hepler, Ph.D.

Local Cohomology and Morse Data

This post is next in my series of posts on Morse theory and its various generalizations. I last talked about “Classical Morse Theory,” (CMT) which studies the how the topology of a (real) smooth manifold is related to critical points of Morse functions defined on that manifold. Say the manifold is called $M$ , and we have some Morse function, $f: M \to \mathbb{R}$ ; (CMT;A) says that the topological type of the set $M_{\leq c} := f^{-1}(-\infty,c]$ remains constant unless $c \in \mathbb{R}$ passes over a critical value of $f$ . In the case where $v$ is a critical value, we study the change in topology of $M_{\leq c}$ with a pair of spaces, $(A,B)$ , which we call local Morse data for for f at p (which I’ll write as $|LMD(f,p)|$ ), defined as follows: let $\epsilon > 0$ be “sufficiently small,” $B_\epsilon(p)$ a small ball centered at $p$ of radius $\epsilon$ (say, with respect to some Riemannian metric on $M$ ). Then, for $\delta > 0$ “sufficiently small,”

$|LMD(f,p)| := (B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta], B_\epsilon(p) \cap f^{-1}(v-\delta) )$

(thankfully, the topological type of $|LMD(f,p)|$ is independent of the choice of metric, and independent of $\epsilon,\delta$ , provided that they’re chosen to be sufficiently small).

Say $p \in M$ is a non-degenerate critical point of $f$ of index $\lambda$ (recall that this means the Hessian of $f$ at $p$ in non-singular, and has $\lambda$ negative eigenvalues), $f(p) =v$ the corresponding critical value. Since critical values are locally isolated in $\mathbb{R}$ , there exists some $\delta > 0$ small so that $v$ is the only critical value of $f$ in the interval $[v-\delta,v+\delta]$ . Then, (CMT;B) says that $M_{\leq v + \delta}$ is obtained as a topological space from $M_{\leq v-\delta}$ by attaching the space $latex B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta]$ along the space $latex B_\epsilon(p) \cap f^{-1}(v-\delta)$. More specifically,

$|LMD(f,p)| \cong (D^\lambda \times D^{n-\lambda}, \partial D^\lambda \times D^{n-\lambda})$

where $n = \dim M$ , and $D^k$ is the $k$ -dimensional disk.

Now, me being me, I need to see how this fits into more general machinery. Thankfully, the way has already been paved for us: the Morse theory for constructible sheaves explored in Topology of Singular Spaces and Constructible Sheaves by J. Schurmann. There, local Morse data is framed, functorially, in terms of local cohomology groups:

(*) $LMD(f,p) := R\Gamma_{\{f \geq v\}}(\mathbb{Z}_M^\bullet)_p$

where $\mathbb{Z}_M^\bullet$ is the constant sheaf with stalk $\mathbb{Z}$ on $M$ , considered as a complex of sheaves concentrated in degree zero. $R\Gamma_{f \geq v}(-)$ is the derived functor “sections with support in $\{f \geq v\} := f^{-1}[v, \infty)$ , and we take the stalk at the point $p$ . This all seems a bit complicated, and it is at first for everybody. Worth investigating though, since understanding the LMD(f,p) construction is instrumental in generalizing the ideas of CMT and SMT to the “microlocal” setting; in particular, to the derived category (of bounded, constructible complexes of sheaves), and the construction of the “microsupport”of a complex of sheaves.

Let’s investigate (*). By constructibility of $R\Gamma_{\{\geq f \geq v\}}(\mathbb{Z}_M^\bullet)$ , there is an $\epsilon > 0$ such that

$R\Gamma_{\{ f \geq v\} }(\mathbb{Z}_M^\bullet)_p \cong R\Gamma (B_\epsilon(p);R\Gamma_{\{ f \geq v\}}(\mathbb{Z}_M^\bullet))$

which is isomorphic to

$R\Gamma(B_\epsilon(p),B_\epsilon(p) \cap f^{-1}[v-\delta,v); \mathbb{Z}_M^\bullet)$

where $\delta > 0$ is such that $f(\Sigma f) \cap [v-\delta,v+\delta] = \{v\}$ . Then, by (CMT;A), there is a homeomorphism of pairs $(B_\epsilon(p),B_\epsilon(p) \cap f^{-1}[v-\delta,v)) \overset{\thicksim}{\to} (B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta],B_\epsilon(p) \cap f^{-1}(v-\delta))$ , inducing the isomorphism

$R\Gamma_{\{f \geq v\} }(\mathbb{Z}_M^\bullet)_p \cong R\Gamma(B_\epsilon(p) \cap f^{-1}[v-\delta,v+\delta],B_\epsilon(p) \cap f^{-1}(v-\delta);\mathbb{Z}_M^\bullet)$

In short,

$LMD(f,p) \cong R\Gamma(|LMD(f,p)|;\mathbb{Z}_M^\bullet)$

Local Triviality -> Locally Cone-Like

So, locally cone-like…what did I mean by that? Let $X \subseteq M$ be a complex analytic subset of a complex manifold, $\mathfrak{S}$ a Whitney stratification of $X$ , $p \in X$ . Suppose we’ve given $M$ a Riemannian metric, r, and denote by $B_\delta(p) = \{ q \in M | r(q,p) \leq \delta \}$ the “ball off radius $\delta$ about $p$ .” Choosing local coordinates about $p$ in $M$ , we might as well assume that we’re dealing with the ordinary Euclidean distance. ANYWAY, for sufficiently small $\delta > 0$ , the “boundary” $\partial B_\delta(p)$ transversely intersects all strata of $\mathfrak{S}$ (this isn’t too hard to show…suppose not, use the local finiteness criterion for $\mathfrak{S}$ , and apply the Curve Selection Lemma to each stratum to achieve a contradiction). Then, there is a homeomorphism (preserving the strata), which I’ll call a $\mathfrak{S}$ -homeomorphism, of germs:

$(B_\delta(p) \cap X, p) \overset{\thicksim}{\to} (Cone(\partial B_\delta(p) \cap X), p)$

This can be rephrased a bit more efficiently. Let $r: X \to \mathbb{R}$ be “distance squared from $p$ .” Then, for $\delta > 0$ sufficiently small, the map $r: X \to [0,\delta]$ is a proper, stratified submersion. Think about it. The “stratified submersion” part tells you that the level sets of $r$ are transverse to strata. Properness allows us to invoke something called “Thom’s first isotopy lemma,” which tells us the cone bit.

