Nonsmooth manifold decompositions
Abstract.
We study the structure induced on a smooth manifold by a continuous selection of smooth functions. In case such selection is suitably generic, it provides a stratification of the manifold, whose strata are algebraically defined smooth submanifolds. When the continuous selection has nondegenerate critical points, stratification descends to the local topological structure. We analyze this structure for the maximum of three smooth functions on a 4-manifold, which provides a new perspective on the theory of trisections.
Key words and phrases:
continuous selection of functions, stratification, trisection1. Introduction
Classical Morse theory is an indispensable tool in the study of low dimensional manifolds. By looking at the fibers of a smooth function, a manifold may be described as a union of fairly simple pieces. Assembling those pieces into few homeomorphic handlebodies yields special decompositions: Heegaard splittings of 3-manifolds, trisections of 4-manifolds [11] and multisections of higher dimensional manifolds [3].
A Heegaard decomposition of a closed, connected and oriented 3-manifold is obtained from a classical Morse function . Gay and Kirby have shown that a trisection of a closed, connected and oriented 4-manifold may similarly be constructed from a suitable Morse 2-function [11]. On the other hand, they described how the handle decomposition, corresponding to a classical Morse function on a 4-manifold, fits into a trisection. Though natural and illuminating, neither of the two perspectives is quite successful in explaining the triple symmetry of a trisected manifold. Continuous selections of smooth functions offer an alternative between the 2-dimensional and the classical 1-dimensional Morse functions. Instead of describing a manifold in a linear fashion – from bottom up – they allow its construction “from inside out” in several directions.
Originally, trisections of closed 4-manifolds were defined in the smooth category. However, some amount of research has lately been focused on trisections of PL manifolds that respect the underlying (singular) triangulations [2, 7, 18]. In another spirit, we study trisections induced by a continuous selection of smooth functions on a smooth manifold. This point of view offers several advantages: the fibers are easier to grasp, the spine and the trisection surface are algebraically defined, while handle decomposition carries a triple symmetry due to the underlying stratification.
A continuous selection of smooth functions on a manifold is a continuous function that coincides with at least one of the functions at each point of . Applying generalized derivation, introduced by Clarke [8], the topology of may be studied by a suitably adapted version of Morse theory. We begin by analyzing the structure, revealed by a continous selection of smooth functions on a manifold. Under some nondegeneracy conditions, such structure defines a stratification of the manifold, whose strata are algebraically defined smooth submanifolds. We show examples of 3-dimensional Heegaard splittings and 4-dimensional trisections as instances of such stratifications. Moreover, the local topological structure around the critical point of a Morse CS function is studied, and a stratified handle decomposition, corresponding to such function, is described.
The paper is organized as follows. In Section 2, we briefly present the basic concepts from nonsmooth analysis that we will need. In Section 3 we study CS functions and show that the structure, induced on a smooth manifold by such function, defines its stratification. In Section 4, we focus on 4-manifolds. In Subsection 4.1, we recall the definition of trisection and describe CS functions that induce trisections of 4-manifolds. In Subsection 4.2, we investigate the local structure of a 4-manifold around a critical point of a CS Morse function , which gives rise to stratified handles with triple symmetry.
Acknowledgements
The author would like to thank Peter Feller for several discussions that led to the ideas presented in this paper. This research was supported by the Slovenian Research Agency grants P1-0292, J1-4031 and N1-0278.
2. Preliminaries from nonsmooth analysis
A classical reference on nonsmooth analysis is Clarke’s book [8]. Morse theory adapted to the context of piecewise smooth functions was studied in [6, 13, 5, 4, 1]. Here we recall some basic definitions and results. For the remainder of this section, we denote by a smooth -manifold.
Definition 2.1.
Let be a continuous function and let be smooth functions. The function is called a continuous selection of the functions if it satisfies the following two conditions:
-
(1)
the set is nonempty for every point ,
-
(2)
for any index , there exists a point with .
