Miyazawa’s 2023 preprint (Miyazawa 2023), now accepted by the Annals of Mathematics, constructs embeddings of the real projective plane in the -sphere that are topologically isotopic but not smoothly isotopic. In fact, it gives infinitely many such smooth embeddings, all equivalent from the topological point of view and all distinct from the smooth point of view.
Knot theory has appeared several times on this blog; see here and here. In those posts, we discussed ordinary knots: embedded copies of in . In dimension , one can also knot entire surfaces. A surface knot in is a smooth embedding of a closed surface . As usual, we often do not distinguish between the embedding and its image .
The simplest closed non-orientable surface is the real projective plane , the space of all lines through the origin in . A -knot is a smooth embedding where stands for “projective”. Two -knots are called smoothly isotopic if there is a continuous family of diffeomorphisms such that is the identity and They are called topologically isotopic if the same condition holds with homeomorphisms in place of diffeomorphisms. In the topological category, the embedded surface is assumed to be locally flat: near every point, it looks topologically like the standard inclusion of in . An exotic pair of embeddings is a pair that is topologically isotopic but not smoothly isotopic.
Before Miyazawa’s work, exotic pairs of non-orientable surface knots in were known, but only for surfaces of larger non-orientable genus. For example, Finashin, Kreck, and Viro (Finashin et al. 1988) constructed exotic embeddings of the connected sum of copies of , and Finashin (Finashin 2009) later gave examples for the connected sum of copies. By contrast, the case of a single projective plane remained mysterious. Miyazawa (Miyazawa 2023) proved that even this smallest case has infinitely many smooth forms.
The theorem says that the embeddings are all the same from the purely topological point of view: a homeomorphism of , continuously deformable to the identity, can move any one of them to any other. Smoothly, however, they are all different. Moreover, no diffeomorphism of , even one not assumed to be isotopic to the identity, can carry to when . Thus the gap between topology and smooth topology already appears for the smallest non-orientable surface that embeds in .