Local Triviality

In a perfect world, all “naturally occurring” geometric objects in mathematics and physics would have a nice manifold structure, together with a well-behaved ring of functions. Unfortunately, this is simply not the case. But how do we proceed? Do we part completely from the safety of the power techniques of differential topology, and descend into the wilds of general topology? What do we keep? For me, the answer is the idea of local triviality. A manifold is “locally trivial” in the sense that the local (ambient) topological type of any particular point is that of ordinary Euclidean space.

One of the basic themes of singularity theory is that of stratifying spaces, or chopping up a singular space into a (disjoint) union of smooth or complex manifolds. Not haphazardly, of course. We require that the pieces, or strata, fit together in precise ways for some semblance of regularity. Another basic theme is to consider these singular spaces as embedded inside some ambient smooth or complex manifold, so as to retain as much differential topology as possible, considering strata as (suitably nice!) collections of submanifolds of the ambient space.

First, a partition. Let $X$ be a closed subset of a smooth manifold, $M$ . This will be our singular space. A (non-empty, duh) collection of submanifolds $\mathfrak{S} = \{S_\alpha\}_\alpha$ is a smooth partition of X if:

$X$ is the disjoint union of the $S_\alpha$ ;
Each $S_\alpha$ is a connected, smooth submanifold of $M$ .
$\mathfrak{S}$ is locally finite. That is, for all $x \in X$ , there is an open neighborhood $U$ of $x$ in $X$ such that $U \cap S_\alpha \neq \emptyset$ for only finitely many indices $\alpha$ .

Analogously, if $X$ is a complex analytic subset of a (connected) complex manifold $M$ , where the submanifolds $S_\alpha$ are now required to be connected complex submanifolds of $M$ . Additionally, we require that each $\overline{S_\alpha}$ is an irreducible complex analytic subset of $M$ , and that $\overline{S_\alpha} \backslash S_\alpha$ is a complex analytic subset of $M$ as well.

Partitions are the basic building block, after strata (in terms of complexity of their definition). What we actually use, however, are stratifications. This additional step serves to partially order the strata of a partition $\mathfrak{S}$ . Precisely, a smooth (resp., complex) partition $\mathfrak{S}$ of $X$ is a smooth (resp., complex analytic) stratification if it satisfies the Condition of the Frontier: If $S_\alpha, S_\beta \in \mathfrak{S}$ ( $S_\alpha \neq S_\beta$ ) are such that $S_\alpha \cap \overline{S_\beta} \neq \emptyset$ , then $S_\alpha \subset \overline{S_\beta}$ and $\dim S_\alpha < \dim S_\beta$ .

So now that $X$ has been chopped up into a bunch of nice submanifolds, how exactly do we “piece” these pieces together to understand the geometry and topology of $X$ ? That is, how do strata meet each other? The Whitney Conditions (after mathematician Hassler Whitney) are one method of imposing regularity on this “piecing together.” Basically, the Whitney conditions (there are two) control the behavior of “limiting tangent planes” of higher dimensional strata to lower dimensional strata.

Whitney’s condition (a) for a pair of strata $(S_\beta,S_\alpha)$ (where $S_\alpha,S_\beta \in \mathfrak{S}, S_\alpha \neq S_\beta$ ) at a point $p \in S_\alpha \cap \overline{S_\beta}$ states that, for any sequence of points $S_\beta \ni p_i \to p \in S_\alpha$ such that the limit $\lim_{i} T_{p_i} S_\beta = \tau$ exists (inside the Grassmannian of $\dim S_\beta$ -planes in the tangent space $T_p M$ , where $M$ is the ambient manifold), one has the inclusion $T_p S_\alpha \subseteq \tau$ .

We’d say $\mathfrak{S}$ is a Whitney (a) stratification if every pair $(S_\beta,S_\alpha)$ of strata satisfies Whitney’s condition (a) at all points $p \in S_\alpha \cap \overline{S_\beta}$ .

In terms of conormal spaces to strata, this has a particularly simple expression: the pair $(S_\beta,S_\alpha)$ satisfies Whitney’s condition (a) at $p \in S_\alpha \cap \overline{S_\beta}$ if there is an inclusion of fibers $\overline{T_{S_\beta,p}^*M} \subseteq T_{S_\alpha,p}^*M$ over $p$ .

Whitney’s condition (b) for a pair $(S_\beta,S_\alpha)$ at a point $p \in S_\alpha \cap \overline{S_\beta}$ states that, (after fixing a local coordinate system) for all sequences of points $S_\beta \ni p_i \to p \in S_\alpha$ such that the limit $\lim_i T_{p_i}S_\beta = \tau$ exists, for all sequences of points $S_\alpha q_i \to p \in S_\alpha$ such that the limiting “secant line” $\lim_i \overline{q_i p_i} = \mathfrak{l}$ exists (remember, we fixed a local coordinate system ahead of time, so this makes sense), there is an inclusion $\mathfrak{l} \subseteq \tau$ as subspaces of $T_pM$ . We’d say $\mathfrak{S}$ is a Whitney (b) stratification if this condition holds for all pairs of incident strata.

Thankfully, it doesn’t matter what local coordinate system you pick at the beginning, the condition is independent of that choice. We pick one for the sole purpose of making sense of these “secant lines.” Also, condition (b) implies condition (a), so it’s a strictly stronger requirement. Henceforth, a Whitney stratification will mean a (complex analytic) Whitney (b) stratification.