Denote by the set of all continuous selections of the functions . The set is called the active index set of at the point . Furthermore, the set
is called the set of essentially active indices (where “ ” and “ ” denote the interior and the closure of a set ). A function is called essentially active at if .
It follows from the above definition that for every . For every piecewise differentiable function on and every there exists a collection of smooth functions that are essentially active in a neighborhood of [17, Proposition 4.1.1]. Moreover, a continuous selection of smooth functions is locally Lipschitz continuous. In this setting, generalized differentiation defined by Clarke has the following form.
Definition 2.2.
Let be a continuous selection of smooth functions . Clarke’s subdifferential of at a point is defined by
where denotes the convex hull of a set . A point is called a critical point of if .
More generally, when is a continuous selection of smooth functions on a smooth manifold , the Clarke’s subdifferential of may be defined in a similar manner using the local coordinate charts on . Let be a local chart with and , then we define .
A version of the first Morse Lemma for locally Lipschitz continuous functions was proved in [1].
Proposition 2.3 ([1]).
Let be a locally Lipschitz continuous function and denote by its lower level set for . Suppose is compact and does not contain any critical point of . Then there exists a Lipschitz continuous mapping such that
In order to study the local behaviour of around a critical point of a piecewise differentiable function, we need the following definition.
Definition 2.4.
Let be smooth functions and . A critical point of is called nondegenerate if the following two conditions hold:
-
(ND1)
For each , the set of differentials is linearly independent;
-
(ND2)
The second differential is regular on
and the numbers are such that
In this case, the quadratic index of the critical point is the dimension of a maximal linear subspace of on which the quadratic form is negative definite.
A version of the second Morse lemma for continuous selections of smooth functions was proved by Jongen and Pallaschke [13]. It requires the following definition.
Definition 2.5.
Two continuous functions are said to be topologically equivalent at if there exist open subsets of and of in and a homeomorphism , for which and on .
Theorem 2.6 ([13]).
Let be twice continuously differentiable functions on an -manifold and let . Choose local coordinates in the neighborhood of a point . Then the following holds:
-
(i)
if is a noncritical point of , then and are locally topologically equivalent at the point ;
-
(ii)
if is a nondegenerate critical point of , then is locally topologically equivalent at to a function of the form
where , , and is the quadratic index of at .
In particular, Theorem 2.6 implies that the nondegenerate critical points of the function are isolated.
Example 1. Define functions by and let denote the continuous selection . Consider the restriction of to the 2-sphere . There are two disjoint circles of points for which , namely . These two circles separate the sphere into an open annulus, where , and two open disks on which .
2pt \pinlabel at 57 42 \pinlabel at 170 73 \pinlabel at 90 182 \endlabellist

The annulus contains two critical points of index 2 and two critical points of index 1. Each of the disks with contains a critical point of index 2. To find the “nonsmooth” critical points of that lie on the circles and , we express in the local coordinates , given by , and obtain . Critical points are the solutions of
At , there are four nondegenerate critical points of quadratic index 0. At each of these points, is locally topologically equivalent to the function . At , there are four nondegenerate critical points of quadratic index 1. At each of these points, is locally topologically equivalent to the function .
3. Stratification of a manifold, induced by a CS function
A smooth function on a manifold reveals important information about its basic constituent parts. What happens if we observe a continuous selection of smooth functions instead? It turns out that “corners” of various dimensions provide an additional structure.
3.1. CS functions on manifolds
In this paper, we will focus on the following two types of functions.
Definition 3.1.
Let be a smooth manifold. A CS function on is a function for which there exist smooth functions such that
-
(1)
,
-
(2)
at any point we have ,
-
(3)
at any point , the gradients of the active functions are affinely independent.
A CS Morse function on is a CS function on whose all critical points are nondegenerate.
Throughout this section, we denote by a smooth -manifold and by a CS function on , given as the continuous selection of smooth functions . For any subset , we denote by
the subset of points where the active index set equals , and let . Note that for any .
