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Nonsmooth manifold decompositions

Eva Horvat University of Ljubljana, Faculty of Education, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia, eva.horvat@pef.uni-lj.si
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, trisection

1. 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 YY is obtained from a classical Morse function f:YB1f\colon Y\to B^{1}. Gay and Kirby have shown that a trisection of a closed, connected and oriented 4-manifold XX may similarly be constructed from a suitable Morse 2-function f:XB2f\colon X\to B^{2} [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 f1,,fmf_{1},\ldots,f_{m} on a manifold MM is a continuous function that coincides with at least one of the functions fif_{i} at each point of MM. Applying generalized derivation, introduced by Clarke [8], the topology of MM 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 f={f1,f2,f3}f=\{f_{1},f_{2},f_{3}\} 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 max{f1,f2,f3}\max\{f_{1},f_{2},f_{3}\}, 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 MM a smooth nn-manifold.

Definition 2.1.

Let f:Mf\colon M\to\mathbb{R} be a continuous function and let f1,f2,,fm:Mf_{1},f_{2},\ldots,f_{m}\colon M\to\mathbb{R} be smooth functions. The function ff is called a continuous selection of the functions f1,f2,,fmf_{1},f_{2},\ldots,f_{m} if it satisfies the following two conditions:

  1. (1)

    the set If(x)={i{1,2,,m}|f(x)=fi(x)}I_{f}(x)=\left\{i\in\{1,2,\ldots,m\}|f(x)=f_{i}(x)\right\} is nonempty for every point xMx\in M,

  2. (2)

    for any index i{1,2,,m}i\in\{1,2,\ldots,m\}, there exists a point xMx\in M with iIf(x)i\in I_{f}(x).

Denote by CS(f1,,fm)CS(f_{1},\ldots,f_{m}) the set of all continuous selections of the functions f1,f2,,fmf_{1},f_{2},\ldots,f_{m}. The set If(x)I_{f}(x) is called the active index set of ff at the point xx. Furthermore, the set

I^f(x0)={i{1,,m}|x0cl(int{x|f(x)=fi(x)})}\widehat{I}_{f}(x_{0})=\left\{i\in\{1,\ldots,m\}\,|\,x_{0}\in\mathop{\rm{cl}}(\mathop{\rm{int}}\{x\,|\,f(x)=f_{i}(x)\})\right\}

is called the set of essentially active indices (where “ intS\mathop{\rm{int}}\,S” and “ clS\mathop{\rm{cl}}S” denote the interior and the closure of a set SS). A function fif_{i} is called essentially active at x0x_{0} if iI^f(x0)i\in\widehat{I}_{f}(x_{0}).

It follows from the above definition that I^f(x)If(x)\widehat{I}_{f}(x)\subseteq I_{f}(x) for every xMx\in M. For every piecewise differentiable function ff on MM and every xMx\in M there exists a collection {f1,f2,,fm}\{f_{1},f_{2},\ldots,f_{m}\} of smooth functions that are essentially active in a neighborhood of xx [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 ff be a continuous selection of smooth functions f1,,fm:nf_{1},\ldots,f_{m}\colon\mathbb{R}^{n}\to\mathbb{R}. Clarke’s subdifferential of ff at a point xnx\in\mathbb{R}^{n} is defined by

f(x)=conv{fi(x)|iI^f(x)},\partial f(x)=\mathop{\textrm{conv}\,}\left\{\nabla f_{i}(x)\,|\,i\in\widehat{I}_{f}(x)\right\}\;,

where convS\mathop{\textrm{conv}\,}S denotes the convex hull of a set SS. A point x0nx_{0}\in\mathbb{R}^{n} is called a critical point of ff if 0f(x0)0\in\partial f(x_{0}).

More generally, when ff is a continuous selection of smooth functions on a smooth manifold MM, the Clarke’s subdifferential of ff may be defined in a similar manner using the local coordinate charts on MM. Let ϕ:UM\phi\colon U\to M be a local chart with UnU\subset\mathbb{R}^{n} and ϕ(0)=x\phi(0)=x, then we define f(x)=conv{(fiϕ)(0)|iI^f(x)}\partial f(x)=\mathop{\textrm{conv}\,}\left\{\nabla(f_{i}\circ\phi)(0)\,|\,i\in\widehat{I}_{f}(x)\right\}.

A version of the first Morse Lemma for locally Lipschitz continuous functions was proved in [1].

Proposition 2.3 ([1]).

Let f:Mf\colon M\to\mathbb{R} be a locally Lipschitz continuous function and denote by Mt=f1((,t])M_{t}=f^{-1}((-\infty,t]) its lower level set for tt\in\mathbb{R}. Suppose MβM_{\beta} is compact and f1([α,β])f^{-1}([\alpha,\beta]) does not contain any critical point of ff. Then there exists a Lipschitz continuous mapping F:Mβ×[0,1]MβF\colon M_{\beta}\times[0,1]\to M_{\beta} such that

F(Mβ,t)Mβ+t(αβ),F(x,t)=xfor all xMα and all t[0,1] .F(M_{\beta},t)\subset M_{\beta+t(\alpha-\beta)}\,,\quad F(x,t)=x\;\textit{for all $x\in M_{\alpha}$ and all $t\in[0,1]$\;.}

In order to study the local behaviour of MM around a critical point of a piecewise differentiable function, we need the following definition.

Definition 2.4.

Let f1,,fm:Mf_{1},\ldots,f_{m}\colon M\to\mathbb{R} be smooth functions and fCS(f1,,fm)f\in CS(f_{1},\ldots,f_{m}). A critical point x0Mx_{0}\in M of ff is called nondegenerate if the following two conditions hold:

  • (ND1)

    For each iI^f(x0)i\in\widehat{I}_{f}(x_{0}), the set of differentials {fj(x0)|jI^f(x0)\{i}}\left\{\nabla f_{j}(x_{0})\,|\,j\in\widehat{I}_{f}(x_{0})\backslash\{i\}\right\} is linearly independent;

  • (ND2)

    The second differential x2L(x,λ)(x0)\nabla_{x}^{2}L(x,\lambda)(x_{0}) is regular on

    T^(x0)=iI^f(x0)Ker(fi(x0)),whereL(x,λ)=iI^f(x0)λifi(x),\widehat{T}(x_{0})=\bigcap_{i\in\widehat{I}_{f}(x_{0})}Ker(\nabla f_{i}(x_{0}))\;,\quad\textrm{where}\quad L(x,\lambda)=\sum_{i\in\widehat{I}_{f}(x_{0})}\lambda_{i}f_{i}(x)\;,

    and the numbers λi\lambda_{i}\in\mathbb{R} are such that

    iI^f(x0)λifi(x0)=0,iI^f(x0)λi=1,λi0 for every iI^f(x0).\sum_{i\in\widehat{I}_{f}(x_{0})}\lambda_{i}\nabla f_{i}(x_{0})=0\,,\quad\sum_{i\in\widehat{I}_{f}(x_{0})}\lambda_{i}=1\,,\quad\lambda_{i}\geq 0\textrm{ for every }i\in\widehat{I}_{f}(x_{0})\;.

In this case, the quadratic index of the critical point x0x_{0} is the dimension of a maximal linear subspace of T^(x0)\widehat{T}(x_{0}) on which the quadratic form yTx2L(x,λ)(x0)yy^{T}\nabla_{x}^{2}L(x,\lambda)(x_{0})y 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 f,g:Mf,g\colon M\to\mathbb{R} are said to be topologically equivalent at (x0,y0)M×M(x_{0},y_{0})\in M\times M if there exist open subsets UU of x0x_{0} and VV of y0y_{0} in MM and a homeomorphism ϕ:UV\phi\colon U\to V, for which ϕ(x0)=y0\phi(x_{0})=y_{0} and fϕ1=gf\circ\phi^{-1}=g on VV.

Theorem 2.6 ([13]).

Let f1,,fm:Mf_{1},\ldots,f_{m}\colon M\to\mathbb{R} be twice continuously differentiable functions on an nn-manifold MM and let fCS(f1,,fm)f\in CS(f_{1},\ldots,f_{m}). Choose local coordinates (y1,y2,,yn)(y_{1},y_{2},\ldots,y_{n}) in the neighborhood of a point x0Mx_{0}\in M. Then the following holds:

  • (i)

    if x0x_{0} is a noncritical point of ff, then ff and f(x0)+y1f(x_{0})+y_{1} are locally topologically equivalent at the point (x0,0)(x_{0},0);

  • (ii)

    if x0x_{0} is a nondegenerate critical point of ff, then ff is locally topologically equivalent at (x0,0)(x_{0},0) to a function of the form

    f(x0)+g(y1,,yk)i=k+1k+myi2+j=k+m+1nyj2,f(x_{0})+g(y_{1},\ldots,y_{k})-\sum_{i=k+1}^{k+m}y_{i}^{2}+\sum_{j=k+m+1}^{n}y_{j}^{2}\;,

    where k=|I^f(x0)|1k=|\widehat{I}_{f}(x_{0})|-1, gCS(y1,,yk,i=1kyi)g\in CS(y_{1},\ldots,y_{k},-\sum_{i=1}^{k}y_{i}), and mm is the quadratic index of ff at x0x_{0}.

In particular, Theorem 2.6 implies that the nondegenerate critical points of the function fCS(f1,,fm)f\in CS(f_{1},\ldots,f_{m}) are isolated.

