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Introduction

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## Publications

Publications (212)

This chapter has three parts. The first one presents rational singularities: the geometric genus of these germs vanishes. We provide several of their topological characterisations, and we list several key examples as well. Then we examine elliptic singularities. We start with Kulikov and minimally elliptic germs, but then we discuss the general (we...

Here we collect some general material regarding local analytic germs, which might help the reader during the reading of the book.

For a fixed resolution of a normal surface singularity we discuss the following objects: the local divisor class group, Q-Cartier divisors and the analytic and topological canonical coverings, natural line bundles, properties of the canonical cycle, the Gorenstein and Q-Gorenstein properties, different vanishing theorems (e.g. the local version of...

We define the multivariable Hilbert and Poincaré series associated with a resolution of a normal surface singularity. They are related with a multi-indexed linear subspace arrangement associated with the analytic type of the germ. We consider their topological analogues as well: a multivariable series defined from the resolution graph and also a to...

We define the link of an isolated singularity and we state several of its properties. The most important one says that it is a complete topological invariant: it characterizes the local germ topologically. This is guaranteed by the real cone structure of a complex analytic set near any given point. We represent the link as a plumber 3-manifold, for...

We list several key examples, which run continuously through the chapters of the book as supporting material of the theory. The first is the family of weighted homogeneous singularities. We present the topological classification of the (underlying) Seifert 3-manifolds and also the analytic classification together with several analytic invariants. T...

First, we define the lattice cohomology associated with the link of a normal surface singularity, whenever this link is a rational homology sphere. This is done via the lattice of a resolution and well-chosen (Riemann-Roch type) weight functions of the lattice points. We prove that it is independent of all the choices and depends only on the link....

We define the purely graph-theoretical notion of cyclic covering of graphs, and in some key cases we classify all the cyclic coverings of a fixed graph. This will be applied to the case of the ramified cyclic covering of a surface singularity via the embedded resolution graph of the germ of an analytic function. In this case certain connections wit...

We fix the link of a normal surface singularity and we assume that it is a rational homology sphere. Via its plumbing graph we introduce and discuss several topological invariants: the Casson invariant (whenever the link is an integral homology sphere), the Casson-Walker invariant, the Turaev torsion and the Seiberg-Witten invariant. Then we recall...

We introduce the notions of modification and resolution. In the presence of a resolution of a surface singularity we also define the exceptional curve, the lattice generated by the irreducible exceptional curves together with its natural intersection form, its dual lattice and the Lipman cone. We prove several of their properties, e.g., the negativ...

We study the Seiberg-Witten invariant via the multivariable (combinatorial) series associated with the resolution graphs and certain quasipolynomials associated with them. In this discussion the methods of the multivariable equivariant Ehrhart theory will have a key role. Firstly, the ‘periodic constant’ of certain series will serve as correction t...

Let ( X , o ) be a complex normal surface singularity with rational homology sphere link and let $$\widetilde{X}$$ X ~ be one of its good resolutions. Consider an effective cycle Z supported on the exceptional curve and the isomorphism classes $$\mathrm{Pic}(Z)$$ Pic ( Z ) of line bundles on Z . The set of possible values $$h^1(Z,\mathcal {L})$$ h...

The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, 2019; Nagy and Némethi in Adv Math 371:20, 2020; Nagy and Némethi in Pure Appl Math Q 16(4):1123–1146, 2020). Besides the genera...

Let ( C , 0 ) (C,0) be a reduced curve germ in a normal surface singularity ( X , 0 ) (X,0) . The main goal is to recover the delta invariant δ ( C ) \delta (C) of the abstract curve ( C , 0 ) (C,0) from the topology of the embedding ( C , 0 ) ⊂ ( X , 0 ) (C,0)\subset (X,0) . We give explicit formulae whenever ( C , 0 ) (C,0) is minimal generic and...

We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincaré series are well–defined and have good properties only if the link is a rational homology sphere. We wish to create a model when this assumption is not valid: we an...