Lemma 3.2.
is an open submanifold of for any . Moreover, is a closed subset of for any .
Proof.
Let us denote . Since is continuous, the set is a closed subset of . Its complement is an open subset of for every , and
is an open subset of (thus a submanifold). For any collection of indices , the set is a closed subset of . ∎
It follows from Lemma 3.2 that is a smooth submanifold of codimension 0 in . More generally, each is a submanifold of . To prove this, we need the following standard result.
Proposition 3.3 ([14, Proposition 5.28]).
Let be a subset of a smooth -manifold . Then is an embedded -submanifold of if and only if every point has a neighborhood in such that is a level set of a submersion .
Proposition 3.4.
Let be any subset such that for some . Then is a smooth submanifold of codimension in .
Proof.
For , the statement follows from Lemma 3.2.
Suppose that for two indices and that . Choose an arbitrary . Let be a neighborhood of in such that for any index , we have (such a neighborhood exists since is a closed subset of that does not contain by Lemma 3.2). Define a map by . Then is a smooth function on and since at any point the gradients and are affinely independent, is a submersion. Since , Proposition 3.3 implies that is an embedded smooth submanifold of codimension 1 in .
Now let be any subset with and choose an arbitrary . Let be a neighborhood of in such that for any index , we have . Define a map by
At any point , the set is affinely independent, thus is a submersion. Since , Proposition 3.3 implies that is an embedded smooth submanifold of codimension in . ∎
For any , we denote by the topological boundary (frontier) of the subset . Note that , since is a submanifold of .
Lemma 3.5.
For any , we have
Proof.
Recall that . By Lemma 3.2, is a closed subset of that contains . Moreover,
is an open subset of whose closure equals , and its boundary is exactly . It follows that for any . ∎
Lemma 3.6.
For any , we have .
Proof.
By Lemma 3.2, is an open subset of that is clearly contained in , thus .
To show the other inclusion, first observe that is a partition of into pairwise disjoint smooth submanifolds by Proposition 3.4. Suppose that for some , then this intersection is an open subset of that intersects at least one of the submanifolds of codimension 0 nontrivially (in fact, in an open set), thus for some , which gives a contradiction. Thus, the subsets are pairwise disjoint for . Now choose any , and let be a neighborhood of in such that . Suppose that for some . It follows from our reasoning above that , therefore . By the Condition (2) of Definition 3.1, we have and thus . But then the neighborhood intersects , which gives a contradiction. It follows that and thus . ∎
Lemma 3.7.
For any we have
Proof.
Suppose that . For every , we have and it follows by continuity that . Thus, and since , is a proper subset of . This shows that .
Let us show the other inclusion by induction on . To begin, let and choose an element with . By Condition (2) of Definition 3.1, we have and thus by Lemma 3.6. Since , it follows that .
Now suppose that for some and every with we have . Choose any subset with and write . By Proposition 3.4, is a submanifold of codimension and is a submanifold of codimension 0. Thus, is a closed topological submanifold of codimension and is a closed topological submanifold of codimension 1. It follows by the induction hypothesis that and , therefore
(1) |
is a closed topological submanifold of codimension . In the decomposition (1), is a submanifold of codimension and each with is a submanifold of codimension . Thus, for every and every neighborhood of in , we have . It follows that and since , we have . ∎
Proposition 3.8.
For any and any , the following statements are equivalent:
-
(a)
,
-
(b)
for every ,
-
(c)
,
-
(d)
for every neighborhood of and every we have .
Proof.
(a) (b) Choose an . Since for every , it follows by continuity that for every .
(b) (a) Suppose that for every , then . If , it follows that . Otherwise, we have and then Lemma 3.7 implies that .
(b) (d) Suppose that for every and let be a neighborhood of in . Then and it follows by Lemma 3.6 that , therefore .
