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Counting Morse functions on the 2-sphere

Part of: Low-dimensional topology Combinatorics Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  01 September 2008

Liviu I. Nicolaescu
Liviu I. Nicolaescu*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA (email: [email protected])
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Abstract

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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.

Keywords

Morse functionsequivalence of smooth functionsReeb graphslattice walksCatalan numbersYoung latticefirst-order quasilinear partial differential equationselliptic functions

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 05A17: Partitions of integers 57M15: Relations with graph theory 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 5 , September 2008 , pp. 1081 - 1106
Copyright
Copyright © Foundation Compositio Mathematica 2008