Equivariant Trisections for Group Actions on Four-Manifolds

Explore advanced concepts in 4-dimensional topology through this mathematical lecture that introduces G-equivariant trisections for finite group actions on 4-manifolds. Learn about the fundamental theory where G represents a finite group acting on a smooth, orientable, connected, closed 4-dimensional manifold X, with particular focus on G-invariant surfaces S embedded within X. Discover the innovative concept of G-equivariant trisections and understand how smooth embedded G-invariant surfaces can be positioned in equivariant bridge trisected form. Master the main theoretical result demonstrating that any 4-dimensional G-manifold admits a G-equivariant trisection where the invariant surface S achieves equivariant bridge trisected position. Examine how this theory elegantly reduces complex 4-dimensional equivariant topology problems to manageable 2-dimensional data through G-invariant shadow diagrams, since G-equivariant trisections are completely determined by their spines. Understand the proof techniques involving an equivariant version of the Laudenbach and Poénaru theorem, which provides the theoretical foundation for this dimensional reduction. Gain insights into collaborative research methodologies through work developed jointly with Evan Scott, and appreciate how this framework opens new pathways for analyzing the topological structure of group actions on 4-manifolds through combinatorial and diagrammatic methods.