Physical Explanation due to Witten for a Vector Bundle Description While the work of Minasian and Moore gave a hint that K-theory is the more natural description for D-brane charges, Witten gave a physical explanation why K-theory should be the correct framework.

This was suggested by Witten in the case of a torsion H-field and by Bouwknegt and Mathai in the nontorsion case. The Freed—Witten anomaly is a global world-sheet anomaly of string theory in the presence of D-branes and a nontrivial H-field. The strategy of Maldacena, Moore, and Seiberg was again to start with all allowed here, anomaly-free D-branes and then identify configurations that can be dynamically transformed into each other. This leads to the following two conditions: 1. If the H-field is trivial, then the first condition just says that the D-brane must be spinC.

These two conditions lead naturally to the twisted K-theory classification of D-brane charges. The twisted K-theory class on the spacetime comes as a twisted K-theory, see Part 4 push-forward of an ordinary untwisted K-theory class on the D-brane. Finally, both the unphysical and unstable branes are nicely interpreted within the Atiyah—Hirzebruch spectral sequence see Chap. D-Branes as Boundary Conditions for Open Strings As already mentioned, open strings end on D-branes, which in turn means that D-branes can be characterized by conformally invariant boundary conditions on the open string worldsheet.

From this worldsheet point of view, the D-brane charge group is again obtained along a similar strategy as before. Firstly, we classify all boundary conditions for a given closed string background and then identify those which are connected by renormalization group RG flows on the world-sheet boundary.

Unfortunately, in most situations, one is neither able to classify all bound- ary conditions nor to classify all RG flows between them. In some cases, however, this strategy was nevertheless successfully pursued as will be discussed in the fol- lowing paragraph. One should stress that this classification is in a way comple- mentary to the ones already discussed, because it does not rely on any geometrical structures of the target space.

Suggested Reading 5 For a two-dimensional topological field theory, where the whole content of the theory is encoded in a finite-dimensional Frobenius algebra, this problem was ad- dressed by Moore and Segal By using sewing constraints, they obtained the complete classification of D-branes in these theories in terms of K-theory. Alekseev and Schomerus worked out the charge group from the worldsheet approach for the SU 2 WZW model, which led to precise agreement with the twisted K-theory.

A more structural connection between twisted K-theory and conformal field theory data was shown by Freed et al. Consider a simple, simply connected, compact Lie group G of dimension d. Central exten- sions of its smooth loop group LG by the circle group T are classified by their level k. Positive energy representations of LG at fixed level are the ones which are im- portant in string theory.

## The Topology of Fibre Bundles. (PMS) - Norman Steenrod - Google Livres

The free abelian group Rk LG of irreducible isomorphism classes with the multiplication given by fusion rules of conformal field theory is called the Verlinde ring of G at level k see Chap. Here, G acts on itself by the adjoint action and the ring structure on K-theory is the convolution Pontryagin product. Suggested Reading A good point to start studying the use of K-theory in string theory is the original paper by Witten There is also a good review by Olsen and Szabo For the classification of D-brane charges in the presence of a nontrivial H-field in terms of twisted K-theory, one might consult the paper by Maldacena, Moore and Seiberg and the review by Moore A more recent review on K-theory in string theory that also discusses the limitations of the K-theory classification is the one by Evslin Lecture Note Ser.

Since fibre space had already a precise meaning in the thesis of Serre, we fol- lowed the Grothendieck idea but changed the terminology to bundle as in Fibre Bundles. The term bundle fits with the idea that a bundle with additional structure would be compatible with the usual terminology for such notions as vector bundle, principal bundle, and fibre bundle as explained in Chaps. The idea of the category of morphisms is found in many contexts in mathematics. In the case of a compact X, we can recover X from the algebra C X as the space of maximal ideals.

This is the Gelfand—Naimark theorem. For compact B, we can describe vector bundles over B as finitely generated projective modules over C B. This is the Serre—Swan theorem which is proved in Chap. Isomorphism classes of projective modules correspond to algebraic equivalence classes of idempotents in all the possible matrix algebras. In Chap. These are the three versions of K-theory discussed in the first part.

As preparation for the homotopy version of K-theory considered in Part II, we introduce the basic properties of principal fibre bundles in Chap. Every vector bundle is the fibre bundle of a principal GL n -bundle, and frequently, the special extra features of a vector bundle can be described in terms of the related principal bundle. Local triviality is considered, and the related notion of descent which plays a basic role in algebraic geometry is introduced for principal bundles. The principal bundle is also the natural domain for the study of the gauge group of a bundle.

