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1 results

  • Counting Morse functions on the 2-sphere

  • Part of
  • Liviu I. Nicolaescu
  • Journal:
    Compositio Mathematica / Volume 144 / Issue 5 / September 2008
    Published online by Cambridge University Press:
    01 September 2008, pp. 1081-1106
    Print publication:
    September 2008
    • Article
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  • We count how many ‘different’ Morse functions exist on the 2-sphere. There are several ways of declaring that two Morse functions f and g are ‘indistinguishable’ but we concentrate only on two natural equivalence relations: homological (when the regular sublevel sets f and g have identical Betti numbers) and geometric (when f is obtained from g via global, orientation-preserving changes of coordinates on S2 and ℝ). The count of homological classes is reduced to a count of lattice paths confined to the first quadrant. The count of geometric classes is reduced to a count of certain labeled trees, which is encoded by certain elliptic integrals.