The main purpose (the only one I’ve seen or cared about, so far) for introducing Whitney stratifications is that the local, ambient topological type of $X$ is locally trivial along strata. This is intimately related to the locally cone-like nature of complex (and real!) analytic sets that I (briefly) mention in this post https://brainhelper.wordpress.com/2013/09/26/the-milnor-fibration-and-why-you-should-care/. It’s not exactly “locally Euclidean,” but it’s something!

But, I’m tired, and don’t feel like talking anymore. Until next time.

Over the past few decades, Morse theory has undergone many generalizations, into many different fields. At the moment, I only know of a few, and I understand even fewer. Well, let’s begin at the beginning:

Classical Morse theory (CMT)
Stratified Morse theory (SMT)
Micro-local Morse theory (MMT)

The core of these theories is, of course, the study of Morse functions on suitable spaces and generalizations/interpretations of theorems in CMT to these spaces. For CMT, the spaces are smooth manifolds (or, compact manifolds, if your definition of Morse function doesn’t require properness). SMT looks at Morse functions on (Whitney) stratified spaces, usually real/complex varieties (either algebraic or analytic), and more generally, subanalytic subsets of smooth manifolds. MMT deals with both cases, but from a more “meta” perspective that I’m not going to tell you about right now.

The overarching theme is pretty simple: one can investigate the (co)homology of $X$ by examining the behavior of level sets of Morse functions as they “pass through” critical values. First, we’ll need some notation. Let $M$ be a smooth manifold, $a < b \in \mathbb{R}$ , and let $f: M \to \mathbb{R}$ be a smooth function. Then, set

$M_{\leq a} := f^{-1}(-\infty,a]$
$M_{< a} := f^{-1}(-\infty,a)$
$M_{[a,b]} := f^{-1}[a,b]$

In CMT, this overarching idea is described by two “fundamental” theorems:

Fundamental Theorem of Classical Morse theory, A (CMT;A):

Suppose $f$ has no critical values on the interval $[a,b] \subseteq \mathbb{R}$ . Then, $M_{\leq a}$ is diffeomorphic to $M_{\leq b}$ , and the inclusion $M_{\leq a} \hookrightarrow M_{\leq b}$ is a homotopy equivalence (that is, $M_{\leq a}$ is a deformation-retract of $M_{\leq b}$ ).

Homologically speaking, this last point can be rephrased as $H_*(M_{\leq b},M_{\leq a}) = 0$ (for singular homology with $\mathbb{Z}$ coefficients).

Fundamental Theorem of Classical Morse theory, B (CMT;B):

Suppose that $f$ has a unique critical value $v$ in the interior of the interval $[a,b] \subseteq \mathbb{R}$ , corresponding to the isolated critical point $p \in M$ of index $\lambda$ . Then, $H_k(M_{\leq b},M_{\leq a})$ is non-zero only in degree $k = \lambda$ , in which case $latex H_\lambda(M_{\leq b},M_{\leq a}) \cong \mathbb{Z}$.

So, if $c \in \mathbb{R}$ varies across a critical value $a < v < b$ of $f$ , the topological type of $M_{\leq c}$ “jumps” somehow. If we want to compare how topological type of $M_{\leq b}$ differs from that of $M_{\leq a}$ , the obvious thing to do is consider them together as a pair of spaces $(M_{\leq b}, M_{\leq a})$ and look at the relative (co)homology of this pair. CMT;A and CMT;B together tell us that we’re only going to get non-zero relative homology of this pair when there is a critical value between $a$ and $b$ , and in that case, the homology is non-zero only in degree $\lambda$ .

But HOW does the topological type change, specifically, as we cross the critical value?

Stability and Genericity

Before I begin, I want to give credit where credit is due: much of the exposition (especially the proofs) of my last post was paraphrased from Guillemin and Pollack’s Differential Topology [1]. One of my favorites.

Okay, moving on.

We saw last time that Morse functions are pretty neat, and are abundant; “almost all” smooth functions are actually Morse functions. I’d like to take a minute to talk about this type of property (along with a related notion, “genericity“), as well as a notion called stability. “Almost all” is usually a phrase one comes across in analysis (or, as we saw, fields that use analysis, of which there are tons), and it means “the set of ‘bad choices’ is a set of (some suitable) measure zero.” “Genericity,” or, “being generic” is more or less the algebro-geometric counterpart to “almost all” (although it isn’t uncommon to use “generic” to mean “almost all”). Something is generic in something else if it is true on an open dense set. In algebraic geometry, we’d usually say “the set of bad choices lies in a subset of strictly smaller dimension.” Anyway, the basic idea is that such properties/objects are what you’d expect to find if you picked one “at random.” For example: draw a curve on a piece of paper (and pretend it’s $\mathbb{R}^2$ ). If you were to close your eyes, and put your finger on the paper, you’d basically always miss the curve, and land on blank space, illustrating that a generic point of $\mathbb{R}^2$ isn’t on the curve.

What do I mean by stability? What is “stable?” If you recall the sketch of a proof I gave last time for “almost all functions are Morse functions,” given some smooth functions $f: U \to \mathbb{R}$ (where $U \subseteq \mathbb{R}^n$ is an open subset), we “deformed,” or “perturbed” $f$ into a Morse function, $f_a := f + a_1 x_1 + \cdots +a_n x_n$ , by adding a generic linear form. If I deform some smooth map $f_0: M \to N$ to another map $f_1 : M \to N$ , I’m invoking one the fundamental operations in (differential) topology: homotopy. We’d say $f_0$ and $f_1$ are homotopic, usually written $f_0 \thicksim f_1$ , if there exists a smooth map $F: M \times [0,1] \to N$ such that, for all $x \in M$ , $F(x,0) = f_0(x)$ and $F(x,1) = f_1(x)$ . Smoothness of $F$ then ensures that all the “in between” maps $f_t(x) := F(x,t)$ are smooth as well. Here’s a simple example to illustrate that this is really what we mean when we say $f_0$ is deformed to $f_1$ .