Example 1. Define functions f1,f2:3f_{1},f_{2}\colon\mathbb{R}^{3}\to\mathbb{R} by f1(x,y,z)=x2y2,f2(x,y,z)=y2z2f_{1}(x,y,z)=x^{2}-y^{2},f_{2}(x,y,z)=y^{2}-z^{2} and let ff denote the continuous selection f=max{f1,f2}f=\max\{f_{1},f_{2}\}. Consider the restriction of ff to the 2-sphere S2={(x,y,z)3|x2+y2+z2=1}S^{2}=\{(x,y,z)\in\mathbb{R}^{3}\,|\,x^{2}+y^{2}+z^{2}=1\}. There are two disjoint circles of points xS2x\in S^{2} for which I^f(x)={1,2}\widehat{I}_{f}(x)=\{1,2\}, namely C±={(x,±13,z)3|x2+z2=23}C_{\pm}=\left\{\left(x,\pm\frac{1}{\sqrt{3}},z\right)\in\mathbb{R}^{3}\,|\,x^{2}+z^{2}=\frac{2}{3}\right\}. These two circles separate the sphere into an open annulus, where If={1}I_{f}=\{1\}, and two open disks on which If={2}I_{f}=\{2\}.

\labellist
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2pt \pinlabelxx at 57 42 \pinlabelyy at 170 73 \pinlabelzz at 90 182 \endlabellist

Refer to caption
Figure 1. Critical points of the function ff on S2S^{2} from Example 2.6 are indicated by red (quadratic index 0), green (quadratic index 1) and yellow (index 2) dots.

The annulus contains two critical points (±1,0,0)(\pm 1,0,0) of index 2 and two critical points (0,0,±1)(0,0,\pm 1) of index 1. Each of the disks with If={2}I_{f}=\{2\} contains a critical point (0,±1,0)(0,\pm 1,0) of index 2. To find the “nonsmooth” critical points of ff that lie on the circles C+C_{+} and CC_{-}, we express ff in the local coordinates ϕ±:2S2\phi_{\pm}\colon\mathbb{R}^{2}\to S^{2}, given by ϕ±(x,z)=(x,±1x2z2,z)\phi_{\pm}(x,z)=(x,\pm\sqrt{1-x^{2}-z^{2}},z), and obtain (f1ϕ±)(x,z)=2x2+z21,(f2ϕ±)(x,z)=1x22z2(f_{1}\circ\phi_{\pm})(x,z)=2x^{2}+z^{2}-1,(f_{2}\circ\phi_{\pm})(x,z)=1-x^{2}-2z^{2}. Critical points are the solutions of

t(f1ϕ±)+(1t)(f2ϕ±)=(6t2)xx+(6t4)zz=0.\displaystyle t\,\nabla(f_{1}\circ\phi_{\pm})+(1-t)\nabla(f_{2}\circ\phi_{\pm})=(6t-2)x\frac{\partial}{\partial x}+(6t-4)z\frac{\partial}{\partial z}=0\;.

At t=23t=\frac{2}{3}, there are four nondegenerate critical points (0,±13,±23)\left(0,\pm\sqrt{\frac{1}{3}},\pm\sqrt{\frac{2}{3}}\right) of quadratic index 0. At each of these points, ff is locally topologically equivalent to the function 13+|y|+x2-\frac{1}{3}+|y|+x^{2}. At t=13t=\frac{1}{3}, there are four nondegenerate critical points (±23,±13,0)\left(\pm\sqrt{\frac{2}{3}},\pm\sqrt{\frac{1}{3}},0\right) of quadratic index 1. At each of these points, ff is locally topologically equivalent to the function 13+|y|z2\frac{1}{3}+|y|-z^{2}.

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 MM be a smooth manifold. A CS function on MM is a function f:Mf\colon M\to\mathbb{R} for which there exist smooth functions f1,,fm:Mf_{1},\ldots,f_{m}\colon M\to\mathbb{R} such that

  1. (1)

    fCS(f1,f2,,fm)f\in CS(f_{1},f_{2},\ldots,f_{m}),

  2. (2)

    at any point xMx\in M we have If(x)=I^f(x)I_{f}(x)=\widehat{I}_{f}(x),

  3. (3)

    at any point xMx\in M, the gradients of the active functions {fj|jIf(x)}\{f_{j}\,|\,j\in I_{f}(x)\} are affinely independent.

A CS Morse function on MM is a CS function on MM whose all critical points are nondegenerate.

Throughout this section, we denote by MM a smooth nn-manifold and by ff a CS function on MM, given as the continuous selection of smooth functions f1,f2,,fm:Mf_{1},f_{2},\ldots,f_{m}\colon M\to\mathbb{R}. For any subset J{1,2,,m}J\subset\{1,2,\ldots,m\}, we denote by

MJ={xM|If(x)=J}M_{J}=\left\{x\in M\,|\,I_{f}(x)=J\right\}

the subset of points where the active index set equals JJ, and let NJ={xM|JIf(x)}N_{J}=\{x\in M\,|\,J\subseteq I_{f}(x)\}. Note that MJNJM_{J}\subseteq N_{J} for any J{1,2,,m}J\subset\{1,2,\ldots,m\}.

Lemma 3.2.

M{i}M_{\{i\}} is an open submanifold of MM for any i{1,2,,m}i\in\{1,2,\ldots,m\}. Moreover, NJN_{J} is a closed subset of MM for any J{1,2,,m}J\subseteq\{1,2,\ldots,m\}.

Proof.

Let us denote Mfi={xM|fi(x)=f(x)}M_{f_{i}}=\left\{x\in M\,|\,f_{i}(x)=f(x)\right\}. Since fiff_{i}-f is continuous, the set Mfi=(fif)1(0)M_{f_{i}}=(f_{i}-f)^{-1}(0) is a closed subset of MM. Its complement M\MfiM\backslash M_{f_{i}} is an open subset of MM for every i{1,2,,m}i\in\{1,2,\ldots,m\}, and

M{i}=j{1,2,m}\{i}(M\Mfj)M_{\{i\}}=\bigcap_{j\in\{1,2\ldots,m\}\backslash\{i\}}(M\backslash M_{f_{j}})

is an open subset of MM (thus a submanifold). For any collection of indices J{1,2,,m}J\subseteq\{1,2,\ldots,m\}, the set NJ={xM|JIf(x)}=jJMfjN_{J}=\left\{x\in M\,|\,J\subseteq I_{f}(x)\right\}=\bigcap_{j\in J}M_{f_{j}} is a closed subset of MM. ∎

It follows from Lemma 3.2 that M{i}M_{\{i\}} is a smooth submanifold of codimension 0 in MM. More generally, each MJM_{J} is a submanifold of MM. To prove this, we need the following standard result.

Proposition 3.3 ([14, Proposition 5.28]).

Let SS be a subset of a smooth nn-manifold MM. Then SS is an embedded kk-submanifold of MM if and only if every point pSp\in S has a neighborhood UU in MM such that USU\cap S is a level set of a submersion F:UnkF\colon U\to\mathbb{R}^{n-k}.

Proposition 3.4.

Let J{1,2,,m}J\subset\{1,2,\ldots,m\} be any subset such that If(x)=JI_{f}(x)=J for some xMx\in M. Then MJM_{J} is a smooth submanifold of codimension |J|1|J|-1 in MM.

Proof.

For |J|=1|J|=1, the statement follows from Lemma 3.2.

Suppose that J={i,j}J=\{i,j\} for two indices ij{1,2,,m}i\neq j\in\{1,2,\ldots,m\} and that MJM_{J}\neq\emptyset. Choose an arbitrary xMJx\in M_{J}. Let UU be a neighborhood of xx in MM such that for any index r{1,2,,m}\Jr\in\{1,2,\ldots,m\}\backslash J, we have UN{r}=U\cap N_{\{r\}}=\emptyset (such a neighborhood exists since N{r}N_{\{r\}} is a closed subset of MM that does not contain xx by Lemma 3.2). Define a map F:UF\colon U\to\mathbb{R} by F(w)=fi(w)fj(w)F(w)=f_{i}(w)-f_{j}(w). Then FF is a smooth function on UU and since at any point wUw\in U the gradients fi(w)\nabla f_{i}(w) and fj(w)\nabla f_{j}(w) are affinely independent, FF is a submersion. Since UM{i,j}=F1(0)U\cap M_{\{i,j\}}=F^{-1}(0), Proposition 3.3 implies that M{i,j}=MJM_{\{i,j\}}=M_{J} is an embedded smooth submanifold of codimension 1 in MM.

Now let J={i1,i2,,ik}{1,2,,m}J=\{i_{1},i_{2},\ldots,i_{k}\}\subset\{1,2,\ldots,m\} be any subset with MJM_{J}\neq\emptyset and choose an arbitrary xMJx\in M_{J}. Let UU be a neighborhood of xx in MM such that for any index r{1,2,,m}\Jr\in\{1,2,\ldots,m\}\backslash J, we have UN{r}=U\cap N_{\{r\}}=\emptyset. Define a map F:Uk1F\colon U\to\mathbb{R}^{k-1} by

F(w)=(fi1(w)fik(w),fi2(w)fik(w),,fik1(w)fik(w)).F(w)=\left(f_{i_{1}}(w)-f_{i_{k}}(w),f_{i_{2}}(w)-f_{i_{k}}(w),\ldots,f_{i_{k-1}}(w)-f_{i_{k}}(w)\right)\;.

At any point wUw\in U, the set {fij(w)|j{1,2,,k}}\{\nabla f_{i_{j}}(w)\,|\,j\in\{1,2,\ldots,k\}\} is affinely independent, thus FF is a submersion. Since UMJ=F1(0)U\cap M_{J}=F^{-1}(0), Proposition 3.3 implies that MJM_{J} is an embedded smooth submanifold of codimension k1k-1 in MM. ∎

For any J{1,2,,m}J\subseteq\{1,2,\ldots,m\}, we denote by MJ\partial M_{J} the topological boundary (frontier) of the subset MJMM_{J}\subset M. Note that MJ(MJ)=M_{J}\cap(\partial M_{J})=\emptyset, since MJM_{J} is a submanifold of MM.

