Topology of closed one-forms.

*(English)*Zbl 1052.58016
Mathematical Surveys and Monographs 108. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3531-9/pbk). xi, 246 p. (2004).

This excellent monograph deals with an important topic of modern mathematics: the study of geometrical, topological and dynamical properties of closed 1-forms on a manifold. The book is organized into ten chapters and in what follows we will briefly present each of them.

Chapters 1 and 2 present a detailed introduction into Novikov theory concerning a natural generalization of Morse theory in which instead of critical points of smooth functions on a manifold are considered closed 1-forms and their zeros. The geometric ideas which motivate this important theory as well as explicite computations in some examples are also given.

Chapters 3 and 4 study the universal chain complex and are based on the author’s paper [Commun. Contemp. Math. 1, No. 4, 467–495 (1999; Zbl 0964.57030)]. These two chapters play a central role in the book and give a general answer to the problem of constructing the Novikov complexes over different extensions of the group ring of the manifold. The construction of the universal complex described in these chapters uses a new method of algebraic collapse originally introduced in the author’s work with A. Ranicki [Proc. Steklov Inst. Math. 225, 363–371 (1999; Zbl 0983.58007)]. The universal complex gives many different Novikov complexes and many different inequalities comparing the number of zeros to the Betti numbers of certain local coefficient systems. The author shows by example that these new inequalities are sometimes stroger than the Novikov inequalities.

In Chapter 5 the author presents several different generalizations of the Novikov inequalities some of them obtained with M. Braverman [C. R. Acad. Sci., Paris, Sér. I 321, No. 7, 897–902 (1995; Zbl 0846.58016), Math. Proc. Camb. Philos. Soc. 122, No. 2, 357–375 (1997; Zbl 0894.58012)]. They relate the Poincaré polynomials of different connected components of the set of zeros to the Novikov counting polynomial of the manifold.

In Chapter 6, the author describes Novikov-type inequalities by using the von Neumann Betti numbers instead of the Novikov numbers. These interesting results are based on the author’s paper [Proc. Am. Math. Soc. 128, No. 9, 2819–2827 (2000; Zbl 0952.58017)].

The main purpose of Chapter 7 is to suggest an equivariant version of the critical point theory for closed 1-forms. The author describes relations in an equivariant setting between the topology of the set of zeros of a closed equivariant basic 1-form and suitable cohomological invariants of the manifold. As an interesting application, the equivariant generalization of Novikov theory is used to compute the cohomology of the fixed point set of a symplectic circle action.

In Chapter 8, the author describes the main result of his paper [Funct. Anal. Appl. 19, 40–48 (1985; Zbl 0603.58030)] about the exactness of the Novikov inequalities for manifolds with infinite cyclic fundamental group. This strong result states that for the considered category of manifolds in any nonzero cohomology one may find a closed 1-form for which the Novikov inequalities become equalities. Some other interesting consequences of this result are also included.

The problem of improving the Novikov inequalities for closed 1-forms with Morse-type zeros with the additionally property that they are harmonic with respect to a Riemannian metric is discussed in Chapter 9. The author presents his results obtained with G. Katz and J. Levine [Topology 37, No. 3, 469–483 (1998; Zbl 0911.58001)]. This chapter also contains a detailed study of the geometric properties of singular foliations of closed 1-forms.

In Chapter 10, the author suggests a Lusternik-Schnirelmann-type critical point theory for closed 1-forms by indicating a generalization of the Lusternik-Schnirelmann category. The results contained in this chapter are based on the author’s recent paper [Topol. Methods Nonlinear Anal. 19, No. 1, 123–152 (2002; Zbl 1098.57020)].

The book ends with four appendices containing some additionally expositions on manifolds with corners, Morse-Bolt functions on manifolds with corners, Morse-Bolt inequalities, relative Morse theory. The book also contains a suggestive bibliography and a useful index.

The book is written in a very clear and rigorous manner and it is recommended for researchers and graduated students which are interested in differential topology and its applications, global analysis, differential geometry and nonlinear analysis.