(d) (b) Suppose that every neighborhood of intersects for every . Then and since , it follows by continuity that for every .
(c) (d) Let and choose a neighborhood of in . Then for every and it follows by Lemma 3.6 that .
(d) (c) Suppose that every neighborhood of intersects for every ; then , therefore for every .
∎
3.2. Stratification
We have shown that a CS function on a smooth manifold induces a decomposition of into a union
of pairwise disjoint smooth submanifolds. Moreover, the closures of these submanifolds carry an extra structure, described in Lemma 3.7 and Proposition 3.8. This is an instance of a well-known and studied phenomenon, called stratification.
Definition 3.9.
[9] A filtered space is a Hausdorff topological space , equipped by a sequence of closed subspaces
for some integer . The space is called the -skeleton, and the index is called the formal dimension of the skeleton. The connected components of are called the strata of of formal dimension . If is a filtered space of formal dimension , then the components of are called the regular strata of and all the other strata are called singular strata.
Definition 3.10.
[9] Let be a filtered space. The filtered set satisfies the frontier condition if for any two strata of the following holds:
if , then . |
A stratified space is a filtered space that satisfies the frontier condition.
Consider our manifold , endowed by the CS function , and define the skeleta of by
for .
Proposition 3.11.
The sequence
defines a stratification of .
Proof.
We have and by definition. More generally, it follows from Lemma 3.7 and Proposition 3.8 that is a closed subspace of for every . The regular strata of are submanifolds for . More generally, we have
Thus, any submanifold with represents a stratum of formal dimension . Any two strata and with are disjoint, while Lemma 3.7 implies that the boundary of any stratum is a disjoint union of strata. If we have for two strata and , then it follows that either or . Therefore the above filtration of satisfies the frontier condition. ∎
What can we say about the fibers of the CS function ? Let us denote for .
Corollary 3.12.
The sequence of subspaces
defines a stratification of the fiber for any .
Proof.
Since is a closed subspace of , also is a closed subspace of . We have , therefore the subsets represent the strata of formal dimension .
Remark. Observe that the restriction of a CS function to every stratum is a smooth function, since for any . We might thus think of a CS function as a “piecewise smooth” function, where the smoothness condition holds on every stratum of our stratification (instead of a simplex in a triangulation).
Example 1. Define a CS function on the 3-sphere
by . Its image contains two critical values; the critical fiber is homeomorphic to the 2-torus, while the critical fiber looks like a Hopf link. Every regular fiber
for is a link of two tori that represent the boundary of a regular neighborhood of the Hopf link. The regular strata of the stratification induced by are the interiors of two solid tori and , whose common boundary is the singular stratum . Thus, stratification induced by gives the Heegaard decomposition of the 3-sphere of genus 1.
Example 2. Note that the function from Example 3.12 is not a CS Morse function. On the other hand, the linear function on is a CS Morse function with 4 critical points , , and . A regular fiber for is a union of two disks, glued along their boundary, and the critical fiber is a wedge of two spheres. A regular fiber for is a disjoint union of two spheres. The stratification, induced by , gives a Heegaard decomposition of genus 0.
4. CS functions inducing trisections
4.1. Trisections
Our main motivation for studying CS functions is to understand their role in the theory of trisections. Recall the following definition from [11].
Definition 4.1.
Let be a closed, connected, oriented 4-manifold. Given two integers , a -trisection of is a decomposition such that:
-
•
is a 4-dimensional 1-handlebody obtained by attaching 1-handles to one 0-handle for each ,
-
•
is a 3-dimensional genus handlebody for each and
-
•
is a closed surface of genus .
The surface is called the trisection surface, while the union of the 3-dimensional handlebodies is called the spine of the trisection.