Grothendieck, A. We use the term bundle in the most general context, and then in the next chapters, we define the main concepts of our study, that is, vector bundles, principal bundles, and fibre bundles, as bundles with additional structure. The reader with a background in category theory and topology will see only a slightly different approach from the usual one. We will introduce several notations used through the book, for example, set , top , gr , and k denote, respectively, the categories of sets, spaces, groups, and k-modules for a commutative ring k.

These are explained in the context of the defini- tion of a category in Sect. Definition Let B be a space. Many bundles arise as restrictions of a product bundle. In Sect. In fact, we can relate bundles over distinct base spaces with morphisms general- izing 1. We have an important functor associated with bundles over a space which we introduce directly. The formal definition is in a later section. The formal algebraic properties of the set of cross sections are explained in two remarks 5. Now, we consider the special case of an induced bundle where the general con- cept is considered in Sect.

With the restriction, we can introduce an important concept. In general, the definitions and examples extend to the case of an infinite dimensional vector space with a given topology as a fibre, for example, an infinite dimensional Hilbert space. See Chaps. Such a cross section is called a vector field, and most vector fields in mathematics and physics are of this simple form. Example Tangent and normal bundle to the sphere.

As in 2. This is an example of a fibre product considered in 3. We consider one basic example. Again, the fibre is isomorphic to the vector space V.

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On the other hand, E V is very far from being the product bundle as we shall see as the theory develops in the first two parts. Remark The Whitney sum in 2. Example Returning to the n-sphere Sn and the examples of 2. Returning to the n-dimensional real projective space Pn R and the example of 2. Definition A category C consists of three sets of data. I A class whose elements are called objects of C.

When we say X is in C we mean X is an object in C. Before considering examples of categories, we introduce the notion of isomor- phism in a category. It is a general concept in category theory, and in a specific category it is something which has to be identified, not defined, in special examples. Definition Let C be a category. The composition of two isomorphisms and the inverse of an isomorphism are iso- morphisms.

The relation two objects are isomorphic is an equivalence relation. Remark The isomorphisms in set are the bijective functions, the isomor- phisms in top are the bijective continuous maps whose inverse function is continuous, the isomorphisms in gr are the bijective group morphisms, and the isomorphisms in k for a commutative ring k are the bijective k-linear maps. Using the previous section as a model, we are able to define bundles in any cate- gory C. The category bun is just the category Mor top. For a functor, the following axioms are satisfied.

This follows directly from the axioms for a functor. If we think of categories as objects and functors as morphisms between categories, then we have the first idea about the category of categories. Unfortunately, the concept has to be modified, and this we do in later chapters. These are the two axiomatic properties of a functor in the definition 5. The composition of two equivalences of categories is again an equivalence of categories. Observe that equivalence of categories is a weaker relation than isomorphism of categories.

It almost never happens that two categories are isomorphic, but equiva- lent categories will have a bijection between isomorphism classes and related mor- phism sets. In fact, this is the way of recognizing that a functor is an equivalence by starting with its mapping properties on the morphism sets. In the next proposition, we have a useful criterion for a functor to be an equivalence from one category to another. Hence, T is fully faithful.

This proves the proposition. Remark A space Y is separated or Hausdorff provided the following equiva- lent conditions are satisfied. This is a basic definition and assertion in general topology. For the proof of the equivalence of 1 and 2 , we assume firstly 1. Conversely, we assume 2. This establishes the equivalence of 1 and 2. Example Let D be a discrete space. This is called the product bundle with discrete fibre.

Now, we are in position to define coverings in the mature context and describe the category of coverings as a full subcategory of the category of bundles. References Grothendieck, A. Given a space X, we take a real or complex finite dimensional vector space V and make V the fibre of a bundle over X, where each fibre is isomorphic to this vector space. This is the product vector bundle with base X and fibre V. On first sight, the product bundle appears to have no special features, but it con- tains other vector bundles which often reflect the topology of X in a strong way.

This happens, for example, for the tangent bundle and the normal bundle to the spheres which are discussed in 1 2. All bundles of vector spaces that we will consider will have the local triviality property, namely, they are locally isomorphic to a product bundle. The product bundle is also basic because most of the vector bundles we will be considering will be subvector bundles of a product vector bundle of higher dimension.