Here’s a really, really nice description of this notion in [1]:

In the real world of sense perceptions and physical measurements, no continuous quantity or functional relationship is ever perfectly determined. The only physically meaningful properties of a mapping, consequently, are those that remain valid when the map is slightly deformed. Such properties are stable properties, and the collection of maps that posses a particular stable property may be referred to as a stable class of maps. Specifically, a property is stable provided that whenever $f_0: X \to Y$ possesses the property and $f_t : X \to Y$ is a homotopy of $f_0$ , then, for some $\epsilon > 0$ , each $f_\epsilon$ with $t < \epsilon$ also possesses the property.

In this vein, the idea is that stable properties are “observable.” These are the types of things we want to look for when playing around with functions.

I said before that the ideas of stability and genericity were related. Suppose I want to find a Morse function $f: M \to \mathbb{R}$ . We know already that almost all smooth, real valued functions on $M$ are Morse. But what happens if I happen to pick a bad one? Never fear; we deform $f$ by adding some generic linear form $\ell_a = \sum_{i=1}^n a_i x_i$ . Moreover, by the genericity of the choice of $\ell_a$ , we can pick good choices of “deformation vector” $a = (a_1,\cdots,a_n)$ such that $\| a \|$ is arbitrarily small. Hence, even if we end up picking a bad function $f$ , for any $\epsilon > 0$ , we can find a Morse function $f_a$ such that $\|f - f_a\| < \epsilon$ . This is a common occurrence for stable properties: even if you happen to find a bad function, there are arbitrarily close good functions (in the space of smooth maps with, say, the supremum norm). Some of the most common stable properties for a smooth map $f: M \to N$ are:

local diffeomorphisms
immersions
submersions
maps transversal to a given submanifold $Q \subseteq N$
embeddings
diffeomorphisms.

Morse functions are also stable, with the caveat that we require our domain to be compact. Let $f$ is a Morse function on a compact manifold $X$ , and let $f_t$ be a homotopic family of functions with $f_0 = f$ . Then, $f_t$ is Morse for all $t$ sufficiently small.

In upcoming events, we’ll want to analyze the topology of a manifold by studying the level sets of Morse functions on the them, and these notions of genericity and stability will ensure that the selection of such functions is never in short supply.

A Tour de Morse (theory)

Morse theory is amazing. Very geometric, more-or-less very intuitive. You don’t really explore it in detail until you’ve seen a fair bit of differential topology, but if you look closely, you start getting exposed to its core ideas as early as multivariate Calculus.

As is the fashion in modern geometry (specifically, algebraic geometry), we study geometric objects by studying the behavior of (appropriate classes of ) functions on them. But which functions? In algebraic geometry, if you’ve got some nice affine variety, you’ve got a set god-given functions to use: the coordinate ring of the variety. Here, in the affine case, this is a finitely generated, reduced (=no nilpotents) algebra over a field. Basically, a quotient of a polynomial ring by a radical ideal. Not too bad, quite manageable.

However, for a smooth manifold $M$ , the class of smooth functions on $M$ is really big. To make it “worse”, the existence of bump functions makes it hard to obtain too much cohomology info from the sheaf of smooth functions. Of course, we can use differential forms to obtain geometric (cohomological) info; this is known as the de Rham cohomology of $M$ , and it’s actually isomorphic to the singular cohomology of $M$ , which is also isomorphic to the Cech cohomology of the constant sheaf, $\mathbb{R}_M$ .

Cue Morse functions.

But first (I lied), we need to recall some basic terminology. Let $f: M \to N$ be a smooth function between the smooth manifolds $M$ and $N$ . For every point $x \in M$ , $f$ induces a linear transformation $d_xf : T_x M \to T_{f(x)}N$ between tangent spaces. We say that $x$ is a regular point of $f$ if the map $d_xf$ is surjective (this means “ $f$ is a submersion at $x$ “). If $d_xf$ is not surjective, we say $x$ is a critical point of $f$ . Suppose $f(x) = y$ . We say $y$ is a regular value of $f$ if, for all $x \in f^{-1}(y)$ , $x$ is a regular point of $f$ . If this is not the case (i.e., some point in the preimage of $y$ is a critical point), we say that $y$ is a critical value of $f$ . If you’ve been good and remember your basic Calculus, regularity of a point/value tells us a lot (topologically) about $M$ near $x$ . Via the Implicit Function Theorem, if $y$ is a regular value, the set $f^{-1}(y)$ is a smooth submanifold of $M$ , of pure codimension one. If $x$ is a regular point of $f$ , there is an open neighborhood, $U$ of $x$ in $M$ such that $f^{-1}(f(x)) \cap U$ is a smooth submanifold of $M$ of pure codimension one.

But what happens at critical points? Critical values? How much do we have to worry? How abundant are they? Fortunately, we have

Sard’s Theorem: the set of critical values of $f: M \to N$ has measure zero in $N$ .

So, “almost all” points of $N$ are regular values of $f$ . But, let’s go deeper: what happens at critical points?

Okay, so this is where you start seeing this stuff in early Calculus. Say we’ve got a smooth function $f: \mathbb{R} \to \mathbb{R}$ , and we look at its graph, $M$ , in $\mathbb{R}^2$ . One of the first things we investigate are the “tangent lines” to points on the graph; here, these are the tangent spaces to $M$ . Using this we can answer the question “where does $f$ achieve extreme values?” Every Calc student knows (or, should know) that these can only happen (at smooth points of the domain of $f$ ) when the tangent line to $f$ at some point $x \in \mathbb{R}$ is “horizontal”, that is, when $f'(x) = 0$ . Equivalently, when $d_xf : T_x\mathbb{R} \to T_{f(x)}\mathbb{R}$ is not surjective (since in the one dim. case, $d_xf(v) = f'(x)v$ , and $d_xf$ is surjective iff $f'(x) \neq 0$ ).

