Cohomology and Differential Forms

By Izu Vaisman

This monograph explores the cohomological theory of manifolds with various sheaves and its application to differential geometry. Based on lectures given by author Izu Vaisman at Romania's University of Iasi, the treatment is suitable for advanced undergraduates and graduate students of mathematics as well as mathematical researchers in differential geometry, global analysis, and topology.
A self-contained development of cohomological theory constitutes the central part of the book. Topics include categories and functors, the Čech cohomology with coefficients in sheaves, the theory of fiber bundles, and differentiable, foliated, and complex analytic manifolds. The final chapter covers the theorems of de Rham and Dolbeault-Serre and examines the theorem of Allendoerfer and Eells, with applications of these theorems to characteristic classes and the general theory of harmonic forms.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 28, 2016
• ISBN: 9780486815121

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Index

PREFACE

There are in differential geometry three important structures with specific sheaves of germs of differentiable functions: the differentiable manifolds with the sheaf of germs of locally constant functions, the foliated manifolds with the sheaf of germs of differentiable functions which are constant on the leaves, and the complex analytic manifolds with the sheaf of germs of holomorphic functions. In each case, there is a corresponding cohomology theory which plays an essential role.

In the present monograph, which is a result of lectures given at the University of Iasi (Romania) and of some personal research, we wish to expose the known method of calculating the cohomology spaces of the manifold with coefficients in the above mentioned sheaves using globally defined differential forms, and to give some applications.

It is assumed that the reader knows enough algebra, analysis, topology, and classical differential geometry, and that he has sufficient training in abstract mathematical reasoning.

In the first part, our principal aim is to expose the Čech cohomology with coefficients in a sheaf of Abelian and non-Abelian groups. For a general setting of the problems and to avoid repetition we begin with a chapter on categories and functors, where we only intend to acquire the corresponding language. The second chapter develops Čech cohomology theory, including the non-Abelian case. The geometric theory of the nonabelian Čech cohomology, i.e., the theory of fiber bundles with structure group is exposed in the third chapter. In the fourth chapter we review the principal subjects of differential geometry, considering the general case of manifolds modeled on Banach spaces. Finally, the last chapter contains the solution of the problems announced, the theorems of de Rham and Dolbeault–Serre and the corresponding theorem which we gave for foliated manifolds. An informative treatment of the theorem of Allendoerfer and Eells is also given. In the same chapter, applications of these theorems to characteristic classes and the general theory of harmonic forms are treated.

We consider it useful for the reader to have in the preliminary chapters a more complete development than would be strictly necessary for our cohomology problems. In the book, there is a basic text where all things are proved, with the exception of technical details, and a supplementary text where we give, without proof, information which is less important here but still worthy of consideration by the reader.

Let us mention some new features (in form or content) to be found in this book : the introduction of pseudocategories and strengthened categories ; the derivative functors ; the theory of atlases ; a new definition of sheaves with values in a category; the study of foliated Riemann manifolds and all things related to them (in the fifth chapter); etc.

The limitations of our work did not permit us to develop other de Rham theorems, e.g., those arising in the deformation theory of pseudo-group structures, but the reader will find here all that is necessary for an independent study of them.

The literature at the end of the book contains only works which we employed in writing it and the references are not always to those who introduced the notions or obtained the results but rather to those whose papers we employ. Cross references are given in the usual manner, e.g., "Proposition 2.3.7, Formula 2.4(8)", etc., where in the same chapter (or section), the chapter (section) number is omitted. Notation the meaning of which is clear from the context is not explained.

We hope that this monograph will be useful to both graduate students and research workers in differential geometry and global analysis.

I should like to express here my gratitude to the Mathematical Faculty of the University of Iasi for providing me the opportunity to teach courses on the subject of this book, to Prof. R. Miron, Prof. V. Cruceanu, Dr. L. Maxim, and Dr. V. Oproiu for reading the manuscript, and to the Department of Mathematics of the University of Illinois for inviting me to visit, an invitation which led me to write this book in English (I am sorry that, for reasons beyond my control, this visit could not be undertaken).

It is a pleasure for me to acknowledge my special gratitude to Prof. Samuel I. Goldberg of the University of Illinois for his interest in my mathematical activity, for reviewing this book, and correcting its English, and for his invaluable help in publication.