Lemma 3.5.

For any i{1,2,,m}i\in\{1,2,\ldots,m\}, we have

M{1,2,,m}=M{1,2,,m}\{i}=j=1mM{1,2,,m}\{j}.M_{\{1,2,\ldots,m\}}=\partial M_{\{1,2,\ldots,m\}\backslash\{i\}}=\bigcap_{j=1}^{m}\partial M_{\{1,2,\ldots,m\}\backslash\{j\}}\;.
Proof.

Recall that N{1,2,,m}\{i}={xM|{1,2,,m}\{i}If(x)}N_{\{1,2,\ldots,m\}\backslash\{i\}}=\{x\in M\,|\,\{1,2,\ldots,m\}\backslash\{i\}\subseteq I_{f}(x)\}. By Lemma 3.2, N{1,2,,m}\{i}N_{\{1,2,\ldots,m\}\backslash\{i\}} is a closed subset of MM that contains M{1,2,,m}\{i}M_{\{1,2,\ldots,m\}\backslash\{i\}}. Moreover,

M{1,2,,m}\{i}={xN{1,2,,m}\{i}|f(x)fi(x)}=N{1,2,,m}\{i}(ffi)1(\{0})M_{\{1,2,\ldots,m\}\backslash\{i\}}=\{x\in N_{\{1,2,\ldots,m\}\backslash\{i\}}\,|\,f(x)\neq f_{i}(x)\}=N_{\{1,2,\ldots,m\}\backslash\{i\}}\cap(f-f_{i})^{-1}(\mathbb{R}\backslash\{0\})

is an open subset of N{1,2,,m}\{i}N_{\{1,2,\ldots,m\}\backslash\{i\}} whose closure equals N{1,2,,m}\{i}N_{\{1,2,\ldots,m\}\backslash\{i\}}, and its boundary is exactly {xN{1,2,,m}\{i}|f(x)=fi(x)}=M{1,2,,m}\{x\in N_{\{1,2,\ldots,m\}\backslash\{i\}}\,|\,f(x)=f_{i}(x)\}=M_{\{1,2,\ldots,m\}}. It follows that M{1,2,,m}\{i}=M{1,2,,m}\{j}\partial M_{\{1,2,\ldots,m\}\backslash\{i\}}=\partial M_{\{1,2,\ldots,m\}\backslash\{j\}} for any i,j{1,2,,m}i,j\in\{1,2,\ldots,m\}. ∎

Lemma 3.6.

For any i{1,2,,m}i\in\{1,2,\ldots,m\}, we have M{i}=intN{i}M_{\{i\}}=\mathop{\rm{int}}N_{\{i\}}.

Proof.

By Lemma 3.2, M{i}M_{\{i\}} is an open subset of MM that is clearly contained in N{i}N_{\{i\}}, thus M{i}intN{i}M_{\{i\}}\subseteq\mathop{\rm{int}}N_{\{i\}}.

To show the other inclusion, first observe that {MJ|J{1,2,,m}}\{M_{J}\,|\,J\subseteq\{1,2,\ldots,m\}\} is a partition of MM into pairwise disjoint smooth submanifolds by Proposition 3.4. Suppose that intN{i}intN{j}\mathop{\rm{int}}N_{\{i\}}\cap\mathop{\rm{int}}N_{\{j\}}\neq\emptyset for some jij\neq i, then this intersection is an open subset of MM that intersects at least one of the submanifolds M{r}M_{\{r\}} of codimension 0 nontrivially (in fact, in an open set), thus fi(z)=fj(z)=f(z)f_{i}(z)=f_{j}(z)=f(z) for some zM{r}z\in M_{\{r\}}, which gives a contradiction. Thus, the subsets intN{r}\mathop{\rm{int}}N_{\{r\}} are pairwise disjoint for r{1,2,,m}r\in\{1,2,\ldots,m\}. Now choose any yintN{i}y\in\mathop{\rm{int}}N_{\{i\}}, and let UU be a neighborhood of yy in MM such that UintN{i}U\subset\mathop{\rm{int}}N_{\{i\}}. Suppose that fj(y)=f(y)f_{j}(y)=f(y) for some j{1,2,,m}\{i}j\in\{1,2,\ldots,m\}\backslash\{i\}. It follows from our reasoning above that yintN{j}y\notin\mathop{\rm{int}}N_{\{j\}}, therefore yN{j}\(intN{j})y\in N_{\{j\}}\backslash(\mathop{\rm{int}}N_{\{j\}}). By the Condition (2) of Definition 3.1, we have jIf(y)=I^f(y)j\in I_{f}(y)=\widehat{I}_{f}(y) and thus ycl(intN{j})y\in\mathop{\rm{cl}}(\mathop{\rm{int}}N_{\{j\}}). But then the neighborhood UU intersects intN{j}\mathop{\rm{int}}N_{\{j\}}, which gives a contradiction. It follows that If(y)={i}I_{f}(y)=\{i\} and thus yM{i}y\in M_{\{i\}}. ∎

Lemma 3.7.

For any J{1,2,,m}J\subseteq\{1,2,\ldots,m\} we have

MJ=JIMI.\partial M_{J}=\bigcup_{J\subsetneq I}M_{I}\;.
Proof.

Suppose that xMJx\in\partial M_{J}. For every jJj\in J, we have fj|MJ=f|MJf_{j}|_{M_{J}}=f|_{M_{J}} and it follows by continuity that fj(x)=f(x)f_{j}(x)=f(x). Thus, JIf(x)J\subset I_{f}(x) and since xMJx\notin M_{J}, JJ is a proper subset of If(x)I_{f}(x). This shows that MJJIMI\partial M_{J}\subseteq\bigcup_{J\subsetneq I}M_{I}.

Let us show the other inclusion by induction on |J||J|. To begin, let J={j}J=\{j\} and choose an element xMx\in M with JIf(x)J\subsetneq I_{f}(x). By Condition (2) of Definition 3.1, we have jI^f(x)j\in\widehat{I}_{f}(x) and thus xcl(intN{j})=cl(M{j})x\in\mathop{\rm{cl}}(\mathop{\rm{int}}N_{\{j\}})=\mathop{\rm{cl}}(M_{\{j\}}) by Lemma 3.6. Since xM{j}x\notin M_{\{j\}}, it follows that xM{j}x\in\partial M_{\{j\}}.

Now suppose that for some n1n\geq 1 and every J{1,2,,m}J\subseteq\{1,2,\ldots,m\} with |J|=n|J|=n we have JIMIMJ\bigcup_{J\subsetneq I}M_{I}\subseteq\partial M_{J}. Choose any subset J{1,2,,m}J^{\prime}\subseteq\{1,2,\ldots,m\} with |J|=n+1|J^{\prime}|=n+1 and write J=J{j}J^{\prime}=J\sqcup\{j\}. By Proposition 3.4, MJM_{J} is a submanifold of codimension n1n-1 and M{j}M_{\{j\}} is a submanifold of codimension 0. Thus, MJ\partial M_{J} is a closed topological submanifold of codimension nn and M{j}\partial M_{\{j\}} is a closed topological submanifold of codimension 1. It follows by the induction hypothesis that MJ=JIMI\partial M_{J}=\bigcup_{J\subsetneq I}M_{I} and M{j}={j}IMI\partial M_{\{j\}}=\bigcup_{\{j\}\subsetneq I}M_{I}, therefore

(1) MJM{j}=JIMI=MJJIMI\displaystyle\partial M_{J}\cap\partial M_{\{j\}}=\bigcup_{J^{\prime}\subseteq I}M_{I}=M_{J^{\prime}}\cup\bigcup_{J^{\prime}\subsetneq I}M_{I}

is a closed topological submanifold of codimension nn. In the decomposition (1), MJM_{J}^{\prime} is a submanifold of codimension nn and each MIM_{I} with JIJ^{\prime}\subsetneq I is a submanifold of codimension n+1\geq n+1. Thus, for every xJIMIx\in\bigcup_{J^{\prime}\subsetneq I}M_{I} and every neighborhood UU of xx in MM, we have UMJU\cap M_{J^{\prime}}\neq\emptyset. It follows that JIMIcl(MJ)\bigcup_{J^{\prime}\subsetneq I}M_{I}\subseteq\mathop{\rm{cl}}(M_{J^{\prime}}) and since (JIMI)MJ=\left(\bigcup_{J^{\prime}\subsetneq I}M_{I}\right)\cap M_{J^{\prime}}=\emptyset, we have JIMIMJ\bigcup_{J^{\prime}\subsetneq I}M_{I}\subseteq\partial M_{J^{\prime}}. ∎

Proposition 3.8.

For any xMx\in M and any J{1,2,,m}J\subseteq\{1,2,\ldots,m\}, the following statements are equivalent:

  • (a)

    xcl(MJ)x\in\mathop{\rm{cl}}(M_{J}),

  • (b)

    f(x)=fj(x)f(x)=f_{j}(x) for every jJj\in J,

  • (c)

    JI^f(x)J\subseteq\widehat{I}_{f}(x),

  • (d)

    for every neighborhood UU of xx and every jJj\in J we have UM{j}U\cap M_{\{j\}}\neq\emptyset.

Proof.

