arXiv - MATH - Geometric Topology最新文献_第3页

Exotically knotted closed surfaces from Donaldson's diagonalization for families 从唐纳森对角线化的族的外结闭曲面

arXiv - MATH - Geometric Topology Pub Date : 2024-09-11 DOI: arxiv-2409.07287

Hokuto Konno, Abhishek Mallick, Masaki Taniguchi

{"title":"Exotically knotted closed surfaces from Donaldson's diagonalization for families","authors":"Hokuto Konno, Abhishek Mallick, Masaki Taniguchi","doi":"arxiv-2409.07287","DOIUrl":"https://doi.org/arxiv-2409.07287","url":null,"abstract":"We introduce a method to detect exotic surfaces without explicitly using a\u0000smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the\u0000construction. Our main tools are two versions of families (Seiberg-Witten)\u0000generalizations of Donaldson's diagonalization theorem, including a real and\u0000families version of the diagonalization. This leads to an example of a pair of\u0000exotically knotted $mathbb{R}P^2$'s embedded in a closed 4-manifold whose\u0000complements are diffeomorphic, making it the first example of a non-orientable\u0000surface with this property. In particular, any invariant of a\u00004-manifold-surface pair (including invariants from real Seiberg-Witten theory\u0000such as Miyazawa's invariant) fails to detect such an exotic $mathbb{R} P^2$.\u0000One consequence of our construction reveals that non-effective embeddings of\u0000corks can still be useful in pursuit of exotica. Precisely, starting with an\u0000embedding of a cork $C$ in certain a 4-manifold $X$ where the cork-twist does\u0000not change the diffeomorphism type of $X$, we give a construction that provides\u0000examples of exotically knotted spheres and $mathbb{R}P^2$'s with diffeomorphic\u0000complements in $ C # S^2 times S^2 subset X # S^2 times S^2$ or $C #\u0000mathbb{C}P^2 subset X # mathbb{C}P^2 $. In another direction, we provide\u0000infinitely many exotically knotted embeddings of orientable surfaces, closed\u0000surface links, and 3-spheres with diffeomorphic complements in once stabilized\u0000corks, and show some of these surfaces survive arbitrarily many internal\u0000stabilizations. By combining similar methods with Gabai's 4D light-bulb\u0000theorem, we also exhibit arbitrarily large difference between algebraic and\u0000geometric intersections of certain family of 2-spheres, embedded in a\u00004-manifold.","PeriodicalId":501271,"journal":{"name":"arXiv - MATH - Geometric Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}