2pt \pinlabel at 1540 210 \pinlabel at 620 220 \pinlabel at 1540 560 \pinlabel at 1540 1010 \pinlabel at 600 630 \pinlabel at 1540 1420 \pinlabel at 380 1440 \pinlabel at 880 1440 \pinlabel at 1390 1440 \pinlabel at 1540 1645 \pinlabel at 690 1120 \pinlabel at 660 900 \pinlabel at 920 1010 \endlabellist

Example 1. Define a CS function on the complex projective plane by
The regular strata induced by are given by
each of which is diffeomorphic to . A singular stratum
is diffeomorphic to , and the final stratum
is diffeomorphic to the torus. Thus, induces the standard (1,0)-trisection of .
The fibers of are sketched in Figure 2. The image contains three critical values: , and . The restriction of to the regular strata contains one critical point in each stratum; these are the points
whose union represents the critical fiber . These three critical points are degenerate (the Hessian of is a singular matrix). Therefore, is not a CS Morse function.
A regular fiber for is a disjoint union of three copies of , given by
The first of these represents a 3-sphere with a Heegaard decomposition of genus 1 and Heegaard surface , while the circles and represent the cores of the two handlebodies.
The critical fiber is a union of three copies of , glued together pairwisely along three circles of critical points. Each of the critical circles lives in a singular stratum :
Each critical circle is the core of one of the 3-dimensional handlebodies that build the spine of the trisection.
A regular fiber for is a 3-torus , given by
where each of the above subsets represents a product of a 2-torus with an interval, and they are glued together pairwisely along their boundary components. The critical fiber is a 2-torus, representing the singular stratum , all of whose points are critical.
Based on the above example, we formulate the following definition.
Definition 4.2.
Let be a closed connected oriented 4-manifold and let be a CS function on . We say that induces a trisection if
defines a trisection of .
The topology of a manifold , revealed by a CS Morse function , can also be seen through restrictions of to the submanifolds for .
Lemma 4.3.
Let be a CS Morse function. Then is a classical Morse function for every .
Proof.
Lemma 4.4.
Let be a CS Morse function on a manifold . If is a critical point of with , then is a nondegenerate critical point of .
Proof.
Suppose is a critical point of and . Let be such that and . By Proposition 3.4, is a submanifold of . Denote by the associated embedding. Then
for . Here we denoted by the projection of a tangent vector to the subspace . It follows that , thus is a critical point of the restriction . Since is a nondegenerate critical point of , it is a nondegenerate critical point of the restriction . ∎
The following theorem describes the general setting in which a CS function does induce a trisection of a 4-manifold.
Theorem 4.5.
Let be a CS Morse function on a closed, connected and oriented manifold . Suppose that in each regular stratum , has a single critical point of index 4, critical points of index 3 and no other critical points for . Moreover, suppose that in each singular stratum , has a single critical point of index 3, critical points of index 2 and no other critical points for all . Then induces a -trisection of the 4-manifold .
Proof.
By Proposition 3.4 and Lemma 3.5, is a 3-manifold with boundary . Observe that the restriction of to every stratum is a smooth function. Since is a CS Morse function, the restriction of to every stratum is a Morse function by Lemma 4.4.
The function has a single critical point of index 0 and critical points of index 1, thus is a 3-dimensional handlebody of genus and is a closed orientable surface of genus .
The function has a single critical point of index 0 and critical points of index 1, thus is a 4-dimensional handlebody of genus for . ∎
4.2. The local structure of trisections, induced by a CS Morse function
In this section, we investigate the local structure of a manifold around a nondegenerate critical point of a CS function. After some general observations, we focus on the case of a CS Morse function, inducing a trisection of a 4-manifold.
The local topology of a manifold around a critical point in the case of the maximum (resp. minimum) function is particularly simple.
Corollary 4.6 ([13, Corollary 3.4]).
Let be a nondegenerate critical point with quadratic index of . Assume and let .
-
(1)
If , then is topologically equivalent to the function
at .
-
(2)
If , then is topologically equivalent to the function
at .
In the case of the maximum function, the index of a critical point is completely defined by the restriction of to the submanifold . Denote by the index of a critical point of the function .
