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We notice that a generic nonsingular gradient field $v = \nabla f$ on a compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$ of $X$. We study the transformations of $K(f, v)$ that are induced by deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$ with counting the \emph{double-tangent} trajectories of the $v$-flow, i.e. the trajectories that are tangent to the boundary $\d X$ at a pair of distinct points. Let $gc(X)$ be the minimum number of such trajectories, minimum being taken over all nonsingular $v$'s. We call $gc(X)$ the \emph{gradient complexity} of $X$. Next, we prove that there are only finitely many $X$ of bounded gradient complexity, provided that $X$ is irreducible and boundary irreducible with no essential annuli. In particular, there exists only finitely many hyperbolic manifolds $X$ with bounded $gc(X)$. For such $X$, their normalized hyperbolic volume gives an upper bound of $gc(X)$. If an irreducible and boundary irreducible $X$ with no essential annuli admits a nonsingular gradient flow with no double-tangent trajectories, then $X$ is a standard ball. All these and many other results of the paper rely on a careful study of the stratified geometry of $\d X$ relative to the $v$-flow. It is characterized by failure of $\d X$ to be \emph{convex} with respect to a generic flow $v$. It turns out, that convexity or its lack have profound influence on the topology of $X$. This interplay between intrinsic concavity of $\d X$ with respect to any gradient-like flow and complexity of $X$ is in the focus of the paper.
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arXiv:math/0611005v2 [math.GT] 19 Feb 2008
CONVEXITY OF MORSE STRATIFICATIONS AND GRADIENT
SPINES OF 3-MANIFOLDS
GABRIEL KATZ
In memory of Jerry Levine, friend and mentor
Abstract. We notice that a generic nonsingular gradient field v = f on a
compact 3-fold X with boundary canonically generates a simple spine K(f, v)
of X. We study the transformations of K(f, v) that are induced by defor-
mations of the data (f, v). We link the Matveev complexity c(X) of X with
counting the double-tangent trajectories of the v-flow, i.e. the trajectories that
are tangent to the boundary X at a pair of distinct points. Let gc(X) be the
minimum number of such trajectories, minimum being taken over all nonsin-
gular v’s. We call gc(X) the g radient complexity of X. Next, we prove that
there are only finitely m any X of bounded gradient complexity, provided that
X is irreducible and boundary irreducible with no essential annuli. In par-
ticular, there exists only finitely many hyperbolic manifolds X with bounded
gc(X). For such X, their normalized hyperbolic volume gives an upper bound
of gc(X). If an irreducible and boundary irreducible X with no essential annuli
admits a nonsingular gradient flow with no double-tangent trajectories, then
X is a standard ball. All these and many other results of the paper rely on
a careful study of the stratified geometry of X relative to the v-flow. It is
characterized by fai lure of X to be convex with respect to a generic flow v. It
turns out, that convexity or its lack have profound influence on the topology
of X. This interplay between intrinsic concavity of X with r espect to any
gradient-like flow and complexity of X is in the focus of the paper.
1. Introduction
Classical Morse theo ry links singularities of Morse functions with topology of
closed manifolds. Specifically, singularities of Morse functions f : X R cause
interruptions of the f-gradient flow, and the homology or even the topological type
of a manifold X can be expr e ssed in terms of such interruptions (see [C]). These
terms include descending and ascending disks, attaching maps , a nd spaces of flow
trajectories which connect the singularities.
On manifolds with b oundary, an additional source of the flow interruption occurs:
it comes from a particular geometry of the boundary X, o r rather from the failure
of the boundary to b e convex with respect to the flow (see Definition 4.1). In fact,
on manifolds with boundary, one can trade the f-singularities in the interior of X
for these boundary effects. In our a pproach, the boundary effects take the central
stage, while the singularities themselves remain in the background. In the pape r, we
apply this philoso phy to 3-manifolds. Many of our results allow for str aightforward
multidimensional generalizations, the other are specifically three-dimensional.
Date: February 19, 2008.
1
2 GABRIEL KATZ
Some of our theorems are in the spirit of the pioneering work of I. Ishii on, so-
called, flow-spines [I], [I1] (see also a recent paper by Y. Koda [Ko] and an excellent
monogra ph ”Branched Standard Spines of 3-manifolds” by Benedetti and Petronio
[BP], followed by [BP1]). In an earlier version of this paper [K], we managed to
overlo ok all this line of re search... As we will intro duce the relevant constructions,
we will descr ibe some technical differences between the flow-spines of [I] and the
branched spines of [GR1], [BP], on one hand, and the gradient spines on the other.
For now, it is sufficient to say that any generic gradient flow defines its gradient
spine in a canonical way, and that we allow for fields v that are not nece ssarily
concave with respect to the boundary X.
Our motivation comes from the desire to understand better the interplay b e -
tween the intrinsic concavity of X with respect to gener ic gradient flows and the
topology of the underlining 3-fold X. We conjecture that there exists a numerical
topological invariant that measures the failure of convexity with respect to any non-
singular gradient flow—“some manifolds intrinsically are just more concave than
others... In a sense, the gradient complexity gc(X), introduced in this paper, can
serve as a crude measure of intrinsic concavity of X. In fact, a 3-manifold X with a
connected boundary which admits a convex gradient-like field v is a handleb ody; so
a random manifold does not admit convex no nsingular gradient flows. For instance,
H
2
(X; Z) 6= 0 constitutes an obstruction to the convexity for any nonsingular gra-
dient flow. At the same time, any manifold with boundar y admits a strictly concave
traversing (but not necessarily a gradient!) flow [B P].
The c ombinatorial complexity theory of Matveev ([M]) helps us to uncover the
behavior of generic nonsing ular gradient-type flows on 3-folds or, rather, the in-
teractions of such flows with the boundary. Before describing these results in full
generality, let us give to the reader their taste. For example, we prove that on
a manifold X, obtained from the Poincar`e homology sphere by removing an o pen
disk, any nonsingular gradient-like flow has at least five trajectories that are tangent
to the boundary X, each one at a pair of distinct points; moreover X admits a
gradient-like flow with not more than 6 · 5 = 30 such trajectories. Another example
is provided by the remarkable hyperbolic manifold M
1
that has the minimal (among
hyperbolic manifolds) volume V 0.94272. By removing an open disk from M
1
we get a manifold X on which any nonsingula r gradient-like flow has at least nine
trajectories, each one tangent to the sphere X at a pair of distinct points; more-
over X admits a gradient-like flow w ith not more than 6 · 9 = 54 double-tange nt
trajectories.
