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ξ) ∈ H n(B; R) of an R-oriented n-plane bundle ξ over the base space B by e(ξ) = Φ−1µ2, where µ ∈ H n(T ξ; R) is the Thom class. Giving the universal oriented n-plane bundle over BSO(n) the R-orientation induced by its integral orientation, this deﬁnes the Euler class e ∈ H n(BSO(n); R). 200 CHARACTERISTIC CLASSES OF VECTOR BUNDLES If n is odd, then 2µ2 = 0 and thus 2e = 0. If R = Z2, then Sqn(µ) = µ2 and thus e = wn. The name “Euler class” is justiﬁed by the following classical result, which well illustrates the kind of information that characteristic numbers can encode.1 Theorem. If M is a smooth closed oriented manifold, then the characteristic number e[M ] = he(τ (M )), zi ∈ Z is the Euler characteristic of M. The evident inclusion T n ∼= SO(2)n −→ SO(2n) is a maximal torus, and it induces a map BT n −→ BSO(2n). A calculation shows that e restricts to the nth elementary symmetric polynomial β1 · · · βn. The cited inclusion factors through the homomorphism U (n) −→ SO(2n), hence BT n −→ BSO(2n) factors through r : BU (n) −→ BSO(2n). This implies another basic fact about the Euler class. Proposition. r∗ : H ∗(BSO(2n); Z) −→ H ∗(BU (n); Z) sends e to cn. The presence of 2-torsion makes the description of the integral cohomology rings of BO(n) and BSO(n) quite complicated, and these rings are almost never used in applications. Rather, one uses the mod 2 cohomology rings and the following description of the cohomology rings that result by elimination of 2-torsion. Theorem. Take coeﬃcients in a ring R in which 2 is a unit. Then H ∗(BO(2n)) ∼= H
∗(BO(2n + 1)) ∼= H ∗(BSO(2n + 1)) ∼= R[p1,..., pn] and H ∗(BSO(2n)) ∼= R[p1,..., pn−1, e], with e2 = pn. 8. A glimpse at the general theory We should place the theory of vector bundles in a more general context. We have written BO(n), BU (n), and BSO(n) for certain “classifying spaces” in this chapter, but we deﬁned a classifying space BG for any topological group G in Chapter 16 §5. In fact, the spaces here are homotopy equivalent to the spaces of the same name that we deﬁned there, and we here explain why. Consider bundles ξ : Y −→ B with ﬁber G. For spaces U in a numerable open cover O of B, there are homeomorphisms φ : U ×G −→ p−1(U ) such that p◦φ = π1. We say that Y is a principal G-bundle if Y has a free right action by G, B is the orbit space Y /G, ξ is the quotient map, and the φ are maps of right G-spaces. We say that ξ : Y −→ B is a universal principal G-bundle if Y is a contractible space. In particular, for any topological group G whose identity element is a nondegenerate basepoint, such as any Lie group G, the map p : EG −→ BG constructed in Chapter 16 §5 is a universal principal G-bundle. The classiﬁcation theorem below implies that the base spaces of any two universal principal G-bundles are homotopy equivalent, and it is usual to write BG for any space in this homotopy type. Observe that the long exact sequence of homotopy groups of a universal principal G-bundle gives isomorphisms πq(BG) ∼= πq−1(G) for q ≥ 1. We have implicitly constructed other examples of universal principal G-bundles when G is O(n), U (n), or SO(n). To see this, consider Vn(Rq). Write Rq = Rn
× Rq−n and note that this ﬁxes embeddings of O(n) and O(q − n) in the orthogonal group O(q). Of course, O(q) acts on vectors in Rq and thus on n-frames. Consider the ﬁxed n-frame x0 = {e1,..., en}. Any other n-frame can be obtained from 1See Corollary 11.12 of Milnor and Stasheﬀ Characteristic Classes for a proof. 8. A GLIMPSE AT THE GENERAL THEORY 201 this one by the action of an element of O(q), and the isotropy group of x0 is O(q − n). Thus the action of O(q) is transitive, and evaluation on x0 induces a homeomorphism O(q)/O(q − n) −→ Vn(Rq) of O(q)-spaces. The action of O(n) ⊂ O(q) is free, and passage to orbits gives a homeomorphism O(q)/O(n)×O(q−n) −→ Gn(Rq). It is intuitively clear and not hard to prove that the colimit over q of the inclusions O(q − n) −→ O(q) is a homotopy equivalence and that this implies the contractibility of Vn(R∞). We deduce that Vn(R∞) is a universal principal O(n)bundle. We have analogous universal principal U (n)-bundles and SO(n)-bundles. There is a classiﬁcation theorem for principal G-bundles. Let PG(B) denote the set of equivalence classes of principal G-bundles over B, where two principal G-bundles over B are equivalent if there is a G-homeomorphism over B between them. Via pullback of bundles, this is a contravariant set-valued functor on the homotopy category of spaces. Theorem. Let γ : Y −→ Y /G be any universal principal G-bundle. The natural transformation Φ : [−, Y /G] −→ PG(−) obtained by sending the homotopy class of a map f : B −→ Y /G to the equivalence class of the
principal G-bundle f ∗Y is a natural isomorphism of functors. Now let F be any space on which G acts eﬀectively from the left. Here an action is eﬀective if gf = f for every f ∈ F implies g = e. For a principal Gbundle Y, let G act on Y × F by g(y, f ) = (yg−1, gf ) and let Y ×G F be the orbit space (Y × F )/G. With the correct formal deﬁnition of a ﬁber bundle with group G and ﬁber F, every such ﬁber bundle p : E −→ B is equivalent to one of the form Y ×G F −→ Y /G ∼= B for some principal G-bundle Y over B; moreover Y is uniquely determined up to equivalence. In fact, the “associated principal G-bundle” Y can be constructed as the function space of all maps ψ : F −→ E such that ψ is an admissible homeomorphism onto some ﬁber Fb = p−1(b). Here admissibility means that the composite of ψ with the homeomorphism Fb −→ F determined by a coordinate chart ∼=−→ p−1(U ), b ∈ U, coincides with action by some element of G. The φ : U × F left action of G on F induces a right action of G on Y ; this action is free because the given action on F is eﬀective. The projection Y −→ B sends ψ to b when ∼=−→ Fb, and it factors through a homeomorphism Y /G −→ B. Y inherits ψ : F local triviality from p, and the evaluation map Y × F −→ E induces an equivalence of bundles Y ×G F −→ E. We conclude that, for any F, PG(B) is naturally isomorphic to the set of equivalence classes of bundles with group G and ﬁber F over B. Fiber bundles with group O(n) and ﬁber Rn are real n-plane bundles, ﬁber bundles with group U (n) and ﬁber Cn are complex n-plane bundles, and ﬁber bundles with group SO(n) and ﬁ
ber Rn are oriented real n-plane bundles. Thus the classiﬁcation theorems of the previous sections could all be rederived as special cases of the general classiﬁcation theorem for principal G-bundles stated in this section. In our discussion of Stiefel-Whitney and Chern classes, we used that passage to classifying spaces is a product-preserving functor, at least up to homotopy. For the functoriality, if f : G −→ H is a homomorphism of topological groups, then consideration of the way bundles are constructed by gluing together coordinate charts shows that a principal G-bundle ξ : Y −→ B naturally gives rise to a 202 CHARACTERISTIC CLASSES OF VECTOR BUNDLES principal H-bundle f∗Y −→ B. This construction is represented on the classifying space level by a map Bf : BG −→ BH. In fact, if EG −→ BG and EH −→ BH are universal principal bundles, then any map ˜f : EG −→ EH such that ˜f (xg) = ˜f (x)f (g) for all x ∈ EG and g ∈ G induces a map in the homotopy class Bf on passage to orbits. For example, if f : G −→ G is given by conjugation by γ ∈ G, f (g) = γ−1gγ, then ˜f (x) = xγ satisﬁes this equivariance property and therefore Bf is homotopic to the identity. This explains why inner conjugations induce the identity map on passage to classifying spaces, as we used in our discussion of Stiefel-Whitney and Chern classes. If EG −→ BG and EG′ −→ BG′ are universal principal G and G′ bundles, then EG × EG′ is a contractible space with a free action by G × G′. The orbit space is BG × BG′, and this shows that BG × BG′ is a choice for the classifying space B(G × G′) and is therefore homotopy equivalent to any other choice. The explicit construction of BG given in Chapter 16 §5 is functorial in G on the point-set level and not just up to homotopy, and it is product preserving in the strong
sense that the projections induce a homeomorphism B(H × G) ∼= BH × BG. PROBLEMS (1) Verify that w(RP q) = 1 if and only if q = 2k − 1 for some k. (2) Prove that RP 2k cannot immerse in R2k+1−2. (By the Whitney embedding theorem, any smooth closed n-manifold immerses in R2n−1, so this is a best possible non-immersion result.) (3) Prove that all tangential Stiefel-Whitney numbers of RP q are zero if and only if q is odd. (4) * Try to construct a smooth compact manifold whose boundary is RP 3. (5) Prove that a smooth closed n-manifold M is R-orientable if and only its tangent bundle is R-orientable. CHAPTER 24 An introduction to K-theory The ﬁrst generalized cohomology theory to be discovered was K-theory, and it plays a vital role in the connection of algebraic topology to analysis and algebraic geometry. The fact that it is a generalized cohomology theory is a consequence of the Bott periodicity theorem, which is one of the most important and inﬂuential theorems in all of topology. We give some basic information about K-theory and, following Adams and Atiyah, we explain how the Adams operations in K-theory allow a quick solution to the “Hopf invariant one problem.” One implication is the purely algebraic theorem that the only possible dimensions of a real (not necessarily associative) division algebra are 1, 2, 4, and 8. We shall only discuss complex Ktheory, although there is a precisely analogous construction of real K-theory KO. From the point of view of algebraic topology, real K-theory is a substantially more powerful invariant, but complex K-theory is usually more relevant to applications in other ﬁelds. 1. The deﬁnition of K-theory Except where otherwise noted, we work with complex vector bundles throughout this chapter. Dimension will mean complex dimension and line bundles will mean complex line bundles. We consider the set V ect(X) of equivalence classes of vector bundles over a space X. We assume unless otherwise speciﬁed that X
is compact. We remind the reader that vector bundles can have diﬀerent dimension over diﬀerent components of X. The set V ect(X) forms an Abelian monoid (= semi-group) under Whitney sum, and it forms a semi-ring with multiplication given by the (internal) tensor product of vector bundles over X. There is a standard construction, called the Grothendieck construction, of an Abelian group G(M ) associated to an Abelian monoid M : one takes the quotient of the free Abelian group generated by the elements of M by the subgroup generated by the set of elements of the form m + n − m ⊕ n, where ⊕ is the sum in M. The evident morphism of Abelian monoids i : M −→ G(M ) is universal: for any homomorphism of monoids f : M −→ G, where G is an Abelian group, there is a unique homomorphism of groups ˜f : G(M ) −→ G such that ˜f ◦ i = f. If M is a semi-ring, then its multiplication induces a multiplication on G(M ) such that G(M ) is a ring, called the Grothendieck ring of M. If the semi-ring M is commutative, then the ring G(M ) is commutative. Definition. The K-theory of X, denoted K(X), is the Grothendieck ring of the semi-ring V ect(X). An element of K(X) is called a virtual bundle over X. We write [ξ] for the element of K(X) determined by a vector bundle ξ. Since ε is the identity element for the product in K(X), it is standard to write q = [εq], where εq is the q-dimensional trivial bundle. For vector bundles over a 203 204 AN INTRODUCTION TO K-THEORY based space X, we have the function d : V ect(X) −→ Z that sends a vector bundle to the dimension of its restriction to the component of the basepoint ∗. Since d is a homomorphism of semi-rings, it induces a dimension function d : K(X) −→ Z, which is a homomorphism of rings. Since d is an isomorphism when X is a point, d
can be identiﬁed with the induced map K(X) −→ K(∗). Definition. The reduced K-theory ˜K(X) of a based space X is the kernel of d : K(X) −→ Z. It is an ideal of K(X) and thus a ring without identity. Clearly K(X) ∼= ˜K(X) × Z. We have a homotopical interpretation of these deﬁnitions, and it is for this that we need X to be compact. By the classiﬁcation theorem, we know that E Un(X) is naturally isomorphic to [X+, BU (n)]; we have adjoined a disjoint basepoint because we are thinking cohomologically and want the brackets to denote based homotopy classes of maps. We have maps in : BU (n) −→ BU (n + 1). With our construction of classifying spaces via Grassmannians, these maps are inclusions, and we deﬁne BU to be the colimit of the BU (n), with the topology of the union. We say that bundles ζ and ξ are stably equivalent if, for a suﬃciently large q, the bundles ζ ⊕ εq−m and ξ ⊕ εq−n are equivalent, where m = d(ζ) and n = d(ξ). Let E U (X) be the set of stable equivalence classes of vector bundles over X. If X is connected, or if we restrict attention to vector bundles that are n-plane bundles for some n, then E U is isomorphic to colim E Un(X), where the colimit is taken over the maps E Un(X) −→ E Un+1(X) obtained by sending a bundle ξ to ξ ⊕ ε. Since a map from a compact space X into BU has image in one of the BU (n), and similarly for homotopies, we see that in this case [X+, BU ] ∼= colim[X+, BU (n)] and therefore E U (X) ∼= [X+, BU ]. A deeper use of compactness gives the following basic fact. Proposition. If ξ : E −→ X is a vector bundle over X, then there is a bundle η over X
such that ξ ⊕ η is equivalent to εq for some q. Sketch proof. The space ΓE of sections of E is a vector space under ﬁberwise addition and scalar multiplication. Using a partition of unity argument, one can show that there is a ﬁnite dimensional vector subspace V of Γ(E) such that the map g : X × V −→ E speciﬁed by g(x, s) = s(x) is an epimorphism of bundles over X. The resulting short exact sequence of vector bundles, like any other short exact sequence of vector bundles, splits as a direct sum, and the conclusion follows. Corollary. Every virtual bundle over X can be written in the form [ξ] − q for some bundle ξ and non-negative integer q. Proof. Given a virtual bundle [ω] − [ζ], where ω and ζ are bundles, choose η such that ζ ⊕ η ∼= εq and let ξ = ω ⊕ η. Then [ω] − [ζ] = [ξ] − q in K(X). Corollary. There is a natural isomorphism E U (X) −→ ˜K(X). Proof. Writing {ξ} for the stable equivalence class of a bundle ξ, the required isomorphism is given by the correspondence {ξ} ↔ [ξ] − d(ξ). Corollary. Give Z the discrete topology. For compact spaces X, there is a natural isomorphism K(X) ∼= [X+, BU × Z]. 1. THE DEFINITION OF K-THEORY 205 For nondegenerately based compact spaces X, there is a natural isomorphism ˜K(X) ∼= [X, BU × Z]. Proof. When X is connected, the ﬁrst isomorphism sends [ξ] − q to (f, n − q), where ξ is an n-plane bundle with classifying map f : X −→ BU (n) ⊂ BU. The isomorphism for non-connected spaces follows since both functors send disjoint unions to Cartesian products. The second isomorphism follows from the ﬁrst since d : K(X) −→ Z can be
