2025/01/29 by Meier, Jeffrey, Scott, Evan · 1 citation
#57K40 #57K45 #57M12 #57M60 #57R65 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2501.17999
Let G be a finite group, and let X be a smooth, orientable, connected, closed 4-dimensional G-manifold. Let S be a smooth, embedded, G-invariant surface in X. We introduce the concept of a G-equivariant trisection of X and the notion of G-equivariant bridge trisected position for S and establish that any such X admits a G-equivariant trisection such that S is in equivariant bridge trisected position. Our definitions are designed so that G-equivariant (bridge) trisections are determined by their spines; hence, the 4-dimensional equivariant topology of a G-manifold pair (X,S) can be reduced to the 2-dimensional data of a G-equivariant shadow diagram. As an application, we discuss how equivariant trisections can be used to study quotients of G-manifolds. We also describe many examples of equivariant trisections, paying special attention to branched covering actions, hyperelliptic involutions, and linear actions on familiar manifolds such as S4, S2× S2, and \mathbbCP2. We show that equivariant trisections of genus at most one are geometric, and we give a partial classification for genus-two.