de Gruyter Studies in Mathematics 32


de Gruyter Studies in Mathematics
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

18
19
20
21
22
23
24
25

26
27
28
29
30
31

Riemannian Geometry, 2nd rev. ed., Wilhelm P. A. Klingenberg
Semimartingales, Michel Me´tivier
Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup
Spaces of Measures, Corneliu Constantinescu
Knots, 2nd rev. and ext. ed., Gerhard Burde and Heiner Zieschang
Ergodic Theorems, Ulrich Krengel
Mathematical Theory of Statistics, Helmut Strasser
Transformation Groups, Tammo tom Dieck
Gibbs Measures and Phase Transitions, Hans-Otto Georgii
Analyticity in Infinite Dimensional Spaces, Michel Herve´
Elementary Geometry in Hyperbolic Space, Werner Fenchel
Transcendental Numbers, Andrei B. Shidlovskii
Ordinary Differential Equations, Herbert Amann
Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and
Francis Hirsch
Nevanlinna Theory and Complex Differential Equations, Ilpo Laine
Rational Iteration, Norbert Steinmetz
Korovkin-type Approximation Theory and its Applications, Francesco Altomare
and Michele Campiti
Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev
Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima,
Yoichi Oshima and Masayoshi Takeda
Harmonic Analysis of Probability Measures on Hypergroups, Walter R. Bloom

and Herbert Heyer
Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov
Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii,
Victor E. Shatalov and Boris Yu. Sternin
Probability Theory, Heinz Bauer
Variational Methods for Potential Operator Equations, Jan Chabrowski
The Structure of Compact Groups, Karl H. Hofmann and Sidney A. Morris
Measure and Integration Theory, Heinz Bauer
Stochastic Finance, 2nd rev. and ext. ed., Hans Föllmer and Alexander Schied
Painleve´ Differential Equations in the Complex Plane, Valerii I. Gromak, Ilpo
Laine and Shun Shimomura
Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel
and Jakob Nielsen
The Reidemeister Torsion of 3-Manifolds, Liviu I. Nicolaescu
Elliptic Curves, Susanne Schmitt and Horst G. Zimmer


Andrei V. Pajitnov

Circle-valued Morse Theory

≥
Walter de Gruyter · Berlin · New York


Author
Andrei V. Pajitnov
Laboratoire de mathe´matiques Jean Leray
UMR 6629 du CNRS
Universite´ de Nantes

2, rue de la Houssinie`re
44322 Nantes
France
E-mail:


Mathematics Subject Classification 2000:
Primary: 58E05, 37C10, Secondary: 57M25, 57R70, 37C27, 37C30
Keywords:
Circle-valued Morse functions, the Morse complex, the Novikov complex, cellular gradients,
dynamical zeta functions.

ȍ Printed on acid-free paper which falls within the guidelines of the ANSI
Ț
to ensure permanence and durability.

Library of Congress Ϫ Cataloging-in-Publication Data
A CIP catalogue record for this book is available from the Library of Congress.

ISSN 0179-0986
ISBN-13: 978-3-11-015807-6
ISBN-10: 3-11-015807-8
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at .
Ą Copyright 2006 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
All rights reserved, including those of translation into foreign languages. No part of this book may be
reproduced in any form or by any means, electronic or mechanical, including photocopy, recording,
or any information storage and retrieval system, without permission in writing from the publisher.
Printed in Germany.

Cover design: Rudolf Hübler, Berlin.
Typeset using the author’s TEX files: Matthias Pfizenmaier.
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen.


To the memory of my mother,
Nadejda Vassilievna Pajitnova



Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Part 1.

Morse functions and vector fields on manifolds . .
. . . . . . . . . . . . . .

17

Manifolds without boundary . . . . . . . . . . . . . . . . . . .
Cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

24

Chapter 1.
1.
2.

Chapter 2.
1.
2.
3.

Morse functions and their gradients . . . . . . . . . . .

33

Morse functions and Morse forms . . . . . . . . . . . . . . . .
Gradients of Morse functions and forms . . . . . . . . . . . .
Morse functions on cobordisms . . . . . . . . . . . . . . . . .

35
50
62

Chapter 3.
1.
2.
3.

Vector fields and C 0 topology

15

Gradient flows of real-valued Morse functions

. . . . .

