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arxiv: 2605.18114 · v1 · pith:XOBYJZ3Anew · submitted 2026-05-18 · 🧮 math.DS

Singularity Collisions through Homotopical Dynamical cancellation

Dahisy Lima , Ketty de Rezende , Denilson Ten\'orio This is my paper

Pith reviewed 2026-05-20 00:39 UTC · model grok-4.3

classification 🧮 math.DS
keywords GGS flowshomotopical cancellationspectral sequencessingular manifoldsdynamical systemschain complexesMorse theory
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The pith

A spectral sequence analysis of the GGS chain complex establishes a bijective correspondence between algebraic cancellations and homotopical dynamical cancellations in singular flows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a theory of homotopical dynamical cancellation for generalized Gutierrez-Sotomayor flows on singular manifolds, extending classical Morse theory to handle collisions of invariant sets. It introduces a GGS chain complex to encode the dynamical and topological information of these flows. A filtered version of this complex is then analyzed via spectral sequences, revealing that cancellations in the algebraic modules directly correspond to homotopical cancellations that preserve the manifold's homotopy type. This framework provides a systematic way to study how singularities interact while maintaining essential topological features.

Core claim

We introduce collisions of invariant sets and homotopical dynamical cancellations that preserve the homotopy type of the underlying singular manifold for generalized Gutierrez-Sotomayor flows on GGS manifolds. By developing a GGS chain complex that encodes essential dynamical and algebraic-topological information and providing a spectral sequence analysis of its filtered version, we establish a bijective correspondence between algebraic cancellations of the modules and the homotopical dynamical cancellations in the flow.

What carries the argument

The GGS chain complex and its filtered spectral sequence, which together capture the dynamical and homotopical information to establish the bijective correspondence between algebraic and dynamical cancellations.

If this is right

  • The classical cancellation theory of Morse flows extends to the singular setting via homotopical dynamical cancellations.
  • Collisions of invariant sets can be tracked algebraically without altering the homotopy type of the manifold.
  • The spectral sequence provides a tool to analyze and predict these cancellations in GGS flows.
  • Examples demonstrate how this correspondence applies to specific flows with singularities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could generalize to other types of singular dynamical systems beyond GGS flows.
  • Computational implementations of the chain complex might enable simulation of singularity collisions.
  • The correspondence suggests new ways to compute topological invariants from flow data.
  • Further work could explore higher-dimensional cases or different filtrations on the complex.

Load-bearing premise

The GGS chain complex and its spectral sequence accurately encode the essential dynamical and homotopical information of the flows, so that algebraic cancellations match the actual dynamical ones.

What would settle it

Finding a concrete GGS flow on a manifold where a homotopical cancellation of singularities occurs but the corresponding module cancellation does not appear in the spectral sequence of the GGS chain complex, or vice versa.

Figures

Figures reproduced from arXiv: 2605.18114 by Dahisy Lima, Denilson Ten\'orio, Ketty de Rezende.

