Pathcomponents of Morse map spaces of surfaces
by
S. I. Maksimenko
Institute of National Academy of Sciense of Ukraine

Let M be a smooth closed oriented surface and P be one-dimensional manifold without border, that is a line R or a circle P. Let f:M --> P be a smooth map. A critical point x in M of the map f is nondegenerate if a gessian at x (a matrix whose entries are the second partial derivatives of f at this point) is nondegenerate matrix. The map f: M --> P is Morse if all of its critical points are nondegenerate.

Let C^\infty(M, P) be the space of all C^\infty-smooth maps from M into P. Denote by M(M, P) the subspace of C^\infty(M, P) consisting of all Morse mappings. The subspace M(M, P) is open and everywhere dense in C^\infty(M, P).

Let f: M --> P be a Morse mapping. Denote by c_k, k=0, 1, 2 the number of critical points of F of index k. The triple (c₀, c₁, c₂) is called a critical type of Morse map f.

It is easy to show that if two Morse functions belong to the same component of M(M, P) then they have the same critical type. Thus, the critical type is an invariant of path-component of M(M, P). Another natural invariant is a homotopy type of Morse map, which in the case P=R plays no role since C^\infty(M, P) is connected.

Recently, Matveev S. V. [] and Sharko V. V. [], independently each from other and using different methods obtain a classification of path-components of Morse map space M(M, R).

They prove that two Morse functions f, g: M --> R belong to the same pathcomponent of M(M, R) if and only if they have the same critical type.

When P = S¹ the situation is complicated by the fact that, in general, the space C^\infty(M, S¹) is not connected. Therefore, for classification of components of M(M, S¹), we need at least two invariants of components of M(M, S¹): homotopy class (that is a component of C^\infty(M, S¹) containing given component of M) and critical type of the maps of this component.

In [] (see also  []) author shows that this pair of invarians exactly describes the pathcomponents of M(M, S¹):

Theorem 1 Let f, g: M --> S¹ be two Morse maps. They belong to the same pathcomponent of M(M, S¹) if and only if they are homotopic as continuous maps and have the same critical type.

References

[]

Kudryavtzeva E. A.  Realization of smooth functions on surfaces as height functions. // Mat. sbornik. to appear. (Russian)

[]

Sharko V. V.  Functions on surfaces, 1. // in "Some problems of modern mathematcs", Proceedings of National Academy of Science of Ukraine, 1998, vol. 25 pp. 408-434. (Russian)

[]

Maksimenko S. I.  A components of Morse map spaces. // in "Some problems of modern mathematcs", Proceedings of National Academy of Science of Ukraine, 1998, vol. 25 pp. 135-153. (Russian)

[]

Sergey Maksimenko.  On components of a space of Morse mappings from closed oriented surface into circle. (English)
Internet addres: htpp//xxx.lanl.ru\math.GT/9906031

Paper reference: arXiv:math.GT/9906031

Date received: June 24, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadc-09.