67

Local properties of gradient flows . . . . . . . . . . . . . . . .
Descending discs . . . . . . . . . . . . . . . . . . . . . . . . .
The gradient descent . . . . . . . . . . . . . . . . . . . . . . .

67
81
99

Exercises to Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Part 2.

Transversality, handles, Morse complexes . . . . . . 109

Chapter 4.
1.
2.
3.

Perturbing the Lyapunov discs . . . . . . . . . . . . . . . . . 112
The transverse gradients are generic . . . . . . . . . . . . . . 122
Almost transverse gradients and Rearrangement Lemma . . . 132

Chapter 5.
1.
2.

The Kupka-Smale transversality theory for
gradient flows . . . . . . . . . . . . . . . . . . . . . . . 111

Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Construction of handles . . . . . . . . . . . . . . . . . . . . . 163
Morse functions and the cellular structure of manifolds . . . . 171


viii

Contents

3.
4.

Homology of elementary cobordisms . . . . . . . . . . . . . . 172
Appendix: Orientations, coorientations, fundamental classes
etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Chapter 6. The Morse complex of a Morse function . . .
1. The Morse complex for transverse gradients . . . .
2. The Morse complex for almost transverse gradients
3. The Morse chain equivalence . . . . . . . . . . . . .
4. More about the Morse complex . . . . . . . . . . .

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

195
196
208
212
218

History and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Part 3.

Cellular gradients. . . . . . . . . . . . . . . . . . . . . 229

Chapter 7.
1.
2.

The gradient descent revisited . . . . . . . . . . . . . . . . . . 232
Definition and first properties of cellular gradients . . . . . . . 238

Chapter 8.
1.
2.
3.
4.

Cellular gradients are C 0 -generic . . . . . . . . . . . . . 243

Introduction . . . . . . . . . . . . . . .
The stratified gradient descent . . . .
Quick flows . . . . . . . . . . . . . . .
Proof of the C-approximation theorem

Chapter 9.
1.
2.
3.

Condition (C) . . . . . . . . . . . . . . . . . . . . . . . 231

.
.
.
.

.
.
.
.

.
.
.
.

.

.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.

.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

243
244
254
274

Properties of cellular gradients . . . . . . . . . . . . . . 281

Homological gradient descent . . . . . . . . . . . . . . . . . . 282
Cyclic cobordisms and iterations of the gradient descent map 287
Handle-like filtrations associated with the cellular gradients . 301

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Part 4.

Circle-valued Morse maps and Novikov complexes

Chapter 10.
1.
2.

Completions of rings, modules and complexes . . . . . 325

Chain complexes over A[[t]] . . . . . . . . . . . . . . . . . . . 325
Novikov rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Chapter 11.
1.
2.
3.
4.
5.

323

The Novikov complex of a circle-valued Morse map . . 335

The Novikov complex for transverse gradients . . . . . . . .
Novikov homology . . . . . . . . . . . . . . . . . . . . . . .
The Novikov complex for almost transverse gradients . . . .
Equivariant Morse equivalences . . . . . . . . . . . . . . . .
On the singular chain complex of the infinite cyclic covering

.
.
.
.
.

336
340
346
348
354


Contents

6.
7.

The canonical chain equivalence . . . . . . . . . . . . . . . . . 356
More about the Novikov complex . . . . . . . . . . . . . . . . 359

Chapter 12.
1.
2.
3.
4.

2.
3.

4.
5.
6.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

369

371
375
380

Counting closed orbits of the gradient flow . . . . . . . 383

Lefschetz zeta functions of the gradient flows . . . . . . . . .
Homological and dynamical properties of the cellular gradients
Whitehead groups and Whitehead torsion . . . . . . . . . . .
The Whitehead torsion of the canonical chain equivalence . .

Chapter 14.
1.

Cellular gradients of circle-valued Morse functions and
the Rationality Theorem . . . . . . . . . . . . . . . . . 367

Novikov’s exponential growth conjecture . . . . . . .
The boundary operators in the Novikov complex . .
Cellular gradients of circle-valued Morse functions . .
Gradient-like vector fields and Riemannian gradients

Chapter 13.
1.
2.
3.
4.

ix

383
387
399
407

Selected topics in the Morse-Novikov theory . . . . . . 413

Homology with local coefficients and the de Rham framework
for the Morse-Novikov theory . . . . . . . . . . . . . . . . . .
The universal Novikov complex . . . . . . . . . . . . . . . . .
The Morse-Novikov theory and fibring obstructions . . . . . .
Exactness theorems and localization constructions . . . . . . .
The Morse-Novikov theory of closed 1-forms . . . . . . . . . .
Circle-valued Morse theory for knots and links . . . . . . . . .