identiﬁed with the map [X+, BU × Z] −→ [S0, BU × Z] induced by the coﬁbration S0 −→ X+, and the latter has kernel [X, BU × Z] since X+/S0 = X. For general, non-compact, spaces X, it is best to deﬁne K-theory to mean represented K-theory. Here we implicitly apply CW approximation, or else use the deﬁnition in the following form. Definition. For a space X of the homotopy type of a CW complex, deﬁne K(X) = [X+, BU × Z]. For a nondegenerately based space of the homotopy type of a CW complex, deﬁne ˜K(X) = [X, BU × Z]. When X is compact, we know that K(X) is a ring. It is natural to expect this to remain true for general X. That this is the case is a direct consequence of the following result, which the reader should regard as an aside. Proposition. The space BU × Z is a ring space up to homotopy. That is, there are additive and multiplicative H-space structures on BU × Z such that the associativity, commutativity, and distributivity diagrams required of a ring commute up to homotopy. Indications of proof. By passage to colimits over m and n, the maps pm,n : BU (m) × BU (n) −→ BU (m + n) induce an “addition” ⊕ : BU × BU −→ BU. In fact, we can deﬁne BU in terms of planes in any copy of C∞, and the explicit maps pm,n of Chapter 23 §2 pass to colimits to give G∞(C∞) × G∞(C∞) −→ G∞(C∞ ⊕ C∞); use of an isomorphism C∞ ⊕ C∞ ∼= C∞ gives the required map ⊕, which is well deﬁned, associative, and commutative up to homotopy; the zero-dimensional plane provides a convenient basepoint 0 with which to check that we have a zero element up to homotopy. Using
ordinary addition on Z, we obtain the additive H-space structure on BU × Z. Tensor products of universal bundles give rise to classifying maps qm,n : BU (m) × BU (n) −→ BU (mn). These do not pass to colimits so readily, since one must take into account the bilinearity of the tensor product, for example the relation (γm ⊕ ε) ⊗ γn ∼= (γm ⊗ γn) ⊕ γn, and we merely aﬃrm that, by fairly elaborate arguments, one can pass to colimits to obtain a product on BU × Z. It actually factors through the smash product with respect to the basepoint 0, since that acts as zero for the tensor product, and it restricts to an H-space structure on BO × {1} with basepoint (0, 1). The study of ring spaces such as this is a relatively new, and quite deep, part of algebraic topology. However, the reader should feel reasonably comfortable with the additive H-space structure on BU. 206 AN INTRODUCTION TO K-THEORY 2. The Bott periodicity theorem There are various ways to state, and various ways to prove, this basic result. We describe several versions and implications. One starting point is the following 1 over S2 ∼= CP 1; its points are pairs calculation. We have a canonical line bundle γ2 (L, x), where L is a line in C2 and x is a point on that line. We let H = Hom(γ2 1, ε) denote its dual. Theorem. K(S2) is generated as a ring by [H] subject to the single relation ([H] − 1)2 = 0. Therefore, as Abelian groups, K(S2) is free on the basis {1, [H]} and ˜K(S2) is free on the basis {1 − [H]}. Indication of proof. We think of S2 as the one-point compactiﬁcation of C decomposed as the union of the unit disk D and the complement D′ of the interior of D, so that D ∩ D′ = S1. Any n-plane bundle over S2 restricts to a trivial bundle over D and D′, and these trivial bundles restrict to the same bundle over S
1. Conversely, an isomorphism f from the trivial bundle over S1 to itself gives a way to glue together the trivial bundles over D and D′ to reconstruct a bundle over S2. Say that two such “clutching functions” f are equivalent if the bundles they give rise to are equivalent. A careful analysis of the form of the possible clutching functions f leads to a canonical example in each equivalence class and thus to the required calculation. For any pair of spaces X and Y, we have a K¨unneth-type ring homomorphism speciﬁed by α(x ⊗ y) = π∗ α : K(X) ⊗ K(Y ) −→ K(X × Y ) 1 (x)π∗ 2 (y). Theorem (Bott periodicity). For compact spaces X, α : K(X) ⊗ K(S2) −→ K(X × S2) is an isomorphism. Indication of proof. The restrictions to X × D and X × D′ of a bundle over X×S2 are equivalent to pullbacks of bundles over X, and their further restrictions to S1 are equivalent. Conversely, bundles ζ and ξ over X together with an equivalence f between the restrictions to X × S1 of the pullbacks of ζ and ξ to X × D and X × D′ determine a bundle over X × S2. Again, a careful analysis, which is similar to that in the special case when X = pt, of the equivalence classes of the possible clutching data (ζ, f, ξ) leads to the conclusion. The following useful observation applies to any representable functor, not just K-theory. Lemma. For nondegenerately based spaces X and Y, the projections of X × Y on X and on Y and the quotient map X×Y −→ X∧Y induce a natural isomorphism ˜K(X ∧ Y ) ⊕ ˜K(X) ⊕ ˜K(Y ) ∼= ˜K(X × Y ), and ˜K(X ∧ Y ) is the kernel of the map ˜K(X × Y ) −→ ˜K(X) ⊕ ˜K(Y ) induced by the inclusions of X and Y in X × Y. Proof. The inclusion X �
� Y −→ X × Y is a coﬁbration with quotient X ∧ Y, and X and Y are retracts of X × Y via the inclusions and projections. 2. THE BOTT PERIODICITY THEOREM 207 It follows easily that the K¨unneth map α : K(X)⊗K(Y ) −→ K(X ×Y ) induces a reduced K¨unneth map β : ˜K(X) ⊗ ˜K(Y ) −→ ˜K(X ∧ Y ). We have a splitting ˜K(X) ⊗ ˜K(Y ) ⊕ ˜K(X) ⊕ ˜K(Y ) ⊕ Z ∼= K(X) ⊗ K(Y ) that is compatible with the splitting of the lemma. Therefore the following reduced form of the Bott periodicity theorem is equivalent to the unreduced form that we have already stated. Theorem (Bott periodicity). For nondegenerately based compact spaces X, β : ˜K(X) ⊗ ˜K(S2) −→ ˜K(X ∧ S2) = ˜K(Σ2X) is an isomorphism. Write b = 1 − [H] ∈ ˜K(S2). Since ˜K(S2) ∼= Z with generator b, the theorem implies that multiplication by the “Bott element” b speciﬁes an isomorphism [X, BU × Z] ∼= ˜K(X) −→ ˜K(Σ2X) ∼= [X, Ω2(BU × Z)] for nondegenerately based compact spaces X. Here the addition in the source and target is derived from the natural additive H-space structure on BU × Z on the left and the displayed double loop space on the right. If we had this isomorphism for general non-compact spaces X, we could apply it with X = BU × Z and see that it is induced by a homotopy equivalence of H-spaces β : BU × Z −→ Ω2(BU × Z). In fact, one can deduce such a homotopy equivalence from the Bott periodicity theorem as just stated, but there are more direct proofs. On the right,
the double loop space obviously depends only on the basepoint component BU = BU × {0}. Since π2(BU ) = Z, a little argument with H-spaces shows that Ω2(BU × Z) is 0BU ) × Z, where Ω2 equivalent as an H-space to (Ω2 0BU denotes the component of the basepoint in Ω2BU. Using the identity function on the factor Z, we see that what is needed is an equivalence of H-spaces β : BU −→ Ω2 0BU. In fact, it is easily deduced from the form of Bott periodicity that, up to homotopy, β must be the adjoint of the composite Σ2BU = BU ∧ S2 id ∧b / / BU ∧ BU ⊗ / BU. The inﬁnite unitary group U is deﬁned to be the union of the unitary groups U (n), where U (n) is embedded in U (n + 1) as matrices with last row and column zero except for 1 on the diagonal. Then ΩBU is homotopy equivalent as an Hspace to U. Since π1(U ) = Z and the universal cover of U is the inﬁnite special unitary group SU, ΩU is equivalent as an H-space to (ΩSU ) × Z. Therefore β may be viewed as a map BU −→ ΩSU. Bott’s original proof of the Bott periodicity theorem used the Grassmannian model for BU to write down an explicit map β in the required homotopy class and then used Morse theory to prove that β is a homotopy equivalence. Bott’s map β can also be proved to be a homotopy equivalence using only basic algebraic topology. Since BU and ΩSU are simply connected spaces of the homotopy types of CW complexes, a relative version of the Hurewicz theorem called the Whitehead theorem shows that β will be a weak equivalence and therefore a homotopy equivalence if it induces an isomorphism on integral homology. Since H ∗(BU (n)) = Z[c1,..., cn], H ∗(BU ) ∼= Z[ci\|i ≥ 1]. The H-space
structure on / 208 AN INTRODUCTION TO K-THEORY BU is induced by the maps pm,n, and we ﬁnd that the map ψ : H ∗(BU ) −→ H ∗(BU × BU ) ∼= H ∗(BU ) ⊗ H ∗(BU ) induced by the product is given by ψ(ck) = i+j=k ci ⊗ cj. A purely algebraic dualization argument proves that, as a ring, H∗(BU ) ∼= Z[γi\|i ≥ 1], P where γi is the image of a generator of H2i(CP ∞) under the map induced by the inclusion of CP ∞ = BU (1) in BU. One can calculate H∗(ΩSU ) and see that it too is a polynomial algebra with an explicitly given generator in each even degree. A direct inspection of the map β shows that it carries generators to generators. In any case, it should now be clear that we have a periodic Ω-prespectrum and therefore a generalized cohomology theory represented by it. Definition. The K-theory Ω-prespectrum KU has spaces KU2i = BU ×Z and KU2i+1 = U for all i ≥ 0. The structure maps are given by the canonical homotopy equivalence U ≃ ΩBU = Ω(BU × Z) and the Bott equivalence BU × Z ≃ ΩU. We have a resulting reduced cohomology theory on based spaces such that ˜K 2i(X) = ˜K(X) and ˜K 2i+1(X) = ˜K(ΣX) for all integers i. This theory has products that are induced by tensor products of bundles over compact spaces and that are induced by suitable maps φ : KUi ∧ KUj −→ KUi+j in general, just as for the cup product in ordinary cohomology. It is standard to view this simply as a Z2-graded theory with groups ˜K 0(X) and ˜K 1(X). 3. The splitting principle and the Thom isomorphism Returning to our bundle theoretic construction of K-theory, with X compact, we describe brieﬂy some important general
izations of the Bott periodicity theorem. The reader should recall the Thom isomorphism theorem in ordinary cohomology from Chapter 23 §5. We let ξ : E −→ X be an n-plane bundle over X, ﬁxed throughout this section. (We shall use the letters E and ξ more or less interchangeably.) Results for general vector bundles over non-connected spaces X can be deduced by applying the results to follow to one component of X at a time. Definition. Let E0 be the zero section of E. Deﬁne the projective bundle π : P (E) −→ X by letting the non-zero complex numbers act on E − E0 by scalar multiplication on ﬁbers and taking the orbit space under this action. Equivalently, the ﬁber π−1(x) ⊂ P (E) is the complex projective space of lines through the origin in the ﬁber ξ−1(x) ⊂ E. Deﬁne the canonical line bundle L(E) over P (E) to be the subbundle of the pullback π∗E of ξ along π whose points are the pairs consisting of a line in a ﬁber of E and a point on that line. Let Q(E) be the quotient bundle π∗E/L(E) and let H(E) denote the dual of L(E). Observe that P (ε2) = X×CP 1 is the trivial bundle over X with ﬁber CP 1 ∼= S2. The ﬁrst version of Bott periodicity generalizes, with essentially the same proof by analysis of clutching data, to the following version. Regard K(P (E)) as a K(X)algebra via π∗ : K(X) −→ K(P (E)). Theorem (Bott periodicity). Let L be a line bundle over X and let H = H(L ⊕ ε). Then the K(X)-algebra K(P (L ⊕ ε)) is generated by the single element [H] subject to the single relation ([H] − 1)([L][H] − 1) = 0. There is a further generalization to arbitrary bundles E. To place it in context, we shall ﬁrst
explain a cohomological analogue that expresses a diﬀerent approach 3. THE SPLITTING PRINCIPLE AND THE THOM ISOMORPHISM 209 to the Chern classes than the one that we sketched before. It will be based on a generalization to projective bundles of the calculation of H ∗(CP n). The proofs of both results are intertwined with the proof of the following “splitting principle,” which allows the deduction of explicit formulas about general bundles from formulas about sums of line bundles. Theorem (Splitting principle). There is a compact space F (E) and a map p : F (E) −→ X such that p∗E is a sum of line bundles over F (E) and both p∗ : H ∗(X; Z) −→ H ∗(F (E); Z) and p∗ : K(X) −→ K(F (E)) are monomorphisms. This is an easy inductive consequence of the following result, which we shall refer to as the “splitting lemma.” Lemma (Splitting lemma). Both π∗ : H ∗(X; Z) −→ H ∗(P (E); Z) and π∗ : K(X) −→ K(P (E)) are monomorphisms. Proof of the splitting principle. The pullback π∗E splits as the sum L(E) ⊕ Q(E). (The splitting is canonically determined by a choice of a Hermitian metric on E.) Applying this construction to the bundle Q(E) over P (E), we obtain a map π : P (Q(E)) −→ P (E) with similar properties. We obtain the desired map p : F (E) −→ X by so reapplying the projective bundle construction n times. Explicitly, using a Hermitian metric on E, we ﬁnd that the ﬁber F (E)x is the space of splittings of the ﬁber Ex as a sum of n lines, and the points of the bundle p∗E are n-tuples of vectors in given lines. The splitting lemma implies the desired monomorphisms on cohomology and K-theory. Theorem. Let x = c1(L(E)) ∈ H 2(P (
E); Z). Then H ∗(P (E); Z) is the free, and the Chern classes of ξ are char- 1, x,..., xn−1 H ∗(X; Z)-module on the basis acterized by c0(ξ) = 1 and the formula n (−1)kck(E)xn−k = 0. Xk=0 Sketch proof. This is another case where the Serre spectral sequence shows that the bundle behaves cohomologically as if it were trivial and the K¨unneth theorem applied. This gives the structure of H ∗(P (E)) as an H ∗(X)-module. In particular, it implies the splitting lemma and thus the splitting principle in ordinary cohomology. It also implies that there must be some description of xn as a linear combination of the xk for k < n, and the splitting principle may now be used to help determine that description. Write n xn = (−1)k+1c′ k(E)xn−k. Xk=1 k(E). One deduces that c′ This deﬁnes characteristic classes c′ k(E) = ck(E) by verifying that the c′ k satisfy the axioms that characterize the Chern classes. For a line bundle E, L(E) = E and c1(E) = c′ 1(E) by the deﬁnition of x. One ﬁrst veriﬁes by direct calculation that if E = L1 ⊕ · · · ⊕ Ln is a sum of line bundles, then k(E) is the kth elementary symmetric polynomial in the c1(Lk). By the Whitney sum formula for the Chern classes, this Q implies that c′ k(E) = ck(E) in this case. The general case follows from the splitting principle. Indeed, we have a map P (p∗E) −→ P (E) of projective bundles whose 1≤k≤n(x − c1(Lk)) = 0. This implies that c′ 210 AN INTRODUCTION TO K-THEORY induced map on base spaces is p : F (E) −→ X. Writing p∗E ∼= L1 ⊕ · · · ⊕ Ln
and using the naturality of the classes c′ k, we have p∗(c′ k(E)) = c′ It follows easily that the c′ Since the remaining axioms are clear, this implies that c′ k(L1 ⊕ · · · ⊕ Ln) = σk(c1(L1),..., ck(Ln)). k satisfy the Whitney sum axiom for the Chern classes. k = ck. The following analogue in K-theory of the previous theorem holds. Observe that, since they are continuous operations on complex vector spaces, the exterior powers λk can be applied ﬁberwise to give natural operations on vector bundles. Theorem. Let H = H(E). Then K(P (E)) is the free K(X)-module on the, and the following formula holds: 1, [H],..., [H]n−1 basis n (−1)k[H]k[λkE] = 0. Xk=0 Sketch proof. Suppose ﬁrst that E is the sum of n line bundles. Using the fact that if E is an n-plane bundle and L is a line bundle, then P (E) is canonically isomorphic to P (E ⊗ L), one can reduce to the case when the last line bundle is trivial. One can then argue by induction from the previous form of the Bott periodicity theorem. For a general bundle E, one then deduces the structure of K(P (E)) as a K(X)-module by a patching argument from coordinate charts and the case of trivial bundles. This implies the splitting lemma and thus the splitting principle in K-theory. It also implies that there must be some formula describing [H]n as a polynomial in the [H]k for k < n. One reason that the given formula holds will be indicated shortly. Projective bundles are closely related to Thom spaces. The inclusion of vector bundles ξ ⊂ ξ ⊕ ε induces an inclusion of projective bundles P (E) ⊂ P (E ⊕ ε). We give E a Hermitian metric and regard the Thom space T ξ as the quotient D(E)/S(E) of the unit disk bundle by the unit sphere bundle. The total space of ε is X × C