(a) \Rightarrow (b) Choose an xcl(MJ)x\in\mathop{\rm{cl}}(M_{J}). Since f|MJ=fj|MJf|_{M_{J}}=f_{j}|_{M_{J}} for every jJj\in J, it follows by continuity that f(x)=fj(x)f(x)=f_{j}(x) for every jJj\in J.
(b) \Rightarrow (a) Suppose that f(x)=fj(x)f(x)=f_{j}(x) for every jJj\in J, then JIf(x)J\subseteq I_{f}(x). If If(x)=JI_{f}(x)=J, it follows that xMJx\in M_{J}. Otherwise, we have JIf(x)J\subsetneq I_{f}(x) and then Lemma 3.7 implies that xMJx\in\partial M_{J}.
(b) \Rightarrow (d) Suppose that f(x)=fj(x)f(x)=f_{j}(x) for every jJj\in J and let UU be a neighborhood of xx in MM. Then JIf(x)=I^f(x)J\subseteq I_{f}(x)=\widehat{I}_{f}(x) and it follows by Lemma 3.6 that xcl(int(N{j}))=cl(M{j})x\in\mathop{\rm{cl}}(\mathop{\rm{int}}(N_{\{j\}}))=\mathop{\rm{cl}}(M_{\{j\}}), therefore UM{j}U\cap M_{\{j\}}\neq\emptyset.
(d) \Rightarrow (b) Suppose that every neighborhood of xx intersects M{j}M_{\{j\}} for every jJj\in J. Then xcl(M{j})x\in\mathop{\rm{cl}}(M_{\{j\}}) and since f|M{j}=fj|M{j}f|_{M_{\{j\}}}=f_{j}|_{M_{\{j\}}}, it follows by continuity that f(x)=fj(x)f(x)=f_{j}(x) for every jJj\in J.
(c) \Rightarrow (d) Let JI^f(x)J\subseteq\widehat{I}_{f}(x) and choose a neighborhood UU of xx in MM. Then xcl(intN{j})x\in\mathop{\rm{cl}}(\mathop{\rm{int}}N_{\{j\}}) for every jJj\in J and it follows by Lemma 3.6 that UM{j}U\cap M_{\{j\}}\neq\emptyset.
(d) \Rightarrow (c) Suppose that every neighborhood of xx intersects M{j}M_{\{j\}} for every jJj\in J; then xcl(M{j})=cl(intN{j})x\in\mathop{\rm{cl}}(M_{\{j\}})=\mathop{\rm{cl}}(\mathop{\rm{int}}N_{\{j\}}), therefore jI^f(x)j\in\widehat{I}_{f}(x) for every jJj\in J. ∎

3.2. Stratification

We have shown that a CS function fCS(f1,,fm)f\in CS(f_{1},\ldots,f_{m}) on a smooth manifold MM induces a decomposition of MM into a union

M=J{1,2,,m}MJM=\bigcup_{J\subseteq\{1,2,\ldots,m\}}M_{J}

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 XX, equipped by a sequence of closed subspaces

=X0X1X2Xn1Xn=X\emptyset=X^{0}\subset X^{1}\subset X^{2}\subset\ldots\subset X^{n-1}\subset X^{n}=X

for some integer n0n\geq 0. The space XiX^{i} is called the ii-skeleton, and the index ii is called the formal dimension of the skeleton. The connected components of Xi=XiXi1X_{i}=X^{i}-X^{i-1} are called the strata of XX of formal dimension ii. If XX is a filtered space of formal dimension nn, then the components of Xn=XnXn1X_{n}=X^{n}-X^{n-1} are called the regular strata of XX and all the other strata are called singular strata.

Definition 3.10.

[9] Let XX be a filtered space. The filtered set XX satisfies the frontier condition if for any two strata S,TS,T of XX the following holds:

if Scl(T)S\cap\mathop{\rm{cl}}(T)\neq\emptyset, then Scl(T)S\subset\mathop{\rm{cl}}(T) .

A stratified space is a filtered space that satisfies the frontier condition.

Consider our manifold MM, endowed by the CS function fCS(f1,,fm)f\in CS(f_{1},\ldots,f_{m}), and define the skeleta of MM by

Xi={xM||If(x)|mi+1}X^{i}=\left\{x\in M\,|\,|I_{f}(x)|\geq m-i+1\right\}

for 0im0\leq i\leq m.

Proposition 3.11.

The sequence

=X0X1Xm1Xm=M\emptyset=X^{0}\subset X^{1}\subset\ldots\subset X^{m-1}\subset X^{m}=M

defines a stratification of MM.

Proof.

We have X0=X^{0}=\emptyset and Xm=MX^{m}=M by definition. More generally, it follows from Lemma 3.7 and Proposition 3.8 that Xi=|J|=mi+1cl(MJ)X^{i}=\bigcup_{|J|=m-i+1}\mathop{\rm{cl}}(M_{J}) is a closed subspace of MM for every i{0,1,,m}i\in\{0,1,\ldots,m\}. The regular strata of MM are submanifolds M{i}M_{\{i\}} for i{1,2,,m}i\in\{1,2,\ldots,m\}. More generally, we have

XiXi1={xM|mi+1|If(x)|<mi+2}=|J|=mi+1MJ.X^{i}-X^{i-1}=\left\{x\in M\,|\,m-i+1\leq|I_{f}(x)|<m-i+2\right\}=\bigcup_{|J|=m-i+1}M_{J}\;.

Thus, any submanifold MJM_{J} with |J|=i|J|=i represents a stratum of formal dimension mi+1m-i+1. Any two strata MJM_{J} and MJM_{J^{\prime}} with JJJ\neq J^{\prime} are disjoint, while Lemma 3.7 implies that the boundary of any stratum is a disjoint union of strata. If we have Scl(T)S\cap\mathop{\rm{cl}}(T)\neq\emptyset for two strata SS and TT, then it follows that either S=TS=T or STS\subset\partial T. Therefore the above filtration of MM satisfies the frontier condition. ∎

What can we say about the fibers of the CS function f:Mf\colon M\to\mathbb{R}? Let us denote Yi=Xif1(t)Y^{i}=X^{i}\cap f^{-1}(t) for i=0,1,,mi=0,1,\ldots,m.

Corollary 3.12.

The sequence of subspaces

=Y0Y1Ym1Ym=f1(t)\emptyset=Y^{0}\subset Y^{1}\subset\ldots\subset Y^{m-1}\subset Y^{m}=f^{-1}(t)

defines a stratification of the fiber f1(t)f^{-1}(t) for any tf(M)t\in f(M).

Proof.

Since f1(t)f^{-1}(t) is a closed subspace of MM, also YiY^{i} is a closed subspace of MM. We have YiYi1=Xif1(t)=|J|=mi+1{xMJ|f(x)=t}Y^{i}-Y^{i-1}=X_{i}\cap f^{-1}(t)=\bigcup_{|J|=m-i+1}\{x\in M_{J}\,|\,f(x)=t\}, therefore the subsets {xMJ|f(x)=t}\{x\in M_{J}\,|\,f(x)=t\} represent the strata of formal dimension m|J|+1m-|J|+1.

Let SS and TT be two strata with Scl(T)S\cap\mathop{\rm{cl}}(T)\neq\emptyset. By Lemma 3.7 and Proposition 3.8, there exist subsets J,K{1,2,,m}J,K\subset\{1,2,\ldots,m\} with S={xM|If(x)=J,f(x)=t}S=\{x\in M\,|\,I_{f}(x)=J,f(x)=t\} and cl(T)={xM|KIf(x),f(x)=t}\mathop{\rm{cl}}(T)=\{x\in M\,|\,K\subseteq I_{f}(x),f(x)=t\}, therefore

Scl(T)={xM|KIf(x)=J,f(x)=t}.S\cap\mathop{\rm{cl}}(T)=\{x\in M\,|\,K\subseteq I_{f}(x)=J,f(x)=t\}\neq\emptyset\;.

It follows that either J=KJ=K or KJK\subsetneq J, thus S=TS=T or STS\subseteq\partial T. We have shown that the above filtration of f1(t)f^{-1}(t) satisfies the frontier condition. ∎

Remark. Observe that the restriction of a CS function ff to every stratum is a smooth function, since f|MJ=fj|MJf|_{M_{J}}=f_{j}|_{M_{J}} for any jJj\in J. 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

