Topology Of Real And Angle Valued Maps And Graph Representations (A Brief Survey)

TL;DR

This paper surveys a new topological invariant framework for real and angle-valued maps using graph representations, introducing barcodes and Jordan cells that generalize Morse-Novikov invariants with stability and homotopy invariance.

Contribution

It introduces a novel class of computable topological invariants based on graph representations, providing an alternative to Morse-Novikov theory for analyzing tame maps.

Findings

Stability results for bar codes.

Homotopy invariance of Jordan cells.

Introduction of new polynomials refining Betti numbers.

Abstract

(lecture delivered at the Congress of the Romanian mathematicians, Brasov, June 2011) Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to Morse-Novicov theory for real or angle valued Morse maps. The invariants are "barcodes" and "Jordan cells". From them one can derive all familiar topological invariants which can be derived via Morse-Novikov theory, like the Betti numbers and in the case of angle valued maps also the Novikov Betti numbers and the monodromy. Stability results for (some) bar codes and the homotopy invariance of the Jordan cells are the key results, and two new polynomials for any nonnegative integer (up to the dimension of the source) associated to a continuous nonzero complex valued map provide potentially interesting refinements of the Betti numbers and of the Novikov Betti numbers. In our theory the bar codes which are intervals with ends critical values/angles, the Jordan cells and the " canonical long exact sequence" of a tame map are the analogues of instantons between rest points, closed trajectories and of the Morse-Smale complex of the gradient of a Morse function in the Morse-Novikov theory.