Linear relations, monodromy and Jordan cells of a circle valued map

In this paper we review the definition of the monodromy of an angle valued map based on linear relations as proposed in [3]. This definition provides an alternative treatment of the Jordan cells, topological persistence invariants of a circle valued maps introduced in [2]. We give a new proof that homotopic angle valued maps have the same monodromy, hence the same Jordan cells, and we show that the monodromy is actually a homotopy invariant of a pair consisting of a compact ANR X and a one dimensional integral cohomology class ξ ∈ H(X;Z). We describe an algorithm to calculate the monodromy for a simplicial angle valued map f : X → S, X a finite simplicial complex, providing a new algorithm for the calculation of the Jordan cells of the map f.