A generic vector field v on X gives rise to a natural stratification
X
+
1
X
+
2
X
+
3
X(1.1)
by compact submanifolds, where dim (
+
j
X) = 3 j. Here
+
1
X is the part of the
boundary
1
X := X where v points inside X.
2
X is a 1-dimensional locus where
v is tangent to the boundary X. Its portion
+
2
X X consists of points where
v points inside
+
1
X. Similarly,
3
X is a finite locus where v is tangent to
2
X.
Finally,
+
3
X
3
X consists of points where v points inside
+
2
X.
In his groundbreaking 1929 paper [Mo], Morse discovered some beautiful con-
nections o f this stratification to the index o f the field v.
1
1
Actually, the results of [Mo] apply to compact manifolds X of any dimension.
3
Now, let us describe the content of our pap er section by section.
Section 2 starts with a sketch of main results from [Mo] (see Theorem 2.1 and
Corollary 2.1). It also contains one remark abo ut the role that stratification (1.1)
plays in the Gauss-Bonnet Theor e m (see Theore m 2.2 and [G] for an interesting
general discussio n). More importantly, we notice that, at any point x X, the
v-flow defines a projection of the boundary X into a germ of the constant level
surface f
1
(f(x)). At a generic point x
2
X this projection is a fold, while at
x
3
X it is a cusp. Throughout the paper, these folds and cusps provide us with
crucial measuring devices for probing the topology of X. A significant portion of
the paper is preoccupied with role of the cusps.
Section 3 As in [Mo], the stratificatio n {
+
j
X}
j
is in the focus of our investi-
gation. Here we prove that the surface
+
1
X c an be subjected to 1-sugery via a
deformation of the gra dient-like field v. This allows one to change the topology of
the stratum
+
1
X almost at will (see Lemma 3.1 and Corollary 3.1).
Section 4 For given nonsingular Morse data (f, v), we introduce the notion of
s-convexity, s = 2 or 3. The 2-convexity of v is defined as the property
+
2
X = . I t
puts a severe restrictions on the topology of X (see Theorem 4.2 and Corollary 4.5).
In contract, the 3- c onvexity,
+
3
X = , by itself has no topologic al significance: one
can always deform (f, v) to eliminate
+
3
X to gether with all other cusps (Theorem
9.5).
2
However, when we fix the topology of
+
1
X, some co mbinations of cusps from
3
X acquire topological invariance (Corollary 9.2).
Although convexity o r its lack are defined in terms of the gradient-like fields,
we can arrange for the 2-convexity if we know that singularities of f|
X
admit a
particular ordering induced by f (similar to the self-indexing property). Specifically,
the singularities of f |
X
can be divided into two groups: the positive Σ
+
1
where the
gradient v = f is direc ted inwards X, and the negative Σ
1
where v is directed
outwards (see Fig. 1). Theorem 4.1 claims that when f
1
) is above f
+
1
), then
one can deform the riemannian metric on X so that the convexity of the gradient
flow will be guaranteed. Hence, it is impossible to find a nonsingular function f
with the property f
1
) > f
+
1
) on 3-folds X that ar e not handlebo dies. In
addition, Theorem 4.2 describes an interplay between the dynamics of the flow v
through the “bulk” X and of the v-induced flow v
1
in
1
X, on the one hand, and
the convexity phenomenon, on the other.
In Corollary 4.5, we prove that an acyclic X is a 3-disk if and o nly if one of the
two properties are satisfied: (1) X admits nonsingular 2-convex Morse data (f, v),
(2) X admits nonsingular 3-convex Morse data (f, v) with a connected
+
1
X.
Section 5 is devoted to properties of gradient spines, a constructio n ce ntral to
our investigations. In spirit, but not technically, it represents a special class of flow-
spines [I]. The difference b e tween the two classes reflects the difference be tween the
spaces of nonsingular vector and gradient fields on a given manifold.
Recall that a spine K X is a compact cellular two-dimensional subcomplex
K of the 3-fold X, such that X \ K is homeomorphic to the pro duct
[X \ (X K)] × [0, 1).
2
Note that Theorem 4.1.9 in [BP] implies
3
X = for all, so called, traversing flows.
4 GABRIEL KATZ
The re lation between general spines and ambient 3-folds is subtle: a manifold X
has many non-homeomorphic spines K, and there a re topologically distinct X that
share the same spine. In order to make the rec onstruction of X from K possible,
K has to be rather special (cf. [M], [BP]).
In fact, a generic nonsingular gradient-like field v canonically gives rise to a
spine that we call gradient (see Fig. 10, 12). A gradient spine K is a union of
+
1
X
with the descending v-trajectories that pass through
+
2
X. Like branched spines
(Definition 6.6), gradient spines inherit orientations from the boundary X and
have a preferred side in the ambient X. Exactly these properties of a gradient spine
K allow for its resolution into a surface S homeomorphic to
+
1
X and, eventually,
for a r e c onstruction of X from K (Theorem 6.1). By modifying the field v, we can
arrange for
+
1
X, and thus for S, to be homeomorphic to a disk D
2
. As a result, we
get our Origa mi Theorem 5.2 : any 3-manifold X with a connected boundar y has a
gradient spine K obtained from a disk D
2
by identifying certain arcs in D
2
with
the appropriate arcs in the interior of D
2
(see Fig. 14, 15). This statement is similar
in flavor to Theorem 1 .2 , [I], where disk-shaped sections of generic (non-gradient)
flows on clo sed manifolds are employed for the same goal.
Section 6 deals with combinatorial structures that generalize the notion of a
gradient spine K (see Definitions 6.1—6.4). We start with a 2-complex K whose
local ge ometry is modeled after gradient spines (see Fig. 20). Adding a system of,
so called, T N-markers to K along its singularity set s(K) produces an object which
captures the topology of the ambient X and admits a canonic resolution into an
oriented surface. Such a polyhedron K with markers is called an abstract gradient
spine. Unlike generic 2-complexes, each abstract gradient spine K is a spine of
some manifold (see [BF] and [K], Appendix, Theorem 10.1).
