Gutierrez-Sotomayor Flows on Singular Surfaces
Murilo A. J. Zigart, Ketty A. de Rezende, Nivaldo G. Grulha Jr., Dahisy V. S. Lima
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Gutierrez-Sotomayor Flows on Singular Surfaces
M.A.J. Zigart K.A. de Rezende N.G. Grulha Jr. D.V.S. Lima A BSTRACT
In this work we address the realizability of a Lyapunov graph labeled with GS singularities, namelyregular, cone, Whitney, double crossing and triple crossing singularities, as continuous flow on asingular closed -manifold M . Furthermore, the Euler characteristic is computed with respect to thetypes of GS singularities of the flow on M . Locally, a complete classification theorem for minimalisolating blocks of GS singularities is presented in terms of the branched one manifolds that make upthe boundary. MSC 2020:
Introduction
The study of singular varieties plays an important role in the interface of Algebraic Geometry, Singularity Theory andF-Theory.An aspect that was greatly explored by MacPherson was the existence and uniqueness of Chern classes for singularalgebraic varieties [1]. Later Brasselet and Schwartz [2, 3, 4] applied the study of vector fields on singular varieties toprovide a new framework in which to consider the study of characteristic classes. Around the same period, Gutierrezand Sotomayor used Singularity Theory of differentiable maps to investigate C -structurally stable vector fields onmanifolds with cone, cross-cap, double crossing and triple crossing singularities, and proved their genericity in the setof C -vector fields. In [5], these vector fields were studied by considering their associated continuous flows, makingit possible to use algebraic topological tools, namely, the Conley Index Theory. These flows were named therein asGutierrez-Sotomayor flows, for short, GS flows. This work opened up many questions in Topological Dynamics, morespecifically, on the global qualitative study of continuous flows on singular varieties. In [ ? ], spectral sequences wereused as a tool to understand how a flow undergoes homotopical cancellations on a -dimensional singular variety.In [5], the existence of a Lyapunov function f associated to a GS flow was established, showing the existence ofisolating blocks for the singularities of a GS flow and allowing one to define a Lyapunov graph L f . Furthermore, thePoincaré-Hopf condition was proved to be a necessary local condition on the flow defined on an isolating block, as wellas, a necessary global condition on the flow defined on a closed singular manifold.The results in [5] opened up many questions in Topological Dynamics, more specifically: by imposing the Poincaré-Hopf conditions on an abstract Lyapunov graph L , are they sufficient to ensure the local realizability of L as a GS flowon an isolating block? Does the local realizabilty of L imply the global realizability of L as a GS flow on a closedsingular manifold?In this work, we advance the local and global study of GS flows by addressing these open questions in [5]. On whatconcerns the local realizability of L , that is, the realizability of a semi-graph L v consisting of a single vertex v in L andits incident edges as a GS flow on an isolating block, it turns out that the Poincaré-Hopf condition is not sufficient. Onthe other hand, even if locally the vertices and their incidents edges are realizable as isolating blocks of GS singularities,it may be the case that there is no global realization corresponding to L . The reason for this is the fact that locally therealization of L v as a GS flow on an isolating block N is not unique due to the many choices of distinguished branched -manifolds that can make up the boundary of N . This multiplicity imposes the main difficulty in the global realizationquestion which relies heavily on an assignment that correctly matches up pairs of boundary components of isolatingblocks, allowing the gluing of the blocks according to L and making it possible to obtain a closed singular variety.In Section 1, background material is presented. In Section 2, the local realizability question is addressed. Morespecifically, the realizability of an abstract Lyapunov semi-graph L v consisting of a single vertex and its incident Partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 Partially supported by CNPq under grant 305649/2018-3 and FAPESP under grant 2018/13481-0. Supported by CNPq under grant 303046/2016-3; and FAPESP under grant 2017/09620-2. a r X i v : . [ m a t h . D S ] S e p PREPRINT - O
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1, 2020weighted edges as a GS flow on an isolating block N . Furthermore, a complete characterization of the branched -manifolds that make up the boundary of a minimal isolating block is provided, as well as, the construction ofpassageways for tubular flows that have the effect of increasing the number of branch points on the boundary of N inorder to match the weights on the edges of L v . In Section 3, several theorems on the global realizability of an abstractLyapunov graph L labelled with GS singularities as a continuous flows on a singular manifold M are presented. InSection 4, the Euler characteristic of these manifolds is computed in terms of the types of GS singularities of the flow. In [6], Gutierrez and Sotomayor introduced the -manifolds with simple singularities . They arise when the regularityconditions in the definition of smooth surfaces of R , in terms of implicit functions and immersions, are not satisfied butthere is sitll the presence of some certain stability [7]. The effect is the increase of the types of admissible local charts. Definition 1.
A subset M ⊂ R l is called a -manifold with simple singularities , or a GS -manifold , if for everypoint p ∈ M there is a neighbourhood V p of p in M and a C ∞ -diffeomorphism Ψ : V p → P such that Ψ( p ) = 0 and P is one of the following subsets of R :i) R = { ( x, y, z ) : z = 0 } , plane;ii) C = { ( x, y, z ) : z − y − x = 0 } , cone;iii) W = { ( x, y, z ) : zx − y = 0 , z ≥ } , Whitney’s umbrella ;iv) D = { ( x, y, z ) : xy = 0 } , double crossing;v) T = { ( x, y, z ) : xyz = 0 } , triple crossing; The subsets M ( P ) ⊂ M of the points of M which admit local charts of type P , where P = R , C , W , D , or T , providea decomposition of M , where the regular part M ( R ) of M is a -dimensional manifold, M ( D ) is a -dimensionalmanifold, while M ( C ) , M ( W ) and M ( T ) are discrete sets. Moreover, the collection { M ( P ) , P } is a stratification of M in the sense of Thom, [8], hence M = (cid:91) P M ( P ) , where P = R , C , W , D , T . Definition 2.
A vector field X of class C r on R l is tangent to a manifold M ⊂ R l with simple singularities if it istangent to the smooth submanifolds M ( P ) , for all P = R , C , W , D , T . The space of such vector fields is denoted by X r ( M ) . A flow X t associated to a vector field X ∈ X r ( M ) is called a Gutierrez-Sotomayor flow (GS flow, for short) on M . Definition 3.
Denote by Σ r ( M ) the set of all vector fields X ∈ X r ( M ) satisfying: • X has finitely many hyperbolic fixed points and hyperbolic periodic orbits; • the singular limit cycles of X are simple and X has no saddle connections; • the α -limit and ω -limit sets of every trajectory of X are fixed points, or a periodic orbits or else a singularcycle. In this paper we consider GS flows associated to vector fields in Σ r ( M ) without periodic orbits and singular cycles.The set of such vector fields is denoted by Σ r ( M ) . Definition 4.
Let M be a GS -manifold and X ∈ X r ( M ) a vector field on M . The set of folds on M , denoted by F ( M ) , is defined as: F ( M ) = M ( D ) \ S D , where S D is the set of double crossing points on M which are singularities (stationary points) of X . In this work we follow the terminology given by Gutierrez and Sotomayor by calling these singularities as simple and remarkthat there is a different notion of simple singularities in the classical classification theory of singularities of mappings (see [7]). The locus of the subset W is called cross-cap. In this paper, we chose to keep the nomenclature used by Gutierrez and Sotomayorin [6]. PREPRINT - O
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1, 2020A GS flow defined on a fold F has the property that, given any point x ∈ F , the limit sets, α ( x ) and ω ( x ) , are either aWhitney, double crossing or a triple crossing singularity. Given a flow X t : M → M and a subset S ⊂ M , define the maximal invariant set of S , denoted by Inv ( S ) , as follows:Inv ( S ) = { x ∈ S | X t ( x ) ∈ S, ∀ t ∈ R } . A subset S ⊂ M is called an isolated invariant set with respect to the flow X t if there exists an isolating neighborhood N for S , i.e., a compact set N such that S = Inv ( N ) ⊂ int ( N ) . An index pair for an isolated invariant set S is a pair ofcompact sets L ⊂ N such that N \ L is an isolating neighborhood of S in M , L is positively invariant relative to N and L is the exiting set of the flow in N .Conley proved that, given an isolated invariant set S , there exists an index pair ( N, L ) for S . Also, if ( N, L ) and ( N (cid:48) , L (cid:48) ) are two index pairs for S , then the pointed spaces ( N/L, [ L ]) and ( N (cid:48) /L (cid:48) , [ L (cid:48) ]) have the same homotopy type.For more details, see [9]. Definition 5.
Let ( N, L ) be an index pair for an isolated invariant set S with respect to a flow X t : M → M .i) The homotopy Conley index of S is the homotopy type of the pointed space ( N/L, [ L ]) .ii) The homology Conley index CH k ( S ) of S is defined as the k -th reduced homology group of ( N/L, [ L ]) .iii) The numerical Conley index h k of S is the rank of the homology Conley index CH k ( S ) . In [5], it was computed the homotopy and homology Conley indices for each type of singularities of a GS flow associatedto a vector field in Σ r ( M ) . Definition 6. An isolating block is an isolating neighborhood N such that its entering and exiting sets, given respec-tively by N + = { x ∈ N | φ ([0 , T ) , x ) (cid:54)⊆ N, ∀ T < } N − = { x ∈ N | φ ([0 , T ) , x ) (cid:54)⊆ N, ∀ T > } , are both closed, with the additional property that the flow is transversal to the boundary of N . The existence of GS isolating blocks follows directly from the existence of Lyapunov functions for GS flows withoutperiodic orbits and singular cycles, proved in [5]. Recall that if p is a singularity of a GS flow and f is a Lyapunovfunction with f ( p ) = c , then given (cid:15) > such that their is no critical value of f in [ c − (cid:15), c + (cid:15) ] , the connectedcomponent N of f − ([ c − (cid:15), c + (cid:15) ]) which contains p is an isolating block for p . In this case, the exiting set is N − = N ∩ f − ( c − (cid:15) ) .The GS isolating blocks are a very useful tool on the construction of examples of GS flows. In fact, given a list of GSisolating blocks, we can successively glue the connected components of the entering set of a block with homeomorphicconnected components of the exiting set of another block, until a closed singular manifold is obtained.In what follows, we consider some topological information on the connected components of the boundary N + and N − ,given by the Lyapunov semi-graphs associated to GS isolating blocks. Given a flow X t associated to a vector field X ∈ Σ r ( M ) , there is an associated Lyapunov function f : M → R suchthat f ( p ) (cid:54) = f ( q ) if p and q are different singularities of X t , and for each stratum M ( P ) of M it follows that: • f | M ( P ) is a smooth function, with f continuous in M , • The critical points of f | M ( P ) are non degenerate and coincide with the singularities of X t , • ddt ( f | M ( P ) ( X t x )) < , if x is not a singularity of X t .Note that the Lyapunov function does not need to be smooth globally, only continuous. The existence of this function isproven in [5].In this work we make use of Lyapunov graphs and semi-graphs as a combinatorial tool that keeps track of topologicaland dynamical data of a flow on M . This information on a GS flow X t on M or on an isolating block N is transferredto a Lyapunov graph, respectively, semi-graph associated to X t and f as follows:3 PREPRINT - O
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Definition 7.
