Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus

Abstract

Let f : T 2 → R be a Morse function on a 2-torus, S(f ) and O(f ) be its stabilizer and orbit with respect to the right action of the group D(T 2 ) of diffeomorphisms of T 2 , D id (T 2 ) be the identity path component of D(T 2 ), andIn fact this result holds for a larger class of smooth functions f : T 2 → R having the following property: for every critical point z of f the germ of f at z is smothly equivalent to a homogeneous polynomial R 2 → R without multiple factors.2000 Mathematics Subject Classification. 57S05, 57R45, 37C05.

Deformations of smooth functions on 2-torus

“…(3) In a series of papers [17,9,16,2] S. Maksymenko and the author described groups π 1 O f (f ) for functions on 2-torus. We shortly review these results in Section 6.…”

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-torus

“…The set G loc v = {g| st(v) | g ∈ G v } which contains restrictions of elements of G v onto the star st(v) is a subgroup of Aut(st(v)), we will call it a local stabilizer of v. It is well known that the group G loc v is isomorphic to the product Z n × Z mn for some n, m ≥ 1, see [4,Theorem 2.5]. More information the reader can find in [2,16] Lemma 4.5 (Theorem 3.2 [3]). Let f be a Morse function on T 2 , and Γ f be its graph.…”

“…Note that G(f ) is the holonomy group of the compact manifold (S 1 ) m /G(f ). An algebraic structure of π 1 O f (f ) for the Morse functions on 2-torus was described in the series of papers by S. Maksymenko and the second author [23,16,22,4]. This is one of non-trivial cases since D id (T 2 ) is not contractible, so the image π 1 D id (T 2 ) in π 1 O f (f ) has non-trivial impact on the algebraic structure of π 1 O f (f ), see (1).…”

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

“…Note that G(f ) is the holonomy group of the compact manifold (S 1 ) m /G(f ). An algebraic structure of π 1 O f (f ) for the Morse functions on 2-torus was described in the series of papers by S. Maksymenko and the second author [23,16,22,4]. This is one of non-trivial cases since D id (T 2 ) is not contractible, so the image…”

“…The set G loc v = {g| st(v) | g ∈ G v } which contains restrictions of elements of G v onto the star st(v) is a subgroup of Aut(st(v)), we will call it a local stabilizer of v. It is well known that the group G loc v is isomorphic to the product Z n × Z mn for some n, m ≥ 1, see[4, Theorem 2.5]. More informations the reader can find in[2,16]. 4.4.…”