413
417
419
421
422
424

History and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Selected Symbols and Abbreviations . . . . . . . . . . . . . . . . . . 445
Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Preface
In the early 1920s M. Morse discovered that the number of critical points
of a smooth function on a manifold is closely related to the topology of the
manifold. This became a starting point of the Morse theory which is now
one of the basic parts of differential topology. Reformulated in modern
terms, the geometric essence of Morse theory is as follows. For a C ∞
function on a closed manifold having only non-degenerate critical points (a
Morse function) there is a chain complex M∗ (the Morse complex) freely
generated by the set of all critical points of f , such that the homology of M∗
is isomorphic to the homology of the manifold. The boundary operators
in this complex are related to the geometry of the gradient flow of the
function.
It is natural to consider also circle-valued Morse functions, that is, C ∞
functions with values in S 1 having only non-degenerate critical points. The
study of such functions was initiated by S. P. Novikov in the early 1980s
in relation to a problem in hydrodynamics. The formulation of the circlevalued Morse theory as a new branch of topology with its own problems
and goals was outlined in Novikov’s papers [102], [105].
At present the Morse-Novikov theory is a large and actively developing domain of differential topology, with applications and connections to
many geometrical problems. Without aiming at an exhaustive list, let us
mention here applications to the Arnol’d Conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical
zeta functions, and the theory of knots and links in S 3 . The aim of the
present book is to give a systematic treatment of the geometric foundations
of the subject and of some recent research results.
The central geometrical construction of the circle-valued Morse theory
is the Novikov complex, introduced by Novikov in [102]. It is a generalization to the circle-valued case of its classical predecessor — the Morse
complex. Our approach to the subject is based on this construction.
We begin with a detailed account of several topics of the classical Morse
theory with a special emphasis on the Morse complex. Part 1 is introductory: we discuss Morse functions and their gradients. The contents of the

2

Preface

first chapter of Part 2 is the Kupka-Smale Transversality theory; then we
define and study the Morse complex.
In Part 3 we discuss the notion of cellular gradients of Morse functions,
introduced in the author’s papers [113], [108]. To explain the basic idea,
we recall that for a Morse function f : W → [a, b] on a cobordism W
the gradient descent determines a map (not everywhere defined) from the
upper boundary f −1 (b) to the lower boundary f −1 (a). It turns out that
for a C 0 -generic gradient this map can be endowed with a structure closely
resembling the structure of a cellular map. We work in this part only
with real-valued Morse functions, however the motivation comes from later
applications to the circle-valued Morse theory.
In Part 4 we proceed to circle-valued Morse functions. In Chapter 11
we define the Novikov complex. Similarly to the Morse complex of a realvalued function, the Novikov complex of a circle-valued Morse function is
a chain complex of free modules generated by the critical points of the
function. The difference is that the base ring of the Novikov complex is no
longer the ring of integers, but the ring L of Laurent series in one variable
with integral coefficients and finite negative part. The homology of the
Novikov complex can be interpreted as the homology of the underlying
manifold with suitable local coefficients.
The boundary operators in the Novikov complex are represented by
matrices with coefficients in L (the Novikov incidence coefficients). One
basic direction of research in the Morse-Novikov theory is to understand
the properties of these Laurent series. The Novikov exponential growth
conjecture says that these series always have a non-zero radius of convergence. A theorem due to the author (1995) asserts that for a C 0 -generic
gradient v of a circle-valued Morse function, every Novikov incidence coefficient is the Taylor series of a rational function. This theorem is the basis

for the contents of Chapter 12. The reader will note that in general we
emphasize the C 0 topology in the space of C ∞ vector fields; we believe that
it is the natural framework for studying the Morse and Novikov complexes.
These results are then applied in Chapter 13 to the dynamics of the
gradient flow of the circle-valued Morse functions. We obtain a formula
which expresses the Lefschetz zeta functions of the gradient flow in terms
of the homotopy invariants of the Novikov complex and the underlying
manifold.
The last chapter of the book contains a survey of some further developments in the circle-valued Morse theory. The exposition here is more rapid
and we do not aim at a systematic treatment of the subject. I have chosen
several topics which are close to my recent research: the Witten framework