In some cases, they will be so twisted that they can only live in an infinite dimensional product vector bundle. We will begin by formulating the concept of bundles of vector spaces over X. These will not be necessarily locally trivial, but they form a well-defined concept and category. Then, a vector bundle is a locally trivial bundle of vector spaces. The point of this distinction is that being a vector bundle is a bundle of vector spaces with an additional axiom and not an additional structure. The local charts which result are not new elements of structure but only a property, but to state the property, we need the notion of bundle of vector spaces.

After this is done, we will be dealing with just vector bundles. We need to use the scalars explicitly, and if either number system applies, we will use the symbol F to denote either the field of real or the field of complex numbers. Although it is not so necessary, we will usually require that the fibres are finite dimensional, hence the subspace topology on the fibres will be the usual vector space topology.

In the infinite dimensional case, the main difference is that one has to preassign a topology on the vector space compatible with addition and scalar multiplication. Bundles of infinite dimensional vector spaces are treated in more detail in Chap. Now, this can be generalized as in 1 2. Remark The composition of morphisms of bundles of vector spaces over B is again a morphism of bundles of vector spaces over B. Hence, it defines a category. Now, we are in a position to make the main definition of this chapter. Of course, this could have been done more directly, but in this way, we try to illustrate the difference between structure and axiom.

Remark The composition of morphisms of vector bundles over B is again a morphism of vector bundles over B. Hence, it defines a category of vector bundles over B. A trivial vector bundle is one which is isomorphic to a product vector bundle. Note that this is the usual definition of the tensor product for X a point. Then, u is an isomorphism. Thus, we can apply 2. In order to study this phenomenon, we recall some elementary conditions on rank of matrices. We give these spaces as usual the natural Euclidean topology.

Since the determinant is a polynomial function, these subsets are algebraic varieties. If rank ub is locally constant, then the bundles of vector spaces ker u and im u are vector bundles. Hence, we see that ker w is locally trivial and thus a vector bundle. A very useful case where we know the rank is locally constant and used in the next chapter is in the following proposition. Now apply 3. Another name for the weak topology is the inductive limit topology.

Let Grn F N denote the quotient space of Pn F N which assigns to a linearly independent n-tuple the subspace F N of dimension n of which the n-tuple is basis. The subject of principal bundles is taken up in Chap. Now, we consider the inductive limit spaces and vector bundles as N goes to infinity. Remark The inductive limit construction of 4. We begin with a circle of implications. This proves the theorem. Definition A vector bundle satisfying any of the equivalent conditions of 5. Theorem Every vector bundle over a compact space is finitely generated. Let V1 ,. This is possible since X is normal.

Thus, there exists a Gauss map for E, and hence E is finitely generated. In the case of both of the previous definitions, a direct quotient process gives a bundle of vector spaces, and as for the question of local triviality, we have the following remarks. Here, we must assume that either A is a closed subspace in a compact X and then use the Tietze extension theorem or assume that A is a subcomplex of a CW -complex X. Definition Let V be a left vector space over F. This formula holds both in the finite and in the infinite case where vectors have only finitely many nonzero components.

By restriction, every inner product on a vector bundle gives an inner product on each subbundle. In particular, every bundle which is induced from the universal bundle over Gn F n has a metric. For this, we need a suitable hypothesis on X. This is proved in Sect. Then, the remainder of the argument in Sect.

This is the case for a compact space X, and for a locally compact space, we can modify the assertion in terms of vanishing and triviality at infinity. It is often referred to as the Serre—Swan correspondence between vector bundles over a space X and finitely generated projective modules over the algebra C X. In the process of showing that every projective module comes from a vector bundle, we start with relations between idempotent elements in matrix algebras. The algebraic relation leads to another description of vector bundles and projective modules as classes of idempotents.

Then, there exist cross sections s1 ,. Proposition Let E be a vector bundle on a normal space X. For the opposite inclusion, we proceed in two steps. From 1 1. This proves the other inclusion and the proposition. Then, there exists by 1. Thus, f is continuous, and this proves the full embedding property. An arbitrary sequence of morphisms of vector bundles is exact provided every subthree term sequence is exact. This exactness assertion is the combination of two assertions.

Again, there is an important point where one has to be rather careful. As we remarked in 2 3. Let R be a commutative ring, and let R denote the category of modules over R. Recall that a module M is finitely generated free if and only if there is a finite set x1 ,. The finite set x1 ,. The next most elementary type of module M is a direct summand of Rn for some n. This leads to the following definition. Definition A finite bibasis of a module M is a set of x1 ,.