Figure 1
Figure 1. Figure 1: Cancellation between non degenerate critical points x1 and x2, in dimension 2. The concept of cancellation became a powerful tool when Smale employed it in 1960 as part of his celebrated proof of the generalized Poincar´e conjecture [28]. A more didactic approach to cancellation in the smooth setting can be found in [18]. In classical Morse theory, consecutive critical points can be canceled by suitably pe… view at source ↗
Figure 2
Figure 2. Figure 2: Homotopical dynamical cancellation between x4, x5, x6 and the flow lines connecting them generating a 3-sheet cone singularity x ′ 4 . In this paper, our goal is to perform homotopical deformations of flow lines, and consequently, of the phase space itself. These deformations lead to collisions of singularities, reducing their total number by producing more degenerate ones, while preserving the homotopy ty… view at source ↗
Figure 3
Figure 3. Figure 3: Homotopical dynamical cancellation between x4, x5, x6 and the flow lines connecting them results in the new singularity x ′ 4 . In Section 3, we introduce the class of Generalized Gutierrez–Sotomayor (GGS) manifolds and their associated flows. The singular nature of these spaces presents substantial difficulties, as we no longer operate within a differentiable framework, unlike in the case of smooth manifo… view at source ↗
Figure 4
Figure 4. Figure 4: Examples of different collisions in (M, φ). (2) One can perform a total collision, that is, a collision defined by an equivalence relation ∼3 on N2 that identifies all points of S. This collapses S to a single repelling fixed point x ′′ in the new space, so that S ′′ = Invφ′′(N2/ ∼3) = {x ′′}. (3) A further collision can be achieved by identifying entire trajectories. We define an equivalence relation ∼4 o… view at source ↗
Figure 5
Figure 5. Figure 5: Collisions producing a monkey saddle [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Collision in Neimark-Sacker bifurcation. First, it can be viewed as a homotopical dynamical cancellation acting on the invariant set S, composed of the periodic orbit γ and all bounded trajectories encapsulated by γ (topologically, S is a 2-dimensional closed disk whose boundary is γ). In this interpretation, one can realize a deformation retraction of the disk, collapsing the entire disk, and the periodic… view at source ↗
Figure 7
Figure 7. Figure 7: Basic Singular Regions. Definition 3.3. Given n, m, k, k′ ∈ N with n, m ≥ 2 and k, k′ ≥ 1, define the following mixed singular regions: (i) For any basic singular regions P and Q, let P ∨ Q denote the singular region formed by the wedge sum of a P region and a Q region at their distinguished singular point; (ii) The singular region WnQ, where Q = Dm, T2k+1, is obtained by identifying the disc D1 of a Dm-re… view at source ↗
Figure 8
Figure 8. Figure 8: Mixed singular regions. neighborhood of K. In this context, a map f : K1 → K2 between subsets K1 ⊂ R m and K2 ⊂ R n is called a diffeomorphism of class C r if both f and its inverse f −1 are of class C r . Definition 3.4. A GGS manifold is a subset M ⊂ R ℓ such that for all p ∈ M there is a neighborhood Vp ⊆ M of p and a diffeomorphism of class C∞ ψ : Vp → P such that ψ(p) = 0, where P is either a regular … view at source ↗
Figure 9
Figure 9. Figure 9: Inclusion diagram of the classes of pairs. Note that GS singularities are specific instances of GGS singularities. See [20] for a detailed discussion of the natures of the GS case. Locally, the singularities associated with a GGS flow that satisfy the condition H exhibit in the following natures: • If p ∈ M(Cn) for n > 2, we say that p has n-sheet cone type (Cn-type) and super attractor (resp., repeller ) … view at source ↗
Figure 10
Figure 10. Figure 10: Morsifications of connections through the fold between cross-cap. (c) Finally, consider the case of a repelling cross-cap x and a stable saddle cross-cap y (resp. an unstable saddle and an attractor), with the isolating block N2 for y as shown in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Morsifications of connections through the fold between double crossing. (4) Finally, we present the morsifications for the mixed singular regions. The morsification of these regions proceeds naturally, and we illustrate the following cases: (a) Let p be a singularity of type P ∨ Q. We present a local morsification of p. To do so, let N be an isolating neighborhood of p. By Definition 3.3, we have N = N1 ∨… view at source ↗
Figure 12
Figure 12. Figure 12: Morsification of a saddle cone singularity and its connections with attracting and repelling singularities. where ∂ m ∗ denotes the boundary map of the Morse complex of Mf. Indeed, if both nu ̸= 0 and nv ̸= 0, then it must hold that nue = nue′ and nve = nve′, which implies that nue′ · nve′ + nue · nve ̸= 0. This leads to a contradiction, since we are in a Morse case. Therefore n(h i 2 (x), hj 1 (y)) · n(h… view at source ↗
Figure 13
Figure 13. Figure 13: Examples of morsification of repeller cross-cap singulariries x ∈ M(W4) [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Morsification possibilities for connections between cross-caps through folds. Now, that we have characterized each boundary map via the boundary map of the morsification, we can do the same for their compositions. It follows directly from items (1) -(6) and the linearity of the boundary maps that: △1 ◦ △2⟨h i 2 (x)⟩ = X p l=1 ∂ m 1 ◦ ∂ m 2 (xe l ), where p ∈ N depends on the morsification process of x as … view at source ↗
Figure 15
Figure 15. Figure 15: Morsifications of the saddle cone x and its connections. If nu = 0 then nv = 0. Indeed, by definition, if nu = 0, we have nue ̸= nue′, (3) and in the morsified manifold we also have: nue′ ̸= nve′ (4) nue ̸= nve. (5) From (3) and (4) we deduce nue = nve′ and from (3) and (5) we obtain nue′ = nve. Thus, nve′ = nue ̸= nue′ = nve Hence, nve′ ̸= nve and nv = 0. Therefore, the column corresponding to h 1 1 (x) … view at source ↗
Figure 16
Figure 16. Figure 16: GGS manifold with cone, cross-cap and double crossing singularities, and its morsification. In this example, the GGS chain groups are given by: Ck(M, X;Z) =    Z⟨h 1 2 (x1)⟩ ⊕ Z⟨h 2 2 (x1)⟩ ⊕ Z⟨h 1 2 (x2)⟩, if k = 2; Z⟨h 1 1 (x3)⟩, if k = 1; Z⟨h 1 0 (x4)⟩ ⊕ Z⟨h 2 0 (x4)⟩ ⊕ Z⟨h 1 0 (x5)⟩, if k = 0. The boundary maps are fully determined once their values on the generators of the GGS chain groups are s… view at source ↗
Figure 17
Figure 17. Figure 17: GGS manifold M and its morsification Mf [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Spectral sequence diagram for Example 7.6 [PITH_FULL_IMAGE:figures/full_fig_p037_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Initial GGS manifold M and first cancellation. h 1 0 (x12) h 1 0 (x8) h 1 0 (x11) h 1 0 (x7) h 1 0 (x6) h 1 0 (x10) h 1 1 (x9) h 1 1 (x4) h 1 1 (x5) h 2 1 (x5) h 1 2 (x ′ 1 ) h 2 2 (x ′ 1 ) h 1 0 (x12) 0 0 0 0 0 0 0 0 0 +1 0 0 h 1 0 (x8) 0 0 0 0 0 0 0 0 +1 0 0 0 h 1 0 (x11) 0 0 0 0 0 0 −1 0 0 −1 0 0 h 1 0 (x7) 0 0 0 0 0 0 0 +1 −1 0 0 0 h 1 0 (x6) 0 0 0 0 0 0 0 −1 0 0 0 0 h 1 0 (x10) 0 0 0 0 0 0 +1 0 0 0 0… view at source ↗
Figure 20
Figure 20. Figure 20: GGS manifold M after first cancellation and after second cancellation. (M3, X3) x ′ 1 x5 x ′ 6 x8 x ′ 10 x12 (M4, X4) x ′ 1 x ′ 5 x ′ 10 x12 [PITH_FULL_IMAGE:figures/full_fig_p039_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: GGS manifold M after third cancellation and after fourth cancellation. The differential d 3 7 determines the homotopical dynamical cancellation of h 1 1 (x4) and h 1 0 (x6), which can be done since n(h 1 1 (x4), h1 0 (x6)) = −1. Cancelling these generators together with h 1 0 (x7), gives rise to (M3, X3) which contains an attracting regular singularity x ′ 6 , as shown in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 22
Figure 22. Figure 22: Initial GGS manifold and final manifold with core flow, after 5 cancellations. Finally, the differential d 7 9 determines the homotopical dynamical cancellation of h 1 1 (x ′ 5 ) and h 1 0 (x ′ 10), which can be performed since n(h 1 1 (x ′ 5 ), h1 0 (x ′ 10)) = −1, Thus cancelling these generators together with h 1 0 (x12), yields a GGS manifold equipped with a core flow (M5, X5), in which x ′ 1 and x ′ … view at source ↗
read the original abstract

We introduce collisions of invariant sets and, in particular, consider dynamical homotopical cancellations that preserve the homotopy type of the underlying singular manifold. We develop the theory of homotopical dynamical cancellation for generalized Gutierrez-Sotomayor (GGS) flows defined on GGS manifolds. This framework extends the classical cancellation theory of Morse flows to the singular setting. To effectively capture these homotopical cancellations, we introduce a GGS chain complex, which encodes essential dynamical and algebraic-topological information. Furthermore, we provide a spectral sequence analysis of a filtered GGS chain complex, demonstrating a bijective correspondence between algebraic cancellations of the modules of the spectral sequence and homotopical dynamical cancellations in the GGS flow. Several illustrative examples are presented, highlighting the practical applicability of the proposed framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces collisions of invariant sets and homotopical dynamical cancellations that preserve the homotopy type of the underlying singular manifold for generalized Gutierrez-Sotomayor (GGS) flows on GGS manifolds. It extends classical Morse cancellation theory to this singular setting by defining a GGS chain complex that encodes dynamical and algebraic-topological information, then analyzes a filtered version of this complex via spectral sequences to demonstrate a bijective correspondence between algebraic cancellations of the spectral sequence modules and homotopical dynamical cancellations in the GGS flow, with several illustrative examples.

Significance. If the bijective correspondence is rigorously established without residual extensions or interfering differentials, the work would supply a useful algebraic-topological tool for tracking singularity collisions and homotopy-preserving cancellations in singular dynamical systems, extending Morse theory in a concrete way. The introduction of the GGS chain complex and its spectral sequence filtration represents a potentially reusable framework if the encoding of dynamical information is shown to be faithful.

major comments (1)
  1. [Spectral sequence analysis of the filtered GGS chain complex] The central claim of a bijective correspondence between algebraic cancellations on the spectral sequence modules and homotopical dynamical cancellations requires that the filtration on the GGS chain complex isolates each cancellation so that no non-trivial extensions or higher differentials on later pages identify distinct dynamical events. The manuscript states the correspondence but supplies no explicit computation of the differentials or proof that the associated graded pieces contain no hidden relations; this is load-bearing for the bijection and could fail on manifolds with non-trivial homotopy in the singular strata.
minor comments (1)
  1. The abstract refers to 'several illustrative examples' but does not indicate in the introduction or statement of results how each example explicitly verifies the bijective correspondence rather than merely illustrating the setup.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Spectral sequence analysis of the filtered GGS chain complex] The central claim of a bijective correspondence between algebraic cancellations on the spectral sequence modules and homotopical dynamical cancellations requires that the filtration on the GGS chain complex isolates each cancellation so that no non-trivial extensions or higher differentials on later pages identify distinct dynamical events. The manuscript states the correspondence but supplies no explicit computation of the differentials or proof that the associated graded pieces contain no hidden relations; this is load-bearing for the bijection and could fail on manifolds with non-trivial homotopy in the singular strata.

    Authors: We thank the referee for highlighting this crucial requirement for rigor. The bijective correspondence relies on the filtration of the GGS chain complex isolating cancellations such that extensions and higher differentials do not mix distinct dynamical events. The manuscript establishes the correspondence by showing that the spectral sequence converges to the homology encoding the preserved homotopy type, with the algebraic cancellations corresponding to the dynamical ones via the chain complex construction. However, we agree that explicit computations of the differentials and a proof that the associated graded pieces contain no hidden relations are needed to fully confirm isolation, especially to address potential issues with non-trivial homotopy in singular strata. In the revised manuscript we will add these explicit computations for the relevant pages of the spectral sequence together with a lemma establishing the absence of interfering relations under the GGS manifold and flow hypotheses. We will also include a short discussion clarifying why the controlled nature of singularities in GGS manifolds prevents such homotopy-related interferences from affecting the bijection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external Morse theory and introduces new objects independently

full rationale

The paper introduces the GGS chain complex and filtered spectral sequence as new constructions to capture homotopical cancellations in generalized Gutierrez-Sotomayor flows, then claims a bijective correspondence via spectral sequence analysis. This extends classical Morse cancellation theory without reducing any central equation or prediction to a fitted parameter or self-citation by construction. The definitions of GGS manifolds and flows are presented as extensions of prior literature rather than tautological redefinitions of the claimed bijection. No load-bearing step equates the algebraic cancellation modules directly to dynamical events via self-reference alone; the correspondence is asserted as a theorem to be proven from the filtered complex. The derivation remains self-contained against external benchmarks such as standard spectral sequence theory and Morse homology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard results from algebraic topology and Morse theory for the background cancellation concepts, plus the new definitions of GGS flows and the chain complex introduced here. No free parameters or invented physical entities are evident from the abstract.

axioms (1)
  • standard math Standard results from algebraic topology and classical Morse cancellation theory apply to the underlying singular manifolds and flows.
    The paper explicitly states that the framework extends the classical cancellation theory of Morse flows to the singular setting.
invented entities (1)
  • GGS chain complex no independent evidence
    purpose: Encodes essential dynamical and algebraic-topological information to capture homotopical cancellations
    Introduced in the paper as a new tool for the filtered spectral sequence analysis.

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