Let f be a Lyapunov function for a GS flow X t associated to X ∈ Σ r ( M ) . Two points x and y on M are equivalent if and only if they belong to the same connected component of a level set of f . In this case, denote x ∼ f y .The relation ∼ f defines an equivalence relation on M . The quotient space M / ∼ f is called a Lyapunov graph of M asssociated to X t and f . For some isolating block N ⊂ M , the quotient space N/ ∼ f is called a Lyapunovsemi-graph of N asssociated to X t and f . The quocient space satisfies:i) each connected component of a level set f − ( c ) colapses to a point. Thus, as the value c varies, f − ( c ) / ∼ f describes a finite set of edges , each of which is labelled with a weight , which corresponds to the first Bettinumber of the respective connected component;ii) if c is such that, a connected component of f − ( c ) contains a singularity of X t , then this connectedcomponent is a vertex v in the quotient space labelled with the numerical Conley index of the singularity , ( h , h , h ) P , together with its type P = R , C , W , D or T ;iii) the number of positively ( resp. negatively) incident edges on the vertex v , i.e., the indegree (resp. outdegree)of v , is denoted by e + v ( resp.e − v ) . Moreover, a Lyapunov graph of M as well as a Lyapunov semi-graph of N associated to X t and f is finite, directedand with no oriented cycles. For more details, see [5].Let X t be a GS flow associated to a vector field X ∈ Σ r ( M ) , such that p is a singularity of X t , and N is an isolatingblock for p . Then ( N, N − ) constitutes an index pair for inv ( N ) = { p } . In this case, the next result which is referredto as the Poincaré-Hopf condition for GS flows, see [5], follows:
Theorem 1 (Poincaré-Hopf Condition) . Let ( N , N ) be an index pair for a singularity p on a GS -manifold, M . Let X ∈ Σ r ( M ) and ( h , h , h ) be the numerical Conley indices for p . Then: ( h − h + h ) − ( h − h + h ) ∗ = e + − B + − e − + B − (1) where ∗ denotes the indices of the reverse flow, e + (resp., e − ) is the number of connected components of the entering(resp., exiting) set of N and B + = e + k (cid:88) k =1 b + k (resp., B − = e − k (cid:88) k =1 b − k ), where b + k (resp., b − k ) is the first Betti number of the k -th connected component of the entering (resp., exiting) set of N . On the other hand, note that the boundary of N is nonempty, i.e. ∂N = N + ∪ N − (cid:54) = ∅ . Hence H ( N ) = 0 . Since N is connected, ˜ H ( N ) = 0 . Consequently, from the long exact sequence −→ CH ( p ) ∂ −→ H ( N − ) i −→ H ( N ) p −→ CH ( p ) ∂ −→ ˜ H ( N − ) −→ one has dim ( ˜ H ( N − )) ≤ dim ( CH ( p )) = h (2)Analogously, considering the reverse flow, one has: dim ( ˜ H ( N + )) ≤ dim ( CH ( p )) = h ∗ (3)From (1), (2) and (3), one can describe how the GS isolating blocks look like in terms of their Lyapunov semi-graphs.More specifically, let v be the vertex on the Lyapunov semi-graph of an isolating block for a GS singularity p , then thepositively incidents edges in v ( e + v ) and the negatively incidents edges in v ( e − v ) satisfy: dim ( ˜ H ( N − ) = dim ( H ( N − )) − e − v − ≤ h (4) dim ( ˜ H ( N + ) = dim ( H ( N + )) − e + v − ≤ h ∗ (5)Therefore, considering all the possibilities for e − v in (4) and for e + v in (5), together with the respective Poincaré-Hopfconditions (1), the necessary conditions on a Lyapunov semi-graph for a GS isolating block were obtained in [5] foreach type of GS singularity as well as their nature.Roughly speaking, we can define the nature of a singularity as follows. If a singularity is an attractor (resp. repeller),we say it has attracting (resp. repelling) nature, for short, nature a (resp. nature r ). In the particular case of a singularity Equivalently, the numerical index will be, at times, substituted by the nature of the singularity PREPRINT - O
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1, 2020of type D since it has two attracting (resp. repelling) natures, we say it has nature a (resp. r ). Similarly, in the caseof a singularity of type T , since it has three attracting (resp. repelling) natures, we say it has nature a (resp. r ). Asingularity of type R or C which is neither of attracting or repelling nature is said to have saddle nature, for short,nature s . A regular saddle on a disk identified along the stable (resp. unstable) manifolds so as to produce a Whitneychart is of nature s s (resp. s u ). Two regular saddles on two disjoint disks identified along their stable (resp. unstable)manifold so as to produce a double crossing chart is of type ss s (resp. ss u ). A regular saddle and an attracting (resp.repelling) singularity on two disjoint disks identified along a stable (resp. unstable) direction so as to produce a doublecrossing chart is of nature sa (resp. sr ). Similarly, two regular saddles and an attracting (resp. repelling) singularity onthree disjoint disks identified as follows: the two saddle disks identify along their unstable (resp. stable) manifolds andsubsequently the attracting (resp., repelling) disk identified along stable (resp. unstable) directions so as to produce atriple crossing chart is of nature ssa (resp. ssr ). For more details see [5].In Table 1 these conditions are presented for all GS singularities with natures a, a , a s, s s , sa, ss s and ssa .By reserving the flow, one obtains the conditions on Lyapunov semi-graphs for GS singularities with natures r, r , r , s, s u , sr, ss u and ssr , respectively. Also the corresponding Poincaré-Hopf condition is given by exchanging B + by B − and vice versa. Type Regular ConeNature a s s a s s
Lyapunovsemi-graphPoincaré-Hopfcondition B + = 1 B + = B − B + = B − − b +1 = b +2 = 1 B + = B − B + = B − Type Whitney Double crossingNature a s s a sa sa
Lyapunovsemi-graphPoincaré-Hopfcondition B + = 2 B + = B − + 1 B + = B − B + = 3 B + = B − + 2 B + = B − + 1 Type Double crossingNature ss s ss s ss s ss s ss s ss s Lyapunovsemi-graphPoincaré-Hopfcondition B + = B − + 2 B + − B − B + = B − + 1 B + = B − + 2 B + = B − B + = B − + 1 Type Double crossing Triple crossingNature ss s ss s a ssa ssa Lyapunovsemi-graphPoincaré-Hopfcondition B + = B − − B + = B − b +1 = 7 B + = B − + 2 B + = B − + 1 Table 1: Lyapunov semi-graphs with the respective Poincaré-Hopf condition5
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1, 2020For each semi-graph in Table 1, considering that b ± i is greater or equal to one, we define the respective Lyapunovsemi-graph with minimal weights to be the one with smallest positive integers satisfying the Poincaré-Hopf condition.In the next section we will consider the realizability of an abstract Lypaunov semi-graph satisfying the Poincaré-Hopfcondition.
In [5], the construction of GS isolating blocks was presented in a similar fashion to the construction of compactmanifolds via the classical Handle Theory. Roughly speaking, a GS handle of type R , C , W , D or T is a subspace of R homeomorphic to a chart presented in Definition 1 and with a well-defined local GS flow. Given a GS handle H P η containing a singularity p of type P and nature η , we consider the gluing of the attaching region of the handle H P η on a distinguished branched -manifold . See [5] for more details. Definition 8. A distinguished branched -manifold is a topological space containing at most connected componentseach of which has a finite number of branched charts , where each branched chart is an intersection of two transversearcs. This intersection of two transverse arcs is called a branch point . In what follows, a schematic description of the steps in the construction of GS isolating blocks N for a singularity p ∈ H P η is presented. Steps in the Construction of the Isolating Blocks Identifying the attaching region A k The attaching region denoted by A k is given by the unstable part of H P η , which resembles the attaching spherein the classical Handle Theory;2. Choosing a distinguished branched -manifold N − In order to guarantee that the isolating block is connected, the chosen branched -manifold, denoted by N − ,must not contain more connected components than A k ;3. Gluing A k in N − × [0 , The gluing of the handle H P η on a collar of a distinguished branched - manifold, N − × [0 , , given by anyembedding f : A k → N − × { } with the property that it maps at least one connected component of A k toeach connected component of N − × { } produces N ;4. Stretching N Stretch N in the direction of the time-reversed flow.Note that in the above construction, the distinguished branched -manifold is denoted by N − precisely because itcorresponds to the exiting set of the isolating block N . Also, the reason that the branched -manifolds have at most connected components is because the maximum number of connected components admitted by the attaching region of aGS handle is , [5]. Let N be a GS isolating block for p ∈ H P η , determined by an embedding f : A k → N − × { } , where A k is theunstable part of ∂ H P η and N − × { } is a distinguished branched -manifold. Since f is an embedding, it follows thatthe number of branched charts in N − × { } = N − × { } = N − is greater or equal to the number of branched chartsin A k . Definition 9.
Let p ∈ H P η be a GS singularity and N an isolating block for p . N is a minimal GS isolating block for p if the exiting set N − has the same number of branched charts as the unstable part of ∂ H P η . Theorem 2.
A Lyapunov semi-graph L v with a single vertex v labelled with a GS singularity is associated to a GS flowon a minimal GS isolating block if and only if:a) L v satisfies the Poincaré-Hopf condition with minimal weights;b) if v is labelled with a singularity of type D and nature ss s (resp. ss u ) such that e + v = 2 (resp. e − v = 2 ), then e − v ≤ and b +1 = b +2 (resp. e + v ≤ and b − = b − );c) if v is labelled with a singularity of type T and nature ssa (resp. ssr ) such that e − v = 2 (resp. e + v = 2 ), then b − = b − (resp. b +1 = b +2 ). PREPRINT - O
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Proof.