Preface

3

for the Morse theory, the theory of fibrations of manifolds over a circle and
the circle-valued Morse theory for knots and links.
Brief historical comments can be found in the concluding sections of
Parts 2, 3 and 4, and some more remarks are scattered through the text.
However I did not aim to present a complete historical overview of the
subject, and I apologize for possible oversights.
The book is accessible for 1st year graduate students specializing in
geometry and topology. Knowledge of the first chapters of the textbooks
of M. Hirsch [61] and A. Dold [29] is sufficient for understanding most
of the book. When we need more material, a brief introduction to the
corresponding theory is included. This is the case for the Hadamard-Perron
theorem (Chapter 4) and the theory of Whitehead torsion (Chapter 13).
Acknowledgements.

I am greatly indebted to S. P. Novikov for sharing his insight during
several years starting from 1985.
I am grateful to many people with whom I discussed the Morse-Novikov
theory throughout the years 1985 – 2005, in particular to P. Akhmetev,
V. Arnol’d, M. Farber, V. A. Ginzburg, J.-C. Hausmann, A. Hohlov, F. Laudenbach, S. Marchiafava, G. Minervini, M. M. Postnikov, J. Przytycki,
A. Ranicki, J.-C. Sikorav, V. Turaev, O. Viro.
I am grateful to D. Millionschikov who read Chapters 1-5 of the manuscript. His comments have led to numerous improvements.
Many thanks to A. Ranicki for his constant support and encouragement.
I am indebted to de Gruyter publishing house and in particular to R. Plato
for help and patience.
The pictures in the book were created using J. Hobby’s Metapost programming language. I am grateful to J. Hobby for his advice. Many thanks
to A. Shen and V. Shuvalov for their help with numerous TEX issues.
Moscow, August 2006

Andrei Pajitnov



Introduction
A C ∞ function f : M → R on a closed manifold M must have at least
two critical points, namely maximum and minimum. This lower bound for
the number of critical points is far from exact: the existence of a function
on M with precisely two critical points implies a strong restriction on the
topology of M . Indeed, let v be the gradient of f with respect to some
Riemannian metric, so that
v(x), h = f (x)(h)
for every x ∈ M and every h ∈ Tx M (here , denotes the scalar product
induced by the Riemannian metric). Assuming that f has only two critical
points: the minimum A and the maximum B, the vector field v has only
two equilibrium points: A and B, and it is not difficult to see that every

non-constant integral curve γ of v has the following property:
lim γ(t) = B,

t→∞

lim γ(t) = A.

t→−∞

Therefore the one-point subset {B} is a deformation retract of the subset
M \ {A}. The deformation retraction is shown in the next figure:

B

A


6

Introduction

M \ {A} is deformed onto B along the flow lines of v. In particular M \ {A}
is contractible, and it is not difficult to deduce that M is a homological
sphere.
This example suggests that the homology of M can provide efficient
lower bounds for the number of critical points of a C ∞ function on a manifold. Such estimates were established by M. Morse in his seminal paper
[98]. Here is an outline of his discovery. Recall that a critical point p of a
function f is called non-degenerate if the matrix of the second order partial
derivatives of f at p is non-degenerate. The number of the negative eigenvalues of this matrix is called the index of p. We shall consider only C ∞
functions whose critical points are all non-degenerate (Morse functions).

Let f : M → R be such a function. Put
Ma = {x ∈ M | f (x)

a}.

M. Morse shows that if an interval [a, b] contains no critical values of f ,
then Ma has the same homotopy type as Mb . If f −1 [a, b] contains one
critical point of f of index k, then Mb has the homotopy type of Ma with
one k-cell attached. The classical example below illustrates this principle.
Here M is the 2-dimensional torus T2 embedded in R3 , and f is the height
function.

