Graph Representations and Topology of Real and Angle Valued Maps

Details Graph Representations and Topology of Real and Angle Valued Maps Graph Representations and Topology of Real and Angle Valued Maps

2012-02-06

arxiv.org/abs/1202.1208v2

Mathematics

Algebraic Topology

32 pages, 2 figures

Scientific paper

In this paper we review the definition of the invariants "bar codes" and "Jordan cells" of real and angle valued tame maps as proposed in Burghelea and Dey and Carlsson et al and prove the homotopy invariance of the sum # B^c_r +#B^o_{r-1}$ and of the Jordan cells. Here B^c_r resp. B^o_r denote the sets of closed resp. open bar codes in dimension r and # denotes cardinality. In addition we provide calculation of some familiar topological invariants in terms of bar codes and Jordan cells. The presentation provides a different perspective on Morse-Novikov theory based on critical values, bar codes and Jordan cells rather than on critical points instantons and closed trajectories of a gradient of a real or angle valued map.

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