In particular, u is defined by u a1 ,. These elements x1 ,. The morphism g is called a lifting of the morphism f. The situation in 3. Example Every free module L is projective. Proposition The following are equivalent for M: a The module M is isomorphic to a direct summand of Rn. For b implies c , we see that the x1 ,. We leave it to the reader to prove the following version of the previous proposi- tion which does not refer to finiteness properties of the modules.

Remark A module M is projective if and only if M is isomorphic to a direct summand of a free module. Now, we combine theorems 2. Theorem Let X be a space over which each vector bundle is finitely generated. In view of criteria 1 6. As a corollary, we have the following theorem. Theorem 4. Use 4. There is a functoriality and a naturality related to this equiva- lence of categories. Then, the cross section functor is contained in the following commutative diagram. Then, condition 5. Proposition The relation of algebraic equivalence is an equivalence relation between idempotents in matrix algebras.

Using 5. Definition Let Idem R denote the set of equivalence classes of idempotents in the various Mn R with respect to the algebraic equivalence relation. See also 3 5. Chapter 4 K-Theory of Vector Bundles, of Modules, and of Idempotents We wish to consider the isomorphism classes of vector bundles over a space X or in view of the correspondences in the previous chapter, either the isomorphism classes of finitely generated modules over C X or algebraic equivalence classes of idempotents in the matrix algebras Mn C X for all n.

In each of the three cases, the direct sum leads to a natural addition on this sets of classes, and with the tensor product, there is a second algebraic operation of product. Unfortunately, these sets of isomorphism classes are not a ring under these operations, but only a semiring. The unfulfilled ring axiom is that each element must have a negative. Using the results of the previous chapter, we can show that these semirings and the corre- sponding K-rings are all isomorphic.

## Topics in the Homology Theory of Fibre Bundles

This is an elementary exercise in the study of functors with universal properties, and it is the basic construction needed to go from isomorphism classes to stable classes of vector bundles, projective modules, and idempotents. Remark Stripping off the multiplication on a semiring resp. This leads to a commutative diagram of categories. In the case that A is a semiring, we define a ring structure on T A by the relation [a, x]. If A is a commutative semiring, then T A is a commutative ring.

The zero vector bundle is the zero class, and the trivial line bundle is the unit in the commutative semiring. Now, we consider an elementary property of the K-functor which plays a basic role later. The zero module is the zero class, and the R-module R is the unit in the commutative semiring for R commutative. Definition Recall from 3 5. Definition The Grothendieck idempotent I-functor on the category rg , resp. The functoriality of I follows from the functoriality of Idem and the universal property 1.

Theorem For a commutative ring R, the map im, which we have introduced in 3 5. We have only to check the multiplicative property of the morphism im in case R is commutative.

### Lectures given at the University of Chicago, 1954 Notes by Edward Halpern

We get the same results for the algebra of bounded continuous functions which we also denote by C X. In the algebraic setting, these modules were related to idempotents in matrix algebras over the matrix algebras Mn A for a general algebra in Sect. In the first four sections of this chapter, the class group or K-groups of vector bundles, finitely generated projective modules, and idempotents were introduced.

Remark For the study of class groups, we see that there is a large interface be- tween the topology of bundles and the algebra of certain modules. This leads to the possibility of using only projections, that is, just selfadjoint idempotents. At a very early stage in the development of topological K-theory, Banach algebras played an important role giving alter- native proofs of Bott periodicity, which is a fundamental subject considered in Chap.

Remark At the center of this interface between K-theory coming from topology and the use of analysis in K-theory is the Atiyah—Singer index theorem. In the process, the homotopy description of K-theory given in Chap. In order to carry this theme a little further, we give two definitions related to the first remark 5. For K-theory from an algebraic point of view, it is the role of the matrix algebra and idempotents that plays the basic role.

Remark In the context of topological algebra, we have a new phenomenon that the algebra is so large that there are enough representatives of idempotents or pro- jections in the algebra itself to determine K-theory. This happens for special cases in algebraic K-theory where up to direct summands with finitely generated free mod- ules, every finitely generated projective is a projective submodule of a Dedekind ring R. A norm is complete provided this metric is complete.

To elements in an algebra A over C, we associate a subset of the complex numbers called the spectrum. We finish this section with some K-theory considerations. This theory follows in many ways the lines of the topological theory in the next chapter even though K A is the algebraic K-theory of A. We recommend the following two references. The second is Blackadar In Sects. This book also takes the reader to many considerations taken up in Chap.