3
The notion of a
~
Y -spine (see Definition 6.5) is still another generalization of
gradient spines. It is a very c lose relative of branched spines. In fact, for an
oriented X, the notions of an oriented branched spine K X and of a
~
Y -spine are
equivalent, provided K
being orientable (Lemma 6.3). Unlike abstract gradient or
branched spines, the
~
Y -spines K are defined extrinsically, that is, in terms of an
embedding in X of the vicinity of the singular set s(K) K. By Lemma 6.2, any
gradient spine is a
~
Y -spine. Moreover, according to Theo rem 8.1,
~
Y -spines admit
a nice” approximation by the gradient spines of the same complexity.
Section 7 We a pply ideas and results of [M], which revolve around Matveev’s
notion of combinatorial complexity of simple 2-complexes a nd compact 3-folds, to
the gradient and
~
Y -spines. We introduce the gradient complexity gc (X) of a 3-
fold X with boundar y as the minimal number of double-tangent trajectories that
a nonsingular gr adient-like field o n X c an have. A double-tang e nt trajectory is
tangent to the boundary
1
X at a pair of distinct points. In ge neral, gc(X) c(X),
where c(X), the Matveev combinatorial complexity, is defined to be the minimal
number of special iso lated singularities
4
that a simple spine K X can have. One
can restrict the scope of this definition only to
~
Y -spines (equivalently, to oriented
branched spines) in order to get the notion of
~
Y -complexity c
~
Y
(X). We prove that
3
In a s ense, the category of abstract gradient spines is equivalent to the category of compact
3-manifolds wi th non-empty boundary.
4
called, butterflies in [M] and Q-si ngularities in this paper
5
gc(X) c
~
Y
(X) c(X). In fact, Theorem 8 .2 claims that gc(X) = c
~
Y
(X), and,
for the ge ometrical pieces X of the SJS decomposition, gc(X) 6 · c(X).
The inequality gc(X) c(X) helps us to restate many results from [M] in the
language of double-tangent trajectories. For instance, by Theorem 7.3, fo r any
natural c, there is no more than finitely many irreducible and boundary irre ducible
with no essential annuli 3-folds X that admit nonsingular g radient-like flows with
c double-tangent trajectories. The number N(c) of such 3-folds has a crude upper
bound Γ
4
(c) · 12
c
, where Γ
4
(c) stands for the number of topological types o f regular
four-va le nt graphs with c vertices at most. In particular, there is no more than
Γ
4
(c) · 12
c
hyperbolic manifolds with c double-tangent trajectories.
Let X being obtained fro m a closed hype rbolic 3-fold Y by removing a number
of open balls. By T heorem 7.5, any non-singular gradient-like flow (as well as any
convex traversing flow) on X has at least V (Y )/V
0
double-tangent trajectories.
Here V (Y ) stands for the hyperbolic volume of Y and V
0
for the volume of the
perfect ideal tetrahedron.
Fortunately, all orientable irreducible and closed 3-manifolds of complexity at
most six (there are 74 member s in this fa mily) have been classified and their minimal
spines have been listed [M]
5
. Some partial results are available for the 1155 close d
irreducible manifolds of complexity at most nine. This has been accomplished by
an algorithmic computation coupled with “hands on” ana ly sis of spines that look
different, but share the same values of the Turaev-Viro invaria nts [TV]. The bottom
line is that all X with c(X) 6 are distinguished by their Turaev-Viro invariants!
Thus, for each manifold Y in the Matveev list and any generic nonsingular gradient
flow on X = Y \ D
3
, we g et a lower bound on the number of double-tangent (to
the boundary
1
X S
2
) trajectories.
Consider any ir reducible and orientable 3-manifold X produced from a closed
manifold Y by removing a ball. In Corollar y 7.1, we prove that if X admits a
nonsingular gr adient-like flow with no double-tangent trajectories, then X is the
standard dis k. In view of Perelman’s work [P1], [P2], this is not an exciting fact,
but it shows how far we can get with our Morse-theoretic techniques.
Section 7 contains a few more results about upper and lower es timates of gc(X)
for manifolds obta ined fro m clo sed manifolds Y by removing a number of 3-balls.
Theorem 7.6 provides a lower bound for gc(X) in ter ms of the presentational com-
plexity of the fundamental g roup π
1
(X). At the same time, a ny self-indexing Morse
function h on Y gives rise to an upper estimate of gc(X) (given in terms of the
attaching maps for the unstable 2-disks o f index two h-critical points).
In Theorem 7.4, we notice that gc(X) can incr ease only as a res ult of 2-sur gery
on X.
Section 8 Here we are addre ssing a natural question: Which spines are of
the gradient type? The main res ult of the section, Theorem 8 .1, claims that any
~
Y -spine K X can be approximated by a gradient spine
˜
K; moreover, c(
˜
K)
c(K). Furthermore, one can get K from
˜
K by c ontrolled elementary collapses
of certain 2- c e lls. Theorem 8.1 depends on some results from Section 9 about
possible cancellatio ns of cusps from
3
X. Theorem 8.2 establishes the equality
c
~
Y
(X) = gc(X) and the crucial inequality c(X) gc(X) 6 · c(X).
5
By definition, a spine of a closed manifold Y is a spine of the punctured Y , that is, of Y \ D
3
.
6 GABRIEL KATZ
Section 9 (together with Section 7) contains our main re sults. Here we analyze
the effect of deforming Morse data (f, v) on the gr adient spine they generates.
Theorem 9.3 claims that, in the process, the gr adient spine goes through a number
of elementary e xpansions a nd collapses of two-cells mingled with so called α- and
β-moves (see Fig. 29, 30). These are analogs o f the second and third Reidemeister
moves for link diagr ams. Theorem 9 .1 describes poss ible cancelations of cusps from
3
X that accompa ny g e neric deformations of v. One of our main results, Theorem
9.4, is a combination of Theorem 9.3 with a sp e c ial case of Phillips’ Theorem
[Ph]. We prove that when two nonsingular functions f
0
and f
1
on X produce the
same invariants h(f
0
), h(f
1
) H
2
(X; Z)—the same Spin
c
-structures in the sense
of Turaev [T]—, then their gradient spines are linked by a sequence of elementary
2-expansions , 2-collapses, and α- and β-moves (see the pr oof of Corollary 8.1 for
the definition o f the invariant h(f ) H
2
(X; Z)).
