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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by Oliviero Mal... klokken arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.20249.pdfDypere Spørsmål