Abstract
In this paper one presents a collection of results about the “bar codes” and “Jordan blocks” introduced in Burghelea and Dey (Discret Comput Geom 50: 69–98 2013) as computer friendly invariants of a tame angle-valued map and one relates these invariants to the Betti numbers, Novikov–Betti numbers and the monodromy of the underlying space and map. Among others, one organizes the bar codes as two configurations of points in $$\mathbb C{\setminus } 0$$ and one establishes their main properties: stability property and when the underlying space is a closed topological manifold, Poincare duality property. One also provides an alternative computer friendly definition of the monodromy of an angle valued map based on the algebra of linear relations as well as a refinement of Morse and Morse–Novikov inequalities.
Highlights
The resultsIn this paper a nice space is a friendlier name for a locally compact ANR (Absolute Neighborhood Retract). Finite dimensional simplicial complexes and finite dimensional topological manifolds are nice spaces but the class is considerably larger
Among others, one organizes the bar codes as two configurations of points in C \ 0 and one establishes their main properties: stability property and when the underlying space is a closed topological manifold, Poincare duality property
At least for spaces homeomorphic to simplicial complexes the set of tame maps is residual in the space of all continuous maps and weakly homotopy equivalent to the space of all continuous maps
Summary
In this paper a nice space is a friendlier name for a locally compact ANR (Absolute Neighborhood Retract). Finite dimensional simplicial complexes and finite dimensional topological manifolds are nice spaces but the class is considerably larger. It is interesting to regard the elements (i), (ii), (iii), that is, the critical values, bar codes and Jordan blocks associated to a tame angle valued map f : X → S1, as parallels to the rest points, the isolated trajectories between rest points and the closed trajectories ( Poincare return maps for closed trajectories) of a vector field which has a Morse angle-valued map f : M → S1 as Lyapunov map These last ones are the concepts which enter the Morse–Novikov theory, cf [30, 31], and are related to the topology of (X, ξf ), where ξf denotes the integral cohomology class defined by f , in a similar way as the elements described in (i), (ii) and (iii) are. The present version ows much to his critics and suggestions
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