Cancellation of Critical Points in 2D and 3D Morse and Morse-Smale Complexes

  • Lidija Čomić
  • Leila De Floriani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


Morse theory studies the relationship between the topology of a manifold M and the critical points of a scalar function f defined on M. The Morse-Smale complex associated with f induces a subdivision of M into regions of uniform gradient flow, and represents the topology of M in a compact way. Function f can be simplified by cancelling its critical points in pairs, thus simplifying the topological representation of M, provided by the Morse-Smale complex. Here, we investigate the effect of the cancellation of critical points of f in Morse-Smale complexes in two and three dimensions by showing how the change of connectivity of a Morse-Smale complex induced by a cancellation can be interpreted and understood in a more intuitive and straightforward way as a change of connectivity in the corresponding ascending and descending Morse complexes. We consider a discrete counterpart of the Morse-Smale complex, called a quasi-Morse complex, and we present a compact graph-based representation of such complex and of its associated discrete Morse complexes, showing also how such representation is affected by a cancellation.


Cell Complex Morse Theory Morse Function Integral Line Dual Complex 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Lidija Čomić
    • 1
  • Leila De Floriani
    • 2
  1. 1.FTNUniversity of Novi Sad(Serbia)
  2. 2.University of Genova (Italy) and University of Maryland(USA)

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