Chapters 1 and 2 present a detailed introduction into Novikov theory concerning a natural generalization of Morse theory in which instead of critical points of smooth functions on a manifold are considered closed 1-forms and their zeros. The geometric ideas which motivate this important theory as well as explicite computations in some examples are also given.

Chapters 3 and 4 study the universal chain complex and are based on the author’s paper [Commun. Contemp. Math. 1, No. 4, 467–495 (1999; Zbl 0964.57030)]. These two chapters play a central role in the book and give a general answer to the problem of constructing the Novikov complexes over different extensions of the group ring of the manifold. The construction of the universal complex described in these chapters uses a new method of algebraic collapse originally introduced in the author’s work with A. Ranicki [Proc. Steklov Inst. Math. 225, 363–371 (1999; Zbl 0983.58007)]. The universal complex gives many different Novikov complexes and many different inequalities comparing the number of zeros to the Betti numbers of certain local coefficient systems. The author shows by example that these new inequalities are sometimes stroger than the Novikov inequalities.

In Chapter 5 the author presents several different generalizations of the Novikov inequalities some of them obtained with M. Braverman [C. R. Acad. Sci., Paris, Sér. I 321, No. 7, 897–902 (1995; Zbl 0846.58016), Math. Proc. Camb. Philos. Soc. 122, No. 2, 357–375 (1997; Zbl 0894.58012)]. They relate the Poincaré polynomials of different connected components of the set of zeros to the Novikov counting polynomial of the manifold.

In Chapter 6, the author describes Novikov-type inequalities by using the von Neumann Betti numbers instead of the Novikov numbers. These interesting results are based on the author’s paper [Proc. Am. Math. Soc. 128, No. 9, 2819–2827 (2000; Zbl 0952.58017)].

The main purpose of Chapter 7 is to suggest an equivariant version of the critical point theory for closed 1-forms. The author describes relations in an equivariant setting between the topology of the set of zeros of a closed equivariant basic 1-form and suitable cohomological invariants of the manifold. As an interesting application, the equivariant generalization of Novikov theory is used to compute the cohomology of the fixed point set of a symplectic circle action.

In Chapter 8, the author describes the main result of his paper [Funct. Anal. Appl. 19, 40–48 (1985; Zbl 0603.58030)] about the exactness of the Novikov inequalities for manifolds with infinite cyclic fundamental group. This strong result states that for the considered category of manifolds in any nonzero cohomology one may find a closed 1-form for which the Novikov inequalities become equalities. Some other interesting consequences of this result are also included.

The problem of improving the Novikov inequalities for closed 1-forms with Morse-type zeros with the additionally property that they are harmonic with respect to a Riemannian metric is discussed in Chapter 9. The author presents his results obtained with G. Katz and J. Levine [Topology 37, No. 3, 469–483 (1998; Zbl 0911.58001)]. This chapter also contains a detailed study of the geometric properties of singular foliations of closed 1-forms.

In Chapter 10, the author suggests a Lusternik-Schnirelmann-type critical point theory for closed 1-forms by indicating a generalization of the Lusternik-Schnirelmann category. The results contained in this chapter are based on the author’s recent paper [Topol. Methods Nonlinear Anal. 19, No. 1, 123–152 (2002; Zbl 1098.57020)].

The book ends with four appendices containing some additionally expositions on manifolds with corners, Morse-Bolt functions on manifolds with corners, Morse-Bolt inequalities, relative Morse theory. The book also contains a suggestive bibliography and a useful index.

The book is written in a very clear and rigorous manner and it is recommended for researchers and graduated students which are interested in differential topology and its applications, global analysis, differential geometry and nonlinear analysis.

Reviewer: D. Andrica (Cluj-Napoca)

##### MSC:

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

58A10 | Differential forms in global analysis |

57R70 | Critical points and critical submanifolds in differential topology |

57R30 | Foliations in differential topology; geometric theory |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |