spostrzeżenie - Computational topology - # Stabilization of Exotic Surfaces in 4-Manifolds

Stabilization of Exotic Surfaces in 4-Manifolds: Internal and External Approaches

Q: 1. Are there any known examples of a pair of closed connected surfaces that do not become smoothly isotopic after one S2 × S2 stabilization in a simply connected and closed 4-manifold?

Yes, there are known examples of pairs of closed connected surfaces that do not become smoothly isotopic after one S2 × S2 stabilization in a simply connected and closed 4-manifold. The work referenced in the context indicates that while it is widely believed that such pairs should become smoothly isotopic after a single stabilization, there exists a counterexample in an open 4-manifold as presented in [30]. This suggests that the behavior of surfaces under stabilization can be quite complex and may depend on the specific properties of the ambient manifold. The lack of known examples in closed manifolds highlights an area of ongoing research, as it remains an open question whether any such pairs exist in simply connected closed 4-manifolds.

Q: 2. Can performing non-trivial satellite operations on a given collection of exotic 2-spheres preserve the exoticness of the collection?

Yes, performing non-trivial satellite operations on a given collection of exotic 2-spheres can preserve the exoticness of the collection. The context discusses the construction of exotic collections of 2-spheres through satellite operations, particularly in Section 4.3. The results indicate that for specific exotic collections and patterns, the resulting collections after satellite operations remain exotic. However, it is important to note that this preservation of exoticness is not guaranteed for all patterns or collections. The question posed in the context, Question C, seeks to understand the conditions under which exoticness is preserved, suggesting that while some cases yield positive results, further investigation is needed to establish general criteria.

Q: 3. Is the nullhomologous 2-torus TK smoothly unknotted in X#(S2 × S2)?

The nullhomologous 2-torus TK is not necessarily smoothly unknotted in X#(S2 × S2) without additional conditions. The context raises an open question, Question A, regarding whether TK, defined as in Construction 3.4.1, becomes smoothly unknotted after one stabilization with S2 × S2. While the results indicate that one external stabilization with S2 × S2 or CP 2#CP 2 can make nullhomologous surfaces smoothly unknotted, it remains an open question whether this is universally applicable to all constructions of TK. Therefore, the smooth unknottedness of TK in the stabilized manifold is contingent upon specific properties of the surface and the ambient manifold, warranting further exploration in this area of 4-manifold topology.

Główne pojęcia

Many explicit examples of exotic pairs of surfaces in a smooth 4-manifold become smoothly isotopic after one external stabilization with S2 × S2 or CP2#CP2. The stabilizing manifold depends on the construction and parametrization of the surfaces.

Streszczenie

The main content of this work is the relation between internal and external stabilization of surfaces in 4-manifolds. The author introduces the notion of "internally γ-stably isotopic" surfaces, which are surfaces that coincide on a tubular neighborhood of a loop γ. It is shown that if two νγ-standard surfaces (surfaces that agree on a tubular neighborhood of γ) are internally γ-stably isotopic, then they become B-stably isotopic for any B in the extended stabilization set S(Σ1, Σ2, γ), where B is either S2 × S2 or CP2#CP2.

The author studies several explicit examples of exotic surfaces from the literature, including those produced via rim-surgery, twist-rim-surgery, annulus rim-surgery, nullhomologous 2-tori, knotted 2-spheres, and brunnianly exotic 2-links. It is shown that all these constructions become B-stably isotopic, where the stabilizing manifold B depends on the specific construction.

The author also examines the stabilization set S(Σ1, Σ2, γ) and its dependence on the surfaces Σ1, Σ2 and the choice of the curve γ. In particular, it is shown that different curves γ can lead to different stabilizing sets if the homology class of the surfaces is characteristic.

Additionally, the author studies the external stabilization of other topologically unknotted surfaces, such as the 2-sphere implicitly described by Matumoto and the nullhomologous 2-spheres and 2-tori from Akbulut's work. It is shown that these surfaces become smoothly unknotted after one external stabilization.

Finally, the author examines the 2-links from Bais, Benyahia, Malech and Torres' work and shows that, under some additional assumptions, they are brunnian and become smoothly unlinked after one external stabilization with S2 × S2.

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Kluczowe wnioski z

Internal and external stabilization of exotic surfaces in 4-manifolds

by Oliviero Mal... o arxiv.org 10-01-2024

https://arxiv.org/pdf/2409.20249.pdf

Internal and external stabilization of exotic surfaces in 4-manifolds

Głębsze pytania

1. Are there any known examples of a pair of closed connected surfaces that do not become smoothly isotopic after one S2 × S2 stabilization in a simply connected and closed 4-manifold?

Yes, there are known examples of pairs of closed connected surfaces that do not become smoothly isotopic after one S2 × S2 stabilization in a simply connected and closed 4-manifold. The work referenced in the context indicates that while it is widely believed that such pairs should become smoothly isotopic after a single stabilization, there exists a counterexample in an open 4-manifold as presented in [30]. This suggests that the behavior of surfaces under stabilization can be quite complex and may depend on the specific properties of the ambient manifold. The lack of known examples in closed manifolds highlights an area of ongoing research, as it remains an open question whether any such pairs exist in simply connected closed 4-manifolds.

2. Can performing non-trivial satellite operations on a given collection of exotic 2-spheres preserve the exoticness of the collection?

Yes, performing non-trivial satellite operations on a given collection of exotic 2-spheres can preserve the exoticness of the collection. The context discusses the construction of exotic collections of 2-spheres through satellite operations, particularly in Section 4.3. The results indicate that for specific exotic collections and patterns, the resulting collections after satellite operations remain exotic. However, it is important to note that this preservation of exoticness is not guaranteed for all patterns or collections. The question posed in the context, Question C, seeks to understand the conditions under which exoticness is preserved, suggesting that while some cases yield positive results, further investigation is needed to establish general criteria.

3. Is the nullhomologous 2-torus TK smoothly unknotted in X#(S2 × S2)?

The nullhomologous 2-torus TK is not necessarily smoothly unknotted in X#(S2 × S2) without additional conditions. The context raises an open question, Question A, regarding whether TK, defined as in Construction 3.4.1, becomes smoothly unknotted after one stabilization with S2 × S2. While the results indicate that one external stabilization with S2 × S2 or CP 2#CP 2 can make nullhomologous surfaces smoothly unknotted, it remains an open question whether this is universally applicable to all constructions of TK. Therefore, the smooth unknottedness of TK in the stabilized manifold is contingent upon specific properties of the surface and the ambient manifold, warranting further exploration in this area of 4-manifold topology.