Microsupport and Propagation

So, last time, I briefly mentioned a sheaf-theoretic “Local Morse Datum” for a smooth (Morse) function f : M \to \mathbb{R} at a (non-degenerate) critical point p \in M

LMD(f,p) := R\Gamma_{\{f \geq 0\}}(\mathbb{Z}_M^\bullet)_p

which gives the integral cohomology of the “local Morse datum” of f at p, considered as a pair of spaces.  Confusing, I know.  It seems like complete overkill at this point, and it is.  

So, let’s go deeper. 

First, we’ll consider more general objects: the local Morse data of f at p, with respect to the complex of sheaves \mathcal{A}^\bullet \in D^b(M), denoted

LMD(f,p; \mathcal{A}^\bullet ) := R \Gamma_{\{ f \geq 0 \}}(\mathcal{A}^\bullet )_p

 

Basically, you have the complex of sheaves, \mathcal{A}^\bullet, on M, and you consider sections of this “sheaf” whose support is contained in the subset \{f \geq 0\} := f^{-1}[0,\infty), and take the stalk cohomology at the point p \in f^{-1}(0).  Sort of like: the sections of the sheaf that propagate in the “positive direction” (where “positive” is taken to be with respect to f (really, the covector d_pf, but we’re not quite there yet 🙂 ) ).  If this stalk cohomology vanishes, then the local sections of the sheaf can be “extended” a little bit further away from p, at least in the “positive” direction.  That’s supposed to be what I mean by “propagate” here.  

 

Why is this idea useful?  

 

Let X \subseteq M be a sufficiently nice closed subset, so we can give it a Whitney stratification (by which I mean, “satisfies Whitney’s condition (b) at all appropriate times”), \mathfrak{S}.  To each stratum S \in \mathfrak{S}, we associate the following subset of the cotangent bundle of M, called the conormal space to S in M:

T_S^* M := \{ (p,\eta) \in T^*M | p \in S, \eta ( T_p S) = 0 \}

which consists of covectors in the cotangent bundle which annihilate the various tangent spaces to S, considered as subspaces of the tangent spaces to M.  Equivalently (after perhaps endowing M with a Riemannian metric), you can think of these elements (p,\eta) as hyperplanes in T_pM that contain the subspace T_p S \subseteq T_p M.  

Let \overset{\thicksim}{f} : M \to \mathbb{R} be a smooth function, f := \overset{\thicksim}{f}|_X its restriction, \Sigma_\mathfrak{S} f the associated stratified critical locus of f.  It is any easy exercise to show that, when \mathfrak{S} is a Whitney (a) stratification, \Sigma_\mathfrak{S} f is a closed subset of X.  Similarly, the Whitney (a) condition is equivalent to requiring the equality:

\bigcup_{S \in \mathfrak{S}} T_S^*M = \bigcup_{S \in \mathfrak{S}} \overline{T_S* M}

Hence, when I say p \in \Sigma_\mathfrak{S} f, there is a unique stratum, say S, for which p \in \Sigma (f|_S) (since the strata are disjoint).  And, if you think about it, if p \in \Sigma (f|_S), we must have 

(p,d_p f) \in T_S^*M

as d_p (f|_S) = d_p f(T_p S) = 0.  Remember this.  

 

Local Stratified Acyclicity (LSA)

Say we have our Whitney stratification, \mathfrak{S}, of X \subseteq M.  Then, \mathfrak{S} satisfies something called LSA

for all \mathcal{F}^\bullet \in D_\mathfrak{S}^b(X) (this means the cohomology sheaves of \mathcal{F}^\bullet are all locally constant with respect to the strata of \mathfrak{S}), for all (germs of) stratified submersions f: (M,p) \to (\mathbb{R},0)

R \Gamma_{\{ f|_X \geq 0 \} }(\mathcal{F}^\bullet )_p = 0.  

I’m not going to prove this, but it’s a consequence of a result called the non-characteristic deformation lemma of Kashiwara and Schapira in Sheaves on Manifolds.  This tells us that the cohomology sheaves of complexes of sheaves are locally constant if a certain (similar) vanishing condition occurs with germs of submersions.  

And now, finally, I can talk about the microsupport of a complex of sheaves, \mathcal{F}^\bullet \in D^b(M), which encodes the “directions” at given points of M where sections of \mathcal{F}^\bullet “do not propagate.”  That is, the directions in which we should expect to detect changes in the cohomology (sheaves) of \mathcal{F}^\bullet.  Precisely, the Microsupport of \mathcal{F}^\bullet is the subset \mu supp(\mathcal{F}^\bullet ) \subseteq T^*M, characterized (in the negative…) by: 

(p,\eta) \notin T^*M if and only if there exists an open subset U of (p,\eta) in T^*M such that, for all smooth function germs f : (M,x) \to (\mathbb{R},0) with (x,d_x f) \in U, one has R \Gamma_{\{ f \geq 0 \}}(\mathcal{F}^\bullet )_x = 0.  

 

So, if (p,\eta) \in \mu supp(\mathcal{F}^\bullet), locally, we can say \eta = d_p f for some smooth function germ at p, and if you move in the direction of the “gradient flow” of f in the “positive direction,”  \mathcal{F}^\bullet will detect a change in the cohomology of M.  I like to picture f to be the germ of some linear form (say in a local coordinate system about p), and the gradient flow is like looking a family of cross sections of M near p.  Moreover, p will be a critical point whatever function we pick, by trivial application of LSA.  

I also think this emphasizes the importance of the connection with Morse data: LMD(f,p; \mathcal{F}^\bullet ) \neq 0 if and only if (p,d_p f) \in \mu supp(\mathcal{F}^\bullet).  

 

Now, say we’ve got our sufficiently nice closed subset X \subseteq M, with Whitney stratification \mathfrak{S}, equipped with inclusion map i: X \hookrightarrow M.  Say we’ve got some smooth function f: M \to \mathbb{R}.  We know that, for p \in S \in \mathfrak{S}, p \in \Sigma (f|_S) if and only if (p, d_p f) \in T_S^*M.  Then, via the isomorphism

R \Gamma_{\{ f|_X \geq 0 \}}(\mathcal{F}^\bullet )_p \cong R \Gamma_{\{ f \geq 0 \}} (Ri_* \mathcal{F}^\bullet )_p

and an application of LSA, this quantity vanishes if and only if f|X is a stratified submersion.  Consequently, we see 

\mu supp(Ri_* \mathcal{F}^\bullet) \subset \bigcup_{S \in \mathfrak{S}} T_S^*M

 

Now…what does the microsupport reveal about the local Morse data of functions with stratified critical points? That’s where things will start to get interesting.  Until next time.  

 

References: 

M. Kashiwara and P. Schapira; Sheaves on Manifolds.

J. Schurmann; Topology of Singular Spaces and Constructible Sheaves.

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: 

  1. X is the disjoint union of the S_\alpha;
  2. Each S_\alpha is a connected, smooth submanifold of M.  
  3. \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?