References Blackadar, B. We outline two important aspects within principal bundle theory in this chapter. At first, the reduction of the structure group G of a principal bundle P. A subgroup H of a topological group G with the subspace topology is again a topological group. Definition Let G be a topological group.

Such morphisms are also called G-equivariant maps or G-maps. We will return to other actions later. Definition Let X be a right left G-space. The orbit space has the quotient topology under the projections. Note that in later chapters we consider more general bundles and call them still G-bundles.

These are bundles where G acts on the base B as well and p is equivariant, see 13 2.

Most G-bundles that we will consider come by this functor from G-spaces. Theorem Every morphism of principal bundles over B is an isomorphism. The description of cross sections of a fibre bundle is very basic because the result is in terms of equivariant maps from the principal bundle space to the fibre of the fibre bundle. The restriction of the structure group is not always possible, and it depends on the existence of a cross section for a related fibre bundle.

Remark Using the notation of 5. There are two other ways of looking at gauge transformations. The bijection between the sets defined by the condition 2 and condition 3 is just the description of the set of cross sections of a fibre bundle 5. This sets up the correspon- dence and proves the proposition. Of special importance was the fact that a vector bundle is a particular example of a fibre bundle which comes with a principal bundle in the background. In this part, Chaps. This is the homotopy invariance of bundles.

These homotopy questions lead to fibre space questions. In Chaps. Chapter 6 Homotopy Classes of Maps and the Homotopy Groups In this chapter, we prepare the basic definitions on the homotopy relation between maps. These ideas apply everywhere in geometry, and it is usually the case that invariants of maps which are interesting are those which are the same for two ho- motopic maps. This gives three interpretations of homotopy which are discussed in the first sections.

This is the homotopy invariance property of bundles. This implies that two homotopic maps induce iso- morphic bundles. A space or set is compact provided it satisfies the Heine—Borel covering property and the Hausdorff separation condition. The compact open topology works best for separated Hausdorff spaces, and we will assume that our spaces are separated. It is needed in the following remark. This is a well- defined continuous map by 1.

Proposition Let T be an arbitrary space. Let X and Y be two pointed spaces. Another name for a pointed map is a base-point-preserving map. Clearly, the composition of two pointed maps is again a pointed map. Definition As before, top denotes the category of spaces and continuous func- tions. This follows from the following general continuity result which is also used in bundle theory for many purposes.

Thus, f is continuous in both cases. For this, we use the following proposition. The standard category language has a special terminology when dealing with homotopy categories. This is a property of a functor which is common in geometry. Now, we return briefly to Chap. Viewing the object T in C op just changes the roles of these two functors. Remark The functor [ , T ] arises in bundle theory and in cohomology, where T is a classifying space.

This is considered in the next section. The associative law is a construction of an interesting homotopy. Now, the group axioms are easily checked. This brings up the question of commutativity of the homotopy groups, and for this we go back to the second definition of the homotopy groups introduced in the context of the associative law. This leads to the following algebraic lemma which applies to only homotopy classes, and it is not true on the level of functions since there is no unit property.

This proves the lemma. Proposition If G, e is a pointed space with a continuous multiplication a. This is another application of the Lemma 5. Of course this proposition applies to a topological group. Of special interest are Lie groups, and here, we state the following basic results for a compact Lie group. Example Let G be a compact, connected Lie group. We have also the special cases which we return to in Chap. The open sets V form a covering of B in the sense of the next definition.

Now, we bring in the numerical functions. For establishing the homotopy property of principal G-bundles, we use envelopes of unity, and for the comparison of a bundle with the Milnor construction in the next chapter, we use partitions of unity. A partition of unity defines an envelope of unity and vice versa by just rescaling the functions. Observe that an induced numerable bundle is numerable. Before we sketch the proof of this theorem, we state the main corollary of this theorem which is the homotopy invariance property for principal G-bundles.

For the composition, we consider a well ordering of the indexing set I. For more details, see Fibre Bundles, 3 4. Remark A convenient class of spaces is the class of paracompact spaces. A Hausdorff space is paracompact if each open covering is numerable. As a con- sequence, a locally trivial principal G-bundle over a paracompact space is always numerable. At first, it looks like finding all principal G-bundles over a space might be a great task, but there is a special construction of a principal G-bundle due to Milnor. It has the property that all other numerable principal G-bundles over all possible spaces are induced from this particular bundle.