Deformations of (f, v) that cause jumps in the value of h (f ) (in the v-induced
Spin
c
-structure) manifest themselves as a “disk-supported surgery on the preferred
spine orientation”. We call them mushroom flips (see Fig. 35).
In Theorem 9.5, we prove that, given generic Morse da ta (f, v), it is possible to
deform them so that all the cusps from
3
X will be eliminated, but the number of
double-tangent trajectories gc(f, v) will be preserved.
Finally, it should be said that the Morse theory on stratified spaces, in general,
and on manifolds with boundary, in particular, has been an area of an active ad-
vanced and interesting research. For a variety of perspectives on this topic see [Mo],
[GM], [F], [C], [Ha]. Our intension is to bring the stratified Mor se theory and the
complexity theory of 3-folds under a single roof.
Acknowledgments This paper is shaped by numerous and valuable discussions
I had with Kiyoshi Igusa. My deep gr atitude goes to him. I am grateful to Yakov
Eliashberg for pointing that some propositions below (Theorem 9.6, Lemma 3.1
and Corollary 3.1) are intimately related to and similar in spirit with his ge neral
theory of folding maps [E1], [E2]
6
. I am also very grateful to the referee of [K] who
informed me about the existing results of [I], [I1], [BP], [BP1] and others.
2. The Morse Stratification on Manifolds with Boundary
Let X be a c ompact 3-manifold with boundary X. Let f : X R be a generic
smooth function. Then f has non-degenerate critical points in the interior of X
and the r e striction of f to the boundary X is also a Morse function. Let v be a
gradient-like vector field for f , that is , df(v) > 0 away from the f -critical points.
Instead of working with such pairs (f, v), we can pick a Riemannian metric on X
and choose v = f , the gradient field. Both points of view are equivalent, but we
prefer the first.
The singularities of f|
X
come in two flavors: positive and negative. At a pos-
itive singularity, the field v is directed inward X, and at a negative singula rity,
outward. This distinction between positive and negative critical points of f|
X
depends on f , not on v. At a positive singularity and in an appropriate coordinate
system {x
1
, x
2
, x
3
} with {x
1
= 0} defining X and x
1
> 0 the interior of X,
f(x) = c + x
1
+ a
2
x
2
2
+ a
3
x
2
3
,
6
They also bare resemblance to some r esults of Harold Levine [L].
7
where c and a
i
6= 0 being constants. At a negative s ingularity, one has
f(x) = c x
1
+ a
2
x
2
2
+ a
3
x
2
3
.
Let Σ
±
1
be the set of positive/negative singularities of f|
X
and let Σ
0
—the set
of singularities of f in the interior of X. Denote by X
c
the set {x X| f(x) c} .
Crossing the critical value c
of a positive singularity cause s the topological type
of X
c
to change, while crossing c
of a negative singularity has no effect on the
topology of X
c
as illustr ated in Figure 1.
Σ 1
+
Σ 1
1-handle
relative 2-handle
Figure 1. A positive singularity of index 1 and a negative singularity
of index 2 on the boundary of a solid. The gradient-like field v is horizontal.
For a gener ic field v, the locus L where the field is tangent to X is a
1-dimensional submanifold of the boundary; L divides X into two doma ins :
+
X
where v is directed inwards of X, and in
X, where it is directed outwards.
Morse noticed that, for a ge neric vector field v, the tangent locus L inherits a
structure in relation to
+
X analogous to that of X in relation to X [Mo]. To
explain this point we need to revise our notations in a way which will be amenable
to rec ursive definitions.
Let
0
X := X, and
1
X := X. Denote by
2
X
1
X the locus where v is
tangent to
1
X. For a gener ic v,
2
X divides
1
X into a domain
+
1
X where v is
directed inwards X and a domain
1
X where v is o utwards inwards X. Evidently,
±
1
X Σ
±
1
. Consider the set
3
X where v is ta ngent to
2
X. The set
3
X divides
2
X into a set
+
2
X where v is directed inwards
+
1
X and a set
2
X where v is
directed outwards
+
1
X. Finally,
3
X =
+
3
X
`
3
X, where v is directed inwards
+
2
X at the points of
+
3
X.
From now and on, we call (f, v) generic if 1) all the strata
j
X
1l3
are regularly
embedded s mooth manifolds a nd 2) a ll the restrictions f |
j
X
are Morse functions.
Most of the time, the second property will be irrelevant, but when we need it, we
do not want to modify our definition. At some point, the word generic” will mean
an additional general position requirement imposed on the field v (see Definition
5.2). When we say that a Riemannian metric is gener ic , we imply that (f, f ) is
generic.
We introduce critical s e ts Σ
±
j
±
j
X of f|
j
X
in a way similar to the one we used
to define Σ
±
1
. With some generic metric in place, let v
j
be the or tho gonal projection
of v onto
j
X, and let n
j
denote the normal field to
j
X inside
j1
X that points
inside
+
j1
X. Note that, away fro m the singularities from Σ
j
, df(v
j
) > 0.
8 GABRIEL KATZ
x
x
3
+
1
+
2
+
1
+
2
+
-
3
Figure 2. The patterns of fields v (the 3D-arrows) and v
1
(the par-
abolic flow) in vicinity of a point from
+
3
X (on the left) and a point
from
3
X (on the right).
For a vector field v
k
as above on X
k
with isolated singularities {x
Σ
k
Int(X
k
)}, denote by Ind
x
(v
k
) its index at x
, and by Ind
+
(v
k
) the sum
P
x
Σ
+
k
Ind
x
(v
k
). Then, acc ording to [Mo], o ne has two sets o f equivalent rela-
tions:
Theorem 2.1 . (Morse Law of Vector Fields). For any generic metric and
0 k 3,
χ(
+
k
X) = Ind
+
(v
k
) + Ind
+
(v
k+1
)
7
Ind
+
(v
k
) =
P
3
j=k
(1)
j
χ(
+
j
X).
X
+
1
X
2
+
X
3
X
-
X
1
-
v
0
Figure 3. A more realistic picture of the boundary
1
X in vicinity of
3
X in relation to the horizontal gradient field v.
Corollary 2.1. For generic vector field v and a metric on X,
Ind(v ) =
3
X
k=0
(1)
k
χ(
+
k
X).
