2019/10/25 by Nithin Kavi, Kavi, Nithin
Computer Science · Mathematics · #37F20 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1910.11954
openalex publication_date 2019/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We start with a disk with 2n vertices along its boundary where pairs of vertices are connected with n strips with certain restrictions. This forms a \it pairing. To relate two pairings, we define an operator called a cut-and-glue operation. We show that this operation does not change an invariant of pairings known as the \it signature. Pairings with a signature of 0 are special because they are closely related to a topological construction through cut and glue operations that have other applications in topology. We prove that all balanced pairings for a fixed n are connected on a surface with any number of boundary components. As a topological application, combined with works of Li, this shows that a properly embedded surface induces a well-defined grading on the sutured monopole Floer homology defined by Kronheimer and Mrowka.