and we write 1x = (x, 1). Deﬁne a map η : D(E) −→ P (E ⊕ ε) by sending a point ex in the ﬁber over x to the line generated by ex − (1 − \|ex\|2)1x. Then η maps D(E) − S(E) homeomorphically onto P (E ⊕ ε) − P (E) and maps S(E) onto P (E) by the evident Hopf map. Therefore η induces a homeomorphism T (ξ) ∼= D(E)/S(E) ∼= P (E ⊕ ε)/P (E). Just as in ordinary cohomology, the Thom diagonal gives rise to a product K(X) ⊗ ˜K(T ξ) −→ ˜K(T ξ). The description of K(P (E)) and the exact sequence in K-theory induced by the coﬁbering P (E) −→ P (E ⊕ ε) −→ T (ξ) lead to the Thom isomorphism in K-theory. There is a natural way to associate elements of K(X) to complexes of vector bundles over X, and the exterior algebra of the bundle E gives rise to an element λE ∈ ˜K(T ξ). This element restricts to a generator of ˜K(Sn x ) for each x ∈ X, and these Thom classes are compatible with Whitney sum, in the sense that λE⊕E′ = λE · λE′. Moreover, the image of λE n k=0(−1)k[H]k[λkE]. Therefore this element maps to zero in in K(P (E ⊕ ε)) is K(P (E)), and this gives the formula in the previous theorem. P 4. THE CHERN CHARACTER; ALMOST COMPLEX STRUCTURES ON SPHERES 211 Theorem (Thom isomorphism theorem). Deﬁne Φ : K(X) −→ ˜K(T (ξ)) by Φ(x) = x · λE. Then Φ is an isomorphism. 4. The Chern character; almost complex structures on spheres We have seen above
that ordinary cohomology and K-theory enjoy similar properties. The splitting theorem implies a direct connection between them. Let R aiti ∈ R[[t]]. be any commutative ring and consider a formal power series f (t) = aixi ∈ H ∗∗(X; R). The sums Given an element x ∈ H n(X; R), we let f (x) = P will be ﬁnite in our applications of this formula. Via the splitting principle, we can use f to construct a natural homomorphism of Abelian monoids ˆf : V ect(X) −→ H ∗∗(X; R), where X is any compact space. For a line bundle over X, we set P For a sum E = L1 ⊕ · · · ⊕ Ln of line bundles over X, we set ˆf (L) = f (c1(L)). n ˆf (E) = f (c1(Li)). i=1 X For a general n-plane bundle E over X, we let ˆf (E) be the unique element of H ∗∗(X; R) such that p∗( ˆf (E)) = ˆf (p∗(E)) ∈ H ∗∗(F (E)). More explicitly, writing p∗E = L1 ⊕ · · · ⊕ Ln, we see that ˆf (p∗(E)) is a symmetric polynomial in the c1(Li) and can therefore be written as a polynomial in the elementary symmetric polynomials p∗(ck(E)). Application of this polynomial to the ck(E) gives ˆf (E). (For vector bundles E over non-connected spaces X, we add the elements obtained by restricting E to the components of X.) By the universal property of K(X), ˆf extends to a homomorphism ˆf : K(X) −→ H ∗∗(X; R). There is an analogous multiplicative extension ¯f of f that starts from the deﬁnition n ¯f (E) = f (c1(Li)) on a sum E = L1 ⊕ · · ·
⊕ Ln of line bundles Li. i=1 Y Example. For any R, if f (t) = 1 + t, then ¯f (E) = c(E) is the total Chern class of E. The example we are interested in is the “Chern character,” which gives rise to an isomorphism between rationalized K-theory and rational cohomology. Example. Taking R = Q, deﬁne the Chern character ch(E) ∈ H ∗∗(X; Q) by ch(E) = ˆf (E), where f (t) = et = ti/i!. For line bundles L and L′, we have c1(L ⊗ L′) = c1(L) + c1(L′). One way to see this is to recall that BU (1) ≃ K(Z, 2) and that line bundles are classiﬁed by their Chern classes regarded as elements of P [X+, BU (1)] ∼= H 2(X; Z). The tensor product is represented by a product φ : BU (1) × BU (1) −→ BU (1) that gives BU (1) an H-space structure. We may think of φ as an element of H 2(BU × BU ; Z) ∼= H 2(BU ; Z) ⊕ H 2(BU ; Z). 212 AN INTRODUCTION TO K-THEORY and this element is the sum of the Chern classes in the two copies of H 2(BU ; Z) (since a basepoint of BU is a homotopy identity element for φ). This has the following implication. Lemma. The Chern character speciﬁes a ring homomorphism ch : K(X) −→ H ∗∗(X; Q). Proof. We must check that ch(E ⊗ E) = ch(E) · ch(E′) for bundles E and E′ over X. It suﬃces to check this when E and E′ are sums of line bundles, in which case the result follows directly from the bilinearity of the tensor product and the relation et+t = etet. ′ ′ This leads to the following calculation. Lemma. For n ≥ 1, the Chern character maps ˜K(
S2n) isomorphically onto the image of H 2n(S2n; Z) in H 2n(S2n; Q). Therefore cn : ˜K(S2n) −→ H 2n(S2n; Z) is a monomorphism with cokernel Z(n−1)!. Proof. The ﬁrst statement is clear for n = 1, when ch = c1, and follows by compatibility with external products for n > 1. The deﬁnition of ch implies that the component chn of ch in degree 2n is cn/(n − 1)! plus terms decomposable in terms of the ci for i < n, and the second statement follows. Together with some of the facts given in Chapter 23 §7, this has a remarkable application to the study of almost complex structures on spheres. Recall that a smooth manifold of even dimension admits an almost complex structure if its tangent bundle is the underlying real vector bundle of a complex bundle. Theorem. S2 and S6 are the only spheres that admit an almost complex struc- ture. Proof. It is classical that S2 and S6 admit almost complex structures and that S4 does not. Assume that S2n admits an almost complex structure. We shall show that n ≤ 3. We are given that the tangent bundle τ is the realiﬁcation of a complex bundle. Its nth Chern class is its Euler class: cn(τ ) = χ(τ ). Since the Euler characteristic of S2n is 2, χ(τ ) = 2ι2n, where ι2n ∈ H 2n(S2n, Z) is the canonical generator. However, cn(τ ) must be divisible by (n − 1)!. This can only happen if n ≤ 3. Obviously the image of ch lies in the sum of the even degree elements in H ∗∗(X; Q), which we denote by H even(X; Q). We deﬁne H odd(X; Q) similarly, and we extend ch to Z2-graded reduced cohomology by deﬁning ch on ˜K 1(X) to be the composite ˜K 1(X) ∼= ˜K(ΣX) ch−→ ˜H even(ΣX; Q) ∼=
˜H odd(X; Q). We then have the following basic result, which actually holds for general compact spaces X provided that we replace singular cohomology by ˇCech cohomology. Theorem. For any ﬁnite based CW complex X, ch induces an isomorphism ˜K ∗(X) ⊗ Q −→ ˜H ∗∗(X; Q). 5. THE ADAMS OPERATIONS 213 Sketch proof. We think of both the source and target as Z2-graded. The lemma above implies the conclusion when X = Sn for any n. One can check that the displayed maps for varying X give a map of Z2-graded cohomology theories. The conclusion then follows from the ﬁve lemma and induction on the number of cells of X. 5. The Adams operations There are natural operations in K-theory, called the Adams operations, that are somewhat analogous to the Steenrod operations in mod 2 cohomology. In fact, the analogy can be given content by establishing a precise relationship between the Adams and Steenrod operations, but we will not go into that here. Theorem. For each non-zero integer k, there is a natural homomorphism of rings ψk : K(X) −→ K(X). These operations satisfy the following properties. (1) ψ1 = id and ψ−1 is induced by complex conjugation of bundles. (2) ψkψℓ = ψkℓ = ψℓψk. (3) ψp(x) ≡ xp mod p for any prime p. (4) ψk(ξ) = ξk if ξ is a line bundle. (5) ψk(x) = knx if x ∈ ˜K(S2n). We explain the construction. By property 2, ψ−k = ψkψ−1, hence by property 1 we can concentrate on the case k > 1. The exterior powers of bundles satisfy the relation λk(ξ ⊕ η) = ⊕i+j=kλi(ξ) ⊗ λj(η). It follows formally that the λk extend to operations K(X) −→ K(X). Indeed, form the group G of power series
with constant coeﬃcient 1 in the ring K(X)[[t]] of formal power series in the variable t. We deﬁne a function from (equivalence classes of) vector bundles to this Abelian group by setting Λ(ξ) = 1 + λ1(ξ)t + · · · + λk(ξ)tk + · · ·. Visibly, this is a morphism of monoids, Λ(ξ ⊕ η) = Λ(ξ)Λ(η). It therefore extend to a homomorphism of groups Λ : K(X) −→ G, and we let λk(x) be the coeﬃcient of tk in Λ(x). We deﬁne the ψk as suitable polynomials in the λk. Recall that the subring of symmetric polynomials in the polynomial algebra Z[x1,..., xn] is the polynomial algebra Z[σ1,..., σn], where σi = x1x2 · · · xi + · · · is the ith elementary symmetric function. We may write the power sum πk = xk n as a polynomial 1 + · · · + xk πk = Qk(σ1,..., σk) in the ﬁrst k elementary symmetric functions. Provided n ≥ k, Qk does not depend on n. We deﬁne ψk(x) = Qk(λ1(x),..., λk(x)). For example, π2 = σ2 is clear from the naturality of the λk. 1 − 2σ2, hence ψ2(x) = x2 − 2λ2(x). The naturality of the ψk If ξ is a line bundle, then λ1(ξ) = ξ and λk(ξ) = 0 for k ≥ 2. Clearly σk 1 = πk + other terms and πk does not occur as a summand of any other monomial in 214 AN INTRODUCTION TO K-THEORY the σi. Therefore Qk ≡ σk
1 modulo terms in the ideal generated by the σi for i > 1. This immediately implies property 4. Moreover, if ξ1,..., ξn are line bundles, then Λ(ξ1 ⊕ · · · ⊕ ξn) = (1 + ξ1t) · · · (1 + ξnt) = 1 + σ1(ξ1,..., ξn)t + σ2(ξ1,..., ξn)t2 + · · ·. This implies the generalization of property 4 to sums of line bundles: 4′ ψk(ξ1 ⊕ · · · ⊕ ξn) = πk(ξ1,..., ξn) for line bundles ξi. Now, if x and y are sums of line bundles, the following formulas are immediate: ψk(x + y) = ψk(x) + ψk(y), ψk(xy) = ψk(x)ψk(y), ψkψℓ(x) = ψkℓ(x) and ψp(x) ≡ xp mod p for a prime p. For arbitrary bundles, these formulas follow directly from the splitting principle and naturality, and they then follow formally for arbitrary virtual bundles. This completes the proof of all properties except 5. We have that ˜K(S2) is generated by 1 − [H], where (1 − [H])2 = 0. Clearly ψk(1 − [H]) = 1 − [H]k. By induction on k, 1 − [H]k = k(1 − [H]). Since S2n = S2 ∧ · · · ∧ S2 and ˜K(S2n) is generated by the k-fold external tensor power (1 − [H]) ⊗ · · · ⊗ (1 − [H]), property 5 follows from the fact that ψk preserves products. Remark. By the splitting principle, it is clear that the ψk are the unique natural and additive operations with the speciﬁed behavior on line bundles. Two further properties of the ψk should be mentioned. The ﬁrst is a direct
consequence of the multiplicativity of the ψk and their behavior on spheres. Proposition. The following diagram does not commute for based spaces X, where β is the periodicity isomorphism: ˜K(X) ψk ˜K(X) β β / ˜K(Σ2X) ψk / ˜K(Σ2X). Rather, ψkβ = kβψk. Therefore the ψk do not give stable operations on the Z-graded theory K ∗. Proposition. Deﬁne ψk H on H even(X; Z) by letting ψk H (x) = krx for x ∈ H 2r(X; Z). Then the following diagram commutes: K(X) ch / H even(X; Q) ψk ψk H K(X) ch / H even(X; Q). Proof. It suﬃces to prove this on vector bundles E. By the splitting principle in K-theory and cohomology, we may assume that E is a sum of line bundles. By additivity, we may then assume that E is a line bundle. Here ψk(E) = Ek and c1(Ek) = kc1(E). The conclusion follows readily from the deﬁnition of ch in terms of et. / / / / 6. THE HOPF INVARIANT ONE PROBLEM AND ITS APPLICATIONS 215 Remark. The observant reader will have noticed that, by analogy with the deﬁnition of the Stiefel-Whitney classes, we can deﬁne characteristic classes in Ktheory by use of the Adams operations and the Thom isomorphism, setting ρk(E) = Φ−1ψkΦ(1) for n-plane bundles E. 6. The Hopf invariant one problem and its applications We give one of the most beautiful and impressive illustrations of the philosophy described in the ﬁrst chapter. We deﬁne a numerical invariant, called the “Hopf invariant,” of maps f : S2n−1 −→ Sn and show that it can only rarely take the value one. We then indicate several problems whose solution can be reduced to the question of when such maps f take the value one. Adams’ original
solution to the Hopf invariant one problem used secondary cohomology operations in ordinary cohomology and was a critical starting point of modern algebraic topology. The later realization that a problem that required secondary operations in ordinary cohomology could be solved much more simply using primary operations in Ktheory had a profound impact on the further development of the subject. Take cohomology with integer coeﬃcients unless otherwise speciﬁed. Definition. Let X be the coﬁber of a based map f : S2n−1 −→ Sn, where n ≥ 2. Then X is a CW complex with a single vertex, a single n-cell i, and a single 2n-cell j. The diﬀerential in the cellular chain complex of X is zero for obvious dimensional reasons, hence ˜H ∗(X) is free Abelian on generators x = [i] and y = [j]. Deﬁne an integer h(f ), the Hopf invariant of f, by x2 = h(f )y. We usually regard h(f ) as deﬁned only up to sign (thus ignoring problems of orientations of cells). Note that h(f ) depends only on the homotopy class of f. If n is odd, then 2x2 = 0 and thus x2 = 0. We assume from now on that n is even. Although not essential to the main point of this section, we record the following basic properties of the Hopf invariant. Proposition. The Hopf invariant enjoys the following properties. (1) If g : S2n−1 −→ S2n−1 has degree d, then h(f ◦ g) = dh(f ). (2) If e : Sn −→ Sn has degree d, then h(e ◦ f ) = d2h(f ). (3) The Hopf invariant deﬁnes a homomorphism π2n−1(Sn) −→ Z. (4) There is a map f : S2n−1 −→ Sn such that h(f ) = 2. Proof. We leave the ﬁrst three statements to the reader. For property 4, let π : Dn −→ Dn/Sn−1 ∼= Sn be the quotient map and deﬁne f : S2n−
1 ∼= (Dn × Sn−1) ∪ (Sn−1 × Dn) −→ Sn by f (x, y) = π(x) and f (y, x) = π(x) for x ∈ Dn and y ∈ Sn−1. We leave it to the reader to verify that h(f ) = 2. We have adopted the standard deﬁnition of h(f ), but we could just as well have deﬁned it in terms of K-theory. To see this, consider the coﬁber sequence S2n−1 f −→ Sn i−→ X π−→ S2n Σf −−→ Sn+1. 216 AN INTRODUCTION TO K-THEORY Obviously i∗ : H n(X) −→ H n(Sn) and π∗ : H 2n(S2n) −→ H 2n(X) are isomorphisms. We have the commutative diagram with exact rows 0 0 / ˜K(S2n) ch / ˜H ∗∗(S2n; Q) ∗ π ∗ π / ˜K(X) ch / ˜H ∗∗(X; Q) ∗ i ∗ i / ˜K(Sn) / 0 ch / ˜H ∗∗(Sn; Q) / 0. Here the top row is exact since ˜K 1(Sn) = 0 and ˜K 1(S2n) = 0. The vertical arrows are monomorphisms since they are rational isomorphisms. By a lemma in the previous section, generators in of ˜K(Sn) and i2n of ˜K(S2n) map under ch to generators of H n(Sn) and H 2n(S2n). Choose a ∈ ˜K(X) such that i∗(a) = in and let b = π∗(i2n). Then ˜K(X) is the free Abelian group on the basis {a, b}. Since n = 0, we have a2 = h′(f )b for some integer h′(f ). The diagram implies that, up i2 to sign, ch(b) = y and ch(a) = x + q
y for some rational number q. Since ch is a ring homomorphism and since y2 = 0 and xy = 0, we conclude that h′(f ) = h(f ). Theorem. If h(f ) = ±1, then n = 2, 4, or 8. Proof. Write n = 2m. Since ψk(i2n) = k2mi2n and ψk(in) = kmin, we have ψk(b) = k2mb and ψk(a) = kma + µkb for some integer µk. Since ψ2(a) ≡ a2 mod 2, h(f ) = ±1 implies that µ2 is odd. Now, for any odd k, while ψkψ2(a) = ψk(2ma + µ2b) = km2ma + (2mµk + k2mµ2)b ψ2ψk(a) = ψ2(kma + µkb) = 2mkma + (kmµ2 + 22mµk)b. Since these must be equal, we ﬁnd upon equating the coeﬃcients of b that 2m(2m − 1)µk = km(km − 1)µ2. If µ2 is odd, this implies that 2m divides km − 1. Already with k = 3, an elementary number theoretic argument shows that this implies m = 1, 2, or 4. This allows us to determine which spheres can admit an H-space structure. Recall from a problem in Chapter 18 that S2m cannot be an H-space. Clearly Sn is an H-space for n = 0, 1, 3, and 7: view Sn as the unit sphere in the space of real numbers, complex numbers, quaternions, or Cayley numbers. Theorem. If Sn−1 is an H-space, then n = 1, 2, 4, or 8. The strategy of proof is clear: given an H-space structure on Sn−1, we construct from it a map f : S2n−1 −→ Sn of Hopf invariant one. The following construction and lemma do this and more. Construction (Hopf construction). Let φ : Sn−1 × Sn−1 −→ Sn−1 be a map. Let C
