  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.
