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    block this user Liviu Nicolaescu Trusted member

    Assistant Professor

    University of Notre Dame

    Compositio Math. 144 (2008) 1081�1106 doi:10.1112/S0010437X08003680 Counting Morse functions on the 2-sphere

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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 S 2 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. 1. The main problem Suppose that X is a smooth compact, oriented manifold without boundary. Following Thom, we say that a smooth function f: X ? R is an excellent Morse function if all of its critical points are nondegenerate, and no two of them lie on the same level set. We denote by MX the space of excellent Morse functions on X. In the remainder of this introduction a Morse function will by default be excellent.

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    New Full text for this article was not available? Send a request to the author(s).
    Title : Compositio Math. 144 (2008) 1081�1106 doi:10.1112/S0010437X08003680 Counting Morse functions on the 2-sphere
    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 S 2 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. 1. The main problem Suppose that X is a smooth compact, oriented manifold without boundary. Following Thom, we say that a smooth function f: X ? R is an excellent Morse function if all of its critical points are nondegenerate, and no two of them lie on the same level set. We denote by MX the space of excellent Morse functions on X. In the remainder of this introduction a Morse function will by default be excellent.
    Subject : unspecified
    Area : Mathematics
    Language : English
    Affiliations : 
    Url : http://www.nd.edu/~lnicolae/MorseCountCompv3.pdf
    Doi : 10.1.1.139.7851

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