R

f
b

a


Introduction

7

The homotopy type of Mb is clearly the homotopy type of Ma with a onedimensional cell e1 attached:

Ma

Mb

∼

Ma ∪ e1

Returning to the general case, it is not difficult to deduce that the
manifold M has the homotopy type of a CW complex with the number of
k-cells equal to the number mk (f ) of critical points of f of index k. This
leads to the Morse inequalities:
mk (f )

bk (M ) + qk (M ) + qk−1 (M )

where bk (M ) is the rank of Hk (M ) and qk (M ) is the torsion number of
Hk (M ), that is, the minimal possible number of generators of the torsion
subgroup of Hk (M ). (This version of the Morse inequalities is due to E.
Pitcher [125]; it is slightly different from Morse’s original version.) The
applications of these results are too numerous to cite here; we will mention
only the classical theorem of M. Morse on the infinite number of geodesics
joining two points of a sphere S n (endowed with an arbitrary Riemannian
metric) and the computation by R. Bott of the stable homotopy groups of
the unitary groups.
The construction described above can be developed further. Intuitively,
it is possible not only to obtain the number of cells of a CW complex X
homotopy equivalent to M , but also to compute the boundary operators in
the corresponding cellular chain complex. In more precise terms, starting
with a Morse function f : M → R and an f -gradient v, one can construct
a chain complex M∗ such that Mk is the free abelian group generated by
critical points of f of index k and the homology of M∗ is isomorphic to
H∗ (M ).
The explicit geometric construction of M∗ is a result of a long development of the Morse theory (especially in the works of R. Thom [157],

S. Smale [150] [149], and E. Witten [163]). By definition, Mk is the free
abelian group generated by the set Sk (f ) of all critical points of f of index
k. The boundary operator Mk → Mk−1 is defined as follows. Let v be
the Riemannian gradient for f with respect to a Riemannian metric on M .
For two critical points p, q of f with ind p = ind q + 1, denote by Γ(p, q; v)


8

Introduction

the set of all flow lines of (−v) from p to q. It turns out that under some
natural transversality condition on the gradient flow, this set is finite. The
gradients satisfying this condition are called Kupka-Smale gradients, they
form a dense subset in the space of all gradients of f . One can associate a
sign ε(γ) = ±1 to each flow line γ of (−v) joining p with q (we postpone
all the details to Chapters 4 and 6). Summing up the signs we obtain the
so-called incidence coefficient of p and q:
ε(γ).

n(p, q; v) =
γ ∈ Γ(p,q; v)

Now we define the boundary operator ∂k : Mk → Mk−1 as follows:
∂k (p) =

n(p, q; v)q.
q∈Sk−1 (f )

One can prove that ∂k ◦ ∂k+1 = 0 for every k and the homology of the

resulting complex is isomorphic to H∗ (M ). This chain complex is called
the Morse complex.
Here is a picture which illustrates the 2-torus case, considered above:

There are four critical points: one of index 0 (the minimum), one of index
2 (the maximum), and two critical points of index 1 (saddle points). There
are eight flow lines of (−v) joining the critical points of adjacent indices;
they are shown in the figure by curves with arrows. The Morse complex is
as follows:
Z
Z2
Z
0
0
where all boundary operators are equal to 0.
It is natural to consider also the circle-valued Morse functions, that is,
C ∞ functions with values in S 1 having only non-degenerate critical points.
Identifying the circle with the quotient R/Z we can think of circle-valued


Introduction

9

Morse functions as multi-valued real functions : locally the value of such a
function is a real number defined up to an additive integer.
A systematic study of circle-valued Morse functions was initiated by
S. P. Novikov in 1980 (see [102]).† The motivation came from a problem
in hydrodynamics, where the application of the variational approach led to
a multi-valued Lagrangian (see the papers [101], [104]). The formulation

of the circle-valued Morse theory as a new branch of topology with its own
problems and goals was outlined in S. P. Novikov’s paper [102], and in
more detail in the survey paper [105].
The central geometric construction of the circle-valued Morse theory is
the Novikov complex which is a generalization to the circle-valued case of its
classical predecessor, the Morse complex. To understand the fundamental
difference between the constructions of the Morse complex and the Novikov
complex let us have a look at the following figure.

p

q

The shaded area depicts the manifold M ; p and q are critical points
of f . Contrary to the real-valued case a flow line of (−v) can turn around
several times; it can well happen that the set of flow lines joining p and q
is infinite, and we will not be able to apply the procedure described above
to the present situation. A way to overcome this difficulty was suggested
by S. P. Novikov: for each positive integer m one counts the flow lines of
(−v) joining p and q and intersecting m times a given level surface of f
(generically there is only a finite number of such flow lines). We obtain
†

The first recorded instance of circle-valued Morse theory is most probably the paper
[124] by Everett Pitcher.