But what about the second derivative? After all, we said $f$ was infinitely differentiable. Hopefully these higher derivatives contain more information?

Of course, you already know the answer. Suppose $f'(x) = 0$ , but $f''(x) \neq 0$ . Well, it’s either going to be positive or negative. If $f''(x) > 0$ , then we know $f$ has a local minimum at $x$ . If $f''(x) < 0$ , then $f$ has a local maximum at $x$ . Similarly, we’d say the graph is locally “concave up” in the former case, “concave down” in the latter. Intuitively, the graph “looks like” the parabola $y = \pm x^2$ around $x$ , depending on the sign. We can’t really apply this analysis in the case where $f''(x) = 0$ ; for that, you’d need to use Taylor’s theorem to get more information about $f$ at $x$ .

It isn’t really until we start doing Calculus in several variables that we see the utility of this approach. Let’s move to three variables. Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a smooth function, and let $M = \{(x,y,f(x,y))| x,y \in \mathbb{R}\}$ be the graph of $f$ . Suppose $p= (x_0,y_0)$ is a critical point of $f$ . Recall the differential in this case is given by

$d_pf(a,b) = a \frac{\partial f}{\partial x}(p) + b \frac{\partial f}{\partial y}(p)$

and saying that $p$ is a critical point of $f$ means that $\frac{\partial f}{\partial x}(p) = \frac{\partial f}{\partial y}(p) =0$ . Since we’ve got more than one variable, any kind of “Second derivative test” is going to need to information from all the second partial derivatives, in some way. For example, how do we reinterpret the criterion $f''(x) \neq 0$ in this case?

I’ll save you the trouble and just say it: what we need to examine is something called the Hessian of f at $p$ :

$H(p) = \begin{pmatrix} \frac{\partial^2 f}{\partial x^2}(p) && \frac{\partial^2 f}{\partial x \partial y}(p) \\ \frac{\partial^2 f}{\partial y \partial x}(p) && \frac{\partial^2 f}{\partial y^2}(p) \end{pmatrix}$

The Hessian of $f$ at a point is just the matrix of second partials of $f$ , arranged in a particular way. (In the general case of $\mathbb{R}^n$ , with coordinates $(x_1,\cdots, x_n)$ , the Hessian takes the form $\left ( \frac{\partial^2 f}{\partial x_i \partial x_j} \right )_{1\leq i,j \leq n}$ . Requiring that “ $f''(x) \neq 0$ now becomes $D(p) := \text{det}H(p) \neq 0$ , and in such a case, we say $p$ is a nondegenerate critical point of $f$ . We say

$p$ is a local minimum of $f$ if $D(p) > 0$ , and $\frac{\partial^2 f}{\partial x^2}(p) > 0$ ;
$p$ is a local maximum of $f$ if $D(p) > 0$ , and $\frac{\partial^2 f}{\partial x^2}(p) < 0$ ; and
$p$ is a saddle point of $f$ if $D(p) < 0$ .

Intuitively, this says that the graph of $f$ locally looks like the paraboloid $z= \pm (x^2 + y^2)$ in the first two cases (depending on the sign), and like the hyperbolic paraboloid (= “saddle”) $z = x^2 - y^2$ in the third case.

But what do I mean “looks like”? Is there a formal way to express this? Of course, or I wouldn’t be talking about it.

Might as well do the general case: Let $M$ be a smooth manifold of dimension $n$ , $f: M \to \mathbb{R}$ a smooth function. Let $p \in M$ , and suppose that $p$ is a nondegenerate critical point of $f$ *. Then, there is a smooth system of coordinates about $p$ such that, in these coordinates, $f$ may be written as

$f(y_1,\cdots, y_n) = f(p) - \sum_{i= 1}^\lambda y_i^2 + \sum_{i = \lambda + 1}^n y_i^2$

where $1 \leq \lambda \leq n$ is the index of f at p (= the number of negative eigenvalues of $H(p)$ ). This result is known as the Morse Lemma, and it legitimizes our intuition from the previous examples.

*We had previously defined the Hessian of $f$ at $p$ within a given coordinate system. As it turns out, “nondegeneracy” of a critical point is independent of coordinates, as is the index.*

Nondegeneracy of a critical point is basically the next best thing to requiring regularity of a point. In addition to the Morse lemma, nondegenerate critical points are isolated as well. That is, at such a point $p$ , we can find an open neighborhood $U$ of $p$ such that $p$ is the only critical point of $f|_U$ . This isn’t even that hard to show: if $(x_1,\cdots,x_n)$ are local coordinates about $p$ , define a new function, $F : M \to \mathbb{R}^n$ via

$F(q) = (\frac{\partial f}{\partial x_1}(q),\cdots,\frac{\partial f}{\partial x_n}(q))$

Since $p$ is a critical point of $f$ , $F(p) = (0,\cdots,0) \in \mathbb{R}^n$ . Then, the differential of $F$ at $p$ is equal to the Hessian of $f$ at $p$ , so nondegeneracy of $p$ implies nonsingularity of $d_pF$ . Hence, by the Inverse Function theorem, $F$ carries some open neighborhood $U$ of $p$ in $M$ diffeomorphically onto an open neighborhood of the origin in $\mathbb{R}^n$ . That is, $p$ is the only critical point of $f$ inside $U$ .

In keeping with all these definitions, we say a smooth function $f: M \to \mathbb{R}^n$ is a Morse function if all its critical points are nondegenerate. Some authors impose the additional requirements that every critical value has only one corresponding critical point, or that $f$ be proper (= preimage of a compact set is compact). For now, I’ll stick to my original definition.

