Counting Morse functions on the 2-sphere

Publisher： Cambridge University Press

E-ISSN： 1570-5846|144|5|1081-1106

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.144, Iss.5, 2008-09, pp. : 1081-1106

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Abstract

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 ). 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.

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