10

Introduction

integers nm (p, q; v) and form a power series in one variable
nm (p, q; v)tm ∈ Z[[t]]

N (p, q; v) =
m∈N

(the Novikov incidence coefficient). For technical reasons it is convenient
to consider these series as elements of a larger ring, namely the ring
L = Z((t)) = {λ =

am tm | am ∈ Z}
m m(λ)

of all Laurent series with integral coefficients and finite negative part. Let
Nk (f, v) be the free L-module freely generated by critical points of f of
index k. Introduce the homomorphism
∂k : Nk → Nk−1 ,

∂k p =

N (p, q; v)q.
q∈Sk−1 (f )

One can prove that ∂k ◦ ∂k+1 = 0 for every k. We obtain the Novikov
complex associated to the pair (f, v). The homology of this complex has
a natural geometric meaning. Namely, consider the infinite cyclic covering
¯ → M induced by the map f : M → S 1 from the universal covering
M
R → S 1 . We obtain a commutative diagram

¯
M
π



M

F

/R

f


/ S1 .

Here F is a Morse function such that
F (tx) = F (x) − 1

¯,
x∈M

for every

and t is a generator of the structure group of the covering π. The homol¯ ) is a module over Z[t, t−1 ]. The basic property of the Novikov
ogy H∗ (M
complex is the following:
¯)⊗L
(1)

H∗ N∗ (f, v) ≈ H∗ (M
L

Z[t, t−1 ],

L = Z((t))). This theorem was stated in [102], see
(where L =
[110] for the proof. The ring L = Z((t)) is a principal ideal domain, and
an easy algebraic argument allows us to deduce from the isomorphism (1)
the following Novikov inequalities for the number mk (f ) of critical points
of f of index k:
bk (M ) + qk (M ) + qk−1 (M )
¯ ) = Hk (M
¯ ) ⊗ L and qk (M )
where bk (M ) is the rank of the L-module Hk (M

(2)

mk (f )

is the torsion number of this module.

L


Introduction

11

One can ask whether these inequalities are optimal, or exact: given a

manifold M and a cohomology class ξ ∈ H 1 (M ) ≈ [M, S 1 ] is there a function f : M → S 1 whose homotopy class equals ξ and all these inequalities
are equalities? A theorem of M. Farber [30] says that this is the case for
6 and any ξ = 0. The
any closed manifold M with π1 (M ) ≈ Z, dim M
restriction on the fundamental group is essential for this theorem already
in the case when we expect the existence of a function f without critical
points, that is, a fibration over S 1 . Vanishing of the Novikov homology
¯ ) is in general not sufficient for fibring the manifold. However the
H∗ (M
construction of the Novikov complex can be generalized further to obtain
a more sophisticated version (the universal Novikov complex ) defined over
a certain completion of the group ring Z[π1 (M )]. Using these tools it is
possible to obtain a necessary and sufficient homotopy-theoretic condition
for the existence of a fibration (this was done in the works of J.-C. Sikorav [147], the author [111], F. Latour [82], A. Ranicki [134], [132]). The
problem of existence of a fibration of a manifold over a circle had been
intensively studied in the 1960s and the 1970s and a homotopy-theoretic
criterium for the existence of a fibration was obtained in the works of W.
Browder and J. Levine [17], and T. Farrell [34]. Another approach was
developed by L. Siebenmann [144]. Thus the Novikov theory provides yet
another necessary and sufficient condition for fibring which at first glance is
completely different. A deeper analysis leads to the identification of the two
fibring obstructions; this was done by A. Ranicki [134], [132]. Thus the
Browder-Levine-Farrell-Siebenmann obstruction theory is embedded into
the Morse-Novikov theory as a particular case of “functions without critical points”. See Sections 2 and 3 of Chapter 14 for an introduction to this
subject.
Now let us have a closer look at the incidence coefficients N (p, q; v) =
nk (p, q; v)tk in the Novikov complex. These series contain a lot of information about the gradient flow, and it is natural to ask what are the
asymptotic properties of nk (p, q; v) when k → ∞. S. P. Novikov conjectured that the coefficients nk (p, q; v) have at most exponential growth when
k → ∞. This conjecture remains one of the most challenging problems in
the field. See the works of V. I. Arnold [3], [4], [5], and of D. Burghelea