We associate (under a minor assumption) to any analytic isolated singularity of dimension $n\geq 2$ the `analytic lattice cohomology' ${\mathbb H}^*_{an}=\oplus_{q\geq 0}{\mathbb H}^q_{an}$. Each ${\mathbb H}^q_{an}$ is a graded ${\mathbb Z}[U]$--module. It is the extension to higher dimension of the `analytic lattice cohomology' defined for a norm...

We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it...

Let $(X,o)$ be a complex analytic normal surface singularity and let ${\mathcal O}_{X,o}$ be its local ring. We investigate the normal reduction number of ${\mathcal O}_{X,o}$ and related numerical analytical invariants via resolutions $\widetilde{X}\to X$ of $(X,o)$ and cohomology groups of different line bundles ${\mathcal L}\in {\rm Pic}(\wideti...

We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric genus of the germ. This is the analytic analogue of the topological lattice cohomology, associated with the link o...

We prove that if [Formula: see text] is a reduced curve germ on a rational surface singularity [Formula: see text] then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair [Formula: see text]. Furthermore, we also identify it with another (a priori) embedded analytic invariant, whi...

We prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of $(X,0)$. A key ingredient in our proof is a topological boun...

We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of t...

We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric $P=W$ conjecture in all Painlevé cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list o...

We provide combinatorial/topological formula for the multiplicity of a complex analytic normal surface singularity whenever the analytic structure on the fixed topological type is generic.

Let (C,0) be a reduced curve germ in a normal surface singularity (X,0). The main goal is to recover the delta invariant of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if...

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topolog...

We use plumbing calculus to prove the homotopy commutativity assertion of the Geometric P=W conjecture in all Painlev\'e cases. We discuss the resulting Mixed Hodge structures on Dolbeault and Betti moduli spaces.

Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions \(\widetilde{X}\rightarrow X\), an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class \(l'\in H^2(\widetilde{X},{\mathbb {Z}})\). With these data we define the variety \(\mathrm{ECa}^{l'}(Z)\) of those effective Car...

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint i...

Let $(X,o)$ be a complex normal surface singularity with rational homology sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an effective cycle $Z$ supported on the exceptional curve and also a possible Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. Define ${\rm Eca}^{l'}(Z)$ as the space of effective Cartier divisors on $...

We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincar\'e series are well-defined and have good properties only if the link is a rational homology sphere. We wish to create a model when this assumption is not valid: we...

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated with $\mathcal{T}$ and its counting functions, which encode rich topological information. Using the `periodic constant' of the serie...

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg–Witten invariant can be computed as the ‘periodic constant’ of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of th...

If $(\widetilde{X},E)\to (X,o)$ is the resolution of a complex normal surface singularity and $c_1:{\rm Pic}(\widetilde{X})\to H^2(\widetilde{X},{\mathbb Z})$ is the Chern class map, then ${\rm Pic}^{l'}(\widetilde{X}):= c_1^{-1}(l')$ has a (Brill--Noether type) stratification $W_{l', k}:= \{{\mathcal L}\in {\rm Pic}^{l'}(\widetilde{X})\,:\, h^1({\...

Let $ \Phi: ({\mathbb C}^2, 0) \to ( {\mathbb C}^3, 0) $ be a finitely determined complex analytic germ and let $(\{f=0\},0)$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of $f$ from the double-point-geometry of $\Phi$.

For any elliptic normal surface singularity with rational homology sphere link we consider a new elliptic sequence, which differs from the one introduced by Laufer and S. S.-T. Yau. However, we show that their length coincide. Using the properties of both sequences we succeed to connect the common length with the geometric genus and also with sever...

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with resp...

Let $(X,o)$ be a complex normal surface singularity. We fix one of its good resolutions $\widetilde{X}\to X$, an effective cycle $Z$ supported on the reduced exceptional curve, and any possible (first Chern) class $l'\in H^2(\widetilde{X},\mathbb{Z})$. With these data we define the variety ${\rm ECa}^{l'}(Z)$ of those effective Cartier divisors $D$...

We study the analytic and topological invariants associated with complex normal singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analyt...

We study the geometry and topology of Hilbert schemes of points on the
orbifold surface [C^2/G], respectively the singular quotient surface C^2/G,
where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition
of the (equivariant) Hilbert scheme of the orbifold into affine space strata
indexed by a certain combinatorial set, the se...