It follows from Theorem 3 and Theorem 4.In other words, the boundaries of a minimal GS isolating block are distinguished branched -manifolds that have theminimal number of branch points admitted by the singularity. In the case of attracting or repelling singularities, thechoice of distinguished branched -manifold is unique. Proposition 1.
Let p ∈ H P a (resp. p ∈ H P r ) be a GS singularity of attracting (resp. repelling) nature of type R , C , W , D or T . Then, p admits a unique minimal GS isolating block up to homeomorphism.Proof. In the case of an attracting singularity p ∈ H P a the result is trivial, since A k = ∅ .Given p ∈ H P r , an repelling singularity, since A k = ∂ H P r , then N − = ∂ H P r is the unique choice for a distinguishedbranched -manifold that permits the gluing to occur via the embedding f = Id . Hence, a repelling singularity admitsa unique isolating block which is homotopic to the local chart itself, H P r . p ∈ H R a p ∈ H C a p ∈ H W a p ∈ H D a p ∈ H T a Table 2: GS isolating blocks for attracting singularitiesBy reversing the flow in Table 2, the minimal GS isolating blocks for repelling singularities are obtained.On the other hand, the construction of minimal GS isolating blocks for singularities of different saddle natures,admits different choices of distinguished branched -manifolds and consequently different ways of embedding A k in N − × [0 , . In the next result, the possible embeddings are characterized in terms of the distinguished branched -manifolds of resulting isolating blocks. Theorem 3.
Let H P η be a GS handle for a singularity p of type P and nature n and N a minimal GS isolating blockfor p . Then all possible distinguished branched -manifolds that form the connected components of the entering andexiting sets of N are described in Table 3, in terms of the Lyapunov semi-graph of N . This characterization is up toflow reversal. In Table 3, the semi-graphs that have edges labelled with more than one distinguished branched -manifold indicate thatany combination of choices represents the boundary components of some minimal GS isolating block corresponding tothe given semi-graph.Note that whenever the singularity p ∈ H P η is the ω -limit of all the folds in H P η , the choice that minimizes the branchedcharts in N − are circles. In the case p ∈ H T ssa , there are six folds in H T ssa , but only four of them have p as their ω -limit.Hence, the minimal number of branched charts in N − is two and thus, there are three possible choices for N − as shownin Table 3.In the proof of Theorem 3, the construction of a minimal GS isolating block N is obtained by considering all possibilitiesof distinguished branched -manifolds that are admissible as exiting sets, as well as, the gluing maps of a GS handle.Note that each choice determines a block N with a distinguished branched -manifold N + as its entering set. Theorem3 provides a complete classification of ( N, N − , N + ) . Proof.
Let p ∈ H P a (resp., p ∈ H P r ), the unique minimal isolating block was obtained in Proposition 1.We now analyze GS singularities with different saddle natures and of type R , C , W , D and T .i) p ∈ H R s This case is well known in classical Handle Theory as the pair of pants . The possible choices for N − are oneor two circles. This corresponds to a semi-graph with e + v + e − v = 3 , where v is labelled with (0 , , R . Inthe case of a semi-graph with e + v + e − v = 2 , where v is labelled with (0 , , R , the corresponding isolatingblock N is obtained by gluing a twisted handle on the circle N − . The block N is homotopy equivalent to thenonorientable block formed by a Mobius band minus a disc.7 PREPRINT - O
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1, 2020 p ∈ H R η p ∈ H C η p ∈ H W η p ∈ H D η ( , ) p ∈ H D η ( , ) ( , ) ( , ) p ∈ H T η ( , ) ( , , )( , ) Table 3: Boundary N − and N + of minimal GS isolating blocks according to the associated Lyapunov semi-graph. Pre quotient Local chart Attaching region
Table 4: Representation of a cone handle H C s ii) p ∈ H C s A cone handle H C s corresponds to the gluing of two discs, each of which has a tubular flow with a degeneratesingularity at its center. Once the two centers are identified, each disc corresponds to the upper and lowersheets of the cone. The attaching region of the handle has two connected components with no branched charts,one on each sheet of the cone, see Table 4. Since there are no branched charts, N − is either one or two circles.Hence, the two connected components which make up the attaching region can be glued to one circle, see leftside of Table 5 or glued to two circles, see right side of Table 5.iii) p ∈ H W s s A Whitney handle H W s s corresponds to a regular handle H R s followed by the identification of the two stableorbits, see Table 6. Also in this case, the attaching region has no branched point and hence, N − is either oneor two circles. Thus the two connected components which make up the attaching region can be glued to onecircle, see left side of Table 7 or glued to two circles, see right side of Table 7.8 PREPRINT - O
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Minimal isolating block Lyapunov graph Minimal isolating block Lyapunov graph → →
Table 5: Minimal isolating blocks for a singularity p ∈ H C s Pre quotient Local chart Attaching region
Table 6: Representation of a Whitney handle H W s s iv) p ∈ H D η a) A double handle H D sa corresponds to two regular handles H R s and H R a in which two pairs of stable orbits areidentified, as shown in Table 8. Consequently, by considering the minimal isolating blocks for the regularsaddle and the attractor, and then identifying the respective stable pair of orbits, a minimal isolating blockfor a singularity of type D and nature sa is produced. See Table 9 for the case that the isolating blocks areorientable. For the nonorientable case, the schematic construction is shown in Table 10.b) Analogously, a double handle H D ss s corresponds to two regular handles H R s in which two pairs of stableorbits are identified. The attaching region has four connected components with no branched charts, hence Minimal isolating block Lyapunov graph Minimal isolating block Lyapunov graph → →
Table 7: Minimal isolating blocks for a singularity p ∈ H W s s Pre quotient Local chart Attaching region
Table 8: Representation of a double handle H D sa PREPRINT - O
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Minimal isolating block Lyapunov graph → →→ →
Table 9: Minimal isolating blocks for a singularity p ∈ H D sa Lyapunov graph moves → Table 10: Minimal isolating blocks for a singularity p ∈ H D sa N − is either one, two, three or four circles. In this case, by analyzing the distinct possibilities of gluingthe attaching region to the different cases of N − , the two saddles prior to the identification are either in oneminimal isolating block or in two disjoint minimal isolating blocks. The isolating block for the singularity p ofnature ss s is produced by identifying two pairs of stable orbits of the two regular saddles. See Table 12 for thecase that the isolating blocks are orientable. Pre quotient Local chart Attaching region
Table 11: Representation of a double handle H D ss s v) p ∈ H T η A handle H T ssa corresponds to two regular handles H R s plus one regular handle H R a , in which six pairs ofstable orbits are identified. Equivalently, one can consider a double handle H D sa plus a regular handle H R s , or adouble handle H D ss u plus a regular handle H R a , followed by the identification of four pairs of stable orbits. SeeTables 13 and 14.The attaching region has two connected components each of which has one branched chart, hence N − haseither one connected component with two branched charts, or two connected component with one branchedchart each. In the prior case, we have two possibilities: two circles that intersect in two points, or three circles10 PREPRINT - O
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Minimal isolating block Lyapunov graph moves → → →→ → →→ →→ →→ →→ →
Table 12: Minimal isolating blocks for a singularity p ∈ H D ss s Pre quotient Local chart Attaching region
Table 13: Representation of a triple handle H T ssa by gluing the handles H D sa and H R s that form a figure eight wedge a circle. In the latter case, N − is the disjoint union of two figures eight. Theisolating block for the singularity p of nature ssa is produced by identifying the corresponding pairs of stableorbits of the saddles H D sa and H R s or of the saddle H D ss u and the handle H R a . See Tables 15 and 16 for theschematic representation of the resulting blocks by their corresponding Lyapunov semi-graphs.11 PREPRINT - O
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Pre quotient Local chart Attaching region
Table 14: Representation of a triple handle H T ssa by gluing the handles H D ss u and H R a Lyapunov graph moves ( D sa + R s → T ssa ) → →→ →→ →→ →→ Table 15: Distinguished branched -manifold on the boundary of minimal isolating blocks for p ∈ H T ssa The next theorem proves the non-realizability of certain Lyapunov semi-graphs as a minimal GS isolating block. Thevertex on these semi-graphs are labelled with the Conley indices of a GS singularity and satisfy the Poincaré-Hopfcondition. 12
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Lyapunov graph moves ( D ssu + R a → T ssa ) ( , ) → ( , ) →→ ( , ) → ( , ) → →→ Table 16: Distinguished branched -manifold on the boundary of minimal isolating blocks for p ∈ H T ssa Theorem 4 (Local non-realizability) . Let X t be a GS flow associated to a vector field X ∈ Σ r ( M ) , such that p is asingularity of X t and N is a minimal isolating block for p . Then there is no Lyapunov semi-graph associated to N suchthat: a) e + v = 2 , e − v = 4 , p ∈ H D ss s ;b) e + v = 2 , e − v = 3 , p ∈ H D ss s ;c) e + v = 2 , e − v = 1 , b − = 1 , b +1 (cid:54) = b +2 , p ∈ H D ss s ;d) e + v = 2 , e − v = 2 , b − = b − = 1 , b +1 (cid:54) = b +2 , p ∈ H D ss s ;e) e + v = 1 , e − v = 2 , b − (cid:54) = b − , p ∈ H T ssa .Proof. The idea of the proof is to analyze the connected components of N + in view of the steps of the constructionof GS isolating blocks presented in the beginning of Section 2. This analysis is based on the different possibilities ofembeddings of the attaching regions of handles to N − .In order to prove ( a ) , recall that a double handle H D ss s corresponds to two regular handles H = H = H R s , where twostable orbits of H are identified, biuniquivocally, to two stable orbits of H as shown in Table 11. Furthermore, theattaching region A k of the handle H D ss s is the union of two copies of S × D , where D is the one-dimensional disc,i.e. D (cid:39) [0 , .Considering e − v = 4 , it follows that N − × [0 , has connected components, that is N − × [0 ,
1] = ∪ i =1 N i × [0 , .It follows from the construction of GS isolating blocks that an embedding f : A k → N − × { } maps each disc D i in A k to N i × { } .The attaching map of H connects it to N × { } and N × { } . Also the attaching map of H connects it to N × { } and N × { } . Whether these attaching maps are orientation preserving or not has no effect on the gluing of the handles.13 PREPRINT - O
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1, 2020Thus, N + is the union of two circles given by the stable part of N ×{ }∪ H ∪ N ×{ } and N ×{ }∪ H ∪ N ×{ } ,intersecting in two points which correspond to the intersection of H and H in H D ss s . Hence, it follows that N + isconnected, i.e., e + v = 1 . See Figure 1. Gluing onto N − Stable part N + identification Lyapunov graph αα (cid:48) ββ (cid:48) αα (cid:48) ββ (cid:48) α ∼ βα (cid:48) ∼ β (cid:48) Figure 1: Boundary N − and N + of an isolating block for p ∈ H D ss s with e − v = 4 Therefore, no minimal GS isolating block for p ∈ H D ss s admits a Lyapunov semi-graph with e + v = 2 , e − v = 4 .The proof of items b ) , c ) , d ) and e ) is similar.In Theorem 1, the set of Lyapunov semi-graphs with a single vertex which satisfies the Poncaré-Hopf condition forConley indices that correspond to the Conley indices of GS singularities, were presented. Now, by Theorem 3 togetherwith Theorem 4, we have realized all possible Lyapunov semi-graphs with minimal weight for GS singularities asminimal GS isolating blocks.This classification implies the following corollary. Corollary 1.
