Topology of angle valued maps, bar codes and Jordan blocks

Burghelea, Dan; Haller, Stefan

doi:10.1007/s41468-017-0005-x

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Topology of angle valued maps, bar codes and Jordan blocks

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Burghelea, Dan
Max Planck Institute for Mathematics, Max Planck Society;

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Burghelea, D., & Haller, S. (2017). Topology of angle valued maps, bar codes and Jordan blocks. Journal of Applied and Computational Topology, 1(1), 121-197. doi:10.1007/s41468-017-0005-x.

Cite as: https://hdl.handle.net/21.11116/0000-0004-672E-6

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 C∖0 and one establishes their main properties: stability property and when the underlying space is a closed topological manifold, Poincaré 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.