7
By definition, Ind
+
(v
3
) = #(Σ
+
3
), and Ind
+
(v
4
) = 0.
9
For an eng aging discussion of the Morse Theorem 2.1 see the paper of Gottlieb
[G].
8
In particular, it describes a link between the Morse stra tificatio n {
+
j
X}
j
and
the geometry (normal curvature K) of
1
X:
Theorem 2.2. Let Φ : X R
3
be a smooth map with a nonzero Jacobian on
the boundary X and p : R
3
R a generic linear funct ion, so that the funct ion
f := p Φ has only isolated singularities in Int(X). Then t he degree of the Gauss
map g : X S
2
can be calculated either by integrating the normal curvature K of
Φ(X) R
3
(Gauss-Bonnet Theorem), or in terms of the v-induced stratification
+
3
X
+
2
X
+
1
X X:
deg(g) =
1
4π
Z
X
K = χ(X) Ind(v)(2.1)
= χ(
+
1
X) χ(
+
2
X) + χ(
+
3
X)
We notice that formulas from Theorem 2.1 and Corollary 2.1 admit equivari-
ant generalizations. For any compa c t Lie group G, a G-manifold X, an equi-
variant function f : X R, and a generic eqivariant metric on X (generating
G-equivariant gradient-like fields (v, v
1
, v
2
)), the invaria nts {χ(
+
k
X)}, as well as
the degree deg(g), can be interpreted as taking values in the Burnside ring B(G) of
G (see [TD] for the definitions).
There is another degree-type invariant of (X, f) linked to generic Morse data
(f, v). The set
2
X
+
1
X carries two non-zero vector fields: the norma l field n
2
that points inside
+
1
X and trivializes the oriented tangent bundle of
1
X along
2
X, and the field v = v
1
. Therefore, v defines a map h :
2
X S
1
. We view h as
an element in the one-dimensiona l oriented bordism group
1
(S
1
) of the circle. This
group splits as
1
(pt)
0
(pt)
0
(pt) (see [CF]), i.e., an element h : M
1
S
1
in
1
(S
1
) is determined by the degree class deg(h) = [h
1
(pt)] Z.
Any deformation of v pres erves the class of h :
2
X S
1
in
1
(S
1
) a nd thus the
degree [h
1
(pt)]. Deformations of f that change the s ingularity set Σ
1
do change
the degree c lass. This degree can be easily computed in terms of the cusp sets
+
3
X
and
3
X.
Lemma 2.1. For a fixed f, the number #(
+
3
X) #(
3
X) equals to twice the
degree of the map h :
2
X S
1
and is independent of the fi elds v, n
2
.
Proof Each loo p γ from
2
X either entirely belongs to one of the two sets
+
2
X
and to
2
X, or the arcs of γ belonging to
+
2
X and to
2
X alternate. I n the first
case, the contribution of γ to deg(h) is ze ro. In the second case, the contribution
of each arc with the ends of opposite polarity is also zero. Each arc with two
positive ends c ontributes a rotation of v by +π, while each arc with two negative
ends contributes a rotation by π (see Fig. 7). Hence the total rotation along γ is
π[#(
+
3
X) #(
3
X)].
By Corollary 9.2, a more refined c ount of the cusps from
3
X will produce a
very different formula for the degree of h :
2
X S
1
.
For a given nonsingular f : X R, each choice of a gradient-like field v locally
gives rise to a map p : X R
2
. Let us outline the construction of p. Add
8
That nice paper attracted my attention to the topic of Morse theory on manifolds with
boundary.
10 GABRIEL KATZ
an external collar W to X and extend the Morse data (f, v) into Y := X W
without adding new singularities. At each point x Int(W ) the (v)-flow defines a
surjection p
x
of a neig hborhood U
x
Y onto a neighborhood V
x
of x in f
1
(f(x)).
Consider the restriction p
x
: U
x
1
X V
x
, x
1
X, to the boundary
1
X.
According to Whitney [W], generic smo oth maps R
2
R
2
have only folds and
cusps as their stable singularities. Therefore, for generic Morse data (f, v) and
x
1
X \
2
X, p
x
: U
x
1
X V
x
is a surjection, for x
2
X \
3
X, p
x
is a
folding along an arc of
2
X, and at x
3
X, p
x
is a cusp map with p
x
(
2
X) being
the cuspidal curve. Note that along
+
2
X, p
x
: X V
x
is locally onto, while along
2
X, it is not.
It is especially e asy to visualize the stratificatio n {
j
X} when X is embedded
or immersed in R
3
and f is induced from a generic linear function l on R
3
. In
such a case, a global surjection p : X R
2
is available. Its fibers are pa rallel to
the gradient vector v = l. Now,
2
X can be identified with the folds of the map
p :
1
X R
2
and
3
X with its cusps.
As we deform a nonsingular field v within generic one-parameter families, the
local structure of the projections p
x
can be described in terms of a few canonical
forms. One of them, the cusp,
F (x, y) = (x
3
+ xy, y)(2.2)
is a stable singularity of a map fro m R
2
to itself.
9
The dove tail t-parameter family
F
t
(x, y) = (x
4
+ x
2
t + xy, y)(2.3)
describes a cancellation of two cusps that will play a significant role in Section 9.
10
3. Surgery on the Morse Stratification
Let X be a compact 3-manifold X with boundary
1
X. Given a smooth function
f : X R with isolated singularities, we can construct a new function with no
singularities inside X: just cut from X a number of tunnels. Each tunnel starts at
the boundary
1
X and has a dead end which engulfs a singularity. Denote by T
the interior of the tunnels. Then f, being restricted to X \ T X, is nonsingular,
and its perturbation can be assumed to be of the Morse type on (X \ T ).
Lemma 3.1. Let X be a compact 3-manifold with boundary
1
X. Let f : X R
be a smooth function with no singularities in a regular neighborhood N of
1
X.
Denote by v be its gradient-like field. Let γ
±
1
X be a simple path that connects
two points from
2
X and has an empty int ersection with the critical set Σ
±
1
.
Then one can deform v in N to a new f-gradient-like vector field ˜v for which
the new set
1
X will be obtained from the original one by the one-surgery along γ.
Outside of N, v = ˜v.
A similar statement holds for any field v
11
which is nonsingular along
1
X and
in general position to it.