Proposition 4.7.
Let be a CS Morse function on a manifold . If is a critical point of with , then .
Proof.
For , the statement is obvious.
Suppose that for some . By Proposition 3.4, is a codimension 1 submanifold of . Since is a critical point of , we have and thus for some . Since the gradient vectors are affinely independent, they span a 1-dimensional vector subspace of . When moving from the point in the direction of the gradient vector , the value of increases and the value of decreases, so we end up at a point with , thus (when moving from in the direction of , the conclusion is analogous). Therefore, the subspaces and have trivial intersection and consequently .
Choose a neighborhood of the critical point such that for . It follows that and . The value of increases as we move away from in either direction of or the direction of . It follows from the argument above that all the negative eigenvalues of the second differential come from the restriction .
Finally, let be any subset. Since is a critical point of , we have for some with . By Proposition 3.4, is a submanifold of codimension . By a similar argument as in the previous case, is a -dimensional subspace whose intersection with is trivial. Therefore, we obtain a decomposition
and any eigenvector of the second differential of corresponding to a negative eigenvalue lives in . ∎
For the remainder of this section, let be a closed connected 4-manifold and let be a CS Morse function on . Suppose that is a critical point of with quadratic index . Then Corollary 4.6 implies there exists a neighborhood of in , a neighborhood of and a homeomorphism such that and , where . Thus, the change in the topological type of the lower level sets at a critical point corresponds to the addition of a topological handle, and induces a topological handle decomposition of . Stratification corresponding to the CS function , however, endows this handle decomposition with an additional structure.
Definition 4.8.
A stratified -handle inside the stratified manifold is a topological handle such that
-
(1)
the core is a -dimensional disk, contained in a single stratum ,
-
(2)
,
-
(3)
the restriction is smooth for every .
A stratified handle decomposition of the stratified manifold is a sequence of subsets
where each is obtained from by the attachement of stratified -handles.
Let us show that the CS Morse function induces a stratified handle decomposition of the 4-manifold . For each regular value of , the level set is a closed 3-manifold, trisected into for . Now let be a critical point of with a 4-ball neighborhood . At , the topology of the lower level set changes by the addition of a topological handle. By Lemma 4.4 and Proposition 4.7, is also a critical point of and , where . In other words, the core of the topological handle is a subset of the stratum . By Lemma 3.7, we have , thus . Since for any and is smooth, the restriction of the topological handle to each stratum with is smooth. Thus we obtain a stratified handle, that we will call the stratified handle, induced by the CS Morse function .
The structure of a stratified handle depends on the index of the critical point and on the stratum where it lives. There are three distinct possibilities (see Figure 3):
-
(1)
(critical point inside a regular stratum) If , the stratified handle is a smooth handle, contained in a regular stratum .
-
(2)
(critical point inside a singular stratum ) The handle is bisected along its core into and . By Proposition 4.7, the descending manifold of the critical point is contained inside and we obtain a bisected stratified handle, whose index is bounded by .
-
(3)
(critical point inside the singular stratum ) is a smooth submanifold of codimension 2. The intersection is a 2-disk and by Proposition 4.7, the descending manifold of the critical point is contained in . It follows that the corresponding handle is a trisected stratified handle and the index of is bounded above by . We have three possibilities:
-
(a)
and is the trisection surface of a trisected 0-handle,
-
(b)
and is a 2-dimensional 1-handle (whose cocore times a trisected disk gives the cocore of the 4-dimensional 1-handle),
-
(c)
and is the core of the 4-dimensional 2-handle, while the cocore is a trisected disk.