and S. Haller [19], where different aspects of the problem are discussed.
In 1995 the author proved that C 0 -generically the incidence coefficients
N (p, q; v) are rational functions of the variable t (see [112], [113]). Namely,
for every Morse function f : M → S 1 there is a subset GC (f ) of the space of
all f -gradients which is open and dense in C 0 -topology, such that for every
v ∈ GC (f ) the Novikov incidence coefficients N (p, q; v) are Taylor series of


12

Introduction

rational functions. The elements of GC (f ) are called cellular gradients. In
Chapter 12 we study cellular gradients and prove this rationality theorem.
Let us return now to the isomorphism (1) between the homology of
the Novikov complex and the completed homology of the infinite cyclic
covering. One can strengthen this result and construct a natural chain
equivalence
N∗ (f, v)

¹ Δ∗(M¯ ) ⊗ L,

φ

L

¯ ) is the simplicial chain complex of M
¯ . It turns out that this
where Δ∗ (M
chain equivalence is an important geometric invariant of the pair (f, v).

The source and the target of φ are free finitely generated complexes with
naturally arising free bases. A standard algebro-topological construction
associates to such a chain equivalence its Whitehead torsion. In the particular case of the ring L this torsion can be considered as an element w(f, v)
of the ring Z[[t]]. Define the Lefschetz zeta function of the gradient flow by
the following formula:
iF (γ)tn(γ) ∈ Z[[t]]

ζL (−v) = exp
γ

where the sum is extended over the set of all closed orbits γ of (−v) and
n(γ) ∈ N, iF (γ) ∈ Q are certain numerical invariants associated with each
closed orbit γ (see Chapter 13 for details). We prove that for a generic
f -gradient v we have
w(f, v) = ζL (−v)

−1

.

A relation between the torsion invariants of the Novikov complex and the
zeta function of the gradient flow was discovered in 1996 by M. Hutchings
and Y-J. Lee [64]. They treated the case when the Novikov complex becomes acyclic after taking the tensor product with Q((t)). Our approach is
based on the techniques of cellular gradients, which allows us to get rid of
the acyclicity assumptions and to generalize the formula above to the case
of the universal Novikov complexes. We first prove the result for cellular
gradients, and then deduce the general case by approximation techniques
(Chapter 13).
There is one class of spaces where circle-valued Morse functions appear
in a very natural way. Let K be a classical knot, that is, the image of a C ∞

embedding of S 1 to S 3 . The knot K is called fibred, if there is a C ∞ map
f : S 3 \ K → S 1 such that f has no critical points and the level surfaces of
f form an open book structure in a neighbourhood of K, as shown in the
next picture (here L0 , Lπ/4 , Lπ/2 denote the level surfaces corresponding to
the values respectively 0, π/4, π/2 ∈ S 1 ):


Introduction

13

Lπ/2
Lπ/4
L0

K
Now let K be an arbitrary knot, non-fibred in general, and f : S 3 \K →
be a Morse function such that its level surfaces form the open book
structure in a neighbourhood of K. It is not difficult to show that the
number of critical points m(f ) of such a Morse function is finite. Put

S1

MN (K) = min m(f )
where the minimum is taken over the set of all such Morse functions f (see
[119]). The Novikov inequalities (2) in this case are reduced to one single
inequality:
MN (K) 2q1 (S 3 \ K),
which is not exact in general. To obtain better lower bounds for MN (K),
the best tool would be the universal Novikov complex. However it is very

complicated algebraically and explicit computations with this complex are
at present beyond our reach. In a joint work with H. Goda we introduced
the twisted Novikov homology which is in a sense intermediate between the
Novikov homology (1) and the universal Novikov homology. On one hand
it reflects the essentially non-abelian structure of the knot groups, and on
the other hand it is effectively computable in many cases with the help of
modern software. A discussion of these invariants and related topics is the
contents of the last section of the book.