There are, up to homeomorphism and flow reversal, minimal GS isolating blocks.Proof. Theorem 4 implies that the collection of Lyapunov semi-graphs of Theorem 3 (Figura 3) contains all semi-graphsthat are realizable as minimal GS isolating blocks. On the other hand, Theorem 3 implies that, up to homeomorphimsand flow reversal, the total number of minimal GS isolating blocks for singularities of types R , C , W , D and T are,respectively, , , , and , totalizing blocks.In the next section, the collection of Lyapunov semi-graphs with a single vertex labelled with the Conley indicesof a GS singularity and which satisfies the Poincaré-Hopf condition for GS flows will be analyzed by removing theminimality weight condition on the edges. In other words, arbitrary weights on the edges will be allowed as long as thePoincaré-Hopf condition is satisfied. Let X t be a GS flow associated to a vector field X ∈ Σ r ( M ) such that p is a singularity of X t and N is an isolatingblock for p . Note that each branched chart on the entering boundary of N + (resp., exiting boundary N − ) represents afold within the isolating block. In terms of the Lyapunov semi-graph, b + i − (resp., b − i − ) equals the number of foldsthat enter (resp. exit) through the corresponding connected component of the block.Note that in the case of a minimal GS isolating block N , the total number F of folds that enter and exit N is: F = , if p ∈ M ( R ) ∪ M ( C )1 , if p ∈ M ( W )2 , if p ∈ M ( D )6 , if p ∈ M ( T ) Furthermore, the ω -limit (resp. α -limit) of all folds that enter (resp., exit) through N + (resp. N − ) is the singularity p ∈ N . Definition 10.
Let N be a GS isolating block for a singularity p of a vector field X ∈ Σ r ( M ) . N is a GS isolatingblock with passageways if there exists at least one fold in N for which p is neither the α -limit nor ω -limit. As a direct consequence of the definition, the next result follows.
Corollary 2.
Let p be a singularity of a vector field X ∈ Σ r ( M ) , of attracting (resp. repelling) nature of type R , C , W , D or T . Then p does not admit a GS isolating block with passageways. PREPRINT - O
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Proof.
Note that if p is of attracting (resp. repelling) nature and N is an isolating block for p , then p is ω -limit (resp. α -limit) of all orbits in N . Corollary 3.
Given N a minimal GS isolating block for p ∈ H P η , let Γ = { ( γ , γ ) , ( γ , γ ) , . . . , ( γ k − , γ k ) } bea collection of pairs of orbits in the regular part of N , with γ i (cid:54) = γ j if i (cid:54) = j , and such that p is not the α -limit northe ω -limit of any γ i in Γ . Then the quotient space N/ ∼ obtained by identifying each pair of orbits γ i − ∼ γ i , for i = 1 , . . . , k is an isolating block for p ∈ H P η with k passageways.Proof. Since N is an isolating block for p ∈ H P η , it follows that N/ ∼ is also an isolating block for p , which has a totalof k folds given by the identification of the pairs of orbits γ i − ∼ γ i , for i = 1 , . . . , k . Given that p is not the α -limitnor the ω -limit of any γ i in Γ , it follows that p is not the α -limit nor the ω -limit of any of the k folds in N/ ∼ . Example 1.
In Table 17, there are three example of GS isolating blocks with passageways, constructed from a minimalGS isolating block for a cone type singularity with saddle nature. The pairs of orbits identified in order to obtaineach one of the three blocks are, respectively, given by Γ = { ( γ , γ ) } , Γ = { ( γ , γ ) , ( γ , γ ) , ( γ , γ ) } , and Γ = { ( γ , γ ) } . Minimal isolating block Passageways in isolating blocks γ γ γ γ γ γ Table 17: Examples of GS isolating blocks with passageways for p ∈ H C s Corollary 3 provides a procedure to construct GS isolating blocks with passageways from a minimal GS isolating block.On the other hand, the next lemma shows that any GS isolating block with passageways can be obtained by this method.
Lemma 1.
Let N be a GS isolating block with k passageways for p ∈ H P η . Then there exists a minimal GS isolatingblock N for p ∈ H P η , and a collection Γ = { ( γ , γ ) , ( γ , γ ) , . . . , ( γ k − , γ k ) } of pairs of regular orbits in N , withthe properties that γ i (cid:54) = γ j if i (cid:54) = j , and p is neither the α -limit nor the ω -limit of any γ i in Γ , such that the quotientspace N / ∼ , obtained by the identification of each pair of orbits γ i − ∼ γ i , for i = 1 , . . . , k , is homeomorphic to N .Proof. Given a GS isolating block N with k passageways for p ∈ H P η , denote by F , . . . , F k the k folds in N forwhich p is neither the α -limit nor the ω -limit. A minimal GS isolating block N for p ∈ H P η is constructed as follows.For each i = 1 , . . . , k :i) Consider N − U i where U i is an (cid:15) -neighborhood of F i in N . Then ∂ ( N − U i ) − ( ∂N ∩ ( N − U i )) is theunion of four disjoint discs D i , D i , D i and D i .ii) Identify the pair of discs in N − U i , say γ i − = D i ∼ D i and γ i = D i ∼ D i , so that the resultingblock has the same number of connected components as N . p f i U i D i D i D i D i γ i − γ i Figure 2: Unzipping passageways15
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1, 2020Since we have removed all the folds F , . . . , F k in this process, the resulting block N is a minimal GS isolating blockfor p ∈ H P η .Finally, note that Γ = { ( γ , γ ) , ( γ , γ ) , . . . , ( γ k − , γ k ) } is a collection of pairs of regular orbits in N , such that p is neither the α -limit nor the ω -limit of any γ i in Γ . Hence, the identifications γ i − ∼ γ i , for i = 1 , . . . , k , imply that N / ∼ is homeomorphic to N .One can ask whether an abstract Lyapunov semi-graph with a single vertex v labelled with the numerical Conley indicesof a GS singularity and arbitrary weights satisfying the Poincaré-Hopf condition is realizable or not as a GS flow on anisolating block.The following Theorem presents necessary and sufficient conditions for a positive answer to this question. So thefollowing theorem generalizes Theorem 2. Theorem 5 (Local realization characterization) . A Lyapunov semi-graph L v with a single vertex v labelled with a GSsingularity is associated to a GS flow on a GS isolating block N if and only if:i) the Poincaré-Hopf condition is satisfied;ii) if v is labelled with a singularity of type C and nature s , with e − v = e + v = 2 , then b +1 is equal to either b − or b − ;iii) if v is labelled with a singularity of type D and nature ss s (resp. ss u ), with e + v = 2 , (resp., e − v = 2 ) then e − v ≤ (resp., e + v ≤ ). Moreover, if L is minimal then b +1 = b +2 (resp., b − = b − ).iv) If v is labelled with a singularity of type T with e − v = 2 (resp., e + v = 2 ) and L is minimal then b − = b − (resp., b +1 = b +2 ).Proof. ( ⇒ ) First, suppose L is associated to a GS flow on a GS isolating block N . Then, we have that:i) ( N, N − ) is an index pair for the singularity p in N . And it follows from Theorem 1 that L satisfies thePoincaré-Hopf condition.ii) If L is minimal, the condition is trivially satisfied, since b +1 = b − = b − = 1 . In this case, N is a minimal GSisolating block given by two cylinders intersecting in a single point which is the cone singularity, as shown inTable 5.If L is not minimal, then N is a GS isolating block with passageways. Thus, Lemma 1 states that N is obtainedfrom the identification of pairs of orbits in a minimal GS isolating block N , such that these orbits do notintersect the singular part of N . So, each pair of orbits must enter and exit N through the same cylinder,that is, the number of folds that enter and the number of folds that exit each cylinder of N must be the same.Hence the number b +1 − must be equal to b − − or b − − . Therefore, b +1 is equal to b − or b − .iii) Suppose v is labelled with a singularity of type D and nature ss s , with e + v = 2 . In addition, if N is minimal,Theorem 4 asserts that the exiting set of N has at most two connected components, that is, e − v ≤ . Onthe other hand, if N is not minimal, Lemma 1 implies that N can be obtained via the identification of pairsof orbits on a minimal isolating block N . Since these identifications may lower the number of connectedcomponents of N , it follows that the exiting set of N also has at most two connected components, i.e, we alsohave e − v ≤ in this case. Similarly, we prove the condition for a singularity of type D and nature ss u .iv) This is a direct consequence of item e ) of Theorem 4. ( ⇐ ) Now, suppose L satisfies conditions i ) through iv ) . If L is minimal, then L can be realized with the distinguishedbranched -manifolds of Table 3, as shown in Theorem 3. On the other hand, if L is not minimal, consider the minimalsemi-graph (cid:96) with the same labellings, indegree and outdegree as L . As before, we can realize (cid:96) by a minimal GSisolating block N . Then, we can identify a pair of orbits that enter and exit N through the same connected componentsof the boundary of N . This identification increases the weights of the respective boundary components of N byone, while preserving the number of connected components of N +0 and N − . Hence, such identification of orbits of N results in a GS isolating block N which realizes a semi-graph with same labellings, indegree and outdegree as (cid:96) .Therefore, after performing a finite number k of identifications so that the weights on each boundary component of theresulting block N k are equal to the weights b ± i of L , we have that N k is a realization of L , proving that L is associatedto a GS flow on a GS isolating block. 16 PREPRINT - O
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1, 2020This concludes the characterization of all Lyapunov semi-graphs labelled with numerical Conley indices of GSsingularities and satisfying the Poincaré-Hopf condition. We will denote these semi-graphs as
GS semi-graphs . Definition 11.