9
It comes from the universal unfolding of the A
3
singurality f(x) = x
3
.
10
It is the universal unfoldi ng of the co dimension 1 singularity of a mapping R
2
to R
2
coming
from the universal (two-parameter) unfoldi ng of A
4
singularity.
11
not necessarily of the gradient type
11
U
H
Q
V
Z
W
X
X
+
1
Figure 4. Performing 1-surgery on
+
1
X.
Proof. Let v
1
be the orthogonal projection of v onto
1
X in the metr ic on X in
which v = f . The idea is to perform surgery on
+
1
X by a homotopy of the field
v v
1
, while keeping f and v
1
fixed. We start with “1-s urgery” on the fields a long
a band H which connects two arcs, say A
2
X a nd B
2
X. The ba nd, with
the exception of small neighborhoods of its two ends, resides in
+
1
X (alternatively,
in
1
X) as shown in Fig. 4. The band avoids the singularities of the function
f|
1
X
, so that v
1
6= 0 everywere in the band. Denote by Q a smller band which is
contained in H (see Fig. 4).
Let n denote the interior normal to
1
X. We decompos e the field v as v
1
+ h · n,
where the function h is positive in the open domain U the shaded area without
the handle (it is bounded on the left and right by the two dotted segments)— and
is nega tive in the interior of the complement to U. In fact, we can assume that 0
is a regular value o f h.
At each point x X, the differential df picks a particular open half-space T
+
f,x
in the tangent space T
x
, and v T
+
f
. Along the boundary
1
X, another family
of half-spaces is available: let T
+
x
denote the set of tangent vectors at x X
which point inside of X. Note that, away from the singularities of f |
1
X
, the cone
T
+
f,x
T
+
x
is open.
n
T
+
T
f
+
v
0
v
1
hn
~
hn
v
0
~
X
Figure 5. Changing the eld v = v
1
+ hn at a point x
1
X into a
field ˜v = v
1
+
˜
hn for which x
+
1
X.
Consider a smooth function
˜
h : H R which satisfies the following properties:
1)
˜
h
1
([0, +)) = V , 2) zero is a regular value of
˜
h and
˜
h
1
(0) = V , 3) the field
v
1
+
˜
h·n T
+
f
. The last property can be achieved by starting with any
˜
h subject to
1) and 2) and rescaling it by a variable factor a > 0, so that v
1
+ (a
˜
h)n T
+
f
T
+
(see Fig. 5). At each point, the existence of an appropriate a follows from the fact
that v
1
T
+
f
. Because the cone T
+
f
T
+
is open and the domain H is compact,
12 GABRIEL KATZ
the global existence of such an a, by a partition-of-unity arg ument, follows from its
existance at each point of H.
We extend the field v
1
+
˜
h · n inside X to get a smooth an f-gradient-like field
w in a small regular neighborhhod W of V . We use a smooth partition of unity
1 = α + β subordinate to the cover W , X \ Z. The function α vanishses in Z and
β in X \ W . Now consider the field ˜v := αv + βw. Since T
+
x
is convex, ˜v T
+
f
.
Moreover, in V , ˜v points inside X and, in H \ V , outside X. Also, outside W ,
˜v = v.
Finally, for a fixe d f , the set of all f -gradient-like fields is open a nd convex.
Hence, any modification of a f -gradient-like field can be obtained by its deforma-
tion.
The arguments for generic (non-gradient) fields v are simila r and simpler .
Corollary 3.1. Under hypotheses and notations of Lemma 3.1, the following claims
are valid. There is a deformation of a given gradient-like field in the neighborhood
N of
1
X so that, for the new gradient-like field, both portions
±
1
X
j
of
±
1
X
residing in each connected component
1
X
j
of
1
X are n onempty, and
+
1
X
j
is
homeomorphic to any given domain in
1
X
j
with a nonempty complement.
In particular, for a given generic f and all j’s, t here exists a gradient-like field
v such that anyone of the two properties is satisfied:
+
1
X
j
is homeomorphic to a disk.
+
1
X
j
and
1
X
j
are homeomorphic surfaces.
A similar statement is valid in a category of generic nonsingular vector fields.
Proof. Let
1
X
j
be a component of
1
X. When
1
X
j
=
+
1
X
j
(or
1
X
j
=
1
X
j
), we can pick a point x
1
X where v
1
6= 0. Then, employing an argument
depicted in Fig. 5, we can deform the field v in the vicinity of x so that, with
respect to the modified gradient-like field, x
1
X (x
+
1
X, correspondingly).
Thus we can assume that
1
X
j
,
+
1
X
j
6= .
Now, by one-surgery on both
+
1
X and
1
X, we can change the topology of
+
1
X any way we like, as long as we keep keep the sets of both polarities nonempty.
No matter how we change the two se ts, we must keep Σ
+
1,j
inside
+
1
X and Σ
1,j
inside
1
X. In particular, we ca n deform the field so that
+
1
X
j
is a 2-disk or, say,
to insure that
+
1
X
j
is homeomorphic to
1
X
j
.
Note that surgery on
+
1
X typically will change the sets
±
3
X.
4. Morse Strata and Convexity
Definition 4.1. Given generic Morse data (f, v) on a manifold X with boundary
we say that v is s-convex (concave), if
+
s
X = (
s
X = , correspondingly). In
particular, if
+
2
X = (
2
X = ), we say that v is symply convex (c oncave).
An existence of convex Morse data has strong top ological implications. Let Σ be
a surface with boundar y. We denote by L(Σ) a smooth 3-manifold with boundary
obtained from the product Σ × [1, 1] by rounding its co rners Σ × 1} and by
replacing a narrow cylindrical band Σ × [ǫ, ǫ] with a ”curved parabolic” one as
shown in Fig. 6. The projection L(Σ) [1, 1] defines a nonsingular function f.
The vertical field v in Σ × [1, 1] is of the f-gradient type. With respec t to it,
2
L(Σ) = Σ. We call the triple (L(Σ), f, v) a lense based on Σ.
13
Figure 6. A lense L(Σ) is convex with respect to the vertical field:
+
2
L(Σ) = . The set
2
L(Σ) is the equator of the lense.
Lemma 4 .1. A connected 3-manifold X admits convex nonsingular data (f, v) if
and only if X is diffeomorphic to a handle body L(
+
1
X).