This should be put side by side with the result of the previous chapter which says that homotopic maps induce isomorphic principal bundles. We conclude the chapter with some specific examples of the Milnor construction where the involved spaces En G and E G to be defined in Sect. Chapter 4, Sect. It would be useful to have control over this arbitrary set, and with the next proposition, it is possible to always use a countable set. Proposition Let P be a numerable principal G-bundle over a space B.

Partitions of unity map the space into simplexes. For example, A0 is a point, A1 is a closed segment, A2 is a triangle, and A3 is a tetrahedron. Now, we extend the basic assertion for a finite part of the local trivializing data to the case of a countable family. From 6 6. We sketch this in the next section. Notation Let E G, ev resp. E G, odd be the subspace of E G consisting of t0 : s0 ,. The details we leave to the reader who can find them in the paragraph just before 4 These deformations are used in the following theorem which completes the last step in the homotopy classification theorem announced in the introduction.

The first step is to use the homotopies described in 3. Now, we return to the introduction of this chapter. It is a well-defined function by 6 6. It is injective by 3. This proves the homotopy classification of numerable principal G-bundles in terms of the Milnor construction. Under multiplication of complex numbers, it is a group called the circle group. There is one more sphere which is the topological group of numbers in a number system, and this is S3 , the group of unit quaternions.

Now, we show how to view certain spheres as the total space in the Milnor con- struction in a very concrete way. Then, the map from x0 ,. Then, the map from z0 ,. These mappings are S1 -equivariant. Then, the map from q0 ,. These mappings are S3 - equivariant. In fact, for a given topological group G, this universal prop- erty is true for other principal G-bundles than the Milnor construction, and in this chapter, we investigate which bundles have this property.

The base space of each universal bundle is a new model for the classifying space BG of the group G, and it is homotopy equivalent to the base space of the Milnor construction. For this analy- sis, we introduce the notion of fibre map and fibre mapping sequence. In this chapter, we consider loop spaces and the related path space. These are not principal bundles, but they have important bundle properties relative to homotopy. We relate and compare these loop space bundles to universal principal G-bundles. The key concept of fibre map is common to both path space bundles and the univer- sal bundles.

For a given topological group G, we consider all the principal bundles over a space B which is reduced to homotopy theory by the universal bundle. In fact, we can speak of the category of squares Sq C over a category C , where a morphism is a morphism on each corner giving a commutative cube. We can also speak of k as a lifting of f along p or as an extension of u along i.

Relative to classes of morphisms in C , we can define properties on morphisms with respect to the existence of factorization. These are lifting and extension prop- erties. Definition Let E be a family of morphisms in C. Definition Let F be a family of morphisms in C.

## Algebraic topology

These properties are dual to each other in C and its opposite category C op. Ex- amples are generated by the following two pairs of dual statements. Recall that the induced morphism is the pro- jection from the fibre product to the first factor. Recall that the coinduced morphism is the injection into the cofibre coproduct of the first factor.

Remark We will apply these general lifting properties to homotopy theory. This was first done by Quillen in LN 43 where he presented an axiomatic ver- sion of homotopy theory which seems to be the most promising approach to an axiomatic version of homotopy theory. This has become especially clear in recent years. For this, we need the classes E of elementary cofibrations and F of elementary fibrations.

The natural lifting k is given as follows. This leads to the following assertion. Remark There are essentially two versions of the definition of fibre maps or fibration. Now base points arise naturally when we consider fibres and cofibres of a map. Fibres are related to fibrations and cofibres to cofibrations by the following exact- ness properties. When the space T is a finite complex like a sphere, then the assertion holds also for a Serre fibration. The reverse inclusion uses the fibration and cofibration conditions.

Taking the fibre or the cofibre of a map in special situations leads to new con- structions to which theorem 3. Definition Let Y be a pointed space. Definition Let X be a pointed space. The cofibre of q is the suspension S X of the pointed space X. We sketch this topic in the next section where it allows one to extend the sequences of the theorem 3. Remark The adjunction morphism 6 2. Using the modifications in the pre- vious Sect. As a result, we have the following homotopy level extension of the adjunction of 4.

For the last statement we use 6 5. Moreover, a f is a fibre map. A fundamental property of j is that it is a homotopy equivalence if f is a fibre map. This is used in comparing two possible sequences in the following diagram which also relates E f to the mapping track T f. The following sequence is exact.. Use 5. The cofibre of the injection a f is just the further identification of all of Y to the base point, that is, the suspension S X.