X = (X × I)/(X × {1}) be the unreduced cone functor and note that we have canonical homeomorphisms of pairs (Dn, Sn−1) ∼= (CSn−1, Sn−1. THE HOPF INVARIANT ONE PROBLEM AND ITS APPLICATIONS 217 and (D2n, S2n−1) ∼= (Dn × Dn, (Dn × Sn−1) ∪ (Sn−1 × Dn)) ∼= (CSn−1 × CSn−1, (CSn−1 × Sn−1) ∪ (Sn−1 × CSn−1)). Take Sn to be the unreduced suspension of Sn−1, with the upper and lower hemispheres Dn − corresponding to the points with suspension coordinate 1/2 ≤ t ≤ 1 and 0 ≤ t ≤ 1/2, respectively. Deﬁne + and Dn f : S2n−1 ∼= (CSn−1 × Sn−1) ∪ (Sn−1 × CSn−1) −→ Sn as follows. Let x, y ∈ Sn−1 and t ∈ I. On CSn−1 × Sn−1, f is the composite CSn−1 × Sn−1 α−→ C(Sn−1 × Sn−1) Cφ −−→ CSn−1 β −→ Dn −, where α([x, t], y) = [(x, y), t] and β([x, t]) = [x, (1 − t)/2]. On Sn−1 × CSn−1, f is the composite Sn−1 × CSn−1 α −→ C(Sn−1 × Sn−1) ′ −−→ CSn−1 β Cφ −→ Dn +, ′ where α′(x, [y, t]) = [(x, y), t] and β′([x, t]) = [x, (1 + t)/2]. The map f, or rather the resulting 2-cell complex X = Sn ∪f D2n, is called the Hopf construction on φ. Giving Sn−1 a basepoint, we obtain inclusions of Sn−1 onto the ﬁrst and second copies of Sn−1 in Sn−1 × Sn−1. The bideg
ree of a map φ : Sn−1 × Sn−1 −→ Sn−1 is the pair of integers given by the two resulting composite maps Sn−1 −→ Sn−1. Thus φ gives Sn−1 an H-space structure if its bidegree is (1, 1). Lemma. If the bidegree of φ : Sn−1 × Sn−1 −→ Sn−1 is (d1, d2), then the Hopf invariant of the Hopf construction on φ is ±d1d2. Proof. Making free use of the homeomorphisms of pairs speciﬁed in the construction, we see that the diagonal map of X, its top cell j, evident quotient maps, and projections πi onto ﬁrst and second coordinates give rise to a commutative diagram in which the maps marked ≃ are homotopy equivalences and those marked ∼= are homeomorphisms: X X/Sn j∼= ∆ ∆ X ∧ X ≃ / X/Dn + ∧ X/Dn − j∧j S2n ∼= D2n/S2n−1 ∆ / (Dn × Dn)/(Sn−1 × Dn) ∧ (Dn × Dn)/(Dn × Sn−1) +XXXXXXXXXXXXXXXXXXXXXXX ∼= ≃ π1∧π2 Dn/Sn−1 ∧ Dn/Sn−1 ∼= Sn ∧ Sn. The cup square of x ∈ H n(X) is the image under ∆∗ of the external product of x with itself. The maps on the left induce isomorphisms on H 2n. The inclusions of Dn in the ith factor of Dn × Dn induce homotopy inverses ι1 : Dn/Sn−1 −→ (Dn × Dn)/(Sn−1 × Dn 218 and AN INTRODUCTION TO K-THEORY ι2 : Dn/Sn−1 −→ (Dn × Dn)/(Dn × Sn−1) to the projections πi in the diagram, and it suﬃces to prove that, up to sign, the composites j ◦ ι1 : Dn/Sn−1 −→ X/Dn + and j ◦
ι2 : Dn/Sn−1 −→ X/Dn − induce multiplication by d1 and by d2 on H n. However, by construction, these maps factor as composites Dn/Sn−1 γ1−→ Sn/Dn + and Dn/Sn−1 γ2−→ Sn/Dn − −→ X/Dn −, + −→ X/Dn where, up to signs and identiﬁcations of spheres, γ1 and γ2 are the suspensions of the restrictions of φ to the two copies of Sn−1 in Sn−1 × Sn−1. The determination of which spheres are H-spaces has the following implications. Theorem. Let ω : Rn × Rn −→ Rn be a map with a two-sided identity element e 6= 0 and no zero divisors. Then n = 1, 2, 4, or 8. Proof. The product restricts to give Rn − {0} an H-space structure. Since Sn−1 is homotopy equivalent to Rn − {0}, it inherits an H-space structure. Explicitly, we may assume that e ∈ Sn−1, by rescaling the metric, and we give Sn−1 the product φ : Sn−1 × Sn−1 −→ Sn−1 speciﬁed by φ(x, y) = ω(x, y)/\|ω(x, y)\|. Note that ω need not be bilinear, just continuous. Also, it need not have a strict unit; all that is required is that e be a two-sided unit up to homotopy for the restriction of ω to Rn − {0}. Theorem. If Sn is parallelizable, then n = 0, 1, 3, or 7. Proof. Exclude the trivial case n = 0 and suppose that Sn is parallelizable, so that its tangent bundle τ is trivial. We will show that Sn is an H-space. Deﬁne a map µ : τ −→ Sn as follows. Think of the tangent plane τx as aﬃnely embedded in Rn+1 with origin at x. We have a parallel translate of this plane to an aﬃne plane with origin at −x. Deﬁne µ by sending a tangent vector y
∈ τx to the intersection with Sn of the line from x to the translate of y. Composing with a trivialization Sn × Rn ∼= τ, this gives a map µ : Sn × Rn −→ Sn. Let Sn ∞ be the one-point compactiﬁcation of Rn. Extend µ to a map φ : Sn × Sn ∞ −→ Sn by letting φ(x, ∞) = x; φ is continuous since µ(x, y) approaches x as y approaches ∞. By construction, ∞ is a right unit for this product. For a ﬁxed x, y −→ φ(x, y) is a degree one homeomorphism Sn ∞. The conclusion follows. ∞ −→ Sn CHAPTER 25 An introduction to cobordism Cobordism theories were introduced shortly after K-theory, and their use pervades modern algebraic topology. We shall describe the cobordism of smooth closed manifolds, but this is in fact a particularly elementary example. Other examples include smooth closed manifolds with extra structure on their stable normal bundles: orientation, complex structure, Spin structure, or symplectic structure for example. All of these except the symplectic case have been computed completely. The complex case is particularly important since complex cobordism and theories constructed from it have been of central importance in algebraic topology for the last few decades, quite apart from their geometric origins in the classiﬁcation of manifolds. The area is pervaded by insights from algebraic topology that are quite mysterious geometrically. For example, the complex cobordism groups turn out to be concentrated in even degrees: every smooth closed manifold of odd dimension with a complex structure on its stable normal bundle is the boundary of a compact manifold (with compatible bundle information). However, there is no geometric understanding of why this should be the case. The analogue with “complex” replaced by “symplectic” is false. 1. The cobordism groups of smooth closed manifolds We consider the problem of classifying smooth closed n-manifolds M. One’s ﬁrst thought is to try to classify them up to diﬀeomorphism, but that problem is in principle unsolvable. Thom’s discovery that one can classify such manifolds up to the weaker equivalence relation of “cobordism�
� is one of the most beautiful advances of twentieth century mathematics. We say that two smooth closed nmanifolds M and N are cobordant if there is a smooth compact manifold W whose boundary is the disjoint union of M and N, ∂W = M ∐ N. We write Nn for the set of cobordism classes of smooth closed n-manifolds. It is convenient to allow the empty set ∅ as an n-manifold for every n. Disjoint union gives an addition on the set Nn. This operation is clearly associative and commutative and it has ∅ as a zero element. Since ∂(M × I) = M ∐ M, M ∐ M is cobordant to ∅. Thus M = −M and Nn is a vector space over Z2. Cartesian product of manifolds deﬁnes a multiplication Nm × Nn −→ Nm+n. This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a single point provides an identity element. We conclude that N∗ is a graded Z2-algebra. Theorem (Thom). N∗ is a polynomial algebra over Z2 on generators ui of dimension i for i > 1 and not of the form 2r − 1. 219 220 AN INTRODUCTION TO COBORDISM As already stated in our discussion of Stiefel-Whitney numbers, it follows from the proof of the theorem that a manifold is a boundary if and only if its normal Stiefel-Whitney numbers are zero. We can restate this as follows. Theorem. Two smooth closed n-manifolds are cobordant if and only if their normal Stiefel-Whitney numbers, or equivalently their tangential Stiefel-Whitney numbers, are equal. Explicit generators ui are known. Write [M ] for the cobordism class of a manifold M. Then we can take u2i = [RP 2i]. We have seen that the Stiefel-Whitney numbers of RP 2i−1 are zero, so we need diﬀerent generators in odd dimensions. For m < n, deﬁne Hn,m to be the hypersurface in RP n × RP m consisting of those pairs ([x0,..., xn], [y0,.
.., ym]) such that x0y0 + · · · + xmym = 0; here (x0,..., xn) ∈ Sn and [x0,..., xn] denotes its image in RP n. We may write an odd number i not of the form 2r − 1 in the form i = 2p(2q + 1) − 1 = 2p+1q + 2p − 1, where p ≥ 1 and q ≥ 1. Then we can take ui = [H2p+1q,2p]. The strategy for the proof of Thom’s theorem is to describe Nn as a homotopy group of a certain Thom space. The homotopy group is a stable one, and it turns out to be computable by the methods of generalized homology theory. Consider the universal q-plane bundle γq : Eq −→ Gq(R∞) = BO(q). Let T O(q) be its Thom space. Recall that we have maps iq : BO(q) −→ BO(q + 1) such that i∗ q(γq+1) = γq ⊕ ε. The Thom space T (γq ⊕ ε) is canonically homeomorphic to the suspension ΣT O(q), and the bundle map γq ⊕ ε −→ γq+1 induces a map σq : ΣT O(q) −→ T O(q + 1). Thus the spaces T O(q) and maps σq constitute a prespectrum T O. By deﬁnition, the homotopy groups of a prespectrum T = {Tq} are where the colimit is taken over the maps πn(T ) = colim πn+q(Tq), πn+q(Tq) Σ−→ πn+q+1(ΣTq) σq ∗−−→ πn+q+1(Tq+1). In the case of T O, it turns out that these maps are isomorphisms if q is suﬃciently large, and we have the following translation of our problem in manifold theory to a problem in homotopy theory. We shall sketch the proof in the next section,
where we shall also explain the ring structure on π∗(T O) that makes it a Z2-algebra. Theorem (Thom). For suﬃciently large q, Nn is isomorphic to πn+q(T O(q)). Therefore Moreover, N∗ and π∗(T O) are isomorphic as Z2-algebras. Nn ∼= πn(T O). 2. Sketch proof that N∗ is isomorphic to π∗(T O) Given a smooth closed n-manifold M, we may embed it in Rn+q for q suﬃciently (By the Whitney large, and we let ν be the normal bundle of the embedding. embedding theorem, q = n suﬃces, but the precise estimate is not important to us.) Embed M as the zero section of the total space E(ν). Then a standard result in diﬀerential topology known as the tubular neighborhood theorem implies that the identity map of M extends to an embedding of E(ν) onto an open neighborhood U of M in Rn+q. 2. SKETCH PROOF THAT N∗ IS ISOMORPHIC TO π∗(T O) 221 Think of Sn+q as the one-point compactiﬁcation of Rn+q. The “PontryaginThom construction” associates a map t : Sn+q −→ T (ν) to our tubular neighborhood U. Observing that T ν − {∞} = E(ν), we let t restrict on U to the identiﬁcation U ∼= E(ν) and let t send all points of Rn+q − U to the point at inﬁnity. The Thom space was tailor made to allow this construction. For q large enough, any two embeddings of M in Rn+q are isotopic, and the homotopy class of t is independent of the choice of the embedding of M in Rn+q. Now choose a classifying map f : M −→ BO(q) for ν. The composite T f ◦ t : Sn+q −→ T O(q) represents an element of πn+q(T O(q)). As the reader should think through, it
is intuitively plausible that cobordant manifolds induce homotopic maps Sn+q −→ T O(q), so that this construction gives a well deﬁned function α : Nn −→ πn+q(T O(q)). However, technically, one can arrange the argument so that this fact drops out without explicit veriﬁcation. Given two n-manifolds, we can embed them and their tubular neighborhoods disjointly in Rn+q, and it follows easily that α is a homomorphism. q − {∞}) = g−1(E(γr q ) for a suﬃciently large r > q, where γr We construct an inverse β to α. Any map g : Sn+q −→ T O(q) has image contained in T (γr q is the restriction of the universal bundle γq to the compact manifold Gq(Rr). By an implication of Sard’s theorem known as the transversality theorem, we can deform the restriction of g to g−1(T γr q )) so as to obtain a homotopic map that is smooth and transverse to the zero section. This use of transversality is the crux of the proof of the theorem. It follows that the inverse image g−1(Gq(Rr)) is a smooth closed n-manifold embedded in Rn+q = Sn+q − {∞}. It is intuitively plausible that homotopic maps gi : Sn+q −→ T O(q), i = 0, 1, give rise to cobordant n-manifolds by this construction. Indeed, with the gi smooth and transverse to the zero section, we can approximate a homotopy between them by a homotopy h which is smooth q ) − {∞}) and transverse to the zero section. Then h−1(Gq(Rr)) is a on h−1(T (γr manifold whose boundary is g−1 1 (Gq(Rr)). It is easy to verify that the resulting function β : πn+q(T O(q)) −→ Nn is a homomorphism. 0 (Gq(Rr)) ∐ g−1 If we start with a manifold M embedded in Rn+q and construct the classifying map f for its
normal bundle to be the Gauss map described in our sketch proof of the classiﬁcation theorem in Chapter 23 §1, then the composite T f ◦t is smooth and transverse to the zero section, and the inverse image of the zero section is exactly M. This proves that β is an epimorphism. To complete the proof, it suﬃces to show that β is a monomorphism. It will follow formally that α is well deﬁned and inverse to β. q such that g−1(E(γr Thus suppose given g : Sn+q −→ T γr q )) is smooth and transverse to the zero section and suppose that M = g−1(Gq(Rr)) is a boundary, say M = ∂W. The inclusion of M in Sn+q extends to a embedding of W in Dn+q+1, by the Whitney embedding theorem for manifolds with boundary (assuming as always that q is suﬃciently large). We may assume that U = g−1(T γr q − {∞}) is a tubular neighborhood and that g : U −→ E(γr q ) is a map of vector bundles. A relative version of the tubular neighborhood theorem then shows that U can be extended to a tubular neighborhood V of W in Dn+q+1 and that g extends to a map of vector bundles h : V −→ E(γr q ) by mapping Dn+q+1 − V to ∞. This extension of g to the disk implies that g is null homotopic. q ). We can then extend h to a map Dn+q+1 −→ T (γr 222 AN INTRODUCTION TO COBORDISM We must still deﬁne the ring structure on π∗(T O) and prove that we have an isomorphism of rings and therefore of Z2-algebras. Recall that we have maps pm,n : BO(m)×BO(n) −→ BO(m+n) such that p∗ m,n(γm+n) = γm×γn. The Thom space T (γm × γn) is canonically homeomorphic to the smash product T O(m) ∧ T O(n), and the bundle map γm × �
�n −→ γm+n induces a map φm,n : T O(m) ∧ T O(n) −→ T O(m + n). If we have maps f : Sm+q −→ T O(m) and g : Sn+q −→ T O(n), then we can compose their smash product with φm,n to obtain a composite map φm,n−−−→ T O(m + n). Sm+n+q+r ∼= Sm+q ∧ Sn+r f ∧g −−→ T O(m) ∧ T O(n) We can relate the maps φm,n to the maps σn. In fact, T O is a commutative and associative ring prespectrum in the sense of the following deﬁnition. Definition. Let T be a prespectrum. Then T is a ring prespectrum if there are maps η : S0 −→ T0 and φm,n : Tm ∧ Tn −→ Tm+n such that the following diagrams are homotopy commutative: Tm ∧ ΣTn id ∧σn / / Tm ∧ Tn+1 Σ(Tm ∧ Tn) Σφm,n (−1)n / ΣTm+n / Tm+n+1 (ΣTm) ∧ Tn σm∧id / Tm+1 ∧ Tn σm+n φm,n+1 &MMMMMMMMMM 8qqqqqqqqqqq φm+1,n S0 ∧ Tn η∧id T0 ∧ Tn Tn ∧ T0 id ∧η Tn ∧ S0; %KKKKKKKKKK ∼= φ0,n and φn,0 Tn Tn yrrrrrrrrrr ∼= T is associative if the following diagrams are homotopy commutative: Tm ∧ Tn ∧ Tp / Tm+n ∧ Tp φm,n∧id id ∧φn,p φm+n,p Tm ∧ Tn+p / Tm+n+p; φm,n+p T
is commutative if there are equivalences (−1)mn : Tm+n −→ Tm+n that suspend to (−1)mn on ΣTm+n and if the following diagrams are homotopy commutative: Tm ∧ Tn t / Tn ∧ Tm φm,n φn,m Tm+n (−1)mn / / Tm+n. When T is an Ω-prespectrum, we can restate this as φm,n ≃ (−1)mnφn,mt. For example, the Eilenberg-Mac Lane Ω-prespectrum of a commutative ring R is an associative and commutative ring prespectrum by the arguments in Chapter & / / /. PRESPECTRA AND THE ALGEBRA H∗(T O; Z2) 223 22 §3. It is denoted HR or sometimes, by abuse, K(R, 0). Similarly, the K-theory Ω-prespectrum is an associative and commutative ring prespectrum. The sphere prespectrum, whose nth space is Sn, is another example. For T O, the required maps (−1)mn : T O(m + n) −→ T O(m + n) are obtained by passage to Thom complexes from a map γm+n −→ γm+n of universal bundles given on the domains of coordinate charts by the evident interchange isomorphism Rm+n −→ Rm+n. The following lemma is immediate by passage to colimits. Lemma. If T is an associative ring prespectrum, then π∗(T ) is a graded ring. If T is commutative, then π∗(T ) is commutative in the graded sense. Returning to the case at hand, we show that the maps α for varying n transport products of manifolds to products in π∗(T O). Thus let M be an m-manifold embedded in Rm+q with tubular neighborhood U ∼= E(νM ) and N be an n-manifold embedded in Rn+r with tubular neighborhood V ∼= E(νN ). Then M × N is embedded in Rm+q+n+r with tubular neighborhood U × V ∼= E