In particular, if an acyclic 3 -manifold X admits convex nonsingular data, it is
a 3-disk.
Proof. The first claim is straightforward (see [BP], Proposition 4.2.2). Note
that the convexity on a connected X implies that
1
X is c onnected.
When X is ac yclic, a homological argument implies that
1
X S
2
. Thus
+
1
X
must be a contractible domain in S
2
, that is, a 2-disk. Therefore, the manifold X
must be shaped as a lens , one face of which is tha t disk.
Fig. 7 shows a typical behavior of a vector field v
1
in a neighborhood of
2
X.
The arcs of
+
2
X come in tree flavors: A is bounded by a pair of points from
+
3
X,
B is bounded by a pair of points from
3
X a nd C is bounded by a pair of mixed
polarity.
This time, we play our convexity game in the dimension 2, not 3. At points of
3
X the field v
1
in
+
1
X is convex, at points of
+
3
X it is concave.
A
B
C
1
+
1
-
3
+
3
-
3
+
3
-
3
+
3
-
Figure 7. The arcs from
+
2
X of the types A, B, and C.
According to Theorem 9.5, we ca n deform v (in the space of nonsingular gradient-
like fields) so that
3
X = , in other words, there are no topological obstructions
to the 3-convexity and 3-concavity of (gradient) fields! On the way to establishing
this fact, we need to perform 0-surgery on
3
X
+
2
X.
Lemma 4.2. Let v be a gradient-like field for f : X R, v 6= 0 along
1
X.
Let C
+
2
X be an arc with one of its ends a
+
3
X, the other end b
3
X,
and no other points of
3
X in its interior. Assume also that f |
C
has n o critical
points.
12
Then we can deform v in the vicinity of C X in such a way that:
12
See Fig. 7, arc C. Note that the absence of critical points of f |
C
implies that f(a) < f (b).
14 GABRIEL KATZ
(1) with respect to the new f-gradient-like field ˜v the strata
+
1
X and
2
X
remain the same,
(2) arc C changes its polarity (from being in
+
2
X to being in
2
X), and the
points a, b are eliminated from the set
3
X.
Proof. L e t v
2
be an orthogonal projection of v on
2
X. The argument is
analogous to the one in Lemma 3.1. However, this time, we will keep both the
direction of n
1
-component v v
1
of the field v and the field v
2
6= 0 fixed in the
vicinity of C, while deforming the field v
1
. Because the direction of the normal
component v v
1
remains the unchaged, the stratum
+
1
X and its boundary
2
X
will be preserved, but the arc C will change its polarity.
Corollary 4.1. For any generic function f : X R with no critical points in
1
X,
there exists a gradient-like field v with the following property. Each arc C
2
X,
which connects a minimum x of f|
2
X
with a consecutive maximum y, has:
(1) a single point from
3
X, provided x Σ
+
2
and y Σ
2
,
(2) a single point from
+
3
X, provided x Σ
2
and y Σ
+
2
, and
(3) no points from
3
X when x, y Σ
±
2
. In that case, the polarity of C is the
same as the polarity of x and y in Σ
2
.
Proof. We can assume that f |
2
X
is Morse and its maxima and minima al-
ternate. By Lemma 4.2, one can change the polarity of arcs C
2
X between
consecutive points a, b from
3
X, provided v
2
|
C
6= 0. Note that the polarity of a
and b must be opposite.
The Morse formula fo r the vector fields (Theorem 2.1) helps to link the topology
of
+
1
X with the distribution of arcs from
+
2
X and points from
±
3
X along its
boundary
2
X.
The next lemma is similar in spirit to Theorem 4.8 from [E1]. That theorem is a
very special case of the Eliashberg gener al theory of folding maps surgery (see [E1],
[E2]). However, we cannot apply Eliashberg’s results directly: our v-generated
foldings p
x
:
1
X f
1
(f(x)) have a nice target space only locally; a natural
target space in our setting is a 2-complex, typically with singularities.
Lemma 4.3. Let X be a compact 3- manifold with a generic nonsingular vector
field v. Then the v-generated stratification {
+
k
X}
0k3
of X has the following
properties:
χ(
+
1
X) χ(
1
X) = #(
+
3
X) #(
3
X) = 2[#(B arcs) #(A arcs)]
and
χ(
+
1
X) = χ(X) + [#(
3
X) #(
+
3
X)]/2.
Proof. Since v 6= 0 in X, the index I(v) = 0. By the Morse formula, χ(X)
χ(
+
1
X) + χ(
+
2
X) χ(
+
3
X) = 0. Thus, χ(
+
1
X) = χ(X) + χ(
+
2
X) χ(
+
3
X).
Loops in
+
2
X do not contribute to χ(
+
2
X), so χ(
+
2
X) = #(arcs in
+
2
X). Hence,
χ(
+
1
X) = χ(X) + #(arcs in
+
2
X) χ(
+
3
X). Note, that the C-arcs and their
ends do not contribute to the difference #(arcs in
+
2
X) χ(
+
3
X): such ar cs have
a single end in
+
3
X. On the other hand, the B-arcs do not contribute to
+
3
X.
Therefore the difference χ(
+
2
X) χ(
+
3
X) is equal [#(A ar c s) + #(B arcs)]
2#(A arcs) = #(B arcs) #(A arcs) = [#(
3
X) #(
+
3
X)]/2.
15
Recall that for any 3 -manifold X, χ(X) =
1
2
χ(
1
X) =
1
2
[χ(
+
1
X) + χ(
1
X)].
Replacing χ(X) with
1
2
[χ(
+
1
X) + χ(
1
X)] in the formulas above, leads to the
relation χ(
+
1
X) χ(
1
X) = χ (
+
3
X) χ(
3
X).
Combining L emma 4.3 with Corollary 3.1, we g e t
Corollary 4.2. #(
+
3
X) = #(
3
X) if and only if χ(
+
1
X) = χ(
1
X) = χ(X).
When
1
X is connected, by deforming v, we can arrange for
+
1
X to be a 2-disk.
For any such choice of Morse data (f, v),
[#(
+
3
X) #(
3
X)]/2 = 1 χ(X).
Corollary 4.3. If a 3-manifold X with χ(X) > 0 admits a nonsingular function
f with a 3-convex Morse data, then the restriction f|
+
1
X
must have at least χ(X)
extrema.