In the case that the weights on the edges of a GS semi-graph are the smallest satisfying Poincaré-Hopfit will be called a minimal GS semi-graph . Hence, we say that a Lyapunov graph L is a GS graph (resp., minimal GSgraph ) if for all vertices v of L the semi-graph formed by v and its incident edges is a GS semi-graph (resp., minimalGS semi-graph ). In the next section, we investigate under what conditions a GS graph is realizable as a GS flow on a closed -dimensionalsingular manifold. In this section we consider the realization of GS graphs as singular flows.
Definition 12.
We say that a GS graph L is realizable if there exists a triple ( M , X t , f ), where M is a closed GS -manifold, X t is a flow associated to a vector field X ∈ Σ r ( M ) and f : M → R is a Lyapunov function associated to X t , such that L is the Lyapunov graph of ( M , X t , f ). In this case, we say that the flow X t defined on M is a realization of L . Let L be a GS graph. We would like to investigate whether L is realizable.So far, we have shown that for each vertex v of L , the GS semi-graph formed by v and its incident edges admits at least one realization for the singularity associatedto v as a minimal isolating block or as an isolating block with passageways. The question now is to determine if theseisolating blocks can be glued to each other as indicated by L .In order to glue the exiting boundary of an isolating block onto the entering boundary of another isolating block, thedistinguished branched -manifolds which make up these two boundary components must be homeomorphic.Usually, in a realization, one isolating block B needs to be glued to at least two other blocks, A and C , so that theexiting boundary of A must be homeomorphic to the entering boundary of B , and at the same time, the exiting boundaryof B must be homeomorphic to the entering boundary of C .Notice that the exiting boundary of A and the entering boundary of C are independent from one another. However,the distinguished branched -manifold on the entering boundary of B restricts the possible distinguished branched -manifolds on the exiting boundary of B and vice versa. So it might happen that we are able to glue A and B , or B and C , but not all three of them simultaneously.In other words, a realization of a GS graph L is reduced to the problem of assigning distinguished branched -manifoldsto the edges of L in such a way that the semi-graph of each vertex v of L is realizable as a GS isolating block withprecisely this choice of distinguished branched -manifolds as boundary components.Most of the results in this section, with the exception of Theorem 6, will consider GS flows without triple crossingsingularities. The reason being the complexity of the distinguished branched -manifolds that make up the boundarycomponents of the corresponding isolating blocks and the fact that this choice has a trickle down effect on boundarycomponents of isolating blocks of other singularities. The following theorem will impose the minimality restriction on the weights of a GS graph. This makes it possibleto have a better control on the choices of the distinguished branched -manifolds that make up the boundary of theisolating blocks. Theorem 6 (Minimal case) . Let L be a minimal GS graph containing singularities of type R , C , W , D and T . L admits a realization.Proof. Since L is minimal, the possible weights on the edges of L are , , or . For each edge, consider theassignment of distinguished branched -manifolds given in Table 18.This choice relies on the collection of minimal GS isolating blocks in Theorem 3 (see Table 3). Note that there is aunique choice of distinguished branched -manifolds for weights equal to , and . In the case of weight there aretwo possibilities. However, the chosen one is common to all GS semi-graphs with weight and hence, it is always17 PREPRINT - O
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1, 2020 w = weight w = 1 w = 2 w = 3 w = 5 w = 7 N − Table 18: Distinguished branched -manifolds with weight w realizable. In the case of weight , there are two possible choices of distinguished branched -manifolds common to allGS semi-graphs, anyone of which can be chosen. Example 2.
In Figure 3, a minimal GS graph is realized as a singular flow on a GS -manifold, as asserted in Theorem6. Realization −−−−−−−−→
Figure 3: Realization of a minimal GS graph as a GS flow
Remark:
The only example of a minimal GS graph with a vertex labelled with a singularity of type T and of attractingor repelling nature is the repeller-attractor pair. In other words, a flow made up of two singularities of type T , oneattractor and one repeller.The next theorem will impose a degree restriction on a GS graph. It turns out that GS graphs with no bifurcation verticesare always realizable. Theorem 7 (Linear Graph) . Let L be a GS graph containing singularities of type R , C , W and D such that every vertex v of L has degree less than or equal to . Then L is realizable.Proof. Given an edge of L with weight w , the realization of L can be achieved by choosing distinguished branched -manifolds according to Table 19. w = weight w = 1 w = 2 w = 3 w = 4 . . . w = 2 k − w = 2 kN − Table 19: Distinguished branched -manifolds with weight w Let N + be a distinguished branched -manifold which is the entering boundary of a minimal GS isolating block N .Consider the sequence of identifications of a pair of points in Figure 4, where these pair of points in N + are on orbits in N whose ω -limit does not belong to N .Each identification produces a branch point. After performing k identifications, the GS isolating block will have k passageways. It is easy to verify that the exiting boundaries of any isolating block arising in this way will have asboundary components the distinguished branched -manifolds in Table 19.18 PREPRINT - O
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Figure 4: Identification sequenceEach GS semi-graph L v . with one vertex v ∈ L can be realized by the above procedure starting from the correspondingminimal GS semi-graph. This corresponding minimal GS semi-graph has the same indegree, outdegree and labelling asthe semi-graph L v .Since any given boundary with weight n has a unique distinguished branched -manifold that is realized on the boundaryof isolating blocks, it is trivial to glue one block to the other. Hence, L is realizable. Example 3.
In Figure 5, a GS graph (on the left) satisfying Theorem 7 is realized as a GS flow (on the right).
Realization −−−−−−−−→
Figure 5: Realization of a GS graph as a GS flowIt is interesting to note that bifurcation vertices with incident edges that are not labelled with minimal weights, introducecomplex combinatorial questions on the choice of the distinguished branched -manifolds that are boundary componentsof isolating blocks. The following example illustrates this fact. Example 4 (Non-realizable) . In Figure 6, the GS graph L contains a bifurcation vertex labelled with a singularityof type W whose weights are not minimal. Note that all semi-graphs containing a unique vertex of L admit only onerealization as a GS isolating block. However, the distinguished branched -manifolds of weight that make up theexiting and entering boundaries of the first two blocks can not be glued to each other. Hence, the graph is non realizable. Figure 6: A GS graph that is non realizable as a GS flow19
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1, 2020The next theorem is an attempt to deal with the problem presented in Example 4. It will blend in the results in Theorems6 and 7 by imposing minimal weights only on incident edges of bifurcation vertices.
Theorem 8.
Let L be a GS graph containing singularities of types R , C , W and D . Suppose that all incident edges ofvertices v of L with degree greater than or equal to has minimal weights. Then L is realizable.Proof. The proof is a direct consequence of Theorems 6 and 7.Let L v be a GS semi-graph consisting of a single vertex v ∈ L and its incident edges. Consider the decomposition of L given as follows: L = (cid:91) deg ( v ) < L v ∪ (cid:91) deg ( v ) ≥ L v We have that each connected component of (cid:83) deg ( v ) < L v is realizable by Theorem 7 with the choice of distinguishedbranched -manifolds presented in Table 20. And each connected component of (cid:83) deg ( v ) ≥ L v is realizable by aminimal GS isolating block.Notice that the intersection (cid:91) deg ( v ) < L v ∩ (cid:91) deg ( v ) ≥ L v is made up of edges with weights , and . Since the choice of distinguished branched -manifolds match for all theseweights, the realizations of the connected components of these two unions of L (cid:48) v s can all be glued together. Hence, L isrealizable. As we have seen in the last section, bifurcation vertices can constitute obstructions in the realizability of a GS graph.The simultaneous presence of bifurcation vertices in GS graphs labelled with Whitney and double crossing singularitiesmay in some cases not be realizable. See Example 4.For this reason our next theorem will restrict the labelings of the vertices on a GS graph to singularities of type R , C and W , excluding singularities of type D . Theorem 9 ( RCW -Case) . All GS graphs labelled only with singularities of type R , C and W are realizable.Proof. Consider the choice of distinguished branched -manifolds for the edges of L labelled with weights w given inTable 20. w = weight w = 1 w = 2 w = 3 w = 4 . . . w = nN − (cid:122) (cid:125)(cid:124) (cid:123) Table 20: Distinguished branched -manifolds with weight w Each GS semi-graph L v formed by a single vertex v ∈ L and its incident edges can be realized from the realization ofthe corresponding minimal GS semi-graph (cid:96) v . Indeed, starting from a minimal GS isolating block N with boundarycomponents in Table 20 which realizes (cid:96) v , one may perform a sequence of identifications of pairs of points in N + , asshown in Figure 7, where these pairs of points are orbits in N whose ω -limit sets do not belong to N . → → → → . . . Figure 7: Identification sequenceThese identifications must be performed until the weights on the boundaries of the block match the weights on theedges of L v . 20 PREPRINT - O
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1, 2020Notice that the topological effect of each identification on the boundary N + is to produce a branch point that determinesa fold along which a cylinder is glued to the block. We remark that, for a vertex of degree , the relative position ofthese cylinders with respect to the minimal GS block may produce topologically non-equivalent realizations of the samesemi-graph.Lastly, the realizations of all GS semi-graphs L v can be glued together due to the fact that all boundaries of weight w have the same distinguished branched -manifold. Example 5.
In Figure 8, we present two topologically non equivalent realizations of a GS graph satisfying Theorem 9.
Realizations −−−−−−−−→
Figure 8: Realizations of a GS graph as a GS flowAlthough Theorem 9 was obtained by excluding the presence of vertices labelled with singularities of type D , if we paycloser attention to the family of distinguished branched -manifolds in Table 20, we notice that it is actually possibleto realize almost every GS semi-graph with a single vertex labelled with a singularity of type D having this familyof distinguished branched -manifolds as boundary components, the exceptions being the GS semi-graph of degree labelled with natures a or r , the GS semi-graph of degree labelled with natures sa or sr , and the GS semi-graph ofdegree labelled with natures ss s or ss u , as states the next Lemma. Lemma 2.