(νM×N ). Identifying Sm+q+n+r with Sm+q ∧ Sn+r, we ﬁnd that the Pontryagin-Thom construction for M × N is the smash product of the Pontryagin-Thom constructions for M and N. That is, the left square in the following diagram commutes. The right square commutes up to homotopy by the deﬁnition of φq,r. Sm+q ∧ Sn+r t∧t / T νm ∧ T νN / T O(q) ∧ T O(r) ∼= φq,r Sm+q+n+r t / T (νM×N ) / T O(q + r). This implies the claimed multiplicativity of the maps α. 3. Prespectra and the algebra H∗(T O; Z2) Calculation of the homotopy groups π∗(T O) proceeds by ﬁrst computing the homology groups H∗(T O; Z2) and then showing that the stable Hurewicz homomorphism maps π∗(T O) monomorphically onto an identiﬁable part of H∗(T O; Z2). We explain the calculation of homology groups in this section and the next, connect the calculation with Stiefel-Whitney numbers in §5, and describe how to complete the desired calculation of homotopy groups in §6. We must ﬁrst deﬁne the homology groups of prespectra and the stable Hurewicz homomorphism. Just as we deﬁned the homotopy groups of a prespectrum T by the formula πn(T ) = colim πn+q(Tq), we deﬁne the homology and cohomology groups of T with respect to a homology theory k∗ and cohomology theory k∗ on spaces by the formulas kn(T ) = colim ˜kn+q(Tq), where the colimit is taken over the maps ˜kn+q(Tq) Σ∗−−→ ˜kn+q+1(ΣTq) σq ∗−−→ ˜kn+q+1(Tq+
1), and kn(T ) = lim ˜kn+q(Tq), / / / / 224 AN INTRODUCTION TO COBORDISM where the limit is taken over the maps ∗ σ −1 ˜kn+q+1(Tq+1) −−−→ ˜kn+q(Tq). q−→ ˜kn+q+1(ΣTq) Σ In fact, this deﬁnition of cohomology is inappropriate in general, diﬀering from the appropriate deﬁnition by a lim1 error term. However, the deﬁnition is correct when k∗ is ordinary cohomology with coeﬃcients in a ﬁeld R and each ˜H n+q(Tq; R) is a ﬁnite dimensional vector space over R. This is the only case that we will need in the work of this chapter. In this case, it is clear that H n(T ; R) is the vector space dual of Hn(T ; R), a fact that we shall use repeatedly. Observe that there is no cup product in H ∗(T ; R): the maps in the limit system factor through the reduced cohomologies of suspensions, in which cup products are identically zero (see Problem 5 at the end of Chapter 19). However, if T is an associative and commutative ring prespectrum, then the homology groups H∗(T ; R) form a graded commutative R-algebra. The Hurewicz homomorphisms πn+q(Tq) −→ ˜Hn+q(Tq; Z) pass to colimits to give the stable Hurewicz homomorphism h : πn(T ) −→ Hn(T ; Z). We may compose this with the map Hn(T ; Z) −→ Hn(T ; R) induced by the unit of a ring R, and we continue to denote the composite by h. If T is an associative and commutative ring prespectrum, then h : π∗(T ) −→ H∗(T ; R) is a map of graded commutative rings. We shall write H∗ and H ∗ for homology and cohomology with coe�
�cients in Z2 throughout §§3–6, and we tacitly assume that all homology and cohomology groups in sight are ﬁnite dimensional Z2-vector spaces. Recall that we have Thom isomorphisms Φq : H n(BO(q)) −→ ˜H n+q(T O(q)) obtained by cupping with the Thom class µq ∈ ˜H q(T O(q)). Naturality of the Thom diagonal applied to the map of bundles γq ⊕ ε −→ γq+1 gives the commutative diagram ΣT O(q) ∆ BO(q)+ ∧ ΣT O(q) σq iq ∧σq T O(q + 1) ∆ / BO(q + 1)+ ∧ T O(q + 1). This implies that the following diagram is commutative: H n(BO(q + 1)) Φq+1 ∗ i q H n(BO(q)) Φq ˜H n+q+1(T O(q + 1)) / ˜H n+q+1(ΣT O(q)) −1 Σ ∗ σ q / ˜H n+q(T O(q)). We therefore obtain a “stable Thom isomorphism” Φ : H n(BO) −→ H n(T O) on passage to limits. We have dual homology Thom isomorphisms Φn : ˜Hn+q(T O(q)) −→ Hn(BO(q)) / / / / / / / 3. PRESPECTRA AND THE ALGEBRA H∗(T O; Z2) 225 that pass to colimits to give a stable Thom isomorphism Φ : Hn(T ) −→ Hn(BO). Naturality of the Thom diagonal applied to the map of bundles γq ⊕γr −→ γq+r gives the commutative diagram T O(q) ∧ T O(r) ∆∧∆ / / BO(q)+ ∧ T O(q) ∧ BO(r)+ ∧ T O(r) id ∧t∧id φq,r (BO(q) × BO(r))+ ∧ T O
(q) ∧ T O(r) T O(q + r) ∆ / BO(q + r)+ ∧ T O(q + r). (pq,r )+∧φq,r As we observed for BU in the previous chapter, the maps pq,r pass to colimits to give BO an H-space structure, and it follows that H∗(BO) is a Z2-algebra. On passage to homology and colimits, these diagrams imply the following conclusion. Proposition. The Thom isomorphism Φ : H∗(T O) −→ H∗(BO) is an iso- morphism of Z2-algebras. The description of the H ∗(BO(n)) and the maps i∗ q in Chapter 23 §2 implies that H ∗(BO) = Z2[wi\|i ≥ 1] as an algebra. However, we are more interested in its “coalgebra” structure, which is given by the vector space dual ψ : H ∗(BO) −→ H ∗(BO) ⊗ H ∗(BO) of its product in homology. It is clear from the description of the p∗ q,r that ψ(wk) = wi ⊗ wj. Xi+j=k From here, determination of H∗(BO) and therefore H∗(T O) as an algebra is a purely algebraic, but non-trivial, problem in dualization. Let i : RP ∞ = BO(1) −→ BO be the inclusion. Let xi ∈ Hi(RP ∞) be the unique non-zero element and let bi = i∗(xi). Then the solution of our dualization problem takes the following form. Theorem. H∗(BO) is the polynomial algebra Z2[bi\|i ≥ 1]. Let ai ∈ Hi(T O) be the element characterized by Φ(ai) = bi. Corollary. H∗(T O) is the polynomial algebra Z2[ai\|i ≥ 1]. Using the compatibility of the Thom isomorphisms for BO(1) and BO, we see that the ai come from H∗(T O(1)). Remember that elements of Hi+1(T O(1)) map
to elements of Hi(T O) in the colimit; in particular, the non-zero element of H1(T O(1)) maps to the identity element 1 ∈ H0(T O). Recall from Chapter 23 §6 that we have a homotopy equivalence j : RP ∞ −→ T O(1). Corollary. For i ≥ 0, j∗(xi+1) maps to ai in H∗(T O), where a0 = 1. / 226 AN INTRODUCTION TO COBORDISM 4. The Steenrod algebra and its coaction on H∗(T O) Since the Steenrod operations are stable and natural, they pass to limits to deﬁne natural operations Sqi : H n(T ) −→ H n+i(T ) for i ≥ 0 and prespectra T. Here Sq0 = id, but it is not true that Sqi(x) = 0 for i > deg x. For example, we have the “stable Thom class” Φ(1) = µ ∈ H 0(T O), and it is immediate from the deﬁnition of the Stiefel-Whitney classes that Φ(wi) = Sqi(µ). Of course, Sqi(1) = 0 for i > 0, so that Φ does not commute with Steenrod operations. The homology and cohomology of T O are built up from π∗(T O) and Steenrod operations. We need to make this statement algebraically precise to determine π∗(T O), and we need to assemble the Steenrod operations into an algebra to do this. Definition. The mod 2 Steenrod algebra A is the quotient of the free associative Z2-algebra generated by elements Sqi, i ≥ 1, by the ideal generated by the Adem relations (which are stated in Chapter 22 §5). The following lemmas should be clear. Lemma. For spaces X, H ∗(X) has a natural A-module structure. Lemma. For prespectra T, H ∗(T ) has a natural A-module structure. The elements of A are stable mod 2 cohomology operations, and our description of the cohomology of K(Z2, q)s in Chapter 22 §5 implies that
A is in fact the algebra of all stable mod 2 cohomology operations, with multiplication given by composition. Passage to limits over q leads to the following lemma. Alternatively, with the more formal general deﬁnitions of the next section, it will become yet another application of the Yoneda lemma. Recall that HZ2 denotes the EilenbergMac Lane Ω-prespectrum {K(Z2, q)}. Lemma. As a vector space, A is isomorphic to H ∗(HZ2). We shall see how to describe the composition in A homotopically in the next section. What is more important at the moment is that the lemma allows us to read oﬀ a basis for A. Theorem. A has a basis consisting of the operations SqI = Sqi1 · · · Sqij, where I runs over the sequences {i1,..., ij} of positive integers such that ir ≥ 2ir+1 for 1 ≤ r < j. What is still more important to us is that A not only has the composition product A ⊗ A −→ A, it also has a coproduct ψ : A −→ A ⊗ A. Giving A ⊗ A its natural structure as an algebra, ψ is the unique map of algebras speciﬁed on i+j=k Sqi ⊗ Sqj. The fact that ψ is a well deﬁned map generators by ψ(Sqk) = of algebras is a formal consequence of the Cartan formula. Algebraic structures like this, with compatible products and coproducts, are called “Hopf algebras.” P We write A∗ for the vector space dual of A, and we give it the dual basis to the basis just speciﬁed on A. While A∗ is again a Hopf algebra, we are only interested In contrast with A, the algebra A∗ is in its algebra structure at the moment. commutative, as is apparent from the form of the coproduct on the generators of A. Recall that HZ2 is an associative and commutative ring prespectrum, so that H∗(HZ2) is a commutative Z2-algebra. The deﬁnition
of the product on HZ2 (in Chapter 22 §3) and the Cartan formula directly imply the following observation. Lemma. A∗ is isomorphic as an algebra to H∗(HZ2). 4. THE STEENROD ALGEBRA AND ITS COACTION ON H∗(T O) 227 We need an explicit description of this algebra. In principle, this is a matter of pure algebra from the results already stated, but the algebraic work is non-trivial. Theorem. For r ≥ 1, deﬁne Ir = (2r−1, 2r−2,..., 2, 1) and deﬁne ξr to be the basis element of A∗ dual to SqIr. Then A∗ is the polynomial algebra Z2[ξr\|r ≥ 1]. We need a bit of space level motivation for the particular relevance of the elements ξr. We left the computation of the Steenrod operations in H ∗(RP ∞) as an exercise, and the reader should follow up by proving the following result. Lemma. In H ∗(RP ∞) = Z2[α], SqIr (α) = α2r all other basis elements SqI of A. for r ≥ 1 and SqI (α) = 0 for The A-module structure maps A ⊗ H ∗(X) −→ H ∗(X) and A ⊗ H ∗(T ) −→ H ∗(T ) for spaces X and prespectra T dualize to give “A∗-comodule” structure maps γ : H∗(X) −→ A∗ ⊗ H∗(X) and γ : H∗(T ) −→ A∗ ⊗ H∗(T ). We remind the reader that we are implicitly assuming that all homology and cohomology groups in sight are ﬁnitely generated Z2-vector spaces, although these “coactions” can in fact be deﬁned without this assumption. Formally, the notion of a comodule N over a coalgebra C is deﬁned by reversing the direction of arrows in a diagrammatic deﬁnition of a module over an algebra. For example, for any
vector space V, C ⊗ V is a comodule with action ψ ⊗ id : C ⊗ V −→ C ⊗ C ⊗ V. Note that, dualizing the unit of an algebra, a Z2-coalgebra is required to have a counit ε : C −→ Z2. We understand all of these algebraic structures to be graded, and we say that a coalgebra is connected if Ci = 0 for i < 0 and ε : C0 −→ Z2 is an isomorphism. When considering the Hurewicz homomorphism of π∗(T O), we shall need the following observation. Lemma. Let C be a connected coalgebra and V be a vector space. An element y ∈ C ⊗ V satisﬁes (ψ ⊗ id)(y) = 1 ⊗ y if and only if y ∈ C0 ⊗ V ∼= V. If V is a C-comodule with coaction ν : V −→ C ⊗ V, then ν is a morphism of C-comodules. Therefore the coaction maps γ above are maps of A∗-comodules for any space X or prespectrum T. We also need the following observation, which is implied by the Cartan formula. Lemma. If T is an associative ring prespectrum, then γ : H∗(T ) −→ A∗ ⊗ H∗(T ) is a homomorphism of algebras. The lemma above on Steenrod operations in H ∗(RP ∞) dualizes as follows. Lemma. Write the coaction γ : H∗(RP ∞) −→ A∗ ⊗ H∗(RP ∞) in the form γ(xi) = j ai,j ⊗ xj. Then P ai,1 = ξr 0 if i = 2r for some r ≥ 1 otherwise. Note that ai,i = 1, dualizing Sq0(αi) = αi. Armed with this information, we return to the study of the algebra H∗(T O). We know that it is isomorphic to H∗(BO), but the crux of the matter is to redescribe it in
terms of A∗. 228 AN INTRODUCTION TO COBORDISM Theorem. Let N∗ be the algebra deﬁned abstractly by N∗ = Z2[ui\|i > 1 and i 6= 2r − 1], where deg ui = i. Deﬁne a homomorphism of algebras f : H∗(T O) −→ N∗ by f (ai) = Then the composite ui 0 if i is not of the form 2r − 1 if i = 2r − 1. g : H∗(T O) γ −→ A∗ ⊗ H∗(T O) id ⊗f −−−→ A∗ ⊗ N∗ is an isomorphism of both A-comodules and Z2-algebras. Proof. It is clear from things already stated that g is a map of both Acomodules and Z2-algebras. We must prove that it is an isomorphism. Its source and target are both polynomial algebras with one generator of degree i for each i ≥ 1, hence it suﬃces to show that g takes generators to generators. Recall that ai = j∗(xi+1). This allows us to compute γ(ai). Modulo terms that are decomposable in the algebra A∗ ⊗ H∗(T O), we ﬁnd γ(ai) ≡ 1 ⊗ ai ξr ⊗ 1 + 1 ⊗ a2r−1 if i is not of the form 2r − 1 if i = 2r − 1. Applying id ⊗f to these elements, we obtain 1 ⊗ ui in the ﬁrst case and ξr ⊗ 1 in the second case. Now consider the Hurewicz homomorphism h : π∗(T ) −→ H∗(T ) of a prespectrum T. We have the following observation, which is a direct consequence of the deﬁnition of the Hurewicz homomorphism and the fact that Sqi = 0 for i > 0 in the cohomology of spheres. Lemma. For x ∈ π∗(T ), γ(h(x))
= 1 ⊗ h(x). Therefore, identifying N∗ as the subalgebra Z2 ⊗ N∗ of A∗ ⊗ N∗, we see that g ◦ h maps π∗(T O) to N∗. We shall prove the following result in §6 and so complete the proof of Thom’s theorem. Theorem. h : π∗(T O) −→ H∗(T O) is a monomorphism and g◦h maps π∗(T O) isomorphically onto N∗. 5. The relationship to Stiefel-Whitney numbers We shall prove that a smooth closed n-manifold M is a boundary if and only if all of its normal Stiefel-Whitney numbers are zero. Polynomials in the StiefelWhitney classes are elements of H ∗(BO). We have seen that the normal StiefelWhitney numbers of a boundary are zero, and it follows that cobordant manifolds have the same normal Stiefel-Whitney numbers. The assignment of Stiefel-Whitney numbers to corbordism classes of n-manifolds speciﬁes a homomorphism # : H n(BO) ⊗ Nn −→ Z2. We claim that the following diagram is commutative: H n(BO) ⊗ Nn id ⊗α H n(BO) ⊗ πn(T O) id ⊗h / H n(BO) ⊗ Hn(T O) # Z2 h, i H n(BO) ⊗ Hn(BO). id ⊗Φ / / / o o 5. THE RELATIONSHIP TO STIEFEL-WHITNEY NUMBERS 229 To say that all normal Stiefel-Whitney numbers of M are zero is to say that w#[M ] = 0 for all w ∈ H n(BO). Granted the commutativity of the diagram, this is the same as to say that hw, (Φ◦h◦α)([M ])i = 0 for all w ∈ H n(BO). Since h, i is the evaluation pairing of dual vector spaces, this implies that (Φ◦h◦α)([M ]) = 0. Since �
� and α are isomorphisms and h is a monomorphism, this implies that [M ] = 0 and thus that M is a boundary. Thus we need only prove that the diagram is commutative. Embed M in Rn+q with normal bundle ν and let f : M −→ BO(q) classify ν. Then α([M ]) is T f represented by the composite Sn+q t−→ T ν −−→ T O(q). In homology, we have the commutative diagram ˜Hn+q(Sn+q) t∗ / ˜Hn+q(T ν) (T f )∗ ˜Hn+q(T O(q)) Φ Φ Hn(M ) / Hn(BO(q)). f∗ Let in+q ∈ ˜Hn+q(Sn+q) be the fundamental class. By the diagram and the deﬁnitions of α and the Hurewicz homomorphism, (f∗ ◦ Φ ◦ t∗)(in+q) = (Φ ◦ (T f )∗ ◦ t∗)(in+q) = (Φ ◦ h ◦ α)([M ]) ∈ Hn(BO(q)). Let z = (Φ◦t∗)(in+q) ∈ Hn(M ). We claim that z is the fundamental class. Granting the claim, it follows immediately that, for w ∈ H n(BO(q)), w#[M ] = hw(ν), zi = h(f ∗w(γq)), (Φ ◦ t∗)(in+q)i = hw(γq), (f∗ ◦ Φ ◦ t∗)(in+q)i = hw(γq), (Φ ◦ h ◦ α)([M ])i. Thus we are reduced to proving the claim. It suﬃces to show that z maps to a generator of Hn(M, M − x) for each x ∈ M. Since we must deal with pairs, it is convenient to use the homeomorphism between T ν and the quotient D(ν)/S(ν) of the unit disk bundle by the unit sphere bundle. Recall that we have a
