It was shown recently that the diffeomorphism type of a closed oriented four-manifold can be described by a combinatorial data, consisting of vanishing cycles of a stable mapping from a four-manifold to a surface (cf. a surface diagram by Williams, and a trisection by Gay-Kirby). In this course, we first introduce basic notions concerning stable mappings to surfaces, such as folds, cusps and vanishing cycles. We then explain how to analyse vanishing cycles of stable mappings on four-manifolds and combinatorial data mentioned above relying on the theory of mapping class groups of surfaces.
The aim of this course is to explain the effects of generic homotopies between stable mappings on their vanishing cycles in terms of mapping class groups of surfaces. We would also like to explain how to determine combinatorial data coming from (explicit examples of) stable mappings on four-manifolds.
・Understand local models of critical points in stable mappings from 4-manifolds to surfaces
・Be familier with surgery homomorphisms on mapping class groups of surfaces
・Understand the effects of homotopies between stable mappings on their vanishing cycles in terms of mapping class groups
・Understand how to obtain diagrams of trisections relying on the theory of mapping class groups
stable mapping, fold, cusp, vanishing cycle, mapping class group, surgery homomorphism, monodoromy, parallel transport, trisection
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The following topics will be covered in this order : -- The definition and examples of stable mappings -- Critical points in stable mappings from 4-manifolds to surfaces -- Mapping class groups of surfaces -- Surgery homomorphisms -- Relation between homotopies of stable mappings and surgery homomorphisms -- Trisections of 4-manifolds||Details will be provided during each class.|
K. Hayano, Modification rule of monodromies in an R_2 move, AGT, 14(2014), no. 4, 2181-2222.
S. Behrens and K. Hayano, Elimination of cusps in dimension 4 and its applications, PLMS, (3) 113(2016), 674-724.
K. Hayano, On diagrams of simplified trisections and mapping class groups, to appear in OJM.