Moreover, a f is a cofibre map. A fundamental property of q is that it is a homotopy equivalence if f is a cofibre map. This is used in comparing two possible sequences in the following diagram which also relates C f to the mapping cylinder Z f. Elementary Corollaries We use the notation of 6. This class includes all CW-complexes, see the next chapter. Even if F is not a group, its set of connected components has a natural group structure coming from the fundamen- tal group of the base. This leads to a general idea which ties the homotopy theory considerations to fibre bundle theory.

Any space which is homotopy equivalent to a space which admits the structure of a CW -complex to be defined in 9 1. Moreover, the Milnor construction B G can be equipped with the structure of a CW -complex when G can be given the structure of a CW-complex, which includes most groups of interest in geometry. Now, apply the same theorem that holds for the Milnor construction 7 3.

Remark Now, we can speak of the universal bundle and the classifying space in the more general setting as a principal G-bundle with contractible total space and base space as classifying space. These spaces with Y having a trivial G-action are path connected by the homotopy classification theorem.

Remark The construction of 7 4. Using the cofibre con- structions, we discuss the Whitehead mapping theorem. This characterization of homotopy equivalence was already used in the study of the uniqueness properties of classifying spaces. This completes a question left open in the previous chapter. The usual definition of cohomology arises in terms of the dual of homology ei- ther directly for coefficients in a field or in terms of chains and cochains as linear forms on chain groups.

These chains can arise very geometrically as cell chains or more generally as simplicial chains. The cochains can be algebraic linear func- tionals or as in the case of manifolds they can be differential forms which become linear functionals on chains of simplexes upon integration over the simplexes and summing over the chain. In this chapter, we use the notion of cofibre maps and the cofibre sequence for a map. Spanier is a reference for this chapter. Frequently, this comes as an increasing sequence of subspaces with union the entire space. Here is a basic application. Then, X is contractible.

Idea of the Proof. Remark The construction of the space En G of 7 2. Idea of the proof. For further background see the book by Whitehead , especially Chaps. So we will list properties of homology and cohomology which can be used in an axiomatic definition as given in the book of Eilenberg and Steen- rod. Let k be a field, or more generally a commutative ring, and let k resp. The ring k plays the role of coefficients for the theory.

These are functors on top , the category of topology spaces. In particular, homology and cohomology are defined on a quotient category of pairs. Also Hq f and H q f are isomorphisms for a homotopy equivalence, that is, a map which is an isomorphism in the homotopy category. Furthermore, the homology and cohomology of a contractible space is isomorphic to the homology and cohomology of a point. Further, using the mapping sequences, we can formulate the exact sequence properties. In particular, we have formulated these two basic properties in terms of the re- duced homology and cohomology.

At this point, there is no relation between ho- mology and cohomology for different values of q, an integer. In the next axiom, we have this relation. Again this is for reduced homology and cohomology. Here, k is the one-dimensional k-module. One frequently used tool is the following sequence which is a consequence of the above properties.

In the cohomological case, the homomorphism is given in a corresponding manner. Remark There is a further extension of all the above considerations to homology Hq X, A; M and cohomology H q X, A; M with coefficients in an abelian group M, or more generally, a k-module M which is useful for certain considerations. Singular Theory The most important construction of homology and coho- mology starts with the so-called singular simplices into a space as basis elements of chain groups.

In local coordinates x1 ,. It is a real or complex vector space depending on whether the forms are real or complex. Both play a fundamental role in topology and geometry. For further reading see Spanier or Warner These formulas follow from the cofibre character of the wedge product. In particular, by 4. The corresponding statement is true on the chain level, and under either of these hypothesis, it holds on the homology level. For the proof of 5. This result was extended by Hurewicz to the following theorem.

Now as a sort of dual to the Hurewicz map, we have the following. For the proof of 6. This is the same graded commutativity that we have in a graded exterior alge- bra. This is illustrated by the following examples of cohomology rings. In the next chapter, we will see that example 2 has to do with character- istic classes of real vector bundles, called Stiefel—Whitney classes, and example 3 has to do with characteristic classes of complex vector bundles, called Chern classes. The naturality will be explained further in the context of duality.

Its relation to the cup product is contained in the next remark. Hence, there are cases where there is a down to earth calculation of homology. Morse functions are a class of functions where more information is available about such filtrations defined by positive real-valued functions. The index of f at critical point is the number of negative eigenvalues of the T 2 f. There is a version of this definition for noncompact manifolds and even infinite dimensional ones due to Palais and Smale. This is treated in Schwarz The two basic references for Morse theory are Milnor and Milnor In particular, in these books, one finds a proof of the following theorem.