relative cap product ∩ : H q(D(ν), S(ν)) ⊗ Hi+q(D(ν), S(ν)) −→ Hi(D(ν)). Letting p : D(ν) −→ M be the projection, which of course is a homotopy equivalence, we ﬁnd that the homology Thom isomorphism is given by the explicit formula Φ : Hi+q(D(ν), S(ν)) −→ Hi(M ) Φ(a) = p∗(µ ∩ a). Let x ∈ U ⊂ M, where U ∼= Rn. Let D(U ) and S(U ) be the inverse images in U of the unit disk and unit sphere in Rn and let V = D(U ) − S(U ). Since D(U ) is contractible, ν\|D(U) is trivial and thus isomorphic to D(U ) × Dq. Write ∂(D(U ) × Dq) = (D(U ) × Sq−1) ∪ (S(U ) × Dq) and observe that we obtain a homotopy equivalence t : Sn+q −→ (D(U ) × Dq)/∂(D(U ) × Dq) ∼= Sn+q by letting t be the quotient map on the restriction of the tubular neighborhood of ν to D(ν\|D(U)) and letting t send the complement of this restriction to the basepoint. / / / / 230 AN INTRODUCTION TO COBORDISM Interpreting t : Sn+q −→ D(ν)/S(ν) similarly, we obtain the following commutative diagram: ˜Hn+q(Sn+q) t∗ ∼= Hn+q(D(U ) × Dq, ∂(D(U ) × Dq)) Hn(D(U ), S(U )) ∼= Φ ∼= Φ t∗ Hn+q(D(ν), S(ν) ∪ D(ν\|M −V )) / Hn(M, M − V ) 4hhhhhhhhhhhhhhhhhh 4hhhhhhhhhhhhhhhhhhh ∼= / Hn(M, M − x). Hn+q(D(ν), S(ν)) Φ / Hn
(M ) The unlabeled arrows are induced by inclusions, and the right vertical arrows are excision isomorphisms. The maps Φ are of the general form Φ(a) = p∗(µ ∩ a). For the top map Φ, µ ∈ Hn+q(D(ν\|D(U)), S(ν\|D(U))) ∼= Hn+q(Sn+q), and, up to evident isomorphisms, Φ is just the inverse of the suspension isomorphism ˜Hn(Sn) −→ ˜Hn+q(Sn+q). The diagram shows that z maps to a generator of Hn(M, M − x), as claimed. 6. Spectra and the computation of π∗(T O) = π∗(M O) We must still prove that h : π∗(T O) −→ H∗(T O) is a monomorphism and that g ◦h maps π∗(T O) isomorphically onto N∗. Write N for the dual vector space of N∗. (Of course, N is a coalgebra, but that is not important for this part of our work.) Remember that the Steenrod algebra A is dual to A∗ and that A ∼= H ∗(HZ2). The dual of g : H∗(T O) −→ A∗ ⊗ N∗ is an isomorphism of A-modules (and of coalgebras) g∗ : A ⊗ N −→ H ∗(T O). Thus, if we choose a basis {yi} for N, where deg yi = ni say, then H ∗(T O) is the free graded A-module on the basis {yi}. At this point, we engage in a conceptual thought exercise. We think of prespectra as “stable objects” that have associated homotopy, homology, and cohomology groups. Imagine that we have a good category of stable objects, analogous to the category of based spaces, that is equipped with all of the constructions that we have on based spaces: wedges (= coproducts), colimits, products, limits, suspensions, loops, homotopies, coﬁber sequences, ﬁber sequences, smash products,
function objects, and so forth. Let us call the stable objects in our imagined category “spectra” and call the category of such objects S. We have in mind an analogy with the notions of presheaf and sheaf. Whatever spectra are, there must be a way of constructing a spectrum from a prespectrum without changing its homotopy, homology, and cohomology groups. In turn, a based space X determines the prespectrum Σ∞X = {ΣnX}. The homology and cohomology groups of Σ∞X are the (reduced) homology and cohomology groups of X; the homotopy groups of Σ∞X are the stable homotopy groups of X. Because homotopy groups, homology groups, and cohomology groups on based spaces satisfy the weak equivalence axiom, the real domain of deﬁnition of these invariants is the category ¯hT that is obtained from the homotopy category hT of based spaces by adjoining inverses to the weak equivalences. This category is equivalent to the homotopy category hC of based CW complexes. Explicitly, the morphisms from X to Y in ¯hT can be deﬁned to be the based homotopy classes of maps ΓX −→ ΓY, where ΓX and ΓY are CW approximations of X and Y. Composition is deﬁned in the evident way. SPECTRA AND THE COMPUTATION OF π∗(T O) = π∗(MO) 231 Continuing our thought exercise, we can form the homotopy category hS of spectra and can deﬁne homotopy groups in terms of homotopy classes of maps from sphere spectra to spectra. Reﬂection on the periodic nature of K-theory suggests that we should deﬁne sphere spectra of negative dimension and deﬁne homotopy groups πq(X) for all integers q. We say that a map of spectra is a weak equivalence if it induces an isomorphism on homotopy groups. We can form the “stable category” ¯hS from hS exactly as we formed the category ¯hT from hT. That is, we develop a theory of CW spectra using sphere spectra as
the domains of attaching maps. The Whitehead and cellular approximation theorems hold, and every spectrum X admits a CW approximation ΓX −→ X. We deﬁne the set [X, Y ] of morphisms X −→ Y in ¯hS to be the set of homotopy classes of maps ΓX −→ ΓY. This is a stable category in the sense that the functor Σ : ¯hS −→ ¯hS is an equivalence of categories. More explicitly, the natural maps X −→ ΩΣX and ΣΩX −→ X are isomorphisms in ¯hS. In particular, up to isomorphism, every object in the category ¯hS is a suspension, hence a double suspension. This implies that each [X, Y ] is an Abelian group and composition is bilinear. Moreover, for any map f : X −→ Y, the canonical map F f −→ ΩCf and its adjoint ΣF f −→ Cf (see Chapter 8 §7) are also isomorphisms in ¯hS, so that coﬁber sequences and ﬁber sequences are equivalent. Therefore coﬁber sequences give rise to long exact sequences of homotopy groups. The homotopy groups of wedges and products of spectra are given by π∗( i Xi) = i π∗(Xi) and π∗( i Xi) = i π∗(Xi). P i Xi −→ Therefore, if only ﬁnitely many πq(Xi) are non-zero for each q, then the natural map W i Xi is an isomorphism. We have homology groups and cohomology groups deﬁned on ¯hS. A spectrum W E represents a homology theory E∗ and a cohomology theory E∗ speciﬁed in terms of smash products and function spectra by Q Q Q Eq(X) = πq(X ∧ E) and Eq(X) = π−qF (X, E) ∼= [X, ΣqE]. Veriﬁcations of the exactness, suspension, additivity, and weak equivalence axioms are immediate from the properties of the category ¯hS. Moreover, every homology
or cohomology theory on ¯hS is so represented by some spectrum E. As will become clear later, Ω-prespectra are more like spectra than general prespectra, and we continue to write Hπ for the “Eilenberg-Mac Lane spectrum” that represents ordinary cohomology with coeﬃcients in π. Its only non-zero homotopy group is π0(Hπ) = π, and the Hurewicz homomorphism maps this group isomorphically onto H0(Hπ; Z). When π = Z2, the natural map H0(HZ2; Z) −→ H0(HZ2; Z2) is also an isomorphism. Returning to our motivating example, we write M O for the “Thom spectrum” that arises from the Thom prespectrum T O. The reader may sympathize with a student who claimed that M O stands for “Mythical Object.” We may choose a map ¯yi : M O −→ ΣniHZ2 that represents the element yi. Deﬁne K(N∗) to be the wedge of a copy of Σni HZ2 for each basis element yi and note that K(N∗) is isomorphic in ¯hS to the product of a copy of ΣniHZ2 for each yi. We think of K(N∗) as a “generalized Eilenberg-Mac Lane spectrum.” It satisﬁes π∗(K(N∗)) ∼= N∗ (as Abelian groups and so as Z2-vector spaces), and the mod 2 Hurewicz homomorphism h : π∗(K(N∗)) −→ H∗(K(N∗)) is a monomorphism. 232 AN INTRODUCTION TO COBORDISM Using the ¯yi as coordinates, we obtain a map ω : M O −→ i ΣniHZ2 ≃ K(N∗). Q The induced map ω∗ on mod 2 cohomology is an isomorphism of A-modules: H ∗(M O) and H ∗(K(N∗)) are free A-modules, and we have deﬁned ω so that
ω∗ sends basis elements to basis elements. Therefore the induced map on homology groups is an isomorphism. Here we are using mod 2 homology, but it can be deduced from the fact that both π∗(M O) and π∗(K(N∗)) are Z2-vector spaces that ω induces an isomorphism on integral homology groups. Therefore the integral homology groups of Cω are zero. By the Hurewicz theorem in ¯hS, the homotopy groups of Cω are also zero. Therefore ω induces an isomorphism of homotopy groups. That is, ω is an isomorphism in ¯hS. Therefore π∗(M O) ∼= N∗ and the Hurewicz homomorphism h : π∗(M O) −→ H∗(M O) is a monomorphism. It follows that g ◦ h : π∗(M O) −→ N∗ is an isomorphism since it is a monomorphism between vector spaces of the same ﬁnite dimension in each degree. 7. An introduction to the stable category To give content to the argument just sketched, we should construct a good category of spectra. In fact, no such category was available when Thom ﬁrst proved his theorem in 1960. With motivation from the introduction of K-theory and cobordism, a good stable category was constructed by Boardman (unpublished) around 1964 and an exposition of his category was given by Adams soon after. However, these early constructions were far more primitive than our outline suggests. While they gave a satisfactory stable category, the underlying category of spectra did not have products, limits, and function objects, and its smash product was not associative, commutative, or unital. In fact, a fully satisfactory category of spectra was not constructed until 1995. We give a few deﬁnitions to indicate what is involved. Definition. A spectrum E is a prespectrum such that the adjoints ˜σ : En −→ ΩEn+1 of the structure maps σ : ΣEn −→ En+1 are homeomorphisms. A map f : T −→ T ′ of prespectra is a sequence of maps fn : Tn −→ T ′ n such that σ′
n ◦ Σfn = fn+1 ◦ σn for all n. A map f : E −→ E′ of spectra is a map between E and E′ regarded as prespectra. We have a forgetful functor from the category S of spectra to the category P of prespectra. It has a left adjoint L : P −→ S. In P, we deﬁne wedges, colimits, products, and limits spacewise. For example, (T ∨ T ′)n = Tn ∨ T ′ n, with the evident structure maps. We deﬁne wedges and colimits of spectra by ﬁrst performing the construction on the prespectrum level and then applying the functor L. If we start with spectra and construct products or limits spacewise, then the result is again a spectrum; that is, limits of spectra are the limits of their underlying prespectra. Thus the category S is complete and cocomplete. Similarly, we deﬁne the smash product T ∧X and function prespectrum F (X, T ) of a based space X and a prespectrum T spacewise. For a spectrum E, we deﬁne E ∧ X by applying L to the prespectrum level construction; the prespectrum F (X, E) is already a spectrum. We now have cylinders E ∧ I+ and thus can deﬁne homotopies between maps of spectra. Similarly we have cones CE = E ∧ I (where I has basepoint 1), suspensions ΣE = E ∧ S1, path spectra F (I, E) (where I has 7. AN INTRODUCTION TO THE STABLE CATEGORY 233 basepoint 0), and loop spectra ΩE = F (S1, E). The development of coﬁber and ﬁber sequences proceeds exactly as for based spaces. The left adjoint L can easily be described explicitly on those prespectra T whose adjoint structure maps ˜σn : Tn −→ ΩTn+1 are inclusions: we deﬁne (LT )n to be the union of the expanding sequence Tn ˜σn−−→ ΩTn+1 Ω˜
σn+1−−−−→ Ω2Tn+2 −→ · · ·. We then have Ω(LT )n+1 = Ω( ΩqTn+1+q) ∼= Ωq+1Tn+q+1 ∼= (LT )n. [ [ We have an evident map of prespectra λ : T −→ LT, and a comparison of colimits shows (by a coﬁnality argument) that λ induces isomorphisms on homotopy and homology groups. The essential point is that homotopy and homology commute with colimits. It is not true that cohomology converts colimits to limits in general, because of lim1 error terms, and this is one reason that our deﬁnition of the cohomology of prespectra via limits is inappropriate except under restrictions that guarantee the vanishing of lim1 terms. Observe that there is no problem in the case of Ω-prespectra, for which λ is a spacewise weak equivalence. For a based space X, we deﬁne the suspension spectrum Σ∞X by applying L to the suspension prespectrum Σ∞X = {ΣnX}. The inclusion condition is satisﬁed in this case. We deﬁne QX = ∪ΩqΣqX, and we ﬁnd that the nth space of Σ∞X is QΣnX. It should be apparent that the homotopy groups of the space QX are the stable homotopy groups of X. The adjoint structure maps of the Thom prespectrum T O are also inclusions, and our mythical object is M O = LT O. In general, for a prespectrum T, we can apply an iterated mapping cylinder construction to deﬁne a spacewise equivalent prespectrum KT whose adjoint structure maps are inclusions. The prespectrum level homotopy, homology, and cohomology groups of KT are isomorphic to those of T. Thus, if we have a prespectrum T whose invariants we are interested in, such as an Eilenberg-Mac Lane Ω-prespectrum or the K-theory Ω-prespectrum, then we can
construct a spectrum LKT that has the same invariants. For a based space X and q ≥ 0, we construct a prespectrum Σ∞ q X whose nth space is a point for n < q and is Σn−qX for n ≥ q; its structure maps for n ≥ q are identity maps. We continue to write Σ∞ q X for the spectrum obtained by applying L to this prespectrum. We then deﬁne sphere spectra Sq for all integers q by letting Sq = Σ∞Sq for q ≥ 0 and S−q = Σ∞ q S0 for q > 0. The deﬁnition is appropriate since ΣSq ∼= Sq+1 for all integers q. We can now deﬁne homotopy groups in the obvious way. For example, the homotopy groups of the K-theory spectrum are Z for every even integer and zero for every odd integer. From here, we can go on to deﬁne CW spectra in very much the same way that we deﬁned CW complexes, and we can ﬁll in the rest of the outline in the previous section. The real work involves the smash product of spectra, but this does not belong in our rapid course. While there is a good deal of foundational work involved, there is also considerable payoﬀ in explicit concrete calculations, as the computation of π∗(M O) well illustrates. With the hope that this glimpse into the world of stable homotopy theory has whetted the reader’s appetite for more, we will end at this starting point. Suggestions for further reading Rather than attempt a complete bibliography, I will give a number of basic references. I will begin with historical references and textbooks. I will then give references for speciﬁc topics, more or less in the order in which topics appear in the text. Where material has been collected in one or another book, I have often referred to such books rather than to original articles. However, the importance and quality of exposition of some of the original sources often make them still to be preferred today. The subject in its earlier days was blessed with some of the ﬁnest expositors of mathematics, for example Steenrod, Serre, Milnor, and Adams. Some of the references are intended to give historical perspective,
