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Connected components of spaces of Morse functions with fixed critical points

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Abstract

Let M be a smooth closed orientable surface and F = F p,q,r be the space of Morse functions on M having exactly p points of local minimum, q ≥ 1 saddle critical points, and r points of local maximum, moreover, all the points are fixed. Let F f be a connected component of a function f ∈ F in F.We construct a surjection π 0(F) → ℤp+r−1 by means of the winding number introduced by Reinhart (1960). In particular, |π0(F)| = ∞, and the component F f is not preserved under the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point. Let D be the group of orientation preserving diffeomorphisms of M leaving fixed the critical points, D 0 be the connected component of id M in D, and D f ⊂ D be the set of diffeomorphisms preserving F f . Let H f be the subgroup of D f generated by D 0 and all diffeomorphisms h ∈ D preserving some function f 1 ∈ F f , and let H abs f be its subgroup generated by D 0 and the Dehn twists about the components of level curves of the functions f 1 ∈ F f . We prove thatH abs f ⊊ D f for q ≥ 2 and construct an epimorphism D f /H abs f → ℤ q−12 by means of the winding number. A finite polyhedral complex K = K p,q,r associated with the space F is defined. An epimorphism µ: π 1(K) → D f /H f and finite generating sets for the groups D f /D 0 and D f /H f in terms of the 2-skeleton of the complex K are constructed.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    E. A. Kudryavtseva

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  1. E. A. Kudryavtseva
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Original Russian Text © E.A. Kudryavtseva, 2012, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2012, Vol. 67, No. 1, pp. 3–12.

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Kudryavtseva, E.A. Connected components of spaces of Morse functions with fixed critical points. Moscow Univ. Math. Bull. 67, 1–10 (2012). https://doi.org/10.3103/S0027132212010019

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