Let L v be a GS semi-graph with a single vertex v and its incident edges. Then, L v can be realized by thefamily of distinguished branched -manifolds in Table 20 if and only if L v satisfies:i) v is labelled with a singularity of type R , C , W or D ;ii) if v has degree , then its incident edge has weight equal to or ;iii) if v is labelled with a singularity of type D and nature sa or sr , then v has degree ;iv) v has degree less or equal to ;v) if v is labelled with a singularity of type D and nature ss s (resp. ss u ), with e + v = 1 , e − v = 2 (resp. e + v = 2 , e − v = 1 ), then { b − , b − } (resp. { b +1 , b +2 } ) is equal to { , B + − } (resp. { , B − − } );vi) if e − v = 3 (resp. e + v = 3 ), then { b − , b − , b − } (resp. { b +1 , b +2 , b +3 } ) is equal to { , , B − − } ;Proof. The proof relies heavily on the characterization of possible distinguished branched -manifolds that make upthe boundary components of minimal GS isolating blocks given in Theorem 3. ( ⇒ ) First, suppose L v can be realized as a GS flow on a GS isolating block, which has as boundary components adistinguished branched -manifolds of Table 20. Now we verify that L v satisfies conditions i ) through vi ) :21 PREPRINT - O
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1, 2020i) Theorem 3 states that there is no minimal GS isolating block for a singularity of type T with the choice ofdistinguished branched -manifolds in Table 20 as boundary components. Therefore, there cannot be GSisolating blocks with passageways for singulatiries of type T with this choice either. Hence, v must be labelledwith a singularity of type R , C , W or D ;ii) If v has degree , then the possible labels on v are either attracting or repelling singularities of type R , W or D . However, it follows from Theorem 3 that a realization of a GS isolating block for an attracting or repellingsingularity of type D cannot be achieved with the distinguished branched -manifold of weight pictured inTable 20. Thus, v is either labelled with an attracting or repelling singularity of type R or W , which impliesweight or on its incident edge, respectively;iii) The possible degrees of a vertex v labelled with a singularity of type D and nature sa or sr are or . However,according to Theorem 3, only the cases where v has degree admit realizations matching the choice ofdistinguished branched -manifolds in Table 20;iv) If we suppose by contradiction that the degree of v is greater than , than v must have degree , which impliesthat v is labelled with a singularity of type D and nature ss s or ss u . Either way, Theorem 3 implies that such L v do not admit a realization matching the distinguished branched -manifolds of Table 20. This follows sincethe distinguished branched -manifold of weight on Table 20 cannot be chosen as the boundary of a minimalGS isolating block for such singularities. Furthermore, adding passageways to their minimal isolating blocksproduce boundaries that also do not belong to Table 20. This contradicts our initial hypothesis that L v admitsa realization with boundary components within Table 20. Hence, v must have degree less or equal to ;v) The realization of L v , in the case that v is labelled with a singularity of type D and nature ss s , with minimalweights on its incident edges, is given in Table 12. We recall that turning the block upside down gives aminimal GS isolting block for nature ss u . The minimal GS isolating block shows that, even though e − v = 2 (resp. e + v = 2 ), in order to create a GS isolating block with passageways and boundaries matching Table 20,the folds must exit the block through only one of the exiting (resp. entering) boundaries. Thus the weights { b − , b − } (resp. { b +1 , b +2 } ) must be equal to { , B + − } (resp. { , B − − } );vi) Analogous to item v ) ; ( ⇐ ) Suppose L v satisfies conditions i ) through vi ) and denote by (cid:96) v the minimal GS semi-graph associated to L v ,that is, it has the same labelling, indegree and outdegree, but with minimal weights on its incident edges satisfying thePoincaré-Hopf condition. Then, by Theorem 3, (cid:96) v admits a realization as a GS flow on a minimal GS isolating block N with boundary components given by the distinguished branched -manifolds in Table 20. One can identify orbitsof N as in Figure 7, such that the topological effect of each identification produces a branch point that determines afold along which a cylinder is glued to the block. One can perform a finite number of such identifications until theweights on the boundaries of N become equal to the respective weights on the edges of L v . Moreover, the distinguishedbranched -manifold produced by each of these identifications belong to Table 20, thus completing the proof.It turns out that attractors, repellers and other singularities of type D that do not satisfy Lemma 2 may obstruct therealization of a GS graph, even in the case where all folds have singularities of type D as their α, ω limit set.Recall that an isolating block with passageways arises from the identification of specific orbits on a minimal isolatingblock. Since the identification of orbits may lower the number of connected components on the boundaries of the block,in some cases, we must choose orbits that enter and exit the block through the same connected components.We remark that this fact is one of the causes that may prevent the gluing of two blocks. Indeed, if two blocks have adifferent number of connected components, it may happen that the orbits that must be identified, in order to producethe same distinguished branched -manifold, may enter or exit the block through different connected components,preventing the realization. See Example 6 below. It is worth mentioning that this scenario never occurs in a GS graphlabelled with singularities of type R , C and W as proved in Theorem 9. And also does not occur whenever bifurcationvertices are not present, as proved in Theorem 7. Example 6 (Non-realizable) . In the example presented in Figure 9 (a), the GS graph which is labelled with singularitiesof type R and D , is non-realizable although satisfying the Poincaré-Hopf condition on all edges. The obstruction arisessince there is no common choice of distinguished branched -manifolds of weight that permits its realization. Morespecifically, all vertices of L admit a unique realization as a GS flow on a GS isolating block, but the distinguishedbranched -manifolds of weight of these realizations are not homeomorphic, thus preventing the gluing of all blocks. PREPRINT - O
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On the other hand, if we consider homeomorphic distinguished branched -manifolds of weight , note that the weightson the exiting boundaries of the GS isolating block for a regular saddle do not match the weights on L , see Figure 9 (b). ( a ) ( b ) Figure 9: (a) Non-realizable GS graph; (b) Regular saddle isolating block with N + of weight As seen in Example 6, bifurcation vertices without the minimal weight condition may not be realizable. The problem inthis example resides in the fact that there are two non homeomorphic distinguished branched -manifolds of weight that form the boundary of two isolating blocks that must be glued to each other. In order to obtain a result for bifurcationvertices in general, we must restrict the allowable boundary components that make up the isolating blocks.In what follows, we will consider the family of distinguished branched -manifolds in Table 19 and analyze underwhat conditions bifurcation vertices and its incident edges can be realized with this family of distinguished branched -manifolds. This is the content of the next Lemma. Lemma 3.
Let L v be a GS semi-graph with a single vertex v and its incident edges, belonging to Table 3, such that v islabelled with a singularity of type R , C , W or D . Then, L v can be realized by the family of distinguished branched -manifolds in Table 19 if and only if L v satisfies the following conditions, according to the singularity with which v islabelled:i) (regular or double crossing) if e ± v = 1 , e ∓ v = 2 , |B + − B − | = 1 and B ± is odd, then b ∓ , b ∓ are odd;ii) (Whitney) if e ± v = 1 and e ∓ v = 2 , then B ± is even and { b ∓ , b ∓ } is equal to { , B ± − } ;iii) (double crossing) if e + v = e − v = 2 and B ± > B ∓ , then the weights b ± , b ± are even, and { b ∓ , b ∓ } is eitherequal to { , B ± − } or { b ± − , b ± − } ;iv) (double crossing) if e ∓ v = 3 , then b ∓ i = 1 for at least one index i ∈ { , , } . Moreover, if B ± is odd, then theweights b ∓ , b ∓ , b ∓ are all odd;v) (double crossing) If e ∓ v = 4 , then b ∓ i = 1 for at least two edges. Moreover, if B ± is odd, then the weights b ∓ , b ∓ , b ∓ , b ∓ are all odd;Proof. The proof relies heavily on Lemma 1, which states that all GS isolating blocks with passageways for GSsingularities can be constructed from minimal GS isolating blocks via identifications of pair of orbits. ( ⇒ ) First, suppose L v can be realized as a GS flow on a GS isolating block N with the distinguished branched -manifolds of Table 19 as boundary components. Now we verify that L v satisfies conditions i ) through v ) :i) Let W s ( p ) (resp. W u ( p ) ) be the stable (resp. unstable) manifold of the singularity p ∈ N . Then, whether p isa singularity of type R or D , it follows that W s ( p ) ∩ N + (resp. W u ( p ) ∩ N − ) is formed by two points q and q . In the case B ± is odd, we have that each connected component of N ± \ { q , q } has an even number ofbranch points. Since all folds associated to branch points on the same connected component of N + \ { q , q } PREPRINT - O
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1, 2020(resp. N − \ { q , q } ) must exit (resp. enter) N through the same connected component of N − (resp. N + ), itfollows that each connected component of N − (resp. N + ) has an even number of branch points. Hence, wemust have b ∓ and b ∓ both odd.ii) If N is a minimal GS isolating block, then B ± = 2 and b ∓ = b ∓ = 1 . Hence, { b ∓ , b ∓ } is equal to { , B ± − } .Moreover, N ± is a figure eight. Denote by q the branch point of N ± , we have that N ± \{ q } has two connectedcomponents such that orbits that enter (resp. exit) N through each connected component of N ± \ { q } exit(resp. enter) N through different connected components of N ∓ .If N is not minimal, Lemma 1 asserts that N can be constructed from a minimal isolating block N . Denoteby q the branch point of N ± . Since we cannot identify orbits that enter (resp. exit) N ± through differentconnected components of N ± \ { q } , it follows that N also has a branch point ˜ q such that N ± \ { ˜ q } has twoconnected components.Note that every distinguished branched -manifold of Table 19 with odd weight remains connected after theremoval of one branch point. Thus, B ± cannot be odd. Hence, B ± is even. Furthermore, after removing thebranch point that disconnects a distinguished branched -manifold of Table 19 with even weight, the remaining B ± − branch points all lie in the same connected component. Thus, all folds associated to these branch pointsexit N ∓ through the same connected component. Therefore, we have that { b ∓ , b ∓ } is equal to { , B ± − } .iii) If L v is minimal, then B ± = 4 , where b ± = b ± = 2 and B ∓ = 2 , with b ∓ = b ∓ = 1 . Hence, b ± and b ± areeven and { b ∓ , b ∓ } is equal to { , B ± − } which coincides with { b ± − , b ± − } . Moreover, N ± = N ± ∪ N ± is the disjoint union of two figures eight. Denote by q j the branch point of N ± j for j ∈ { , } . As in item ii ) ,we have that N ± j \ { q } has two connected components such that orbits that enter (resp. exit) N through eachconnected component of N ± j \ { q } exit (resp. enter) N through different connected components of N ∓ . Also,by the same argument used in item ii ) , one shows that if L v is not minimal, then b ± and b ± are even.Now consider the case where L v has weights b ± > and b ± > , i.e., N ± = N ± ∪ N ± is the disjoint unionof two distinguished branched -manifolds of weights b ± and b ± , respectively. Then it follows that N has ( b ± −