some are classical papers in the subject, some are follow-ups to material in the text, and some give an idea of the current state of the subject. In fact, many major parts of algebraic topology are nowhere mentioned in any of the existing textbooks, although several were well established by the mid-1970s. I will indicate particularly accessible references for some of them; the reader can ﬁnd more of the original references in the sources given. 1. A classic book and historical references The axioms for homology and cohomology theories were set out in the classic: S. Eilenberg and N. Steenrod. Foundations of algebraic topology. Princeton University Press. 1952. I believe the only historical monograph on the subject is: J. Dieudonn´e. A history of algebraic and diﬀerential topology, 1900–1960. Birkh¨auser. 1989. A large collection of historical essays will appear soon: I.M. James, editor. The history of topology. Elsevier Science. To appear. Among the contributions, I will advertise one of my own, available on the web: J.P. May. Stable algebraic topology, 1945–1966. http://hopf.math.purdue.edu 2. Textbooks in algebraic topology and homotopy theory These are ordered roughly chronologically (although this is obscured by the fact that the most recent editions or versions are cited). I have included only those texts that I have looked at myself, that are at least at the level of the more elementary chapters here, and that oﬀer signiﬁcant individuality of treatment. There are many other textbooks in algebraic topology. Two classic early textbooks: P.J. Hilton and S. Wylie. Homology theory. Cambridge University Press. 1960. 235 236 SUGGESTIONS FOR FURTHER READING E. Spanier. Algebraic topology. McGraw-Hill. 1966. An idiosyncratic pre-homology level book giving much material about groupoids: R. Brown. Topology. A geometric account of general topology, homotopy types, and the fundamental groupoid. Second edition. Ellis Horwood. 1988. A homotopical introduction close to the spirit of this book: B. Gray. Homotopy theory, an introduction to algebraic topology. Academic Press. 1975. The standard
current textbooks in basic algebraic topology: M.J. Greenberg and J. R. Harper. Algebraic topology, a ﬁrst course. Benjamin/ Cummings. 1981. W.S. Massey. A basic course in algebraic topology. Springer-Verlag. 1991. A. Dold. Lectures on algebraic topology. Reprint of the 1972 edition. SpringerVerlag. 1995. J.W. Vick. Homology theory; an introduction to algebraic topology. Second edition. Springer-Verlag. 1994. J.R. Munkres. Elements of algebraic topology. Addison Wesley. 1984. J.J. Rotman. An introduction to algebraic topology. Springer-Verlag. 1986. G.E. Bredon. Topology and geometry. Springer-Verlag. 1993. Sadly, the following are still the only more advanced textbooks in the subject: R.M. Switzer. Algebraic topology. Homotopy and homology. Springer-Verlag. 1975. G.W. Whitehead. Elements of homotopy theory. Springer-Verlag. 1978. 3. Books on CW complexes Two books giving more detailed studies of CW complexes than are found in textbooks (the second giving a little of the theory of compactly generated spaces): A.T. Lundell and S. Weingram The topology of CW complexes. Van Nostrand Reinhold. 1969. R. Fritsch and R.A. Piccinini. Cellular structures in topology. Cambridge University Press. 1990. 4. Diﬀerential forms and Morse theory Two introductions to algebraic topology starting from de Rham cohomology: R. Bott and L.W. Tu. Diﬀerential forms in algebraic topology. Springer-Verlag. 1982. I. Madsen and J. Tornehave. From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press. 1997. The classic reference on Morse theory, with an exposition of the Bott periodicity theorem: J. Milnor. Morse theory. Annals of Math. Studies No. 51. Princeton University Press. 1963. A modern use of Morse theory for the analytic construction of homology: M. Schwarz. Morse homology. Progress in Math. Vol. 111. Birkh¨auser. 1993. 9. THE EILENBERG-
MOORE SPECTRAL SEQUENCE 237 5. Equivariant algebraic topology Two good basic references on equivariant algebraic topology, classically called the theory of transformation groups (see also §§16, 21 below): G. Bredon. Introduction to compact transformation groups. Academic Press. 1972. T. tom Dieck. Transformation groups. Walter de Gruyter. 1987. A more advanced book, a precursor to much recent work in the area: T. tom Dieck. Transformation groups and representation theory. Lecture Notes in Mathematics Vol. 766. Springer-Verlag. 1979. 6. Category theory and homological algebra A revision of the following classic on basic category theory is in preparation: S. Mac Lane. Categories for the working mathematician. Springer-Verlag. 1971. Two classical treatments and a good modern treatment of homological algebra: H. Cartan and S. Eilenberg. Homological algebra. Princeton University Press. 1956. S. MacLane. Homology. Springer-Verlag. 1963. C.A. Weibel. An introduction to homological algebra. Cambridge University Press. 1994. 7. Simplicial sets in algebraic topology Two older treatments and a comprehensive modern treatment: P. Gabriel and M. Zisman. Calculus of fractions and homotopy theory. SpringerVerlag. 1967. J.P. May. Simplicial objects in algebraic topology. D. Van Nostrand 1967; reprinted by the University of Chicago Press 1982 and 1992. P.G. Goerss and J.F. Jardine. Simplicial homotopy theory. Birkh¨auser. To appear. 8. The Serre spectral sequence and Serre class theory Two classic papers of Serre: J.-P. Serre. Homologie singuli´ere des espaces ﬁbr´es. Applications. Annals of Math. (2)54(1951), 425–505. J.-P. Serre. Groupes d’homotopie et classes de groupes ab´eliens. Annals of Math. (2)58(1953), 198–232. A nice exposition of some basic homotopy theory and of Serre’s work: S.-T. Hu. Homotopy theory. Academic Press. 1959. Many of the textbooks cited in §2 also treat the Serre spectral sequence. 9. The Eilenberg-Moore spectral sequence
There are other important spectral sequences in the context of ﬁbrations, mainly due to Eilenberg and Moore. Three references: S. Eilenberg and J.C. Moore. Homology and ﬁbrations, I. Comm. Math. Helv. 40(1966), 199–236. 238 SUGGESTIONS FOR FURTHER READING L. Smith. Homological algebra and the Eilenberg-Moore spectral sequences. Trans. Amer. Math. Soc. 129(1967), 58–93. V.K.A.M. Gugenheim and J.P. May. On the theory and applications of diﬀerential torsion products. Memoirs Amer. Math. Soc. No. 142. 1974. There is a useful guidebook to spectral sequences: J. McCleary. User’s guide to spectral sequences. Publish or Perish. 1985. 10. Cohomology operations A compendium of the work of Steenrod and others on the construction and analysis of the Steenrod operations: N.E. Steenrod and D.B.A. Epstein. Cohomology operations. Annals of Math. Studies No. 50. Princeton University Press. 1962. A classic paper that ﬁrst formalized cohomology operations, among other things: J.-P. Serre. Cohomologie modulo 2 des complexes d’Eilenberg-Mac Lane. Comm. Math. Helv. 27(1953), 198–232. A general treatment of Steenrod-like operations: J.P. May. A general algebraic approach to Steenrod operations. In Lecture Notes in Mathematics Vol. 168, 153–231. Springer-Verlag. 1970. A nice book on mod 2 Steenrod operations and the Adams spectral sequence: R. Mosher and M. Tangora. Cohomology operations and applications in homotopy theory. Harper and Row. 1968. 11. Vector bundles A classic and a more recent standard treatment that includes K-theory: N.E. Steenrod. Topology of ﬁbre bundles. Princeton University Press. 1951. Fifth printing, 1965. D. Husemoller. Fibre bundles. Springer-Verlag. 1966. Third edition, 1994. A general treatment of classiﬁcation theorems for bundles and ﬁbrations: J.P.
May. Classifying spaces and ﬁbrations. Memoirs Amer. Math. Soc. No. 155. 1975. 12. Characteristic classes The classic introduction to characteristic classes: J. Milnor and J.D. Stasheﬀ. Characteristic classes. Annals of Math. Studies No. 76. Princeton University Press. 1974. A good reference for the basic calculations of characteristic classes: A. Borel. Topology of Lie groups and characteristic classes. Bull. Amer. Math. Soc. 61(1955), 297–432. Two proofs of the Bott periodicity theorem that only use standard techniques of algebraic topology, starting from characteristic class calculations: H. Cartan et al. P´eriodicit´e des groupes d’homotopie stables des groupes classiques, d’apr`es Bott. S´eminaire Henri Cartan, 1959/60. Ecole Normale Sup´erieure. Paris. 14. HOPF ALGEBRAS; THE STEENROD ALGEBRA, ADAMS SPECTRAL SEQUENCE 239 E. Dyer and R.K. Lashof. A topological proof of the Bott periodicity theorems. Ann. Mat. Pure Appl. (4)54(1961), 231–254. 13. K-theory Two classical lecture notes on K-theory: R. Bott. Lectures on K(X). W.A. Benjamin. 1969. This includes a reprint of perhaps the most accessible proof of the complex case of the Bott periodicity theorem, namely: M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles. Acta Math. 112(1994), 229–247. M.F. Atiyah. K-theory. Notes by D.W. Anderson. Second Edition. AddisonWesley. 1967. This includes reprints of two classic papers of Atiyah, one that relates Adams operations in K-theory to Steenrod operations in cohomology and another that sheds insight on the relationship between real and complex K-theory: M.F. Atiyah. Power operations in K-theory. Quart. J. Math. (Oxford) (2)17(1966), 165–193. M.F. Atiyah. K-theory and reality. Quart. J. Math. (Oxford) (2
)17(1966), 367– 386. Another classic paper that greatly illuminates real K-theory: M.F. Atiyah, R. Bott, and A. Shapiro. Cliﬀord algebras. Topology 3(1964), suppl. 1, 3–38. A more recent book on K-theory: M. Karoubi. K-theory. Springer-Verlag. 1978. Some basic papers of Adams and Adams and Atiyah giving applications of K-theory: J.F. Adams. Vector ﬁelds on spheres. Annals of Math. 75(1962), 603–632. J.F. Adams. On the groups J(X) I, II, III, and IV. Topology 2(1963), 181–195; 3(1965), 137-171 and 193–222; 5(1966), 21–71. J.F. Adams and M.F. Atiyah. K-theory and the Hopf invariant. Quart. J. Math. (Oxford) (2)17(1966), 31–38. 14. Hopf algebras; the Steenrod algebra, Adams spectral sequence The basic source for the structure theory of (connected) Hopf algebras: J. Milnor and J.C. Moore. On the structure of Hopf algebras. Annals of Math. 81(1965), 211–264. The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: J. Milnor. The Steenrod algebra and its dual. Annals of Math. 67(1958), 150–171. Two classic papers of Adams; the ﬁrst constructs the Adams spectral sequence relating the Steenrod algebra to stable homotopy groups and the second uses secondary cohomology operations to solve the Hopf invariant one problem: J.F. Adams. On the structure and applications of the Steenrod algebra. Comm. Math. Helv. 32(1958), 180–214. 240 SUGGESTIONS FOR FURTHER READING J.F. Adams. On the non-existence of elements of Hopf invariant one. Annals of Math. 72(1960), 20–104. 15. Cobordism The beautiful classic paper of Thom is still highly recommended: R. Thom. Quelques propri´et´
es globals des vari´et´es diﬀ´erentiables. Comm. Math. Helv. 28(1954), 17–86. Thom computed unoriented cobordism. Oriented and complex cobordism came later. In simplest form, the calculations use the Adams spectral sequence: J. Milnor. On the cobordism ring Ω∗ and a complex analogue. Amer. J. Math. 82(1960), 505–521. C.T.C. Wall. A characterization of simple modules over the Steenrod algebra mod 2. Topology 1(1962), 249–254. A. Liulevicius. A proof of Thom’s theorem. Comm. Math. Helv. 37(1962), 121– 131. A. Liulevicius. Notes on homotopy of Thom spectra. Amer. J. Math. 86(1964), 1–16. A very useful compendium of calculations of cobordism groups: R. Stong. Notes on cobordism theory. Princeton University Press. 1968. 16. Generalized homology theory and stable homotopy theory Two classical references, the second of which also gives detailed information about complex cobordism that is of fundamental importance to the subject. G.W. Whitehead. Generalized homology theories. Trans. Amer. Math. Soc. 102(1962), 227–283. J.F. Adams. Stable homotopy and generalised homology. Chicago Lectures in Mathematics. University of Chicago Press. 1974. Reprinted in 1995. An often overlooked but interesting book on the subject: H.R. Margolis. Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category. North-Holland. 1983. Foundations for equivariant stable homotopy theory are established in: L.G. Lewis, Jr., J.P. May, and M.Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory. Lecture Notes in Mathematics Vol. 1213. Springer-Verlag. 1986. 17. Quillen model categories In the introduction, I alluded to axiomatic treatments of “homotopy theory.” Here are the original and two more recent references: D.G. Quillen. Homotopical algebra. Lecture Notes in Mathematics
Vol. Springer-Verlag. 1967. W.G. Dwyer and J. Spalinski. Homotopy theories and model categories. handbook of algebraic topology, edited by I.M. James. North-Holland. 1995. 43. In A 19. INFINITE LOOP SPACE THEORY 241 The cited “Handbook” (over 1300 pages) contains an uneven but very interesting collection of expository articles on a wide variety of topics in algebraic topology. M. Hovey. Model categories. Amer. Math. Soc. Surveys and Monographs No. 63. 1998. 18. Localization and completion; rational homotopy theory Since the early 1970s, it has been standard practice in algebraic topology to localize and complete topological spaces, and not just their algebraic invariants, at sets of primes and then to study the subject one prime at a time, or rationally. Two of the basic original references are: D. Sullivan. The genetics of homotopy theory and the Adams conjecture. Annals of Math. 100(1974), 1–79. A.K. Bousﬁeld and D.M. Kan. Homotopy limits, completions, and localizations. Lecture Notes in Mathematics Vol. 304. Springer-Verlag. 1972. A more accessible introduction to localization and a readable recent paper on completion are: P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups and spaces. North-Holland. 1975. F. Morel. Quelques remarques sur la cohomologie modulo p continue des pro-pespaces et les resultats de J. Lannes concernent les espaces fonctionnel Hom(BV, X). Ann. Sci. Ecole Norm. Sup. (4)26(1993), 309–360. When spaces are rationalized, there is a completely algebraic description of the result. The main original reference and a more accessible source are: D. Sullivan. Inﬁnitesimal computations in topology. Publ. Math. IHES 47(1978), 269–332. A.K. Bousﬁeld and V.K.A.M. Gugenheim. On PL de Rham theory and rational homotopy type. Memoirs Amer. Math. Soc. No
. 179. 1976. 19. Inﬁnite loop space theory Another area well established by the mid-1970s. The following book is a delightful read, with capsule introductions of many topics other than inﬁnite loop space theory, a very pleasant starting place for learning modern algebraic topology: J.F. Adams. Inﬁnite loop spaces. Annals of Math. Studies No. 90. Princeton University Press. 1978. The following survey article is less easy going, but gives an indication of the applications to high dimensional geometric topology and to algebraic K-theory: J.P. May. Inﬁnite loop space theory. Bull. Amer. Math. Soc. 83(1977), 456–494. Five monographs, each containing a good deal of expository material, that give a variety of theoretical and calculational developments and applications in this area: J.P. May. The geometry of iterated loop spaces. Lecture Notes in Mathematics Vol. 271. Springer-Verlag. 1972. J.M. Boardman and R.M. Vogt. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics Vol. 347. Springer-Verlag. 1973. 242 SUGGESTIONS FOR FURTHER READING F.R. Cohen, T.J. Lada, and J.P. May. The homology of iterated loop spaces. Lecture Notes in Mathematics Vol. 533. Springer-Verlag. 1976. J.P. May (with contributions by F. Quinn, N. Ray, and J. Tornehave). E∞ ring spaces and E∞ ring spectra. Lecture Notes in Mathematics Vol. 577. SpringerVerlag. 1977. R. Bruner, J.P. May, J.E. McClure, and M. Steinberger. H∞ ring spectra and their applications. Lecture Notes in Mathematics Vol. 1176. Springer-Verlag. 1986. 20. Complex cobordism and stable homotopy theory Adams’ book cited in §16 gives a spectral sequence for the computation of stable homotopy groups in terms of generalized cohomology theories. Starting from complex cobordism and related theories, its use has been central to two waves of major developments in stable homotopy theory. A good exposition for the ﬁrst wave: D.
