Characteristic classes Robert R. Bruner Michael Catanzaro J. Peter May Contents Chapter 0. Introduction 5 Chapter 1. Classical groups and bundle theory 7 1. The classical groups 7 2. Fiber bundles 10 3. Principal bundles and homogeneous spaces 13 4. Vector bundles, Stiefel and Grassmann manifolds 16 5. The classification theorem and characteristic classes 19 6. Some homotopical properties of classifying spaces 22 7. Algebraic Tools 24 8. Spectral sequences 24 Chapter 2. Cohomology of the Classical Groups and Stiefel Manifolds 25 1. The complex and quaternionic Stiefel manifolds 25 2. The real Stiefel manifolds 27 Chapter 3. Chern classes 33 Chapter 4. Symplectic classes 39 Chapter 5. Stiefel-Whitney Classes 45 Chapter 6. Steenrod Operations, the Wu formula, and BSpin 51 Chapter 7. Euler and Pontrjagin classes in rings containing 1/2 57 Chapter 8. The Thom Isomorphism 65 Chapter 9. Integral Euler, Pontrjagin and Stiefel-Whitney classes 75 Chapter 10. Applications and Examples 81 Appendix A. Bott periodicity 89 1. Definition of the Maps 89 2. Commutative Diagrams in the φi 93 3. Proof of the Periodicity Theorem 96 Appendix. Bibliography 97 3 CHAPTER 0 Introduction These notes had their genesis in a class Peter May taught in the spring (?) quarter of 1974 at the University of Chicago. Robert Bruner was assigned the task of writing them up in a coherent fashion based on his class notes and Peter’s notes. They were used in this handwritten form for many years at the University of Chicago. (What is the true version of this??) In the summer of 2012 Mike Catanzaro took on the task of TEXing the notes. After that, May and Bruner undertook some reorganization and added a few items to make the notes more self contained. The precipitating event in the decision to publish them was a question from a colleague about the cohomology of a particular homogeneous space. It became clear these basic results in algebraic topology should be available in textbook form. (???????) Compare to Mimura and Toda??? ADAPT THE FOLLOWING INTRODUCTORY SKETCH We develop the classical theory of characteristic classes. Our procedure is simultaneously to compute the cohomology of the relevant classifying spaces and to display the standard axiomatically determined characteristic classes. We first compute the homology and cohomology of Stiefel varieties and classical groups and then use the latter computations to pass to classifying spaces. Along the way, we compute the cohomologies of various homogeneous spaces, such as Sp(n)/U(n),U(2n)/Sp(n),U(n)/O(n), and SO(2n)/U(n). We also obtain the usual intrinsic characterizations, via the Thom isomorphism, of the Stiefel-Whitney and Euler classes. Since we shall have a plethora of explicit calculations, some generic notational conventions will help to keep order. We shall end up with the usual characteristic classes i wi ∈ H (BO(n); F2), the Stiefel-Whitney classes 2i ci ∈ H (BU(n); Z), the Chern classes 4i ki ∈ H (BSp(n); Z), the symplectic classes 4i Pi ∈ H (BO(n); Z), the Pontryagin classes χ ∈ H2n(BSO(2n); Z), the Euler class. The Pi and χ will be studied in coefficient rings containing 1/2 before being intro- duced integrally. We use the same notations for integral characteristic classes and for their images in cohomology with other coefficient rings. Prerequisites: To do. (Just say ”see the next chapter”?) 5 CHAPTER 1 Classical groups and bundle theory We introduce the spaces we shall study and review the fundamentals of bundle theory in this chapter. Aside from a few arguments included for didactic purposes, proofs are generally sketched or omitted. However, Sections 3 and 6 contain some material either hard to find or missing from the literature, and full proofs of such statements have been supplied. We assume once and for all that all spaces we consider are to be of the ho- motopy type of CW-complexes. This ensures that a weak homotopy equivalence, namely a map which induces isomorphisms of homotopy groups for all choices of basepoints, is a homotopy equivalence. By the basic results of Milnor [9] (see also Schon [11]), this is not a very restrictive assumption. We also assume that all spaces are paracompact. This ensures that all bundles are numerable (in the sense speci- fied in Section 2). Since all metric spaces, all countable unions of compact spaces, and all CW-complexes (Miyazaki [10] or Fritsch and Piccinini [6, Thm 1.3.5]), are paracompact, this assumption is also not unduly restrictive. 