Let \( \Phi : (\mathbb {C}^2, 0) \rightarrow ( \mathbb {C}^3, 0) \) be a finitely determined complex analytic germ and let \((\{f=0\},0)\) be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of f from the double-point-geometry of \(\Phi \).

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincar\'e series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of...

Let us fix a normal surface singularity with rational homology sphere link and one of its good resolutions. It is known that each coefficient of the analytic Poincaré series associated with the multivariable divisorial filtration is the topological Euler characteristic of the complement of a certain linear subspace arrangement (determined by the di...

The main question we target is the following: If one fixes a topological type of a complex normal surface singularity then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We answer the question for a specific (in some sense pathological) topological type, which supports rather diffe...

We prove that the link of a complex normal surface singularity is an L--space
if and only if the singularity is rational. This via a recent result of
Hanselman, J. Rasmussen, S. D. Rasmussen and Watson (proving the conjecture of
Boyer, Gordon and Watson), shows that a singularity link is not rational if and
only if its fundamental group is left-ord...

Consider a space X with the singular locus, Z=Sing(X), of positive dimension. Suppose both Z and X are locally complete intersections. The transversal type of X along Z is generically constant but at some points of Z it degenerates. We introduce (under certain conditions) the discriminant of the transversal type, a subscheme of Z, that reflects the...

In 1978 Durfee conjectured various inequalities between the signature σ and the geometric genus pg of a normal surface singularity. Since then a few counter examples have been found and positive results established in some special cases. We prove a 'strong' Durfee-type inequality for any smoothing of a Gorenstein singularity, provided that the inte...

We prove an additivity property for the normalized Seiberg–Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singularities in several steps of th...

The Milnor number, \mu(X,0), and the singularity genus, p_g(X,0), are
fundamental invariants of isolated hypersurface singularities (more generally,
of local complete intersections). The long standing Durfee conjecture (and its
generalization) predicted the inequality \mu(X,0) \geq (n+1)!p_g(X,0), here
n=dim(X,0). Recently we have constructed count...

This is an announcement of conjectures and results concerning the generating
series of Euler characteristics of Hilbert schemes of points on surfaces with
simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite
subgroup of SL(2, C), we conjecture a formula for this generating series in
terms of Lie-theoretic data, which is com...

We study the qualitative structure of the set LS of integral L-space surgery slopes for links with two components. It is known that the set of L-space surgery slopes for a nontrivial L-space knot is always a positive half-line. However, already for two-component torus links the set LS has a very complicated structure, e.g. in some cases it can be u...

We prove an additivity property for the normalized Seiberg-Witten invariants
with respect to the universal abelian cover of those 3-manifolds, which are
obtained via negative rational Dehn surgeries along connected sum of algebraic
knots. Although the statement is purely topological, we use the theory of
complex singularities in several steps of th...

We compute the Heegaard Floer link homology of algebraic links in terms of the multi-variate Hilbert function of the corresponding
plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of
the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the...

We prove a Fortuin-Kasteleyn-Ginibre-type inequality for the lattice of
compositions of the integer n with at most r parts. As an immediate application
we get a wide generalization of the classical Alexandrov-Fenchel inequality for
mixed volumes and of Teissier's inequality for mixed covolumes.

In 1978 Durfee conjectured various inequalities between the signature and the
geometric genus of a normal surface singularity. Since then a few counter
examples have been found and positive results established in some special
cases.
We prove a `strong' Durfee--type inequality for any smoothing of a Gorenstein
singularity, provided that the intersec...

We show a counterexample to a conjecture of de Bobadilla, Luengo,
Melle-Hern\'{a}ndez and N\'{e}methi on rational cuspidal projective plane
curves. The counterexample is a tricuspidal curve of degree 8. On the other
hand, we show that if the number of cusps is at most 2, then the original
conjecture can be deduced from the recent results of Borodzi...

A holomorphic germ \Phi: (C^2, 0) \to (C^3, 0), singular only at the origin,
induces at the links level an immersion of S^3 into S^5. The regular homotopy
type of such immersions are determined by their Smale invariant, defined up to
a sign ambiguity. In this paper we fix a sign of the Smale invariant and we
show that for immersions induced by holo...

