論文の公開元へVERTICAL 3-MANIFOLDS IN SIMPLIFIED GENUS-2 TRISECTIONS OF 4-MANIFOLDS
概要
dimensional handlebodies introduced by Gay and Kirby [5]. They proved that any closed orientable smooth 4-manifold has a trisection. While the trisection has a strong meaning as a 4-dimensional analog of the Heegaard splitting of 3-manifolds, it is also deeply related to the study of stable maps as homotopy-deformations of stable maps are used in their proof. The singular value set of a stable map of a trisection is the union of immersed circles with cusps as in Fig.1, where the singular value set in the white boxes consists of immersed curves with only normal double points and without cusps and radial tangencies. This stable map is called a trisection map. The fiber Σg over the center point p in the figure is a closed orientable surface of genus g, and this has the highest genus among all regular fibers of the trisection map. Set k to be the number of arcs in Fig.1 connecting to neighboring white boxes without cusps. In this case, the trisection is called a (g, k)-trisection. The vanishing cycles of indefinite folds of the trisection map can be represented by simple closed curves on the center fiber Σg . The vanishing cycles of a trisection determines the source 4-manifold up to diffeomorphism. The surface Σg with these simple closed curves is called a trisection diagram. To find the usage of trisections, there are several studies of constructing trisections of given 4-manifolds [10, 14, 12]. It is also used for studies of surfaces embedded in 4manifolds [15, 11]. Classifications of 4-manifolds admitting trisection maps for g = 1 is easy, and that for g = 2 had been done by Meier and Zupan in [16]. The classification for g ≥ 3 is still difficult. On the other hand, it is a long-standing problem in the study of topology of mappings to 2020 Mathematics Subject Classification. Primary 57R35; Secondary 57R65. ...
