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
✔ Specialist skills | Intercultural skills | Communication 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 definition and examples of stable mappings | Details will be provided during each class session |
Class 2 | Critical points in stable mappings from 4-manifolds to surfaces | Details will be provided during each class session |
Class 3 | Mapping class groups of surfaces | Details will be provided during each class session |
Class 4 | Surgery homomorphisms -- Relation between homotopies of stable mappings and surgery homomorphisms | Details will be provided during each class session |
Class 5 | Trisections of 4-manifolds | Details will be provided during each class session |
none required
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.
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manifold theory, calculus of several variables,