S3={(x1,x2,x3,x4)4|x12+x22+x32+x42=1}S^{3}=\{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}\,|\,x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\}

by f(x1,x2,x3,x4)=max{x12+x22,x32+x42}f(x_{1},x_{2},x_{3},x_{4})=\max\{x_{1}^{2}+x_{2}^{2},x_{3}^{2}+x_{4}^{2}\}. Its image f(S3)=[12,1]f(S^{3})=\left[\frac{1}{2},1\right] contains two critical values; the critical fiber f1(12)f^{-1}\left(\frac{1}{2}\right) is homeomorphic to the 2-torus, while the critical fiber f1(1)f^{-1}(1) looks like a Hopf link. Every regular fiber

f1(t)={(x1,x2,x3,x4)|x12+x22=t,x32+x42=1t}\displaystyle f^{-1}(t)=\left\{(x_{1},x_{2},x_{3},x_{4})\,|\,x_{1}^{2}+x_{2}^{2}=t,x_{3}^{2}+x_{4}^{2}=1-t\right\}\cup
{(x1,x2,x3,x4)|x32+x42=t,x12+x22=1t}\displaystyle\left\{(x_{1},x_{2},x_{3},x_{4})\,|\,x_{3}^{2}+x_{4}^{2}=t,x_{1}^{2}+x_{2}^{2}=1-t\right\}

for 12<t<1\frac{1}{2}<t<1 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 ff are the interiors of two solid tori M{1}={(x1,x2,x3,x4)S3|x12+x22>x32+x42}M_{\{1\}}=\left\{(x_{1},x_{2},x_{3},x_{4})\in S^{3}\,|\,x_{1}^{2}+x_{2}^{2}>x_{3}^{2}+x_{4}^{2}\right\} and M{2}={(x1,x2,x3,x4)S3|x12+x22<x32+x42}M_{\{2\}}=\left\{(x_{1},x_{2},x_{3},x_{4})\in S^{3}\,|\,x_{1}^{2}+x_{2}^{2}<x_{3}^{2}+x_{4}^{2}\right\}, whose common boundary is the singular stratum M{1,2}=f1(12)M_{\{1,2\}}=f^{-1}\left(\frac{1}{2}\right). Thus, stratification induced by ff 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 f(x1,x2,x3,x4)=max{x1,x2}f(x_{1},x_{2},x_{3},x_{4})=\max\{x_{1},x_{2}\} on S3S^{3} is a CS Morse function with 4 critical points (12,12,0,0)\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0,0\right), (12,12,0,0)\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0\right), (1,0,0,0)(1,0,0,0) and (0,1,0,0)(0,1,0,0). A regular fiber f1(t)f^{-1}(t) for 12<t<12-\frac{1}{\sqrt{2}}<t<\frac{1}{\sqrt{2}} is a union of two disks, glued along their boundary, and the critical fiber f1(12)f^{-1}(\frac{1}{\sqrt{2}}) is a wedge of two spheres. A regular fiber f1(t)f^{-1}(t) for 12<t<1\frac{1}{\sqrt{2}}<t<1 is a disjoint union of two spheres. The stratification, induced by ff, 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 MM be a closed, connected, oriented 4-manifold. Given two integers 0kg0\leq k\leq g, a (g,k)(g,k)-trisection of MM is a decomposition M=X1X2X3M=X_{1}\cup X_{2}\cup X_{3} such that:

  • XiX_{i} is a 4-dimensional 1-handlebody obtained by attaching kk 1-handles to one 0-handle for each i{1,2,3}i\in\{1,2,3\},

  • XiXjX_{i}\cap X_{j} is a 3-dimensional genus gg handlebody for each ij{1,2,3}i\neq j\in\{1,2,3\} and

  • X1X2X3X_{1}\cap X_{2}\cap X_{3} is a closed surface of genus gg.

The surface X1X2X3X_{1}\cap X_{2}\cap X_{3} is called the trisection surface, while the union of the 3-dimensional handlebodies ij{1,2,3}(XiXj)\cup_{i\neq j\in\{1,2,3\}}(X_{i}\cap X_{j}) is called the spine of the trisection.

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Refer to caption
Figure 2. The fibers of the CS function ff on P2\mathbb{C}P^{2} from Example 2.

Example 1. Define a CS function on the complex projective plane P2\mathbb{C}P^{2} by

f([z0:z1:z2])=1|z0|+|z1|+|z2|max{|z0|,|z1|,|z2|}.f([z_{0}:z_{1}:z_{2}])=\frac{1}{|z_{0}|+|z_{1}|+|z_{2}|}\max\left\{|z_{0}|,|z_{1}|,|z_{2}|\right\}\;.

The regular strata induced by ff are given by

M{i}={[z0:z1:z2]P2|zi=1,|zj|<1,|zk|<1 for {i,j,k}={0,1,2}},M_{\{i\}}=\{[z_{0}:z_{1}:z_{2}]\in\mathbb{C}P^{2}\,|\,z_{i}=1,\,|z_{j}|<1,\,|z_{k}|<1\textrm{ for $\{i,j,k\}=\{0,1,2\}$}\}\;,

each of which is diffeomorphic to int(B2)×int(B2)int(B^{2})\times int(B^{2}). A singular stratum

M{i,j}={[z0:z1:z2]P2|zi=1,|zj|=1,|zk|<1 for {i,j,k}={0,1,2}}M_{\{i,j\}}=\{[z_{0}:z_{1}:z_{2}]\in\mathbb{C}P^{2}\,|\,z_{i}=1,\,|z_{j}|=1,\,|z_{k}|<1\textrm{ for $\{i,j,k\}=\{0,1,2\}$}\}

is diffeomorphic to S1×int(B2)S^{1}\times int(B^{2}), and the final stratum

M{0,1,2}={[z0:z1:z2]P2|z0=1,|z1|=|z2|=1}M_{\{0,1,2\}}=\{[z_{0}:z_{1}:z_{2}]\in\mathbb{C}P^{2}\,|\,z_{0}=1,\,|z_{1}|=|z_{2}|=1\}

is diffeomorphic to the torus. Thus, ff induces the standard (1,0)-trisection of P2\mathbb{C}P^{2}.

The fibers of ff are sketched in Figure 2. The image f(P2)=[13,1]f(\mathbb{C}P^{2})=\left[\frac{1}{3},1\right] contains three critical values: 13\frac{1}{3}, 12\frac{1}{2} and 11. The restriction of ff to the regular strata contains one critical point in each stratum; these are the points

[1:0:0]M{0},[0:1:0]M{1},[0:0:1]M{2},[1:0:0]\in M_{\{0\}},\quad[0:1:0]\in M_{\{1\}},\quad[0:0:1]\in M_{\{2\}}\;,

whose union represents the critical fiber f1(1)f^{-1}(1). These three critical points are degenerate (the Hessian of fif_{i} is a singular matrix). Therefore, ff is not a CS Morse function.

A regular fiber f1(t)f^{-1}(t) for 12<t<1\frac{1}{2}<t<1 is a disjoint union of three copies of S3S^{3}, given by

{[1:z:w]||z|+|w|=1tt}{[z:1:w]||z|+|w|=1tt}{[z:w:1]||z|+|w|=1tt}.\left\{[1\colon z\colon w]\,|\,|z|+|w|=\frac{1-t}{t}\right\}\cup\left\{[z\colon 1\colon w]\,|\,|z|+|w|=\frac{1-t}{t}\right\}\cup\left\{[z\colon w\colon 1]\,|\,|z|+|w|=\frac{1-t}{t}\right\}\;.

The first of these represents a 3-sphere with a Heegaard decomposition of genus 1 and Heegaard surface {[1:z:w]||z|=|w|=1t2t}\left\{[1\colon z\colon w]\,|\,|z|=|w|=\frac{1-t}{2t}\right\}, while the circles {[1:z:0]||z|=1tt}\left\{[1\colon z\colon 0]\,|\,|z|=\frac{1-t}{t}\right\} and {[1:0:w]||w|=1tt}\left\{[1\colon 0\colon w]\,|\,|w|=\frac{1-t}{t}\right\} represent the cores of the two handlebodies.

The critical fiber f1(12)f^{-1}\left(\frac{1}{2}\right) is a union of three copies of S3S^{3}, glued together pairwisely along three circles of critical points. Each of the critical circles lives in a singular stratum M{i,j}M_{\{i,j\}}:

{[1:z:0]||z|=1}M{0,1},{[z:0:1]||z|=1}M{0,2},{[0:1:z]||z|=1}M{1,2}.\left\{[1\colon z\colon 0]\,|\,|z|=1\right\}\subset M_{\{0,1\}},\quad\left\{[z\colon 0\colon 1]\,|\,|z|=1\right\}\subset M_{\{0,2\}},\quad\left\{[0\colon 1\colon z]\,|\,|z|=1\right\}\subset M_{\{1,2\}}\;.

Each critical circle is the core of one of the 3-dimensional handlebodies that build the spine of the trisection.

A regular fiber f1(s)f^{-1}(s) for 13<s<12\frac{1}{3}<s<\frac{1}{2} is a 3-torus T3T^{3}, given by

{[1:z:w]||z|+|w|=1ss,|z|1,|w|1}{[z:1:w]||z|+|w|=1ss,|z|1,|w|1}\displaystyle\left\{[1\colon z\colon w]\,|\,|z|+|w|=\frac{1-s}{s},|z|\leq 1,|w|\leq 1\right\}\cup\left\{[z\colon 1\colon w]\,|\,|z|+|w|=\frac{1-s}{s},|z|\leq 1,|w|\leq 1\right\}
{[z:w:1]||z|+|w|=1ss,|z|1,|w|1},\displaystyle\cup\left\{[z\colon w\colon 1]\,|\,|z|+|w|=\frac{1-s}{s},|z|\leq 1,|w|\leq 1\right\}\;,

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 f1(13)f^{-1}\left(\frac{1}{3}\right) is a 2-torus, representing the singular stratum M{0,1,2}M_{\{0,1,2\}}, all of whose points are critical.

Based on the above example, we formulate the following definition.

Definition 4.2.

Let MM be a closed connected oriented 4-manifold and let fCS(f1,f2,f3)f\in CS(f_{1},f_{2},f_{3}) be a CS function on MM. We say that ff induces a (g,k)(g,k) trisection if

(cl(M{1}),cl(M{2}),cl(M{3}))\left(\mathop{\rm{cl}}(M_{\{1\}}),\mathop{\rm{cl}}(M_{\{2\}}),\mathop{\rm{cl}}(M_{\{3\}})\right)

defines a (g,k)(g,k) trisection of MM.

The topology of a manifold MM, revealed by a CS Morse function fCS{f1,,fm}f\in CS\{f_{1},\ldots,f_{m}\}, can also be seen through restrictions of ff to the submanifolds MJM_{J} for J{1,,m}J\subset\{1,\ldots,m\}.

Lemma 4.3.

