Triangular Bases of Unimodular Bilinear Lattices and Induced Structures: Automorphism Groups, Vanishing Cycles, Monodromy Groups, Braid Group Actions ... Spaces (Lecture Notes in Mathematics, 2398)

This monograph presents a structured and conceptually unified study of upper triangular integer matrices and the rich family of structures they induce.Central objects include unimodular bilinear lattices, vanishing cycles, monodromy groups, braid group actions, and associated moduli spaces.These constructions naturally provide a common framework linking algebraic geometry, representation theory, singularity theory, and the theory of irregular meromorphic connections.To a given matrix is associated a Z-lattice with a unimodular bilinear form (the Seifert form) and a triangular basis.This leads to even and odd intersection forms, reflections and transvections, monodromy groups, and corresponding vanishing cycles.Braid group actions generate orbits of distinguished bases and matrices, which in turn give rise to complex manifolds obtained by gluing Stokes regions.General tools and results are developed throughout, with a systematic analysis of the cases of rank 2 and 3.Already in rank 3 a