Last, but not least, I should like to express my gratitude to my wife Silvia for having created for me very favorable conditions during the writing of this book.

I. Vaisman

Chapter 1

CATEGORIES AND FUNCTORS

1Classes and Sets

The objects with which we usually operate in mathematics are called classes ; they arise by abstraction from the intuitive notion of a collection of things. This intuitive image is unsatisfactory for mathematical operations because it gives rise to contradictions, so we must consider classes as defined by a determined system of axioms. We consider here the same system of axioms as in [7].

In this system, we have one primitive notion, the class, and one primitive relation, belongs to. The classes will be denoted by different letters and the relation A belongs to B will be denoted by A ∈ B. The sign ∉ will be used to deny the relation ∈. A class A is called a set if there is at least one class B such that A ∈ B, and in this case we shall say that A is an element of B. We consider also the equality relation of classes, A = B (and inequality, A ≠ B), as logical identity, i.e., A = B if and only if for any property P(X) of classes, P(A) and P(B) are simultaneously true or false.

The first two axioms of the system which we expose for sets and classes are:

A1. Two classes are equal if and only if they have the same elements.

A2. For any property P(X) of classes, there is a class whose elements are just the sets X for which P(X) is true.

From A1, it follows that the class determined by A2 is unique; it will be denoted by {X | P(X)}. If this class is a set, P(X)—the characteristic property of this set—is called a collectivizing property. For example, the property Χ ∉ X is not a collectivizing property because, A = {X | Χ ∉ X} would be a set and we would derive Russell’s antinomy: A ∈ A if and only if A ∉ A. A is also an example of a class which is not a set.

A class A is called a subclass of the class B if any element of A is also an element of B. We shall write then A ⊆ B or B ⊇ A. In particular, if A is a set, it will be called a subset of B. We give now a third axiom of the system

A3. Every subclass of a set is itself a set.

Using axiom A2, we easily get the existence of the following classes

called respectively the empty class, the union of the elements of A and the power class of A. We now give three new axioms:

A4. Ø is a set.

A5. If A is a setA and are again sets.

A6. If A and B are sets, {A, B} = {X | X = A or X = B} is a set.

As a consequence, if A is a set there is a set {Afor A = Ø and {X | X ∈ {Ø,A} and X ≠ Ø} for A ≠ Ø.

For two classes A and B we define the cartesian product

where (X,Y) = {{X}, {X, Y}} is the ordered pair. A binary relation from A to B is a subclass of A × B and, if the relation is such that to every element of A corresponds only one element of B, it is called a map or a function. Now, we shall introduce a new axiom:

A7. If F is a function from a set A to a class B, the class of elements of B which correspond to elements of A is again a set.

The set given by A7 is the range of F and is denoted by {Xa}a∈A, where X is the element corresponding to a.

An important notion is that of a universal set. This is a set U, satisfying the following conditions:

(a)If X ∈ U, then X ⊆ U.

(b)If X ∈ U.

(c)If X, Y ∈ U, then {X, Y} ∈ U.

(d)If F = {Fi}í∈I and Fi ∈ U, I ∈ U.

We add now to our system the axiom

A8. Every set is an element of some universal set.

As a consequence the existence of infinite sets is implied.

The axiomatic system of sets and classes must still be completed by two special axioms. The first excludes singular sets satisfying the condition X ∈ X, which are not used in classical mathematics. This axiom is:

A9. For every nonempty class A there is an X ∈ A such that there is no set Y for which Y ∈ X and Y ∈ A.

Finally, the last axiom assures the existence of sets which cannot be obtained in a constructive manner from the previous axioms. It is the famous axiom of choice, which can be given in the following form:

A10. For every set A, there is a function defined on the set of nonempty subsets of A which associates with every subset one of its elements.

Using the given system of axioms one can now obtain the usual construction of set theory and the theory of functions and relations. This can be performed as in the so-called naive set theory which we assume known. Hence we shall not go further with the development of the set theory, except for a few comments on partial maps.

Let A, A′ ⊆ A, and B be three nonempty sets and f: A′ → B a map.