-
(a)
2pt \pinlabel at 320 250 \pinlabel at 100 300 \pinlabel at 550 300 \pinlabel at 320 100 \pinlabel at 320 500 \pinlabel at 1080 250 \pinlabel at 1610 300 \pinlabel at 1900 290 \pinlabel at 2040 300 \pinlabel at 1140 310 \pinlabel at 1000 440 \pinlabel at 1280 310 \pinlabel at 1030 150 \endlabellist

Remark. In a similar fashion, we could analyze the local structure around a critical point of the function on , which leads to the dual of handle stratification, described above. In this case, one should define a co-stratified handle as a handle whose cocore is contained inside a single stratum. Thus, we may also obtain handles with bisected core and handles with trisected core. A simple example exhibiting such handles is given by the function on the 4-sphere .
Example 2. Observe the function on the 4-sphere
Each regular stratum is homeomorphic to the open -ball. The singular strata , and are homeomorphic to and their common boundary is homeomorphic to the 2-sphere, thus induces a -trisection of .
2pt \pinlabel at 1150 50 \pinlabel at 1170 250 \pinlabel at 520 190 \pinlabel at 1153 477 \pinlabel at 1153 995 \pinlabel at 1160 740 \pinlabel at 600 710 \pinlabel at 1160 1312 \pinlabel at 120 1312 \pinlabel at 380 1312 \pinlabel at 650 1312 \pinlabel at 1170 1500 \pinlabel at 330 945 \pinlabel at 510 945 \pinlabel at 410 1090 \endlabellist

Let us look at the fibers of the CS Morse function . Its image contains four critical values: and . The critical fiber at is a single point that belongs to . A regular fiber for is homeomorphic to :
In the above decomposition, is homeomorphic to and . Moreover, and is homeomorphic to . The stratification induced by thus gives a 3-dimensional trisection of the regular fiber. The singular fiber has a similar decomposition with a “pinch” in the middle (the common intersection of the three sectors is a single point instead of a circle). The pinching point is a critical point of . A regular fiber above a point is a union of three copies of , touching pairwise along the connected components of their boundaries, which is homeomorphic to .
The singular fiber is homeomorphic to the union of three 3-spheres, touching pairwise in three critical points , and . A regular fiber above that level is a disjoint union of three 3-spheres contained in the three regular strata, and the final critical fiber is a union of three critical points , and . See Figure 4.
Each critical point of corresponds to the attachement of a stratified 4-dimensional handle, which gives a 4-manifold with boundary with a relative trisection. We start with a trisected 0-handle at the critical point , whose boundary is a trisected 3-sphere. The central link of this trisected is an unlink, along which a 2-handle is attached. The core of the 2-handle is a disk, lying in , whose center is the second critical point . The cocore of the 2-handle is a trisected disk. After the attachement of the 0-handle and the 2-handle, the boundary of the resulting manifold is homeomorphic to , split along three copies of by the singular strata . At the critical fiber , a bisected -handle is attached along the tubular neighborhood of each of these 2-spheres. After this simultaneous attachement of three 3-handles, the boundary of the resulting manifold is a disjoint union of three copies of , each in their own regular stratum. At last, three smooth 4-handles are attached at the critical points , and .
Our local analysis of the fibers of a CS Morse function together with Theorem 4.5 implies the following.
Proposition 4.9.
Let be a CS Morse function on a closed, connected and oriented 4-manifold . Suppose that in each regular stratum , has a single critical point of index 4, critical points of index 3 and no other critical points for . Moreover, suppose that in each singular stratum , has a single critical point of index 3, critical points of index 2 and no other critical points for all . Then induces a stratified handle decomposition.
Note that the stratified handles, corresponding to the critical points of inside , are trisected handles. All handles, corresponding to the critical points inside other strata, come in threes: a triple of bisected handles, corresponding to critical points inside for , or a triple of smooth handles contained in for . A typical handle decomposition of , induced by , consists of the following:
We have shown that the triple symmetry of trisections is not restricted to an abstract identification of the sectors, but is also a local phenomenon as we approach the trisection surface. Moreover, the symmetry may be seen on the level of handles, which might provide some new algebraic implications of trisection theory.