1. The classical groups All of our work will deal with the classical Lie groups and related spaces de- fined in this chapter. Good general references for this section are Adams [1] and Chevalley [4]. Let K denote any one of R, C, or H, the real numbers, complex numbers, or quaternions. For α ∈ K, let α denote the conjugate of α.A right inner product space over K is a right K-module W , together with a function ( , ): W × W → K which satisfies the following properties. (i) (x, y + y0) = (x, y) + (x, y0) (ii) (x, yα) = (x, y)α for any α ∈ K (iii) (x, y) = (y, x) (iv) (x, x) ∈ R,(x, x) ≥ 0, and (x, x) = 0 if and only if x = 0. The unmodified term inner product space will mean right inner product space. All inner product spaces will be finite or countably infinite dimensional; we write dim W = ∞ in the latter case. We say that a K-linear transformation T : W → W is of finite type if W contains a finite dimensional subspace V invariant under T such that T restricts to the identity on V ⊥. The classical groups are GL(W ) = {T : W −→ W | T is invertible and of finite type}, U(W ) = {T | T ∈ GL(W ) and T is an isometry}, and, if K = R or K = C, 7 8 1. CLASSICAL GROUPS AND BUNDLE THEORY SL(W ) = {T | T ∈ GL(W ) and det T = 1} SU(W ) = {T | T ∈ U(W ) and det T = 1}. The finite type requirement assures that the determinant is well-defined. By choice of fixed orthonormal basis for W , we can identify GL(W ) with the group of matrices of the form A 0 0 I where A is an invertible n × n matrix with n < ∞. Such a matrix is in U(W ) if and only if A−1 = AT , where A is obtained from A by conjugating each entry and ()T denotes the transpose. Topologize inner product spaces as the union (or colimit) of their finite di- mensional subspaces. By choice of a fixed orthonormal basis and use of matri- 2 ces, the classical groups of W may be topologized as subspaces of Kn when n = dim W < ∞. The same topology may also be specified either in terms of norms of linear transformations or as the compact open topology obtained by re- garding these groups as subsets of the space of maps W → W . With this topology, G(W ) is a Lie group (G = GL, U, SL, or SU) and U(W ) and SU(W ) are compact. When dim W = ∞, G(W ) is topologized as the union of its subgroups G(V ), where V runs through all finite dimensional subspaces of W or through those V in any expanding sequence with union W . A standard theorem of linear algebra states that any element of GL(W ) can be written uniquely as the product of a symmetric positive definite transformation and an element of U(W ), and similarly with GL and U replaced by SL and SU. It follows that the inclusions U(W ) ,→ GL(W ) and SU(W ) ,→ SL(W ) are homotopy equivalences. For our purpose, it suffices to restrict attention to U(W ) and SU(W ). A convenient framework in which to view the classical groups is as follows. Let Ke denote the category of finite or countably infinite dimensional inner product spaces of K with linear isometries as morphisms. Note that isometries need not be surjective. Then U and SU are functors from Ke to the category of topological groups. Obviously if V and W are objects in Ke of the same dimension, then there is an isomorphism V =∼ W in Ke which induces isomorphisms U(V ) =∼ U(W ) and SU(V ) =∼ SU(W ). This formulation has the conceptual clarity common to basis free presentations and will be useful in our proof of Bott periodicity. However, for calculational purposes, it is more convenient to deal with particularly representatives of the classical groups. We define examples as follows, where Kn has its standard inner product. (i) O(n) := U(Rn),O := U(R∞) the orthogonal groups (ii) SO(n) := SU(Rn), SO := SU(R∞) the special orthogonal groups (iii) U(n) := U(Cn),U := U(C∞) the unitary groups (iv) SU(n) := SU(Cn), SU := SU(C∞) the special unitary groups (v) Sp(n) := U(Hn), Sp := U(H∞) the symplectic groups There is another family of classical groups not included in this scheme, namely the spinor groups Spin(n) for n > 2 and Spin = Spin(∞). We define Spin(n) to be the universal covering group of SO(n). Each Spin(n) for n < ∞ is a Lie group and S Spin = n Spin(n). Since π1(SO(n)) = Z/2Z, Spin(n) is a 2-fold cover of SO(n). 1. THE CLASSICAL GROUPS 9 An alternative description of the spinor groups in terms of Clifford algebras is given in Chevalley [4, p.65]. There are forgetful functors ()R : Ce → Re and ( )C : He → Ce. If W is in Ce, then W R is the underlying vector space with inner product the real part of the inner product of W .