Then we call f a partial map from A to B and denote it by f: A − → B. For this map, A′ is the domain, f(A′) is the range or image, A is the source, and B is the target, and they will be denoted respectively by dom f, im f, source f, tar f Two partial maps will be considered as equal if the respective maps are equal and if they have the same source, domain, and target (and, hence, also the same image).

If f: A − → B and g: C − → D, their composition defines a partial map g ∘ f: A − → D given by

whose domain is f−1(im f dom g. If im f dom g = Ø, g ∘ f is not defined.

For partial maps, injectivity and surjectivity have the usual meaning. If f: A − → B is injective, there is a partial map f−1: B − → A such that dom f−1 = im f and im f−1 = dom f and if 1M is the identity map of a set M we have

We note the relations

where f: M − → N, g: N − → M, and the fact that the composition of partial maps is associative when defined.

Generally, calculation with partial maps is performed with the rules of usual maps, taking into account the respective domains and images.

2Pseudocategories and Categories

Usually, the classes studied in mathematics have different structures. In this section we consider a general structure which defines a mathematical language.

1DefinitionA pseudocategory , whose elements are called objects, one for each ordered pair of objects, and partial maps

one for every triple of objects, such that :

(a)the operation given by ∘ is associative when defined;

.

are called morphisms of source A and target B and they are denoted by u: A → B . The operation o is called the composition of morphisms, and 1A, which is obviously unique, is called the identity morphism of A. Another notation for ∘(u, υ) will be υ ∘ u or υu.

We obtain a typical example of a pseudocategory if we take the class of objects consisting of the nonempty sets and the morphisms being the partial maps of sets. The composition of morphisms is the usual composition of maps and the identity morphisms the identity maps. This pseudocategory will be denoted by Ensp (from the French word ensemble, which means set).

2Definitionwill be called a Cantorian pseudocategoryare sets, this pseudocategory will be called quasi-Cantorian.

Ensp is an example of a Cantorian pseudocategory. Another important example is obtained by taking as objects nonempty topological spaces and as morphisms partial continuous maps with open domain. This pseudocategory will be denoted by Topp.

3DefinitionA quasi-Cantorian (Cantorian) pseudocategory whose objects are topological spaces (and whose morphisms are continuous maps with the usual composition) is called a quasitopological (topological) pseudocategory.

4Definitionis always defined, i.e.,

is called a category.

In particular we shall consider Cantorian, quasi-Cantorian, topological, and quasitopological categories.

If the class of objects of a category is a set the category is called small.

The structure of a category is that given at the beginning of this section and it is just the structure which we need for a general mathematical language. We remark that it is also possible to consider more general structures which we call hyperpseudocategories and hyper categories (great categories can be classes. Then, to avoid antinomies, proper universal sets must be chosen [40].

a set consisting of one element for every object A.

The groups and their homomorphisms define the category Grp of groups.

The Abelian groups and their homomorphisms form the category Ab of Abelian groups. In the same manner, we have the category of rings, of A-modules (where A is a commutative and unitary ring) which is denoted by A-Mod, and also categories corresponding to other algebraic structures.

Another category is obtained if the objects are the pointed sets, i.e., sets with a distinguished base point, and the morphisms are maps which preserve base points, This category will be denoted by Ens· and we also have an analogous category Top· of pointed topological spaces and continuous maps preserving base points.

Let us give an important example of a quasitopological category, where the morphisms are classes of maps. We recall that two continuous maps between topological spaces, f, g: X → Y, are homotopic if there is a continuous map F: X × I → Y, I = [0, 1], such that F(x, 0) = f(x) and F(x, 1) = g(x) for x ∈ X. It is easy to verify that homotopy is an equivalence relation on the set YX of all continuous maps from X to Y and that it is compatible with the composition of maps [42]. Hence, if we take the objects the topological spaces and the morphisms the homotopy classes of continuous maps, we get a category. This is the so-called homotopy category of topological spaces.

There are also important categories which are not necessarily cantorian or quasicantorian. Thus, if M is a partial ordering of it, then M is a small category whose objects are the elements a, b, ... of M. The composition of morphisms has an obvious definition. We remark that by a partial ordering we understand a binary relation on M . Hence, we can define a partially ordered set to be a small category with the morphism sets as above.

of subsets of a set M of open subsets of a topological space Tis topological.