Let fCS(f1,f2,,fm)f\in CS(f_{1},f_{2},\ldots,f_{m}) be a CS Morse function. Then f|M{i}=fi|M{i}f|_{M_{\{i\}}}=f_{i}|_{M_{\{i\}}} is a classical Morse function for every i{1,2,,m}i\in\{1,2,\ldots,m\}.

Proof.

M{i}M_{\{i\}} is an open submanifold of MM by Lemma 3.2 and the restriction f|M{i}f|_{M_{\{i\}}} is smooth. For any critical point x0M{i}x_{0}\in M_{\{i\}} we have |I^f(x0)|=1|\widehat{I}_{f}(x_{0})|=1, thus the nondegeneracy conditions (ND1) and (ND2) for ff from Definition 2.4 boil down to the classical nondegeneracy condition for fi|M{i}f_{i}|_{M_{\{i\}}}. ∎

Lemma 4.4.

Let fCS(f1,f2,,fm)f\in CS(f_{1},f_{2},\ldots,f_{m}) be a CS Morse function on a manifold MM. If x0Mx_{0}\in M is a critical point of ff with If(x0)=JI_{f}(x_{0})=J, then x0x_{0} is a nondegenerate critical point of f|MJf|_{M_{J}}.

Proof.

Suppose x0x_{0} is a critical point of ff and If(x0)=J={fi1,fi2,,fik}I_{f}(x_{0})=J=\{f_{i_{1}},f_{i_{2}},\ldots,f_{i_{k}}\}. Let λ1,,λk[0,1]\lambda_{1},\ldots,\lambda_{k}\in[0,1] be such that j=1kλj=1\sum_{j=1}^{k}\lambda_{j}=1 and j=1kλjfij(x0)=0\sum_{j=1}^{k}\lambda_{j}\nabla f_{i_{j}}(x_{0})=0. By Proposition 3.4, MJM_{J} is a submanifold of MM. Denote by ι:MJM\iota\colon M_{J}\to M the associated embedding. Then

(fij|MJ)(x0)=(fijι)(x0)=projTx0MJ(fij(x0))\nabla\left(f_{i_{j}}|_{M_{J}}\right)(x_{0})=\nabla(f_{i_{j}}\circ\iota)(x_{0})=\textrm{proj}_{T_{x_{0}}M_{J}}\left(\nabla f_{i_{j}}(x_{0})\right)

for j=1,2,,kj=1,2,\ldots,k. Here we denoted by projTx0MJ(v)\textrm{proj}_{T_{x_{0}}M_{J}}(v) the projection of a tangent vector vTx0Mv\in T_{x_{0}}M to the subspace Tx0MJTx0MT_{x_{0}}M_{J}\leq T_{x_{0}}M. It follows that j=1kλj(fij|MJ)(x0)=0\sum_{j=1}^{k}\lambda_{j}\nabla\left(f_{i_{j}}|_{M_{J}}\right)(x_{0})=0, thus x0x_{0} is a critical point of the restriction f|MJf|_{M_{J}}. Since x0x_{0} is a nondegenerate critical point of ff, it is a nondegenerate critical point of the restriction f|MJf|_{M_{J}}. ∎

The following theorem describes the general setting in which a CS function does induce a trisection of a 4-manifold.

Theorem 4.5.

Let fCS(f1,f2,f3)f\in CS(f_{1},f_{2},f_{3}) be a CS Morse function on a closed, connected and oriented manifold MM. Suppose that in each regular stratum M{i}M_{\{i\}}, ff has a single critical point of index 4, kk critical points of index 3 and no other critical points for i{1,2,3}i\in\{1,2,3\}. Moreover, suppose that in each singular stratum M{i,j}M_{\{i,j\}}, ff has a single critical point of index 3, gg critical points of index 2 and no other critical points for all ij{1,2,3}i\neq j\in\{1,2,3\}. Then ff induces a (g,k)(g,k)-trisection of the 4-manifold MM.

Proof.

By Proposition 3.4 and Lemma 3.5, M{i,j}M_{\{i,j\}} is a 3-manifold with boundary M{1,2,3}M_{\{1,2,3\}}. Observe that the restriction of ff to every stratum is a smooth function. Since ff is a CS Morse function, the restriction of ff to every stratum is a Morse function by Lemma 4.4.

The function f|M{i,j}-f|_{M_{\{i,j\}}} has a single critical point of index 0 and gg critical points of index 1, thus cl(M{i,j})=cl(M{i})cl(M{j})\mathop{\rm{cl}}(M_{\{i,j\}})=\mathop{\rm{cl}}(M_{\{i\}})\cap\mathop{\rm{cl}}(M_{\{j\}}) is a 3-dimensional handlebody of genus gg and M{1,2,3}=cl(M{1})cl(M{2})cl(M{3})M_{\{1,2,3\}}=\mathop{\rm{cl}}(M_{\{1\}})\cap\mathop{\rm{cl}}(M_{\{2\}})\cap\mathop{\rm{cl}}(M_{\{3\}}) is a closed orientable surface of genus gg.

The function f|M{i}-f|_{M_{\{i\}}} has a single critical point of index 0 and kk critical points of index 1, thus cl(M{i})\mathop{\rm{cl}}(M_{\{i\}}) is a 4-dimensional handlebody of genus kk for i=1,2,3i=1,2,3. ∎

4.2. The local structure of trisections, induced by a CS Morse function max{f1,f2,f3}\max\{f_{1},f_{2},f_{3}\}

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 x0x_{0} be a nondegenerate critical point with quadratic index rr of fCS(f1,f2,,fm)f\in CS(f_{1},f_{2},\ldots,f_{m}). Assume f(x0)=0f(x_{0})=0 and let k=|I^f(x0)|1k=|\widehat{I}_{f}(x_{0})|-1.

  • (1)

    If f=max{f1,f2,,fm}f=\max\{f_{1},f_{2},\ldots,f_{m}\}, then ff is topologically equivalent to the function

    g(y1,y2,,yn)=i=1ryi2+j=r+1nyj2g(y_{1},y_{2},\ldots,y_{n})=-\sum_{i=1}^{r}y_{i}^{2}+\sum_{j=r+1}^{n}y_{j}^{2}

    at (x0,0)(x_{0},0).

  • (2)

    If f=min{f1,f2,,fm}f=\min\{f_{1},f_{2},\ldots,f_{m}\}, then ff is topologically equivalent to the function

    g(y1,y2,,yn)=i=1r+kyi2+j=r+k+1nyj2g(y_{1},y_{2},\ldots,y_{n})=-\sum_{i=1}^{r+k}y_{i}^{2}+\sum_{j=r+k+1}^{n}y_{j}^{2}

    at (x0,0)(x_{0},0).

In the case of the maximum function, the index of a critical point is completely defined by the restriction of ff to the submanifold MJM_{J}. Denote by indf(x0)\mathop{\rm{ind}}_{f}(x_{0}) the index of a critical point x0x_{0} of the function ff.

Proposition 4.7.

Let f=max{f1,f2,,fm}f=\max\{f_{1},f_{2},\ldots,f_{m}\} be a CS Morse function on a manifold MM. If x0x_{0} is a critical point of ff with J=If(x0)J=I_{f}(x_{0}), then indf(x0)=indf|MJ(x0)\mathop{\rm{ind}}_{f}(x_{0})=\mathop{\rm{ind}}_{f|_{M_{J}}}(x_{0}).

Proof.

For |J|=1|J|=1, the statement is obvious.

Suppose that J={i,j}J=\{i,j\} for some ij{1,2,m}i\neq j\in\{1,2,\ldots m\}. By Proposition 3.4, M{i,j}M_{\{i,j\}} is a codimension 1 submanifold of MM. Since x0x_{0} is a critical point of ff, we have 0conv{fi(x0),fj(x0)}0\in conv\{\nabla f_{i}(x_{0}),\nabla f_{j}(x_{0})\} and thus tfi(x0)+(1t)fj(x0)=0t\nabla f_{i}(x_{0})+(1-t)\nabla f_{j}(x_{0})=0 for some 0t10\leq t\leq 1. Since the gradient vectors fi(x0),fj(x0)\nabla f_{i}(x_{0}),\nabla f_{j}(x_{0}) are affinely independent, they span a 1-dimensional vector subspace of Tx0MT_{x_{0}}M. When moving from the point x0x_{0} in the direction of the gradient vector fi(x0)\nabla f_{i}(x_{0}), the value of fif_{i} increases and the value of fjf_{j} decreases, so we end up at a point xx with fi(x)fj(x)f_{i}(x)\neq f_{j}(x), thus xM{i,j}x\notin M_{\{i,j\}} (when moving from x0x_{0} in the direction of fj(x0)\nabla f_{j}(x_{0}), the conclusion is analogous). Therefore, the subspaces Lin{fi(x0),fj(x0)}Lin\{\nabla f_{i}(x_{0}),\nabla f_{j}(x_{0})\} and Tx0M{i,j}T_{x_{0}}M_{\{i,j\}} have trivial intersection and consequently Tx0M=Tx0M{i,j}Lin{fi(x0),fj(x0)}T_{x_{0}}M=T_{x_{0}}M_{\{i,j\}}\oplus Lin\{\nabla f_{i}(x_{0}),\nabla f_{j}(x_{0})\}.