Morse functions are basically as good as it gets for our current approach: Almost all level sets $f^{-1}(c)$ are smooth submanifolds of $M$ of codimension one, and the bad points (=critical points) where our analysis fails are isolated incidents, and even then, we know exactly what $f$ looks like in an open neighborhood of a bad point. But are Morse functions too good to be true? Do we encounter them often? As it turns out, like our worries about regular points/values, “almost all” smooth functions are Morse functions. The core of the proof is actually (again) just Sard’s theorem.

Let’s just examine the case where $f$ is a smooth function on an open subset $U \subseteq \mathbb{R}^n$ to $\mathbb{R}$ . Let $(x_1,\cdots,x_n)$ be a choice of coordinates on $U$ . For $a = (a_1,\cdots,a_n) \in \mathbb{R}^n$ , we define a smooth function

$f_a := f + a_1x_1 + \cdots + a_n x_n$

Theorem: No matter what the function $f$ is, for almost all choices of $a$ , $f_a$ is a Morse function on $U$ .

Again, we use the function $F(q) = (\frac{\partial f}{\partial x_1}(q),\cdots,\frac{\partial f}{\partial x_n}(q))$ . Then, the derivative of $f_a$ at a point $p$ is represented in these coordinates as

$d_p f_a = (\frac{\partial f_a}{\partial x_1}(p),\cdots, \frac{\partial f}{\partial x_n}(p)) = F(p) + a$

So, $p$ is a critical point of $f_a$ if and only if $F(p) = -a$ . Since $f_a$ and $f$ have the same second partials, the Hessian of $f$ at $p$ is the matrix $d_p F$ . If $-a$ is a regular value of $F$ , whenever $F(p) = -a$ , $d_pF$ is nonsingular. Consequently, every critical point of $f_a$ is nondegenerate. Sard’s theorem then implies that $-a$ is a regular value of $F$ for almost all $a \in \mathbb{R}^n$ .

There’s so much more to talk about, but I’ve already rambled on for quite a bit. Until next time.

Goals for the (immediate) future

The next few weeks are going to be very (very) busy, what with qualifying exams, more talks, and extremely hard homeworks, so I’m not sure how much I’ll be posting (but, I won’t bail again, last year (I hope, at least…)).

So, modulo the existence of free time, I’ve been really wanting to do a post that surveys the Milnor fibration in “all” of its various forms, from its conception to modern usage. I’ve already talked about the “classical” case, i.e., Milnor’s original fibration on the sphere (minus the real link), as well as the fibration “in the ball”, from last time. In addition to this, I want to talk about

The “general” Milnor fibration, associated to a complex analytic function on an arbitrary complex analytic space (for this, I’ll need to talk about the notion of a stratification, and various equisingularity conditions we can impose thereon).
A “cohomological” version of the Milnor fibration, which basically says that, for a bounded, constructible complex of sheaves, $\mathcal{F}^\bullet$ , and a complex analytic function $f: (X,0) \to (\mathbb{C},0)$ , there is a fundamental system of compact neighborhoods of 0, $\{K_n \}_n$ , such that all the higher direct image sheaves $R^i f_*(\mathcal{F}^\bullet|_{K_n})$ are locally constant (along with a “stability” result). For this, I’ll probably need to mention some basic results/theory concerning the derived category, constructible sets, etc.
The “vanishing and nearby cycles” of Deligne, associated to a complex analytic function $f: (X,0) \to (\mathbb{C},0)$ . A purely category-theoretic formulation, these produce certain complexes of sheaves whose stalk cohomology at a point is (naturally) isomorphic to the cohomology of the Milnor fiber of $f$ at that point. Really fucking cool, but I’ll need to do quite a bit of explaining to get to these beasts.

In addition to the last point, I want to (later on, after the above stuff) talk about

“Basic” Morse theory
Stratified Morse theory
Microlocal Morse theory

but that might be quite a lofty goal, at least for the time being. Who knows, I might wake up smarter some day.

Until next time.

“The” Milnor Fibration: some proof

As it turns out, the Milnor fibration I talked about last time, i.e., the normalized map

$f/|f| : S_\epsilon - V(f) \cap S_\epsilon \to S^1$

is diffeomorphic to a much “nicer” fibration:

$f: B_\epsilon \cap f^{-1}(\partial D_\eta) \to \partial D_\eta$

where $\partial latex D_\eta$ is the boundary of disk about the origin in $\mathbb{C}$ of radius $\eta$ . And, this is a (smooth) locally trivial fibration for sufficiently small $\epsilon, \eta$ . Intuitively, I think this one makes more sense. You can picture the total space as an open “tubular neighborhood” of the fiber $V(f) = f^{-1}(0)$ inside the closed ball $B_\epsilon$ about the origin.

The proof is basically just an application of

Ehresmann’s Theorem

Let $f: M \to N$ be a smooth map between the smooth manifolds $M$ and $N$ . Then, if $f$ is a proper submersion, it is a smooth locally trivial fibration over $N$ .

In addition, if $Q \subseteq M$ is a closed submanifold such that the restriction $f|_Q$ is still a submersion, then $f|_Q$ is a smooth locally trivial fibration that is compatible with $f$ .

Usually, one takes the closed submanifold $Q$ to be $\partial M$ , in the case where $M$ is a smooth manifold with boundary.

Proof(of the Milnor fibration, isolated critical point)

Not too hard. First, we need to pick epsilon. Choose $\epsilon$ small so that $V(f)$ transversely intersects $S_\varepsilon$ , and such that $\overset{\circ}{B}_\epsilon \cap \Sigma f \subseteq V(f)$ . Now, delta.

For all choices of $\delta > 0$ , the restricted map

$f: B_\epsilon \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*) \to \overset{\circ}{\mathbb{D}}_\delta^*$

is proper, via a quick application of the Heine-Borel theorem. The rest of the proof is just Ehresmann’s theorem.