2) + ( b ± − passageways. By using a two step process, consider the b ± − passageways determinedby N ± . As in item ii ) the folds that intersect N ± \ { p } must exit (resp. enter) either N ∓ or N ∓ . Withoutloss of generality, we can assume that these orbits exit (resp., enter) N ∓ . At this point, this implies that theweight of N ∓ is greater or equal to b ± − . Now consider the folds that intersect N ± \ { q } which must exit(resp., enter) either through N ∓ or N ∓ . If they exit (resp., enter) through N ∓ we are in the case that { b ∓ , b ∓ } is equal to {B ± − , } ; if they exit (resp., enter) through N ∓ we are in the case that { b ∓ , b ∓ } is equal to { b ± − , b ± − } .iv) If N is a minimal GS isolating block, then B ± = 3 and b ∓ = b ∓ = b ∓ = 1 . Moreover, N ± has twobranch points q and q such that N ± \ { q , q } has four connected components. In addition, orbits that enter(resp. exit) N through two of these connected components exit (resp. enter) N through the same connectedcomponent of N − (resp. N + ), while orbits entering (resp. exiting) N through the other two connectedcomponents of N ± \ { q , q } exit (resp. enter) each through a distinct connected component of the remainingones in N ∓ .In the case N is not minimal, let W s ( p ) (resp. W u ( p ) be the stable (resp. unstable) manifold of the singularity p ∈ N . Then, W s ( p ) ∩ N + (resp. W u ( p ) ∩ N − ) is also formed by two branch points { ˜ q , ˜ q } such that N ± \ { ˜ q , ˜ q } has the same properties of N ± \ q , q described in the previous paragraph for a minimalisolating block N . Since all branch points other than ˜ q and ˜ q belong to at most two connected componentsof N ± \ { ˜ q , ˜ q } and N ∓ has three connected components, it follows that at least one connected componentof N ∓ has weight , that is, b ∓ i = 1 for at least one i ∈ { , , } . Furthermore, if B ± is odd, each connectedcomponent of N ± \ { ˜ q , ˜ q } has an even number of branch points. Hence, b ∓ , b ∓ and b ∓ are all odd.v) In this case, the orbits through each one of the four arcs formed by N ± \ { p, q } exit through a distinctconnected component of N ∓ . The remaining argument is analogous to item iv ) . ( ⇐ ) Now, we show that all semi-graphs L v of Table 3 subject to conditions i ) , ii ) , iii ) , iv ) , v ) of Lemma 3 are realizableas a GS flow on a GS isolating block with boundary components in Table 19.Note that, if L v is minimal, then Theorem 3 asserts that there is a minimal GS isolating block with boundary componentsin Table 19 that realizes L v . Some of these blocks can be seen in Tables 5, 7, 9 and 12.24 PREPRINT - O
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1, 2020If L v is not minimal, we proceed as follows:First, we consider a minimal GS isolating block N for a singularity p , such that the Conley index of p is the same thatwhich v is labelled with, N ± has e ± v connected components, and N ± belongs to Table 19.Then in order to create passageways in N , we analyze the possible ways of embedding N +0 in a distinguished branched -manifold N + of Table 19 with weight B + , such that the folds associated to branch points of N + \ N exit theconnected components of N − according to the weights b − i of L v , for i ∈ { , . . . , e − v } .For this purpose, we divide the semi-graphs L v of Table 3 in the following cases:a) if e + v = e − v = 1 , Lemma 3 imposes no restrictions on L v . So, suppose v is labelled with a singularity of type R , C or D and consider any embedding f : N +0 (cid:44) → N + . Then consider a tubular flow on N + \ f ( N +0 ) × [0 , and for each point q i ∈ N + \ f ( N +0 ) ∩ f ( N +0 ) , identify the orbit of N + \ f ( N +0 ) × [0 , that passes throughin q i with the orbit of N that passes through in f − ( q i ) . The resulting block N is a realization of L v suchthat the boundary components of N belong to Table 19.The case where v is labelled with a singularity of type W requires an intermediate step before realizing theembedding f : N +0 (cid:44) → N + only if N + has odd weight. This is due to the fact that N +0 is a figure eightand it cannot be embedded in a distinguished branched -manifold of Table 19 of odd weight. Hence, theidentification shown in Figure 10 must be performed prior to the embedding. → Figure 10: Creating a branch point from a figure eight.b) if e + v < e − v , then L v is subject to one of the conditions i ) , ii ) , iv ) or v ) of Lemma 3.Let W s ( p ) be the stable manifold of the singularity p ∈ N . Note that, if p is a singularity of type R , W s ( p ) ∩ N +0 is the disjoint union of two points. In the case p is a Whitney or double crossing singularity, W s ( p ) ∩ N +0 is exactly the branch points in N +0 , which is equal to or branch points, respectively. Byremoving these points from N +0 we are left with two connected components in the case p is of type R or W ,and four connected components in the case p is of type D . The orbits that pass through each of these connectedcomponents exit N through a connnected component of N − .With this at hand, consider an embedding f : N +0 (cid:44) → N + such that, for each j ∈ { , . . . , e − v } there is aconnected component K + j of N + \ ( f ( W s ( p ) ∩ N +0 ) with b − j − branch points. Now, consider a tubular flowon each connected component of N + \ f ( N +0 ) × [0 , . For each point q i ∈ N + \ f ( N +0 ) ∩ f ( N +0 ) , identifythe orbit of N + \ f ( N +0 ) × [0 , that passes through in q i with the orbit of N that passes through in f − ( q i ) .The resulting block N is a realization of L v such that the boundary components of N belong to Table 19.c) if e + v = 2 , e − v = 1 , then Lemma 3 imposes no restrictions on L v . In this case, N +0 is the disjoint union of twofigures eight, i.e., N +0 = F ∪ F .Let N + = N +1 ∪ N +2 be a distinguished branched -manifold such that N + i belongs to Table 19 and hasweight b + i for i ∈ { , } . In the case that the weights b +1 and b +2 are both odd, or in the case one of them isodd and the other is an even number greater than , we must perform the identification shown in Figure 10 to F and F prior to the embeddings f : F (cid:44) → N +1 and f : F (cid:44) → N +2 . Then one considers the same tubularflow and identifications of orbits of items a ) and b ) . The resulting block N is a realization of L v such that theboundary components of N belong to Table 19.d) if e + v = e − v = 2 , then consider N + = N +1 ∪ N +2 such that N + i belongs to Table 19 and has weight b + i , for i ∈ { , } . According to Table 3, v can be labelled with a singularity of type C or D .If v is labelled with a cone singularity, N +0 is the disjoint union of two circles, i.e. N +0 = S ∪ S and Lemma3 imposes no restrictions on L v . Thus, we proceed as in case a ) for f : S (cid:44) → N +1 and for f : S (cid:44) → N +2 .25 PREPRINT - O
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1, 2020Otherwise, if v is labelled with a singularity of type D , N +0 is the disjoint union of two figures eight, i.e. N +0 = F ∪ F and L v is subject to condition iii ) of Lemma 3. Thus, we proceed as in case b ) for f : F (cid:44) → N +1 and for f : F (cid:44) → N +2 .Hence, the resulting block N is a realization of L v such that the boundary components of N belong to Table19.Cases a ) , b ) , c ) and d ) prove that L v is realizable for all semi-graphs in Table 3 subject to i ) , ii ) , iii ) , iv ) and v ) .The next theorem is a realization theorem which includes bifurcation vertices and labellings with both singularities oftype W and D , and as such will be a generalization of Theorems 7 and 9. Theorem 10.
Let L be a GS graph labelled with singularities of type R , C , W and D such that each GS semi-graph L v belongs to Table 3. Furthermore, suppose that:i) L v satisfies conditions i ) , ii ) , iii ) , iv ) , v ) , vi ) of Lemma 2, for all vertices v ∈ L ; orii) L v satisfies conditions i ) , ii ) , iii ) , iv ) , v ) of Lemma 3, for all vertices v ∈ L ,then L is realizable.Proof. In the case condition i ) is satisfied, it follows from Lemma 2 that all semi-graphs L v can be realized as a GSflow on a GS isolating block with the distinguished branched -manifolds of Table 20 as boundary components. Hence,one glues these isolating blocks according to L .Similarly, if condition ii ) is satisfied, then all semi-graphs L v can be realized as GS flows on GS isolating blocks withboundary components on Table 19, as stated in Lemma 3. Hence, all isolating blocks can be glued together accordingto L . Thus, L is realizable.We remark that the conditions of Theorem 10 are sufficient to ensure the realization of a GS graph under those hypothesisalthough they are far from necessary, as shown in the next example. Example 7.
In figure 11, we present a GS graph L that does not satify Lemma 2, because of the presence of singularitiesof type D and attracting or repelling natures. Also, L does not satisfy Lemma 3 because the bifurcation vertex, which islabelled with a singularity of type W , has an odd weight in its positively incident edge. However, L is realizable as aGS flow on a GS -manifold. As we can see in Figure 11, the distinguished branched -manifold on the boundaries withweight does not belong to Table 19 nor Table 20. Realization −−−−−−−−→
Figure 11: Realization of a GS graph as a GS flow26
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1, 2020Also, we remark that bifurcation vertices labelled with singularities of type T cannot be realized with the distinguishedbranched -manifolds of Table 19 nor of Table 20.The realization theorems presented in this section were obtained subject to some type of simplification. For instance,restrictions on the type of singularities with which the graph is labelled, or restrictions on the weights of the edges or onthe degree of the vertices. Had this not been done, the increase of weights on the edges would imply in a greater numberof choices of distinguished branched -manifolds and the local information on the graph is not sufficient to guaranteethe gluing of the isolating blocks as indicated by the graph. However, even if we can not guarantee the existence of atriple ( M , X t , f ) , we still have control over the Euler characteristic of a possible realization on M . The reason for thisis that the Euler characteristic can be computed in terms of the natures and types of GS singularities, as will be shownin the next section. The Euler characteristic can be defined for any topological space T . and it is denoted by X ( T ) . It is given by: X ( T ) = (cid:88) i ∈ N ( − i β i , (6)where β i is the rank of the i -th homology group H i ( T ) , i.e., the i -th Betti number of T .In [5], a filtration of a closed GS -manifold M was constructed: G ⊂ G ⊂ . . . ⊂ G m = M , such that G k contains exactly k singularities and ( G i , G i − ) is an index pair for the i -th singularity of M . Then, fromthe long exact sequence associated to the pair ( G i , G i − ) : . . . p j −→ H j ( G i , G i − ) ∂ j −→ H j − ( G i − ) i ∗ −→ H j − ( G i ) p j − −→ H j − ( G i , G i − ) ∂ j − −→ . . . it was proven that the Euler characteristic of M is equal to the alternating sum of the numerical Conley indices of thesingularities, i.e.: X ( M ) = (cid:88) p i ∈ Sing ( M ) ( h i − h i + h i ) . (7)In what follows, we give an alternative formula for (7) for closed GS -manifolds, by expressing the Euler characteristicin terms of the total number a , s and r of attracting, saddle and repelling natures, respectively, of the GS singularities.In this sense, we remark that the total number of natures a , s and r of singularities of type D is equal to two. Forinstance, a GS singularity of type D and nature ss s or ss u has two saddle natures, and zero attracting and repellingnatures. On the other hand, the total number of natures a , s and r of singularities of type T is equal to three. A GSsingularity of type T and nature ssa has two saddle natures, one attracting nature and zero repelling nature. Meanwhile,for an attractor singularity of type T , we have that a = 3 and s = r = 0 . Proposition 2.