Choose a neighborhood Ux0U_{x_{0}} of the critical point x0x_{0} such that Ux0cl(M{k})=U_{x_{0}}\cap\mathop{\rm{cl}}(M_{\{k\}})=\emptyset for {i,j,k}={1,2,3}\{i,j,k\}=\{1,2,3\}. It follows that f|Ux0=max{fi,fj}=fi+fj2+|fifj|2f|_{U_{x_{0}}}=\max\{f_{i},f_{j}\}=\frac{f_{i}+f_{j}}{2}+\frac{|f_{i}-f_{j}|}{2} and f|Ux0M{i,j}=fi+fj2f|_{U_{x_{0}}\cap M_{\{i,j\}}}=\frac{f_{i}+f_{j}}{2}. The value of ff increases as we move away from Ux0M{i,j}U_{x_{0}}\cap M_{\{i,j\}} in either direction of fi(x0)\nabla f_{i}(x_{0}) or the direction of fj(x0)\nabla f_{j}(x_{0}) . It follows from the argument above that all the negative eigenvalues of the second differential come from the restriction f|M{i,j}f|_{M_{\{i,j\}}}.

Finally, let J{1,2,,m}J\subseteq\{1,2,\ldots,m\} be any subset. Since x0x_{0} is a critical point of ff, we have jJtjfj(x0)=0\sum_{j\in J}t_{j}\nabla f_{j}(x_{0})=0 for some 0tj10\leq t_{j}\leq 1 with jJtj=1\sum_{j\in J}t_{j}=1. By Proposition 3.4, MJM_{J} is a submanifold of codimension |J|1|J|-1. By a similar argument as in the previous case, Lin{fj(x0)|jJ}Tx0MLin\{\nabla f_{j}(x_{0})\,|\,j\in J\}\leq T_{x_{0}}M is a (|J|1)(|J|-1)-dimensional subspace whose intersection with Tx0MJT_{x_{0}}M_{J} is trivial. Therefore, we obtain a decomposition

Tx0M=Tx0MJLin{fj(x0)|jJ}T_{x_{0}}M=T_{x_{0}}M_{J}\oplus Lin\{\nabla f_{j}(x_{0})\,|\,j\in J\}

and any eigenvector of the second differential of ff corresponding to a negative eigenvalue lives in Tx0MJTx0MT_{x_{0}}M_{J}\leq T_{x_{0}}M. ∎

For the remainder of this section, let MM be a closed connected 4-manifold and let f=max{f1,f2,f3}f=\max\{f_{1},f_{2},f_{3}\} be a CS Morse function on MM. Suppose that x0x_{0} is a critical point of ff with quadratic index rr. Then Corollary 4.6 implies there exists a neighborhood UU of x0x_{0} in MM, a neighborhood VV of 040\in\mathbb{R}^{4} and a homeomorphism ϕ:UV\phi\colon U\to V such that ϕ(x0)=0\phi(x_{0})=0 and fϕ1=gf\circ\phi^{-1}=g, where g(y1,y2,y3,y4)=i=1ryi2+j=r+14yj2g(y_{1},y_{2},y_{3},y_{4})=-\sum_{i=1}^{r}y_{i}^{2}+\sum_{j=r+1}^{4}y_{j}^{2}. 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 ff induces a topological handle decomposition of MM. Stratification corresponding to the CS function ff, however, endows this handle decomposition with an additional structure.

Definition 4.8.

A stratified kk-handle inside the stratified manifold MM is a topological handle j:Bk×BmkhMj\colon B^{k}\times B^{m-k}\to h\subset M such that

  1. (1)

    the core j(Bk×{0})j(B^{k}\times\{0\}) is a kk-dimensional disk, contained in a single stratum MJM_{J},

  2. (2)

    h=KJ(hMK)h=\bigcup_{K\subseteq J}(h\cap M_{K}),

  3. (3)

    the restriction j|j1(hMK)j|_{j^{-1}(h\cap M_{K})} is smooth for every KJK\subseteq J.

A stratified handle decomposition of the stratified manifold MM is a sequence of subsets

=M(1)M(0)M(1)M(2)M(3)M(4)=M,\emptyset=M^{(-1)}\subset M^{(0)}\subset M^{(1)}\subset M^{(2)}\subset M^{(3)}\subset M^{(4)}=M\;,

where each M(k)M^{(k)} is obtained from M(k1)M^{(k-1)} by the attachement of stratified kk-handles.

Let us show that the CS Morse function f=max{f1,f2,f3}f=\max\{f_{1},f_{2},f_{3}\} induces a stratified handle decomposition of the 4-manifold MM. For each regular value of ff, the level set {xM|f(x)=t}\{x\in M\,|\,f(x)=t\} is a closed 3-manifold, trisected into {xM|fi(x)=t}\{x\in M\,|\,f_{i}(x)=t\} for i=1,2,3i=1,2,3. Now let x0Mx_{0}\in M be a critical point of ff with a 4-ball neighborhood Ux0U_{x_{0}}. At x0x_{0}, the topology of the lower level set {xM|f(x)t}\{x\in M\,|\,f(x)\leq t\} changes by the addition of a topological handle. By Lemma 4.4 and Proposition 4.7, x0x_{0} is also a critical point of f|MJf|_{M_{J}} and indf(x0)=indf|MJ(x0)\mathop{\rm{ind}}_{f}(x_{0})=\mathop{\rm{ind}}_{f|_{M_{J}}}(x_{0}), where J=If(x0)J=I_{f}(x_{0}). In other words, the core of the topological handle is a subset of the stratum MJM_{J}. By Lemma 3.7, we have MJcl(MK)KJM_{J}\subset\mathop{\rm{cl}}(M_{K})\Leftrightarrow K\subseteq J, thus Ux0MKKJU_{x_{0}}\cap M_{K}\neq\emptyset\Leftrightarrow K\subseteq J. Since f|MK=fi|MKf|_{M_{K}}=f_{i}|_{M_{K}} for any iKi\in K and fi|MKf_{i}|_{M_{K}} is smooth, the restriction of the topological handle to each stratum MKM_{K} with KJK\subseteq J is smooth. Thus we obtain a stratified handle, that we will call the stratified handle, induced by the CS Morse function ff.

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. (1)

    (critical point inside a regular stratum) If |J|=1|J|=1, the stratified handle is a smooth handle, contained in a regular stratum M{i}M_{\{i\}}.

  2. (2)

    (critical point inside a singular stratum M{i,j}M_{\{i,j\}}) The handle h=j(Bk×Bmk)h=j(B^{k}\times B^{m-k}) is bisected along its core j(Bk×{0})j(B^{k}\times\{0\}) into hM{i}h\cap M_{\{i\}} and hM{j}h\cap M_{\{j\}}. By Proposition 4.7, the descending manifold of the critical point x0x_{0} is contained inside M{i,j}M_{\{i,j\}} and we obtain a bisected stratified handle, whose index is bounded by indf(x0)3\mathop{\rm{ind}}_{f}(x_{0})\leq 3.

  3. (3)

    (critical point inside the singular stratum M{1,2,3}M_{\{1,2,3\}}) M{1,2,3}M_{\{1,2,3\}} is a smooth submanifold of codimension 2. The intersection Ux0M{1,2,3}=DU_{x_{0}}\cap M_{\{1,2,3\}}=D is a 2-disk and by Proposition 4.7, the descending manifold of the critical point x0x_{0} is contained in M{1,2,3}M_{\{1,2,3\}}. It follows that the corresponding handle is a trisected stratified handle and the index of x0x_{0} is bounded above by indf(x0)2\mathop{\rm{ind}}_{f}(x_{0})\leq 2. We have three possibilities:

    1. (a)

      k=0k=0 and DD is the trisection surface of a trisected 0-handle,

    2. (b)

      k=1k=1 and DD is a 2-dimensional 1-handle (whose cocore times a trisected disk gives the cocore of the 4-dimensional 1-handle),

    3. (c)

      k=2k=2 and DD is the core of the 4-dimensional 2-handle, while the cocore is a trisected disk.

\labellist
\hair

2pt \pinlabelx0x_{0} at 320 250 \pinlabelM{i,j}M_{\{i,j\}} at 100 300 \pinlabelM{i,j}M_{\{i,j\}} at 550 300 \pinlabelM{i}M_{\{i\}} at 320 100 \pinlabelM{j}M_{\{j\}} at 320 500 \pinlabelx0x_{0} at 1080 250 \pinlabelDD at 1610 300 \pinlabelx0x_{0} at 1900 290 \pinlabelDD at 2040 300 \pinlabelDD at 1140 310 \pinlabelM{1}M_{\{1\}} at 1000 440 \pinlabelM{3}M_{\{3\}} at 1280 310 \pinlabelM{2}M_{\{2\}} at 1030 150 \endlabellist

Refer to caption
Figure 3. The local structure of MM around a critical point x0x_{0}:     (left) x0M{i,j}x_{0}\in M_{\{i,j\}}, (middle) x0M{1,2,3}x_{0}\in M_{\{1,2,3\}}, case (a), (right) x0M{1,2,3}x_{0}\in M_{\{1,2,3\}}, cases (b) and (c).

Remark. In a similar fashion, we could analyze the local structure around a critical point of the function f=min{f1,f2,f3}f=\min\{f_{1},f_{2},f_{3}\} on MM, 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 f(x1,x2,x3,x4,x5)=min{x1,x2,x3}f(x_{1},x_{2},x_{3},x_{4},x_{5})=\min\{x_{1},x_{2},x_{3}\} on the 4-sphere S4={(x1,x2,x3,x4,x5)5|i=15xi2=1}S^{4}=\left\{(x_{1},x_{2},x_{3},x_{4},x_{5})\in\mathbb{R}^{5}\,|\,\sum_{i=1}^{5}x_{i}^{2}=1\right\}.

Example 2. Observe the function f(x1,x2,x3,x4,x5)=max{x1,x2,x3}f(x_{1},x_{2},x_{3},x_{4},x_{5})=\max\{x_{1},x_{2},x_{3}\} on the 4-sphere

S4={(x1,x2,x3,x4,x5)5|i=15xi2=1}.S^{4}=\left\{(x_{1},x_{2},x_{3},x_{4},x_{5})\in\mathbb{R}^{5}\,|\,\sum_{i=1}^{5}x_{i}^{2}=1\right\}\;.