By our choice of $\epsilon$ , $f$ has no critical points on the interior $B_\epsilon^\circ \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*)$ . Therefore, any critical that occur must be on the “bounding sphere”, $S_\epsilon \cap f^{-1}(\overset{\circ}{\mathbb{D}}_\delta^*)$ . So, we consider the critical points of the function $f|_{S_\epsilon}$ .

BUT, by our choice of $\epsilon$ , $V(f)$ transversely intersects $S_\epsilon$ , and, by “stability” of transversality, there is an open neighborhood of the origin in $\mathbb{C}$ consisting entirely of regular values of $f|_{S_\epsilon}$ . WLOG, we can suppose this neighborhood is of the form $\overset{\circ}{\mathbb{D}}_\delta$ for some $\delta > 0$ . Throw away the origin, and the claim follows by Ehresmann’s theorem.

“The” Milnor Fibration: Classical Case

Where were we?

We had just equated the “innocent” question

“How does $V(f)$ ‘sit inside’ $\mathcal{U}$ at $0$ ?”

with the more precise(ish)

“How is the real link of $V(f)$ at 0 embedded in $S_\varepsilon$ ?”

Milnor’s genius idea was to realize the complement, $S_\varepsilon - K$ , as the total space of a smooth, locally trivial fibration over the unit circle:

Milnor’s Fibration Theorem:

For $\varepsilon > 0$ sufficiently small, the map

$\frac{f}{\| f\|}: S_\varepsilon - K \to S^1$

is the projection of a smooth, locally trivial fibration. In addition, “the” fiber is a smooth, $2n$ -dimensional, parallelizable manifold.

In reality, there are a few other objects called “the” Milnor fibration:

(Inside the ball) For $1 >> \varepsilon >> \delta > 0$ , the restriction

$f: \overset{\circ}{B}_\varepsilon \cap f^{-1}(\partial \mathbb{D}_\delta) \to \partial \mathbb{D}_\delta$

is a smooth, locally trivial fibration.

(the compact fibration) For $1>> \varepsilon >> \delta > 0$ , the restriction

$f: B_\varepsilon \cap f^{-1}(\partial \mathbb{D}_\delta) \to \partial \mathbb{D}_\delta$

is a topological locally trivial fibration.

And here’s the kicker: topological (resp., smooth) locally trivial fibrations over $S^1$ are completely classified by the fiber and the so-called characteristic homeomorphism (resp., diffeomorphism) of the fiber.

I’ll refer to the first two fibrations as “the” Milnor fibration (for now…), and the Milnor fiber as $F_f$ . The characteristic diffeomorphism $h : F_f \to F_f$ is induced by the action of the fundamental group $\pi_1(S^1,1)$ , and is defined upto pseudo-isotopy (endow the total space with a Riemannian metric, horizontally lift $[\gamma(t)] \in \pi_1(S^1,1)$ to the total space, and parallel transport “around” the circle. ) HOWEVER, we do get a unique Monodromy automorphism

$h_* : H_*(F_f;\mathbb{Z}) \overset{\thicksim}{\to} H_*(F_f;\mathbb{Z})$

and this correspondance yields a group representation

$\rho: \pi_1(S^1,1) \to Aut(H_*(F_f;\mathbb{Z}))$

called the local monodromy.

Some more on the fiber:

Theorem (Milnor):

$F_f$ is a complex $n$ -dimensional manifold.
$F_f$ has the homotopy type of a finite CW-complex.

If 0 is an isolated critical point of $f$ , then

$F_f$ is homotopy equivalent to a finite bouquet of $n$ -spheres.

The number of $n$ -spheres, denoted $\mu := \mu(f,0)$ , is called the Milnor number for $f$ at 0, and is the $n$ th Betti number of $F_f$ . Actually, $\mu$ may be calculated quite easily, via an alternative description:

$\mu(f,0) = \text{dim}_\mathbb{C} \left ( \frac{\mathcal{O}_{\mathcal{U},0}}{\mathcal{J}_f} \right )$

where $J_f$ is the Jacobian Ideal of $f$ : the ideal generated by the partial derivatives of $f$ inside the local ring $\mathcal{O}_{\mathcal{U},0}$ of germs of holomorphic functions at 0 (which is isomorphic to the ring of convergent power series in $n+1$ complex variables).

I don’t have it in me now to include examples, but I’ll update this post later on (after I’ve actually slept) with some good ones.

For a closing comment, some reassurance that this all is ACTUALLY useful:

Theorem(Topological Invariance) [Le-Tessier]

For a reduced hypersurface, the homotopy type of the Milnor fiber is an invariant of the local, ambient, topological type of the hypersurface. That is, if $f,g : (\mathcal{U},0) \to (\mathbb{C},0)$ are two reduced complex analytic function germs such that $(V(f),\{0\})$ is homeomorphic to $(V(g),\{0\})$ , then there exists a homotopy-equivalence $\alpha : F_f \to F_g$ such that $\alpha$ commutes with the respective monodromy automorphisms.

Note that, when $0$ is an isolated critical point, this implies that the Milnor number is ALSO an invariant of the local, ambient, topological type of the hypersurface at the origin.

Stay tuned.

“The” Milnor Fibration, and Why-You-Should-Care.

I’m back! After a long period of laziness, I’m back. Mainly, because the past week, I’ve been kicking myself in the ass for losing basically all my notes over the past few months, and I have to present at the math department’s new seminar in singularity theory. Aren’t I smart?

The topic? A really useful technique/object used to study the topology of (complex) analytic spaces, called the Milnor fibration. The reason for the quotes in the title is simple: there are several objects and manifold manifestations of this so-called “Milnor fibration.” Hence, I’ll do my best to introduce “the” main idea, and hopefully walk both myself and whatever readers are out there through its different forms.

