A test for compactness of a foliation. (English. Russian original) Zbl 0857.57030

Mel’nikova, I. A.

Math. Notes 58, No. 6, 1302-1305 (1995); translation from Mat. Zametki 58, No. 6, 872-877 (1995).

We investigate foliations on smooth manifolds that are determined by a closed 1-form with Morse singularities. The problem of investigating the topological structure of level surfaces for such a form was posed by S. P. Novikov [Russ. Math. Surv. 37, No. 5, 1-56 (1982); translation from Usp. Mat. Nauk 37, No. 5(227), 3-49 (1982; Zbl 0571.58011)]. This problem was treated in [S. P. Novikov, Tr. Mat. Inst. Steklova 166, 201-209 (1984; Zbl 0553.58005); A. V. Zorich, Math. USSR, Izv. 31, No. 3, 635-655 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 6, 1322-1344 (1987; Zbl 0668.58004); Le Ty Kuok Tkhang, Math. Notes 44, No. 1/2, 556-562 (1988); translation from Mat. Zametki 44, No. 1, 124-133 (1988; Zbl 0656.58007); L. A. Alaniya, Russ. Math. Surv. 46, No. 3, 211-212 (1991); translation from Usp. Mat. Nauk 46, No. 3(279), 179-180 (1991; Zbl 0736.58001)]. The present paper is devoted to the compactness problem for level surfaces. We introduce the notion of degree of compactness and prove a test for compactness expressed in the terms of the degree. The present paper is a natural continuation of [the author, Math. Notes 53, No. 3, 356-358 (1993); translation from Mat. Zametki 53, No. 3, 158-160 (1993; Zbl 0809.57018)].

MSC:

57R30

Foliations in differential topology; geometric theory

Keywords:

compactness; foliations; smooth manifolds; 1-form; Morse singularities

Citations:

Zbl 0694.58002; Zbl 0666.58013; Zbl 0818.58003; Zbl 0571.58011; Zbl 0553.58005; Zbl 0668.58004; Zbl 0656.58007; Zbl 0736.58001; Zbl 0809.57018

References:

[1]	S. P. Novikov, ”The Hamiltonian formalism and a multivalued version of Morse theory,”Uspekhi Mat. Nauk [Russian Math. Surveys],37, No. 5, 3–49 (1982).
[2]	S. P. Novikov, ”Critical points and level surfaces of multivalued functions,” in:Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],166, 201–209 (1984). · Zbl 0553.58005
[3]	A. V. Zorich, ”Quasiperiodic structure of level surfaces of a 1-form close to a rational one; Novikov’s problem,”Izv. Akad. Nauk SSSR, Ser. Mat. [Math. USSR-Izv.],51, No. 6, 1322–1344 (1987). · Zbl 0668.58004
[4]	T. K. T. Le, ”On the structure of level surfaces of a Morse form,”Mat. Zametki. [Math. Notes],44, No. 1, 124–133 (1988). · Zbl 0656.58007
[5]	L. A. Alaniya, ”On the topological structure of the level surfaces of Morse 1-forms,”Uspekhi Mat. Nauk [Russian Math. Surveys],46, No. 3, 179–180 (1991). · Zbl 0818.58003
[6]	I. A. Mel’nikova, ”A noncompactness test for a foliation onM g 2,”Mat. Zametki. [Math. Notes],53, No. 3, 158–160 (1993).
[7]	Henc Damir, ”Ergodicity of foliations with singularities,”J. Funct. Anal.,75, No. 2 (1987). · Zbl 0637.58021
[8]	A. Dold,Lectures on Algebraic Topology, 2nd ed., Berlin, (1980). · Zbl 0434.55001