Each regular stratum M{i}M_{\{i\}} is homeomorphic to the open 44-ball. The singular strata M{1,2}M_{\{1,2\}}, M{1,3}M_{\{1,3\}} and M{2,3}M_{\{2,3\}} are homeomorphic to int(B3)\mathop{\rm{int}}(B^{3}) and their common boundary M{1,2,3}M_{\{1,2,3\}} is homeomorphic to the 2-sphere, thus ff induces a (0,0)(0,0)-trisection of S4S^{4}.

\labellist
\hair

2pt \pinlabel13-\frac{1}{\sqrt{3}} at 1150 50 \pinlabel0 at 1170 250 \pinlabelS3S^{3} at 520 190 \pinlabel13\frac{1}{\sqrt{3}} at 1153 477 \pinlabel12\frac{1}{\sqrt{2}} at 1153 995 \pinlabel35\frac{3}{5} at 1160 740 \pinlabelS2×S1S^{2}\times S^{1} at 600 710 \pinlabel45\frac{4}{5} at 1160 1312 \pinlabelS3S^{3} at 120 1312 \pinlabelS3S^{3} at 380 1312 \pinlabelS3S^{3} at 650 1312 \pinlabel11 at 1170 1500 \pinlabelS3S^{3} at 330 945 \pinlabelS3S^{3} at 510 945 \pinlabelS3S^{3} at 410 1090 \endlabellist

Refer to caption
Figure 4. The fibers of the CS function ff on S4S^{4} from Example 3. Three different shades in every fiber indicate the decomposition into the three regular strata and critical points are represented by the red dots.

Let us look at the fibers of the CS Morse function ff. Its image f(S4)=[13,1]f(S^{4})=\left[-\frac{1}{\sqrt{3}},1\right] contains four critical values: 13,13,12-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{2}} and 11. The critical fiber at 13-\frac{1}{\sqrt{3}} is a single point that belongs to M{1,2,3}M_{\{1,2,3\}}. A regular fiber f1(t)f^{-1}(t) for t(13,13)t\in\left(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right) is homeomorphic to S3S^{3}:

f1(0)={(0,x2,x3,x4,x5)|x22+x32+x42+x52=1,x20,x30}\displaystyle f^{-1}(0)=\left\{(0,x_{2},x_{3},x_{4},x_{5})\,|\,x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=1,x_{2}\leq 0,x_{3}\leq 0\right\}\cup
{(x1,0,x3,x4,x5)|x12+x32+x42+x52=1,x10,x30}\displaystyle\left\{(x_{1},0,x_{3},x_{4},x_{5})\,|\,x_{1}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=1,x_{1}\leq 0,x_{3}\leq 0\right\}\cup
{(x1,x2,0,x4,x5)|x12+x22+x42+x52=1,x10,x20}=A1A2A3\displaystyle\left\{(x_{1},x_{2},0,x_{4},x_{5})\,|\,x_{1}^{2}+x_{2}^{2}+x_{4}^{2}+x_{5}^{2}=1,x_{1}\leq 0,x_{2}\leq 0\right\}=A_{1}\cup A_{2}\cup A_{3}

In the above decomposition, AiA_{i} is homeomorphic to B3B^{3} and int(Ai)=f1(0)M{i}\mathop{\rm{int}}(A_{i})=f^{-1}(0)\cap M_{\{i\}}. Moreover, AiAjcl(M{i,j})B2A_{i}\cap A_{j}\subseteq\mathop{\rm{cl}}(M_{\{i,j\}})\approx B^{2} and A1A2A3=A1A2A3M{1,2,3}A_{1}\cap A_{2}\cap A_{3}=\partial A_{1}\cap\partial A_{2}\cap\partial A_{3}\subseteq M_{\{1,2,3\}} is homeomorphic to S1S^{1}. The stratification induced by ff thus gives a 3-dimensional trisection of the regular fiber. The singular fiber f1(13)f^{-1}\left(\frac{1}{\sqrt{3}}\right) 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 (13,13,13,0,0)M{1,2,3}\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0\right)\in M_{\{1,2,3\}} is a critical point of ff. A regular fiber above a point t(13,12)t\in\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{2}}\right) is a union of three copies of S2×B1S^{2}\times B^{1}, touching pairwise along the connected components of their boundaries, which is homeomorphic to S2×S1S^{2}\times S^{1}.

The singular fiber f1(12)f^{-1}\left(\frac{1}{\sqrt{2}}\right) is homeomorphic to the union of three 3-spheres, touching pairwise in three critical points (12,12,0,0,0)\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0,0\right), (12,0,12,0,0)\left(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}},0,0\right) and (0,12,12,0,0)\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0\right). 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 f1(1)f^{-1}(1) is a union of three critical points (1,0,0,0,0)(1,0,0,0,0), (0,1,0,0,0)(0,1,0,0,0) and (0,0,1,0,0)(0,0,1,0,0). See Figure 4.

Each critical point of ff 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 (13,13,13,0,0)\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},0,0\right), whose boundary is a trisected 3-sphere. The central link of this trisected S3S^{3} is an unlink, along which a 2-handle is attached. The core of the 2-handle is a disk, lying in M{1,2,3}M_{\{1,2,3\}}, whose center is the second critical point (13,13,13,0,0)\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},0,0\right). 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 S2×S1S^{2}\times S^{1}, split along three copies of S2S^{2} by the singular strata M{i,j}M_{\{i,j\}}. At the critical fiber f1(12)f^{-1}\left(\frac{1}{\sqrt{2}}\right), a bisected 33-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 S3S^{3}, each in their own regular stratum. At last, three smooth 4-handles are attached at the critical points (1,0,0,0,0)(1,0,0,0,0), (0,1,0,0,0)(0,1,0,0,0) and (0,0,1,0,0)(0,0,1,0,0).

Our local analysis of the fibers of a CS Morse function max{f1,f2,f3}\max\{f_{1},f_{2},f_{3}\} together with Theorem 4.5 implies the following.

Proposition 4.9.

Let f=max{f1,f2,f3}f=\max\{f_{1},f_{2},f_{3}\} be a CS Morse function on a closed, connected and oriented 4-manifold MM. Suppose that in each regular stratum M{i}M_{\{i\}}, ff has a single critical point of index 4, kk critical points of index 3 and no other critical points for i{1,2,3}i\in\{1,2,3\}. Moreover, suppose that in each singular stratum M{i,j}M_{\{i,j\}}, ff has a single critical point of index 3, gg critical points of index 2 and no other critical points for all ij{1,2,3}i\neq j\in\{1,2,3\}. Then ff induces a stratified handle decomposition.

Note that the stratified handles, corresponding to the critical points of ff inside M{1,2,3}M_{\{1,2,3\}}, 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 M{i,j}M_{\{i,j\}} for ij{1,2,3}i\neq j\in\{1,2,3\}, or a triple of smooth handles contained in M{i}M_{\{i\}} for i{1,2,3}i\in\{1,2,3\}. A typical handle decomposition of MM, induced by ff, consists of the following:
a trisected 0-handle2g trisected 1-handlesa trisected 2-handle}Σg×B23g bisected 2-handles3 bisected 3-handles}tubular neighborhood of the spine\left.\begin{tabular}[]{ll}$\left.\begin{tabular}[]{lll}a trisected 0-handle\\ $2g$ trisected 1-handles\\ a trisected 2-handle\\ \end{tabular}\right\}\Sigma_{g}\times B^{2}$\\ $3g$ bisected 2-handles\\ $3$ bisected 3-handles\\ \end{tabular}\right\}\textrm{tubular neighborhood of the spine}
3k smooth 3-handles3 smooth 4-handles}sector interiors\left.\begin{tabular}[]{ll}$3k$ smooth 3-handles\\ $3$ smooth 4-handles\\ \end{tabular}\right\}\textrm{sector interiors}

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 ff from Example 3 may be restricted to any embedded surface in S4S^{4}. Restriction to the standardly embedded 2-sphere S2S4S^{2}\subset S^{4} 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].

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\labellist
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2pt \pinlabel13-\frac{1}{\sqrt{3}} at 1150 50 \pinlabel0 at 1170 250 \pinlabelS3S^{3} at 580 240 \pinlabel13\frac{1}{\sqrt{3}} at 1153 477 \pinlabel12\frac{1}{\sqrt{2}} at 1153 995 \pinlabel35\frac{3}{5} at 1160 740 \pinlabelS2×S1S^{2}\times S^{1} at 680 730 \pinlabel45\frac{4}{5} at 1160 1312 \pinlabelS3S^{3} at 120 1312 \pinlabelS3S^{3} at 380 1312 \pinlabelS3S^{3} at 650 1312 \pinlabel11 at 1170 1500 \pinlabelS3S^{3} at 330 945 \pinlabelS3S^{3} at 510 945 \pinlabelS3S^{3} at 410 1090 \endlabellist

Refer to caption
Figure 5. A bridge decomposition of the unknotted 2-sphere by the CS Morse function from Example 3. The sphere intersects the trisection surface in two critical points ±(13,13,13)\pm\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right). The triple of critical points (12,12,0),(0,12,12),(12,0,12)\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right),\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\left(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}\right) represent the maxima of the arcs in the triplane diagram, while the triple of critical points (1,0,0),(0,1,0),(0,0,1)(1,0,0),(0,1,0),(0,0,1) represent the centres of the three disks in the bridge decomposition.