Deformations of circle-valued Morse functions on surfaces

Abstract

Abstract. Let M be a smooth connected orientable compact surface. Denote by F cov (M, S 1 ) the space of all Morse functions f : M → S 1 having no critical points on ∂M and such that for every connected component V of ∂M , the restriction f : V → S 1 is either a constant map or a covering map. Endow F cov (M, S 1 ) with C ∞ -topology. In this note the connected components of F cov (M, S 1 ) are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on ∂M .

Deformations of circle-valued Morse functions on 2-torus

“…On the other hand, if a smooth function f : M Ñ P has only isolated critical points, then by theorem proved by P. T. Church and J. G. Timourian [1] and independently by O. Prishlyak [36], the local topological structure of level sets near any critical point can be realized by level sets of homogeneous polynomial without multiple factors. So the space F(M, P ) consists of "generic" maps with "topologically generic" critical points, see [26,Section 3,4].…”

“…where each A i is either a critical point of f , or a regular leaf of f , or an f -regular neighborhood of some (regular or critical) leaf of f . Note that if X is an f -adapted subsurface, then f | X : X Ñ P belongs to F(X, P ), see [26].…”

“…We restrict out attention to the class of smooth P -valued functions F on a smooth compact surface which satisfies two conditions: functions take constant values on each boundary component and near every critical point can be represented as homogeneous polynomial R 2 Ñ R of degree ě 2 without multiple factors (for definition of this class of functions F, see subsection 2.2). It is well-known that the class F consists of "generic" functions with "topologically generic" singularities ( [26,Section 3,4], subsection 2.2). Thus our restriction is insignificant, since the class F is "wide" enough.…”

“…It is also known that if f : M Ñ P has exactly n critical points, then O f (f ) is homotopy equivalent to some covering space of n th configuration space of M and π 1 O f (f ) is a subgroup of n th braid group B n (M ) [21,Theorem 2,Corollary 4]. A good overview the reader can find in [26,27] where these results are presented in the form of so-called crystallographic and Bieberbach sequences.…”