The CS Morse function from Example 3 may be restricted to any embedded surface in . Restriction to the standardly embedded 2-sphere is also a CS Morse function and gives an analogous handle decomposition in dimension 2, see Figure 5. The induced trisection is called a bridge decomposition of the unknotted 2-sphere. Bridge decompositions of embedded surfaces were introduced by Meier and Zupan and have been extensively studied in the last few years [12, 15, 16].
References
- [1] A. A. Agrachev, D. Pallaschke, S. Scholtes, On Morse Theory for Piecewise Smooth Functions, Journal of Dynamical and Control Systems 3, 449–469, 1997.
- [2] M. Bell, J. Hass, J. H. Rubinstein, S. Tillmann, Computing trisections of 4-manifolds, Proc. Nat. Acad. Sci. USA, 115(43), 10901-10907, 2018.
- [3] F. Ben Aribi, S. Courte, M. Golla, D. Moussard, Multisections of higher-dimensional manifolds, arXiv:2303.08779, 2023.
- [4] S. G. Bartels, L. Kuntz, S. Scholtes, Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis, Theory, Methods & Applications 24 (3), 385–407, 1995.
- [5] J. M. Bonnisseau, B. Cornet, Fixed-point theorems and Morse’s lemma for Lipchitzian functions, J. Math. Analysis Applic. 146, 318–332, 1990.
- [6] M. L. Bougeard, Morse theory for some lower- functions in finite dimension, Mathematical Programming 41, 141–159, 1988.
- [7] M. R. Casali, P. Cristofori, Gem-induced trisections of compact PL 4-manifolds, preprint, arXiv:1910.08777, 2019.
- [8] F. H. Clarke, Optimization and nonsmooth analysis, Society for Industrial and Applied Mathematics, Philadelphia, 1990.
- [9] G. Friedman, Singular intersection homology, http://faculty.tcu.edu/gfriedman/IHbook.pdf, 2019.
- [10] D. Gay, R. Kirby, Indefinite Morse 2-functions; broken fibrations and generalizations, Geom. & Topol. 19, 2465–2534, 2015.
- [11] D. Gay, R. Kirby, Trisecting 4-manifolds, Geom. Topol. 20, 3097–3132, 2016.
- [12] J. Joseph, J. Meier, M. Miller, A. Zupan, Bridge trisections and classical knotted surface theory, arXiv: 2112.11557.
- [13] H. T. Jongen, D. Pallaschke, On linearization and continuous selections of functions, Optimization 19 (3), 343–353, 1988.
- [14] J. M. Lee, Introduction to smooth manifolds, Springer, New York, 2003.
- [15] J. Meier, A. Zupan, Bridge trisections of knotted surfaces in , Trans. Amer. Math. Soc. 369, no. 10, 7343–7386, 2017.
- [16] J. Meier, A. Zupan, Bridge trisections of knotted surfaces in 4-manifolds, - Proceedings of the National Academy of Sciences 115, 10.1073/pnas.1717171115, 2017.
- [17] S. Scholtes, Introduction to piecewise differentiable equations, Springer, ISBN 978-1-4614-4340-7, 2012.
- [18] J. Spreer, S. Tillmann, The trisection genus of standard simply connected PL 4-manifolds, 34th International Symposium on Computational Geometry (SoCG 2018). In Leibniz International Proceedings in Informatics (LIPICS), vl. 99, pp. 71:1-71:13, 2018.
2pt \pinlabel at 1150 50 \pinlabel at 1170 250 \pinlabel at 580 240 \pinlabel at 1153 477 \pinlabel at 1153 995 \pinlabel at 1160 740 \pinlabel at 680 730 \pinlabel at 1160 1312 \pinlabel at 120 1312 \pinlabel at 380 1312 \pinlabel at 650 1312 \pinlabel at 1170 1500 \pinlabel at 330 945 \pinlabel at 510 945 \pinlabel at 410 1090 \endlabellist