Let L be a GS graph. If M is a realization of L , then: X ( M ) = a − s + r + W T, where W and T are the number of vertices in L labelled with singularities of type W and T , respectively, while a , s and r are, respectively, the total number of attracting, saddle and repelling natures of the singularities with which thevertices are labeled.Proof. One has that X ( M ) = (cid:88) p i ∈ Sing ( M ) ( h i − h i + h i ) . (8)Denote by P η the total number of vertices in L labeled with a GS singularities of type P , where P ∈ {R , C , W , D , T } ,with nature η , that is, a singularity p ∈ H P η .In what follows, we summarize the numerical Conley index ( h , h , h ) of p ∈ H P η , which was computed in [5]:Using the numerical Conley index of a GS singularity of type P with nature η and substituting this in (8), we have that:27 PREPRINT - O
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R C
Nature a s r a s r ( h , h , h ) (1 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
0) (0 , ,
0) (0 , , W T
Nature a s s s u r a ssa ssr r ( h , h , h ) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
1) (1 , ,
0) (0 , ,
0) (0 , ,
2) (0 , , X ( M ) = ( R a − R s + R r ) + ( C a − C s + C r ) + ( W a − W s s + 0 W s u + 2 W r ) (9) + ( D a − D sa − D ss s − D ss u + D sr + 3 D r ) + ( T a − T ssa + T ssr + 7 T r ) Now, adding and subtracting terms in (9) corresponding to the number of attracting, saddle and repelling natures withineach GS singularity, one has: X ( M ) = ( R a − R s + R r ) + ( C a − C s + C r ) + ( W a − W s s − W s u + W r )+ (2 D a − D sa − D ss s − D ss u + 0 D sr + 2 D r ) + (3 T a − T ssa − T ssr + 3 T r )+ [ ( W s u + W r ) + ( D ss u + D sr + D r ) − ( D ss s + D sa + D a ) + (4 T r − T a + 2 T ssr ) ] Hence, one has: X ( M ) = a − s + r + [ ( W s u + W r ) + ( D ss u + D sr + D r ) − ( D ss s + D sa + D a )+ (4 T r − T a + 2 T ssr ) ] (10)Since M is a closed manifold, any fold in M admits α -limit and ω -limit. Counting the total number of folds in M in terms of the GS singularities that are α -limit of each fold and then ω -limit of each fold, one obtains the followingequalities:number of folds in M = ( W s u + W r ) + 2( D ss u + D sr + D r ) + (6 T r + 4 T ssr + 2 T ssa )= ( W s s + W a ) + 2( D ss s + D sa + D a ) + (6 T a + 2 T ssr + 4 T ssa ) (11)It follows from the equality (11) that: ( W s u + W r ) + ( D ss u + D sr + D r ) − ( D ss s + D sa + D a ) + (4 T r − T a + 2 T ssr )= ( W s s + W a ) + ( D ss s + D sa + D a ) − ( D ss u + D sr + D r ) + (4 T a + 2 T ssa − T r ) (12) D Nature a sa ss s ss u sr r ( h , h , h ) (1 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
0) (0 , ,
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1, 2020By adding the two sides of equality (12), one has: W s u + W r ) + ( D ss u + D sr + D r ) − ( D ss s + D sa + D a ) + (4 T r − T a + 2 T ssr )]= ( W s u + W r + W s s + W a ) + (2 T r + 2 T a + 2 T ssr + 2 T ssa )= W + 2 T (13)And finally, by substituting (13) in (10), the proof is concluded.We remark that a similar formula for the Euler characteristic of singular surfaces with cross-caps and triple crossingsingularities was proved in [10] in a different setting.In what follows, we use the formula given in Proposition 2 to compute the Euler characteristic of the GS -manifoldspresented in the examples throughout section 3. Example 8.
Let M be the GS -manifold of Example 2. One has that, X ( M ) = 3 − + 0 = 1 .In the case M is the GS -manifold of Example 3, one has that X ( M ) = 2 − + 0 = 1 .On the other hand, if M is the GS -manifold of Example 5, then X ( M ) = 2 − + 0 = 0 .Lastly, if M is the GS -manifold of Example 11, it follows that X ( M ) = 3 − + 0 = 5 . In this work we determine sufficient conditions for the realization of abstract Lyapunov graphs labelled with singularitiesof type regular ( R ), cone ( C ), Whitney ( W ), double crossing ( D ) and triple crossing ( T ) as Gutierrez-Sotomayor flowson closed singular two-manifolds. We show here that for a graph which strictly satisfies the set of necessary conditionsof Theorem 1, its realization is not always possible. At times, not even locally as we show in Theorem 4.Locally, that is, for a semi-graph L v consisting of a single vertex v and its incident edges, sufficient conditions for therealization of L v as a GS flow on a GS isolating block are presented. Moreover, in the case the weights on the incidentedges are the lowest satisfying the Poincaré-Hopf condition, we provide in Theorem 3 a complete characterization ofthe realizations of L v by describing the distinguished branched -manifolds that make up the boundary componentsof all minimal GS isolating blocks for each singularity type. Furthermore, for higher weights on the incident edgessatisfying the Poincaré-Hopf condition, we show in Theorem 5 that GS isolating blocks with passageways which realize L v arise from a process of identification of pairs of orbits on minimal GS isolating blocks.Concerning the global realization question, it can be posed as a problem of assigning distinguished branched -manifoldsto the edges of a graph, so that the induced assignment on the incident edges of each vertex is obtainable as boundarycomponents of an isolating block for the singularity with which the vertex is labelled. Since the set of isolating blocksfor a singularity is given by all the possible ways of adding passageways to a minimal isolating block, this question isequivalent to determining reachability from minimal boundaries to distinguished branched -manifolds with higherweights.The sufficient conditions for global realization presented in Theorem 10 are obtained from two families of distinguishedbranched -manifolds for which the increase of branch points is given in a controlled fashion. Moreover, each of thesetwo families assigns the same distinguished branched -manifold to every edge of the graph with a common weight,thus ensuring that boundary components with common weights of two isolating blocks are always homeomorphic.Thus, all boundaries of the isolating blocks are glued according to the graph guaranteeing the global realization. By nomeans are these families unique since the increase of branch points could be attained by many more identifications thanthe ones considered in this work.One may hope to obtain stronger global results by considering families of distinguished branched -manifolds withmore than one choice of assignment for each weight. Nevertheless, we remark that the number of non-homeomorphicdistinguished branched -manifolds increases quickly as one goes subsequently from one weight to the next. Forinstance, for weights , , , and one has , , , and at least non-homeomorphic distinguished branched -manifolds, respectively. Verifying whether two distinguished branched -manifolds with arbitrary weight arehomeomorphic or not is a difficult question pertaining to the theory of -regular pseudo-graphs. Thus, determiningsufficient conditions for the global realization of Lyapunov graphs is more challenging in this setting.29 PREPRINT - O
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1, 2020Mostly, the global results presented in this work include singularities of types R , C , W and D , leaving out singularitiesof type T due to their intrinsic complexity. With the exception of the realization of graphs with minimal weights on alledges proved in Theorem 6, determining sufficient conditions for the global realization of graphs including R , C , W , D and T singularities remains an open question. References [1] R. D. MacPherson, “Chern classes for singular algebraic varieties,”
Annals of Mathematics , pp. 423–432, 1974.[2] J.-P. Brasselet, “Sur les classes de chern d’un ensemble analytique complexe, caracteristique d’euler-poincare,”
Astérisque , vol. 82, pp. 93–147, 1981.[3] J.-P. Brasselet, J. Seade, and T. Suwa,
Vector fields on singular varieties , vol. 1987. Springer Science & BusinessMedia, 2009.[4] M.-H. Schwartz, “Classes caractéristiques définies par une stratification d’une variété analytique complexe,”
CRAcad. Sci. Paris , vol. 260, pp. 3262–3264, 1965.[5] H. R. M. López and K. A. de Rezende, “Conley theory for gutierrez-sotomayor fields,” 2019.[6] C. Gutierrez and J. Sotomayor, “Stable vector fields on manifolds with simple singularities,”
Proceedings of theLondon Mathematical Society , vol. 3, no. 1, pp. 97–112, 1982.[7] C. G. Gibson,
Singular points of smooth mappings , vol. 25. Pitman publishing, 1979.[8] R. Thom, “Ensembles et morphismes stratifiés,”
Bulletin of the American Mathematical Society , vol. 75, no. 2,pp. 240–284, 1969.[9] C. C. Conley,
Isolated invariant sets and the Morse index . No. 38, American Mathematical Soc., 1978.[10] S. Izumiya and W. L. Marar, “On topologically stable singular surfaces in a 3-manifold,”
Journal of Geometry ,vol. 52, no. 1-2, pp. 108–119, 1995.[11] P. Aluffi and M. Esole, “Chern class identities from tadpole matching in type iib and f-theory,”
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Journal of Topology and Analysis , vol. 2, no. 01, pp. 1–55, 2010.[13] M. Esole, P. Jefferson, and M. J. Kang, “Euler characteristics of crepant resolutions of weierstrass models,”