Let’s start simple, with an innocent question: “Given a complex analytic hypersurface $Y$ in a complex manifold $M$ , and a point $p \in Y$ , how does $Y$ ‘sit inside’ $M$ at $p$ ?”

Since we only care about the local behavior (here) of $Y$ at $p$ in $M$ , without any loss of generality, we can rephrase this as follows: $M = \mathcal{U}$ is a connected open neighborhood of $p = 0 \in \mathbb{C}^{n+1}$ , $Y = V(f) := f^{-1}(0)$ for some (non-constant) complex analytic function germ $f : (\mathcal{U},0) \to (\mathbb{C},0)$ . In this form, our innocent question becomes less hand-wavy:

“How does $V(f)$ (topologically) ‘sit inside’ $\mathcal{U}$ at $0$ ?”

Wtf do we mean by “sit inside”? We want to know about the “topological type” (i.e., the homeomorphism class of the pair $(V(f),\{0\})$ ), but more importantly, we want to know how $V(f)$ is embedded as a subset of $\mathcal{U}$ . For this, we mean the local, ambient, topological type of $V(f)$ in $\mathcal{U}$ at 0, given by the datum of a triple $(\mathcal{W},\mathcal{W} \cap V(f),\{0\})$ , where $\mathcal{W}$ is an open neighborhood of 0 in $\mathcal{W}$ . In our case, it suffices to consider triples of the form $(\overset{\circ}{B}_\varepsilon, \overset{\circ}{B}_\varepsilon \cap V(f), \{0\})$ , where $\overset{\circ}{B}_\varepsilon$ denotes the open ball centered at the origin in $\mathbb{C}^{n+1} \cong \mathbb{R}^{2n+2}$ . For the sake of horribly cramped notation, I’ll suppress the dimension, and hope you can keep up.

From general topological stuff, if we can determine the structure of a triple $(B_\varepsilon, B_\varepsilon \cap V(f),\{0\})$ , we can understand $(\overset{\circ}{B}_\varepsilon, \overset{\circ}{B}_\varepsilon \cap V(f), \{0\})$ .

Now, the question becomes:

“What is the topological type of the triple $(B_\varepsilon, B_\varepsilon \cap V(f),\{0\})$ ?”

But wait, it gets better (due to a result of Milnor):

Theorem (Milnor):

For $\varepsilon > 0$ sufficiently small, there is a homeomorphism of triples $(B_\varepsilon, B_\varepsilon \cap V(f),\{0\}) \cong (Cone(S_\varepsilon),Cone(S_\varepsilon \cap V(f)), \{0\})$ . Moreover, the topological type of this triple is independent of $\varepsilon$ (for $\varepsilon$ sufficiently small).

Where “Cone” is the topological cone. That is, if $X$ is a topological space, $Cone(X) := X \times [0,1] / X \times \{0\}$ . This is just a fancy way of saying “all the interesting topological behavior occurs on the boundary of the ball.” So…if we can figure out how this space $K := S_\varepsilon \cap V(f)$ (called the Real Link of $V(f)$ at 0) is embedded inside $S_\varepsilon$ , then we’ve (more or less) answered our question. It’s important to note that the real link of $V(f)$ at 0 is actually a well-defined object, since the topological type of the above triple is independent of epsilon, when epsilon is chosen “small enough”.

This is basically what Milnor strove to answer with his Fibration. But, you’ll have to wait for the sequel to see that in action.

Before I leave, I wanted to talk about some nice cases/descriptions of the real link of $V(f), K$ :

Lemma:

Let $X$ be a real analytic subset of $\mathbb{R}^{n+1}$ containing the origin, such that $X - \{0\}$ is a real analytic submanifold of $\mathbb{R}^{n+1}$ . Then, for $\varepsilon > 0$ sufficiently small, $X$ transversely intersects the $n$ -sphere $S_\varepsilon.$

proof:

Suppose, for the sake of contradiction, that no such $\varepsilon$ exists. Then, for all $\varepsilon > 0$ , there is some $p \in S_\varepsilon \cap X$ such that

$T_p S_\varepsilon + T_p X \neq T_p \mathbb{R}^{n+1}$

Let $r(x) = \|x\|^2$ be the (real analytic) “norm-squared” function on $\mathbb{R}^{n+1}$ , so that $r^{-1}(\varepsilon^2) = S_\varepsilon$ . In addition, since the origin is the only critical point of $r$ , $T_p S_\varepsilon = T_p r^{-1}(r(p))$ . Then, the above implies $p \in Z := \Sigma(r|_X)$ (where $\Sigma(r|_X)$ is the critical locus of $r$ , restricted to $X$ , which is a real analytic subset of $\mathbb{R}^{n+1}$ ). Since we get such a $p$ for all $\varepsilon > 0$ , it follows that $0 \in \overline{Z}$ , so we can apply the Curve Selection Lemma: there exists a real analytic curve $\gamma : [0,\eta) \to \mathbb{R}^{n+1}$ such that $\gamma(0) = 0$ , and for $t \in (0,\eta)$ , $\gamma(t) \in Z$ . What then?

When in doubt, take the derivative of something: for $t \neq 0$ ,

$r(\gamma(t))' = d_{\gamma(t)} r(\gamma(t)') = 0$

since $\gamma(t)' \in \text{Ker}(d_{\gamma(t)}r)$ . That is, $r(\gamma(t)) = \text{ constant}$ . BUT, $r(\gamma(0)) = 0$ , and $r \circ \gamma$ is a real analytic function, so we actually get $r(\gamma(t)) = 0$ for ALL $t \in [0,\eta)$ . BUT, $r(\gamma(t)) =0$ iff $\gamma(t) = 0$ , CONTRADICTION!

And the lemma follows.

Done.

So…here, if 0 is an isolated critical point of $f$ , this tells us that (for small enough $\varepsilon > 0$ ), that the real link $K$ is a smooth submanifold of $S_\varepsilon$ , of real dimension $2n-1$ .

More later.