Proof. The hypotheses
+
3
X = implies that only B-arcs could be pre sent in
+
2
X. The positive contribution to χ(
+
1
X) comes from the components of
+
1
X
shaped as disk s. We divide disks into two types: 1) disks with no B-arcs in their
boundary (which entirely belongs to
+
2
X or to
2
X) and 2) the rest o f the disks .
Any disk of the first typ e must contain at least one local extremum of f|
+
1
X
.
Any disk of the second type contains at least one B-arc. Since
+
3
X = , we get
χ(
+
1
X) = χ(X)+#(B a rcs). Now the lemma follows from writing down χ(
+
1
X)
as the Euler class of all disks of the first type plus the Euler class of the rest of
+
1
X.
Corollary 4.4. For any connected 3-manifold X with a connected boundary and
Euler number χ, there exit nonsingular Morse data (f, v) so that
+
1
X is a disk
D
2
. For such data, we get #(
+
3
X) 2χ 2 and #(
3
X) 2 2χ. As a result,
when χ > 1, the disk cannot be convex with respect to the field v
1
; as χ grows, the
disk
+
1
X becomes m ore “wavy”. Similarly, when χ < 1, the disk cannot be concave
with respect to v
1
, that is,
3
X 6= .
Proof. By Lemma 3.1, appropriate defor mations of v will produce
+
1
X D
2
.
In view of Lemma 4.3, the claim follows.
The following theorem shows that convexity of Morse data is equivalent to the
possibility of special ordering of the f|
1
X
-critical points by their critical values,
and thus, in general, fails. However, if we formally attach index i+1 to each critical
points x Σ
1
of classical index i, the new self-indexing of f |
1
X
becomes possible.
Theorem 4.1 . Let (f, v) be Morse data whose rest riction (f
1
, v
1
) to the bound-
ary
1
X is also of the Morse type. If
+
2
X = φ, then there is no ascending
trajectory γ(t)
1
X of the vector field v
1
, such that [l im
t+
γ(t)] Σ
+
1
and
[l im
t→−∞
γ(t)] Σ
1
.
Conversely, if no such γ(t) exists, one can deform the gradient-like vector fields
{v, v
1
} (equivalently, the metric g in which v = f ) to a new gradient-like pair
{˜v, ˜v
1
} (to a new metric ˜g), in such a way that, with respect to the new fields,
+
2
X = φ. In particular, if f
+
1
) < f
1
), then f admits convex Morse data
(convex metric ˜g).
In contrast, no nonsingular Morse data (f, v) are 2-concave:
+
2
X 6=
13
.
13
Note that any X admi ts a field v 6= 0 with respect to which
+
2
X = (cf. [BP])
16 GABRIEL KATZ
Proof. For a generic metric g, consider the vector field v = f and its orthogonal
projection v
1
on
1
X. The function h
1
:
1
X R, defined via the formula v =
v
1
+ h
1
· n
1
, where n
1
is the inward normal field, has zero as a regular value. Then
+
1
X = h
1
1
([0, +)),
1
X = h
1
1
((−∞, 0]) and
2
X = h
1
1
(0). Now, if the
ascending trajectory γ(t) which links Σ
1
with Σ
+
1
exists, it must cross somewhere
the boundary
2
X of
1
X. By definition, such crossing belongs to
+
2
X which
must non-empty.
On the other hand, if no such γ(t) exists, then we claim the e xistence of codimen-
sion 1 clo sed submanifold N
1
X, which separates
1
X in two domains A Σ
+
1
and B Σ
1
(A = N = B) and, in addition, has the following property. The
vector field v
1
is transversal to N and points outward of A. Indeed, one can take a
small regular neighborhood (in
1
X) of the union of descending trajectories of all
critical points from Σ
+
1
for the role of A. Here we are employing the fact that no
descending trajectory o riginating at Σ
+
1
reaches Σ
1
.
Since, away from Σ
+
1
Σ
1
, v
1
6= 0, in the tangent space T
x
of X (x
1
X),
there is an open c one T
+
f,x
containing v
1
and comprised of gradient-like vectors.
With such a separator N in pla c e , consider a smooth function
˜
h
1
:
1
X R with
the properties: 1) zero is a regular value of
˜
h
1
and
˜
h
1
1
(0) = N ; 2)
˜
h
1
1
((−∞, 0) =
A,
˜
h
1
1
([0, +) = B; 3 )
˜
h
1
= h
1
in the vicinity of Σ
+
1
Σ
1
; and 4) v
1
+
˜
h
1
·n T
+
f
.
Note that the field ˜v := v
1
+
˜
h
1
· n points inside X along A and outside along B.
Now we can find a metric ˜g, in which ˜v is the gradient of f . In ˜g, ˜v is orthogonal
to the plane K
x
:= Ker(df). Denote by ˜v
1
the ˜g-orthogonal projection of ˜v on T
x
.
Since
˜
h
1
vanishes on N, ˜v
1
= v
1
along N , and therefore, is transversal to N and
points outward A. As a result, with respect to ˜g,
+
2
X = φ.
We ca n defor m the original metric g into ˜g, thus deforming the gradient fields
v, v
1
into the gradient fields ˜v, ˜v
1
.
The last claim follows fr om the o bs e rvation that since v 6= 0 the absolute maxi-
mum (minimum) of f on X must be realized at a point from Σ
1
(from Σ
+
1
).
In view of Theorem 4.1 and Lemma 4.1, we get
Theorem 4.2. A connected 3-manifold X with a connect ed boundary is a handle-
body if and only if one of the following properties is valid:
X admits a smooth nonsingular function f whose restriction on the bound-
ary
1
X is Morse; moreover, no ascending trajectory of a gradient-like field
v
1
links in
1
X a singularity from Σ
1
to a singularity from Σ
+
1
.
X admits a smooth nonsingular function f whose boundary
1
X is convex
with respect to a gradient-like field v, that is,
+
2
X = .
Given the remar kable proo f of the Geometrization Conjecture [P1], [P2], the
proposition below must be viewed just as an illustr ation. It shows what advances
towards the Poincar`e C onjecture are possible by modest means of the Morse Theory
alone. We get the following criteria for recognizing standard 3-dis ks in terms of
Morse data:
Corollary 4.5. An acyclic 3 -manifold X is a 3-disk if and only if one of the
following properties is valid:
(1) X