We now define some pseudocategories (categories) associated with given pseudocategories (categories).

is a pseudocategory, we define the dual pseudocategory , i.e., υ ∘* u = u ∘ υ is also a category.

The existence of a dual category gives a duality principle in the theory of categories and we can obtain the dual of any notion or theorem by taking it in the dual category, i.e., by changing the sense of morphisms.

the objects of which are the ordered pairs (A1, A2) where Aand A, and the morphism sets

is componentwise, i.e.,

if, of course, the components of the right member exist. The identity morphisms will be of the form (1A, 1B).

In the same manner, we can define the product of n pseudocategories. Clearly, the product of categories is also a category.

to be the commutative diagrams of the form

such that

where it is assumed that these composites exist.

The composition of two diagrams (1) is defined by the composition of their vertical arrows, when possible. The identity diagrams will have the vertical arrows (1A, 1B).

is called the arrow pseudocategory .

A generalization is obtained if the arrows are replaced by other diagrams of a given type, which gives general categories of diagrams [40].

5Definitionis a subpseudocategory if it satisfies the following conditions

is a full subpseudocategory . In particular, we shall talk of subcategories and full sub categories. For example, Ens is a subcategory of Ensp, and Top is a subcategory of Topp, but they are not full. Ab is a full subcategory of Grp, the category of Hausdorff spaces and continuous maps is a full subcategory of Top, etc.

be a topological pseudocategory and f: A → B , i.e., a continuous mapping from an open subset D ⊆ A to B.

Let x be a point in D. We define the germ of f at x, and denote by [f]xwhich are defined on open neighborhoods of x , whose objects are pairs (A, a), where a is a point in A, and whose morphism sets are

and the identity morphism is the germ of the identity map. The conditions of is a quasitopological category.

6Definitionis called the local category .

Local categories associated with topological categories play an important role in differential geometry.

3Morphisms, Objects and Operations

The definitions and results which we give here are considered for categories, but some of them are also valid for pseudocategories. As an exercise the reader should make the proper generalizations. These notions are inspired by set theory and the case of general categories is obtained by looking for definitions of the respective notions which do not make use of the elements of the sets.

be a category and u: A → B . Then, for any object X we have the mappings

defined by

.

1DefinitionThe morphism u is an injection (respectively a surjectionif the mapping ′u (respectively u′) is univalent. A morphism which is both an injection and a surjection is called a bijection.

Clearly, injection and surjection are dual notions.

2PropositionIf u and υ are morphisms in a category and υ ∘ u exists and is an injection (surjection) then u is an injection (υ is a surjection). The identity morphisms are bijections.

The first part is a consequence of the following immediate relations

and the second part fpllows directly from the definition of a bijection.

In Ens, injections, surjections, and bijections given by Definition 1 are the same as the classical ones. If u: A → B is an injective map between two sets and υi: X → A (i = 1, 2), υ1 ≠ υ2, then there is an x ∈ X with υ1(x) ≠ υ2(x), hence uυ1(x) ≠ uυ2(x) and it follows that ′u is univalent. Conversely, if ′u is univalent u is also univalent; for otherwise we would have ai ∈ A (i = 1, 2) with u(a1) = u(a2) and a1 ≠ a2, and by defining υ1(X) = a1, υ2(X) = a2 the univalence of ′u is contradicted. Hence, injections in Ens in the sense of Definition 1 and in the classical sense are the same. The proof for surjections and bijections is left to the reader.

The previous results do not hold in any cantorian category. For instance, if u: A → B is some map of sets, u is an injection in the category with objects {A, B} and morphisms {1A, 1B, u} but it may not be an injection in Ens. In the category of rings, the inclusion J ⊂ Q of the integers in the rational numbers is obviously a surjection, but this is not a surjection in Ens. In the category of topological Hausdorff spaces and continuous maps, u: A → B is a surjection if u(A) is dense in B, even if u is not a surjective map. The last two cases provide examples of bijections in cantorian categories which are not bijections in Ens. In the category A-Mod we again have a situation where injections and surjections are injective and surjective mappings, respectively. Hence these notions must be used carefully in the categorical and set theoretical languages. In any case, we have the following obvious result.

3PropositionIf is a cantorian