2024/09/11 by Hokuto Konno, Konno, Hokuto, Abhishek Mallick +3
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2409.07287
openalex publication_date 2024/09/11 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28
We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two versions of families (Seiberg-Witten) generalizations of Donaldson's diagonalization theorem, including a real and families version of the diagonalization. This leads to an example of a pair of exotically knotted ℝP2's embedded in a closed 4-manifold whose complements are diffeomorphic, making it the first example of a non-orientable surface with this property. In particular, any invariant of a 4-manifold-surface pair (including invariants from real Seiberg-Witten theory such as Miyazawa's invariant) fails to detect such an exotic ℝ P2. One consequence of our construction reveals that non-effective embeddings of corks can still be useful in pursuit of exotica. Precisely, starting with an embedding of a cork C in certain a 4-manifold X where the cork-twist does not change the diffeomorphism type of X, we give a construction that provides examples of exotically knotted spheres and ℝP2's with diffeomorphic complements in C # S2 × S2 ⊂ X # S2 × S2 or C # ℂP2 ⊂ X # ℂP2 . In another direction, we provide infinitely many exotically knotted embeddings of orientable surfaces, closed surface links, and 3-spheres with diffeomorphic complements in once stabilized corks, and show some of these surfaces survive arbitrarily many internal stabilizations. By combining similar methods with Gabai's 4D light-bulb theorem, we also exhibit arbitrarily large difference between algebraic and geometric intersections of certain family of 2-spheres, embedded in a 4-manifold.