Functions with isolated singularities on surfaces
by
Sergiy Maksymenko
Institute of Mathematics, NAS of Ukraine, Tereshchenkivs'ka str., 3, Kyiv, 01601, Ukraine

Let M be a smooth connected compact surface, P be either a real line R¹ or a circle S¹, and f:M → P a smooth mapping. We introduce five axioms for f under which one is able to describe the homotopy types of connected components of the stabilizers and orbits of f with respect to the right and left-right actions of the groups of diffeomorphisms of M and P. This result extends the analogous calculations concerning Morse functions [1, 2, 5] and is based on the results of [3, 4].

In particular, suppose that f:M → P has the following properties:

(1) f is constant on every connected component of ∂M and S_f ⊂ Int(M);

(2) for every critical point z ∈ S_f there is a local presentation f:R² → R¹ of f such that z=(0, 0) ∈ R² and f(x, y) is a homogeneous polynomial of some degree p_z ≥ 2 without multiple factors.

Then f satisfies the mentioned above axioms.

Also notice that p_z=2 for all z ∈ S_f, then f is Morse and the calculations of the homotopy types are given in [1, 2, 5].

References.

[1] S. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Annals of Global Analysis and Geometry, 29 no. 3, (2006) 241-285, http://xxx.lanl.gov/abs/math/0310067

[2] S. Maksymenko, Stabilizers and orbits of smooth functions, Bulletin des Sciences Mathematiques, 130 (2006) 279-311, http://xxx.lanl.gov/abs/math/0411612

[3] S. Maksymenko, ∞-jets of diffeomorphisms preserving orbits of vector fields, http://xxx.lanl.gov/abs/0708.0737

[4] S. Maksymenko, Hamiltonian vector fields of homogeneous polynomials in two variables, Proceedings of the Institute of Mathematics of NAS of Ukraine, 2006 3 no. 3, 269-308, http://xxx.lanl.gov/abs/0709.2511

[5] S. Maksymenko, Homotopy dimension of the orbits of Morse functions on surfaces, http://xxx.lanl.gov/abs/0709.2511

Date received: February 29, 2008