The study of formal arcs was initiated by Nash in a 1967 preprint. Arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities. The Nash conjecture on the irreducible components of such arc spaces was proved for surfaces by Fernández de Bobadilla and Pe Pereira and for toric...

We prove that a sufficiently large surgery on any algebraic link is an
L-space.

The abstract link L_d of the complex isolated singularity x^2 + y^2 + z^2 +
v^{2d} = 0 is diffeomorphic to S^3 \times S^2. We classify the embedded links
of these singularities up to regular homotopies precomposed with
diffeomorphisms of S^3 \times S^2. Let us denote by i_d the inclusion of L_d in
S^7. We show that for arbitrary diffeomorphisms \va...

Using the path lattice cohomology we provide a conceptual topological
characterization of the geometric genus for certain complex normal
surface singularities with rational homology sphere links, which is
uniformly valid for all superisolated and Newton non--degenerate
hypersurface singularities.

Consider a space X with a non-isolated singular locus Z. The transversal type
of X along Z is generically constant but at some points of Z it degenerates. We
introduce the discriminant of the transversal type, the subscheme of Z that
reflects these degenerations whenever the generic transversal type is
`ordinary'. The scheme structure is imposed by...

We use topological methods to prove a semicontinuity property of the Hodge
spectra for analytic germs defined on an isolated surface singularity. For this
we introduce an analogue of the Seifert matrix (the fractured Seifert matrix),
and of the Levine--Tristram signatures associated with it, defined for
null-homologous links in arbitrary three dime...

Let X be a complex analytic space. A short analytic arc is a holomorphic map
of the closed unit disc to X such that only the origin is mapped to a singular
point. In contrast with the space of formal arcs studied by Nash, the moduli
space of short analytic arcs usually has infinitely many connected components.
We describe these for surface singular...

The lattice cohomology of a plumbed 3--manifold $M$ associated with a
connected negative definite plumbing graph is an important tool in the study of
topological properties of $M$, and in the comparison of the topological
properties with analytic ones when $M$ is realized as complex analytic
singularity link. By definition, its computation is based...

We construct a version of the lattice homology for plane curve singularities
using the normalization of their components. We prove that the Poincare series
of the associated graded homologies can be identified by an algebraic procedure
with the motivic Poincare series. Hence, for a plane curve singularity the
following objects carry the same inform...

We review some basic facts which connect the deformation theory of normal surface singularities with the topology of their links. The presentation contains some explicit descriptions for certain families of singularities (cyclic quotients, sandwiched singularities).

We study the cobordism of manifolds with boundary, and its applications to
codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the
algebraic theory of surgery. The first main result is a splitting theorem for
cobordisms of algebraic Poincar\'e pairs, which is then applied to describe the
behaviour on the chain level of Seifert surfac...

Let M be a rational homology sphere plumbed 3-manifold associated with a
connected negative definite plumbing graph. We show that its Seiberg-Witten
invariants equal certain coefficients of an equivariant multivariable Ehrhart
polynomial. For this, we construct the corresponding polytopes from the
plumbing graphs together with an action of the firs...

We use purely topological methods to prove the semicontinuity of the mod 2
spectrum of local isolated hypersurface singularities in $\mathbb{C}^{n+1}$,
using Seifert forms of high-dimensional non-spherical links, the
Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality
obtained in earlier work for cobordisms of links.

The algorithm and its proof is a highly generalized version of the algorithm which determines the resolution graph of cyclic coverings. Its origin goes back to the case of suspensions, when one starts with an isolated plane curve singularity \(f^{\prime}\) and a positive integer n, and one determines the resolution graph of the hypersurface singula...

Let \( f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)\, \) be the germ of a complex analytic function and set \( (V_{f},0)=(f^{-1}(0),0).\) Its singular locus \((Sing(V_{f}),0)\) consists of points \(\sum:=\{x:\partial {f}(x)=0\}.\)

The origins of the present work go back to some milestones marking the birth of singularity theory of complex dimension≥2. They include the Thesis of Hirzebruch (1950) containing, among others, the modern theory of cyclic quotient singularities; Milnor’s construction of the exotic 7-spheres as plumbed manifolds associated with “plumbing graphs”; Mu...