Theorem Every compact smooth manifold M admits a nice Morse function. The existence of a Morse function is a partition of unity argument while to change it to a nice Morse function is a much deeper result. Applying the scheme in 8. References 8. Remark In fact, a Morse function not only can be used to control the homotopy type of M but also its diffeomorphism type. This is also explained very well in the books of Milnor cited above. References Milnor, J. Springer-Verlag, New York reprint of the edition.

Whitehead, G. Springer Graduate Texts, Vol. It was natural to try and make calculations with bundles in terms of cohomology for two reasons. With homology and cohomology, there were combinatorial tools for com- putation. Then, cohomology and bundles each had contravariant properties under continuous mappings.

The first definitions of characteristic classes were given by obstructions to existence to cross sections of a bundle or related fibre bundle. Ex- amples of this can be found in the book by Steenrod With the understanding of the cohomology of fibre spaces including the Leray spectral sequence, the theory of characteristic classes was developed along more intrinsic lines. It was clear that the separate theories of Chern classes for complex vector bundles and of Stiefel—Whitney classes of real vector bundles had a parallel structure.

In the first sections, we carry this out using an approach of Grothendieck which also works in the context of algebraic geometry. Then, we have the Euler class and Pontrjagin classes which are introduced also by elementary fibre space methods. Pontrjagin classes can be related nicely to Chern classes. Then, it becomes clear that with splitting principles for vector bundles into line bundles that families of characteristic classes can be introduced from power series by their properties on line bundles.

This is considered in the next chapter. The topic is also treated in Milnor and Stasheff There are two cases of this coincidence. The first is related with real vector bundles while the second is related to complex vector bundles. Notation Let PicC X resp. PicR X denote the group of isomorphism classes of complex resp. The trivial line bundle is the unit, and the dual is the negative.

This is the essentially unique situation where the characteristic class com- pletely determines the bundle. In higher dimensions, there are more characteristic classes of a vector bundle, but they are usually not enough characteristic classes to classify the isomorphism class of the vector bundles.

In the next section, we introduce the fibre space results which are used to define general characteristic classes in terms of the characteristic class of line bundles. Definition Let E be a vector bundle over X.

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The next two theorems, the first for complex vector bundles and the second for real vector bundles, are easy consequences of the cohomology spectral sequence for a fibre map q, but they can also be proved for bundles of finite type with an inductive Mayer—Vietoris argument i.

The following two theorems are useful for uniqueness results and for doing cer- tain calculations. For the proofs of theorems 2. For line bundles, there is nothing to prove. Here, we use 2. The Chern classes exist and are unique by the projective bundle theorem and the splitting principle. Uniqueness of the Chern Classes The first Chern class of a line bundle is unique by axiom 4. In particular, c1 LE n is a unique linear combination of these basis elements with coefficients which we define to be the Chern classes ci E of E.

Remark Now, we have to check the axioms 1 — 4 in 3. For the Whitney sum axiom, we begin with the following space case. Theorem The Chern classes for complex vector bundles exist satisfying axioms 1 — 4 of 3. They are uniquely determined by the axioms. It remains to check the Whitney sum axiom 3. They have the following axiomatic characterization which parallels the axioms of the Chern classes.

The Stiefel—Whitney classes exist and are unique by an argument completely parallel to that for the Chern classes. These are commutative monoids of units with the multiplication as monoid operation. The same can be done for the tangent bundle Whitney sum with the normal bundle.

Thus, the top characteristic class must be zero for the existence of an everywhere nonzero cross section of the vector bundle.

In this sense, we speak of the top characteristic class as an obstruction to the existence of an everywhere nonzero cross section of the bundle. There are inter- esting cases where one can assert that the vanishing of the top characteristic class implies the existence of a everywhere nonzero cross section. Here are some of the topics we may cover: Higher homotopy groups, cofibrations, fibrations, fiber bundles, homotopy sequences, homotopy groups of Lie groups and associated manifolds, cellular approximation, Hurewicz theorem, Whitehead theorem, Eilenberg-MacLane spaces, obstruction theory, Postnikov towers, cohomology of fiber bundles, characteristic classes, spectral sequences, Steenrod sqaures.

And here are some useful textbooks including some olden goldies : Algebraic Topology , by Allen Hatcher , Cambridge University Press, Math Review. Elements of Homotopy Theory , by George W. Whitehead, GTM No. Characteristic Classes , by John W. Milnor and James Stasheff , Ann.