Report Title：Topology of 4-manifolds
Reporter：Byeorhi Kim Pohang University of Science and Technology
Report time：October 20 - October 29, 2025
Report location：Zoom ID: 821 6588 8291, Passcode: 3UTCb6
Specific arrangements:
October 20th 13：30-15：00
Title: Topology of 4-manifolds : 1. Basic tools
Content:
This talk will be based primarily on the textbook of Freedman and Quinn. I will explain several key techniques for handling immersions of surfaces. The central tool is the immersion lemma, which is taken axiomatically as a principle for dealing with embeddings of disks. I will also discuss the twisting operation, which is used in the construction of Whitney disks. These Whitney disks are in turn employed to define the Whitney move, one of the fundamental geometric operations in topology across all dimensions.
October 22nd 13：00-14：30
Title: Topology of 4-manifolds : 2. Eliminating intersections
Content:
This talk will be based primarily on the textbook of Freedman and Quinn. I will discuss several techniques for eliminating intersections, including the Whitney move, the finger move, and the connected sum. We will also examine the algebraic condition provided by the intersection number, which determines when these geometric operations can be applied. In addition, I will introduce the notion of a transverse sphere, which offers special cases where intersections can be removed.
October 27th 13：30-15：00
Title: Topology of 4-manifolds : 3. Capped surfaces
Content:
This talk will be based primarily on the textbook of Freedman and Quinn. In this talk, we focus on the notion of capped surface, a key tool in the disk embedding theorem. Roughly speaking, a capped surface is obtained from an immersed surface by attaching two–dimensional caps, such as Whitney disks or dual disks. These caps, typically in the form of immersed disks, are attached along the boundary of the original surface (or along the boundary of a Whitney disk) and serve to eliminate intersections or control algebraic intersection numbers.
October 29th 13：30-15：00
Title: Topology of 4-manifolds : 4. Capped grope
Content:
This talk will be based primarily on the textbook of Freedman and Quinn. In this talk, we introduce capped groups, obtained by iterating the construction of capped surfaces. The key property of gropes is that once a certain height has been obtained they can “grow.” It is used to disengage the image of gropes from the fundamental group of the manifold, and in a controlled setting to disengage images from each other. Together with the use of capped surfaces, capped gropes provide a powerful tool to control algebraic intersections while simultaneously resolving geometric intersections.