In many cases it is convenient to add to the germ f another germ, say g, such that the pair (f, g) forms an isolated complete intersection singularity (ICIS in short).Traditionally, one studies the g-polar geometry of f in this way, generalizing the classical polar geometry,when g is a generic linear form.

The first goal of the present work is to provide a plumbing representation of the 3-manifold əF, where F is the Milnor fiber of a hypersurface singularity \(f:(\mathbb{C}^3,0)\rightarrow(\mathbb{C},0)\,\)with 1-dimensional singular locus.

Consider \(f(x,y,z)=f^{\prime}(x,y)\,\)and \(g(x,y,z)=z,\,\)where \(f^{\prime}:(\mathbb{C}^2,0)\rightarrow(\mathbb{C},0)\) is an isolated plane curve singularity. It is well-known (see e.g. [16, 45, 137]) that the embedded resolution of \((\mathbb{C}^2,{V}_{f^{\prime}})\) can be obtained by a sequence of quadratic transformations.

The main tool of the present book is the weighted graph Гe introduced and studied in [92]. It has two types of vertices, non-arrowheads and arrowheads. The non-arrowhead vertices have two types of decorations:

Assume that \(f:(\mathbb{C}^3,0)\rightarrow(\mathbb{C},0)\) is the germof a homogeneous polynomial of degree d, and we choose g to be a generic linear function with respect to f.

In the formulation and the proof of the Main Algorithm 10.2 the absence of “vanishing 2-edges” in \( \lceil_\mathcal{C}\)is essential.

The algorithm presented in this chapter provides 4 the plumbing representations of the 3-manifolds \( \partial{F},\,\,\partial_{1}{{F}}\,{\rm{and}}\,\partial_{2}{{F}},\) and the 5 multiplicity systems of the open book decomposition of \( \partial{F},\,{V}_{g}\) as well as the 6 generalized Milnor fibrations \( \partial{F},\, \backslash {V}_{g}\,\pa...

Let us fix again an ICIS (f; g).

We start with general facts regarding the rank of the first homology group of plumbed 3-manifolds. The statements are known, at least for negative definite graphs; see Propositions 4.4.2 and 4.4.5, which serve as models for the next discussion.

Our goal is to prove Theorem 16.2.3. The proof is based on a specific construction. The presentation is written for the graph G (later adapted to b Ĝ as well), but it can be reformulated for G2,j as well.

We believe that a substantial part of the numerical identities and inequalities obtained in the previous chapters are closely related with general properties of mixed Hodge structures (in the sequel abbreviated by MHS) supported by different (co)homology groups involved in the constructions.

Assume that \(f(x,y,z)= f^{\prime}(x,y),\) as in 9.1. Assume that \(f^{\prime}\, \rm{has}\, \sharp=\sharp(f^{\prime})\) local irreducible components, and let \(\mu=\mu(f^{\prime})\) be its Milnor number.

In this section we assume that \(f(x,y,z)= z f^{\prime}(x,y),\) where \(f^{\prime}\) is an isolated plane curve singularity.

We start our list of examples with the series associated with \( T_{\infty,\infty,\infty},\) where \( T_{\infty,\infty,\infty}\) denotes the germ f = xyz.

In this chapter we again provide an alternative way to identify the boundary of the Milnor fiber for a special class of germs.

In this chapter we list some topics that are closely related with the oriented 3-manifold əF, and are natural extensions of the present work.With this we plan to generate some research in this direction.

In order to continue our discussion regarding the polynomials P# and \(p^\sharp_{j},1\leq \rm{j} \leq \rm{s}\) (cf. 14.2.9 and 14.2.10), we have to consider some natural “combinatorial” characteristic polynomials associated with the graph coverings involved.

We develop Morse theory for manifolds with boundary. Besides standard and
expected facts like the handle cancellation theorem and the Morse lemma for
manifolds with boundary, we prove that, under a topological assumption, a
critical point in the interior of a Morse function can be moved to the
boundary, where it splits into a pair of boundary criti...

In this section we review a graph-theoretical construction from [86].