C. Ravenel. Complex cobordism and stable homotopy groups of spheres. Academic Press. 1986. The essential original paper and a very nice survey article on the second wave: E. Devinatz, M.J. Hopkins, and J.H. Smith. Nilpotence and stable homotopy theory. Annals of Math. 128(1988), 207–242. In Proceedings of the 1985 M.J. Hopkins. Global methods in homotopy theory. LMS Symposium on homotopy theory, edited by J.D.S. Jones and E. Rees. London Mathematical Society. 1987. The cited Proceedings contain good introductory survey articles on several other topics in algebraic topology. A larger scale exposition of the second wave is: D.C. Ravenel. Nilpotence and periodicity in stable homotopy theory. Annals of Math. Studies No. 128. Princeton University Press. 1992. 21. Follow-ups to this book There is a leap from the level of this introductory book to that of the most recent work in the subject. One recent book that helps ﬁll the gap is: P. Selick. Introduction to homotopy theory. Fields Institute Monographs No. 9. American Mathematical Society. 1997. There is a recent expository book for the reader who would like to jump right in and see the current state of algebraic topology; although it focuses on equivariant theory, it contains introductions and discussions of many non-equivariant topics: J.P. May et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Monograph. 1996. For the reader of the last section of this book whose appetite has been whetted for more stable homotopy theory, there is an expository article that motivates and explains the properties that a satisfactory category of spectra should have: J.P. May. Stable algebraic topology and stable topological algebra. Bulletin London Math. Soc. 30(1998), 225–234. The following monograph gives such a category, with many applications; more readable accounts appear in the Handbook cited in §17 and in the book just cited: 21. FOLLOW-UPS TO THIS BOOK 243 A. Elmendorf, I. Kriz, M.A. Mandell, and J.P. May, with an appendix by M. Cole. Rings, modules, and algebras
Y Þ such that 'jA D a and g' D f. (1.3.3) Proposition. Let f W X! Y be surjective, continuous, and open. Then Y is separated if and only if R D f.x1; x2/ j f.x1/ D f.x2/g is closed in X X. Let.Xj j j 2 J / be a family of non-empty pairwise disjoint spaces. The set O D fU qXj j U \ Xj Xj open for all j g is a topology on the disjoint union qXj. We call.qXj ; O/ the topological sum of the Xj. A sum of two spaces is denoted X1 C X2. The following assertions are easily veriﬁed from the deﬁnitions. They show that the topological sum together with the canonical inclusions Xj! qXj is a categorical sum in TOP. Given maps fj W Xj! Z we denote by hfj i W q Xj! Z the map with restriction fj to Xj. (1.3.4) Proposition. A topological sum has the following properties: The subspace topology of Xj in qXj is the original topology. Let the space X be the union of the family.Xj j j 2 J / of pairwise disjoint subsets. Then X is the topological sum of the subspaces Xj if and only if the Xj are open. f W q Xj! Y is continuous if each f jXj W Xj! Y is continuous. 1.3.5 Pushout. Let j W A! X and f W A! B be continuous maps and form a pushout diagram A j X f F B J Y in the category SET of sets. Then Y is obtainable as a quotient of X C B. We give Y the quotient topology via hF; J i W X C B! Y. Then the resulting diagram is a pushout in TOP. The space Y is sometimes written X CA B and called the sum Þ of X and B under A (the sum in the category TOPA of spaces under A). 1.3.6 Clutching. An important method for the construction of spaces is to “paste” open subsets; see the
example (1.3.8) for the simplest case. Let.Uj j j 2 J / be a family of sets. Assume that for each pair.i; j / 2 J J a subset U j i Ui is given as well as a bijection gj j ; gk j / a clutching datum if: j. We call the families.Uj ; 1) Uj D U j (2) For each triple.i; j; k/ 2 J J J the map gj j and gj j D id. i induces a bijection i W U j gj 10 Chapter 1. Topological Spaces and gk j ı gj i holds, considered as maps from U j Given a clutching datum, we have an equivalence relation on the disjoint sum k \ U i k. i \ U k i i D gk to U j qj 2J Uj : x 2 Ui y 2 Uj, x 2 U j i and gj i.x/ D y: Let X denote the set of equivalence classes and let hi W Ui! X be the map which sends x 2 Ui to its class. Then hi is injective. Set U.i/ D image hi, then U.i/ \ U.j / D hi.U j i /. Conversely, assume that X is a quotient of qj 2J Uj such that each hi W Ui! X i D h1 ı i / are a clutching datum. is injective with image U.i/. Let U j i ; gj hi W U j construction above to this datum, we get back X and the hi. i D h1 If we apply the Þ j. Then the.Ui ; U j.U.i/ \ U.j // and gj i! U i j i (1.3.7) Proposition. Let.Ui ; U j i / be a clutching datum. Assume that the Ui are topological spaces, the U j j homeomorphisms. Let X carry the quotient topology with respect to the quotient map p W qj 2J Uj! X. Then the following holds: i Ui open subsets, and the gj i! U i i W U j i ; gj (1) The map hi is a homeomorphism onto an open subset of X and p
is open. (2) Suppose the Ui are Hausdorff spaces. Then X is a Hausdorff space if and i.x// is only if for each pair.i; j / the map j a closed embedding. i! Ui Uj, x 7!.x; gj i W U j 1.3.8 Euclidean space with two origins. The simplest case is obtained from open subsets Vj Uj, j D 1; 2, and a homeomorphism'W V1! V2. Then X D U1 [' U2 is obtained from the topological sum U1 C U2 by identifying v 2 V1 with '.v/ 2 V2. Let U1 D U2 D Rn and V1 D V2 D Rn X 0. Let'D id. Then the graph of'in Rn Rn is not closed. The resulting locally Euclidean space is not Hausdorff. If we use '.x/ D x kxk2, then the result is homeomorphic to Þ S n (see (2.3.2)). Suppose a space X is the union of subspaces.Xj j j 2 J /. We say X carries the colimit topology with respect to this family if one of the equivalent statements hold: j 2J Xj! X (the inclusion on each summand) is a (1) The canonical map quotient map. ` (2) C is closed in X if and only if Xj \ C is closed in Xj for each j. (3) A set map f W X! Z into a space Z is continuous if and only if the restric- tions f jXj W Xj! Z are continuous. (1.3.9) Example. Let X be a set which is covered by a family.Xj j j 2 J / of subsets. Suppose each Xj carries a topology such that the subspace topologies of Xi \ Xj in Xi and Xj coincide and these subspaces are closed. Then there is a unique topology on X which induces on Xj the given topology. The space X has Þ the colimit topology with respect to the Xj. 1.4. Compact Spaces 11 Problems Q xAj D j 2J 1. Let.Xj j j 2 J / be spaces and Aj Xj non-
empty subspaces. Then Q Q j 2J Aj is closed if and only if the Aj are closed. Q j Xj! Xk are open maps, and in particular quotient maps. j 2J Aj. The product 2. The projections prk W (The Xj are non-empty.) 3. A space X is separated if and only if the diagonal D D f.x; x/ j x 2 Xg is closed in X X. Let f; g W X! Y be continuous maps into a Hausdorff space. Then the coincidence set A D fx j f.x/ D g.x/g is closed in X. Hint: Use (1.3.3). 4. A discrete space is the topological sum of its points. There is always a canonical homeomorphism X qj Yj Š qj.X Yj /. For each y 2 Y the map X! X Y, x 7!.x; y/ is an embedding. If f W X! Y is continuous, then W X! X Y, x 7!.x; f.x// is an embedding. If Y is a Hausdorff space, then is closed. 1.4 Compact Spaces A family A D.Aj j j 2 J / of subsets of X is a covering of X if X is the union of the Aj. A covering B D.Bk j k 2 K/ of X is a reﬁnement of A if for each k 2 K there exists j 2 J such that Bk Aj. If X is a topological space, a covering A D.Aj j j 2 J / is called open (closed) if each Aj is open (closed). A covering B D.Bk j k 2 K/ is a subcovering of A if K J and Bk D Ak for k 2 K. We say B is ﬁnite or countable if K is ﬁnite or countable. A covering A is locally ﬁnite if each x 2 U has a neighbourhood U such that U \ Aj ¤ ; only for a ﬁnite number of j 2 J. It is called point-ﬁnite if each x 2 X is contained only in a ﬁnite number of Aj. A space X is compact if each open covering has a ﬁnite sub
covering. (In some texts this property is called quasi-compact.) By passage to complements we see: If X is compact, then any family of closed sets with empty intersection contains a ﬁnite family with empty intersection. A set A in a space X is relatively compact if its closure is compact. We recall from calculus the fundamental Heine–Borel Theorem: The unit interval I D Œ0; 1 is compact. A space X is compact if and only if each net in X has a convergent subnet (an accumulation value). A discrete closed set in a compact space is ﬁnite. Let X be compact, A X closed and f W X! Y continuous; then A and f.X/ are compact. (1.4.1) Proposition. Let B, C be compact subsets of spaces X, Y, respectively. Let U be a family of open subsets of X Y which cover B C. Then there exist 12 Chapter 1. Topological Spaces open neighbourhoods U of B and V of C such that U V is covered by a ﬁnite subfamily of U. In particular the product of two compact spaces is compact. One can show that an arbitrary product of compact spaces is compact (Theorem of Tychonoff ). (1.4.2) Proposition. Let B and C be disjoint compact subsets of a Hausdorff space X. Then B and C have disjoint open neighbourhoods. A compact Hausdorff space is normal. A compact subset C of a Hausdorff space X is closed. (1.4.3) Proposition. A continuous map f W X! Y from a compact space into a Hausdorff space is closed. If, moreover, f is injective (bijective), then f is an embedding (homeomorphism). If f is surjective, then it is a quotient map. (1.4.4) Proposition. Let X be a compact Hausdorff space and f W X! Y a quotient map. The following assertions are equivalent: (1) Y is a Hausdorff space. (2) f is closed. (3) R D f.x1; x2/ j f.x1/ D f.x2/g is closed in X X. Let X be a union of subspaces X1 X2. Recall that X carries the
colimit-topology with respect to the ﬁltration.Xi / if A X is open (closed) if and only if each intersection A \ Xn is open (closed) in Xn. We then call X the colimit of the ascending sequence.Xi /. (This is a colimit in the categorical sense.) (1.4.5) Proposition. Suppose X is the colimit of the sequence X1 X2. Suppose points in Xi are closed. Then each compact subset K of X is contained in some Xk. A space is locally compact if each neighbourhood of a point x contains a compact neighbourhood. An open subset of a locally compact space is again locally compact. Let X be a Hausdorff space and assume that each point has a compact neighbourhood. Let U be a neighbourhood of x and K a compact neighbourhood. Since K is normal, K \ U contains a closed neighbourhood L of x in K. Then L is compact and a neighbourhood of x in X. Therefore X is locally compact. In particular, a compact Hausdorff space is locally compact. If X and Y are locally compact, then X Y is locally compact. Let X be a topological space. An embedding f W X! Y is a compactiﬁcation of X if Y is compact and f.X/ dense in Y. The following theorem yields a compactiﬁcation by a single point. It is called the Alexandroff compactiﬁcation or the one-point compactiﬁcation. The additional point is the point at inﬁnity. In a general compactiﬁcation f W X! Y, one calls the points in Y X f.X/ the points at inﬁnity. 1.4. Compact Spaces 13 (1.4.6) Theorem. Let X be a locally compact Hausdorff space. Up to homeomorphism, there exists a unique compactiﬁcation f W X! Y by a compact Hausdorff space such that Y X f.X/ consists of a single point. (1.4.7) Proposition. Let the locally compact space be a union of compact subsets.Ki j i 2 N/. Then there exists a sequence.Ui j i 2 N/ of open subsets with union X such that each SUi is compact and contained in UiC
1. (1.4.8) Theorem. Let the locally compact Hausdorff space M ¤ ; be a union of closed subsets Mn, n 2 N. Then at least one of the Mn contains an interior point. A subset H of a space G is called locally closed, if each x 2 H has a neigh- bourhood Vx in G such that H \ Vx is closed in G. (1.4.9) Proposition..1/ Let A be locally closed in X. Then A D U \ C with U open and C closed. Conversely, if X is regular, then an intersection U \ C, U open, C closed, is locally closed..2/ A locally compact set A in a Hausdorff space X is locally closed..3/ A locally closed set A in a locally compact space is locally compact. Problems 1. Dn=S n1 is homeomorphic to S n. For the proof verify that q Dn! S n; x 7! 2 1 kxk2x; 2kxk2 1 induces a bijection Dn=S n1! S n. 2. Let f W X C! R be continuous. Assume that C is compact and set g.x/ D supff.x; c/ j c 2 C g. Then g W X! R is continuous. 3. Let X be the colimit of an ascending sequence of spaces X1 X2. Then the Xi are subspaces of X. If Xi XiC1 is always closed, then the Xj are closed in X. 4. Let R1 be the vector space of all sequences.x1; x2; : : : / of real numbers which are eventually zero. Let Rn be the subspace of sequences with xj D 0 for j > n. Give R1 the colimit topology with respect to the subspaces Rn. Then addition of vectors is a continuous map R1 R1! R1. Scalar multiplication is a continuous map R R1! R1. (Thus R1 is a topological vector space.) A neighbourhood basis of zero consists of the intersection of R1 with products of the form i1 "i ; "i Œ. The space R1 with this topology is not metrizable. The space has also the colimit topology with respect to the set of ﬁnitedimensional linear sub
spaces. One can also consider the metric topology with respect to the i.xi yi /2 metric d..xi /;.yi // D d is continuous. The space R1 is separated. d. The identity R1! R1 ; denote it by R1 1=2 P Q 14 Chapter 1. Topological Spaces 1.5 Proper Maps A continuous map f W X! Y is called proper if it is closed and the pre-images f 1.y/; y 2 Y are compact. (1.5.1) Proposition. Let K be compact. Then pr W X K! X is proper. If f W X! Y is proper and K Y compact, then f 1.K/ is compact. Let f and g be proper; then f g is proper. As a generalization of the theorem of Tychonoff one can show that an arbitrary product of proper maps is proper. (1.5.2) Proposition. Let f W X! X 0 and g W X 0! X 00 be continuous. (1) If f and g are proper, then g ı f is proper. (2) If g ı f is proper and f surjective, then g is proper. (3) If g ı f is proper and g injective, then f is proper. (1.5.3) Proposition. Let f W X! Y be injective. Then the following are equivalent: (1) f is proper. (2) f is closed. (3) f is a homeomorphism onto a closed subspace. (1.5.4) Proposition. Let f W X! Y be continuous. (1) If f is proper, then for each B Y the restriction fB W f 1.B/! B of f is proper. ` (2) Let.Uj p W is proper, then f is proper. j j 2 J / be a covering of Y such that the canonical map j 2J Uj! Y is a quotient map. If each restriction fj W f 1.Uj /! Uj (1.5.5) Proposition. Let f be a continuous map of a Hausdorff space X into a locally compact Hausdorff space Y. Then f is proper if and only if each compact set K Y has a compact pre-image. If f is proper, then X is locally compact. (1.5.6
) Proposition. Let f W X! X 0 and g W X 0! X 00 be continuous and assume that gf is proper. If X 0 is a Hausdorff space, then f is proper. (1.5.7) Theorem. A continuous map f W X! Y is proper if and only if for each space T the product f id W X T! Y T is closed. Problems 1. A map f W X! Y is proper if and only if the following holds: For each net.xj / in X and each accumulation value y of.f.xj // there exists an accumulation value x of.xj / such 1.6. Paracompact Spaces 15 that f.x/ D y. 2. Let X and Y be locally compact Hausdorff spaces, let f W X! Y be continuous and f C W X C! Y C the extension to the one-point compactiﬁcation. Then f C is continuous, if f is proper. 3. The restriction of a proper map to a closed subset is proper. 4. Let f W X! Y be proper and X a Hausdorff space. Then the subspace f.X/ of Y is a Hausdorff space. 5. Let f W X! Y be continuous. Let R be the equivalence relation on X induced by f, and denote by p W X! X=R the quotient map, by h W X=R! f.X/ the canonical bijection, and let i W f.X/ Y. Then f D i ı h ı p is the canonical decomposition of f. The map f is proper if and only if p is proper, h a homeomorphism, and f.X/ Y closed. 1.6 Paracompact Spaces Let A D.Uj j j 2 J / be an open covering of the space X. An open covering B D.Bj j j 2 J / is called a shrinking of A if for each j 2 J we have the inclusion xBj Uj. A point-ﬁnite open covering of a normal space has a shrinking. A space X is called paracompact if it is a Hausdorff space and if each open covering has an open, locally ﬁnite reﬁnement. A closed subset of a paracompact space is paracompact
. A compact space is paracompact. A paracompact space is normal. Suppose the locally compact Hausdorff space X is a countable union of compact sets. Then X is paracompact. Let X be paracompact and K be compact Hausdorff. Then X K is paracompact. A metric space is paracompact. 1.7 Topological Groups A topological group.G; m; O/ consists of a group.G; m/ with multiplication m W G G! G,.g; h/ 7! m.g; h/ D gh and a topology O on G such that the multiplication m and the inverse W G! G, g 7! g1 are continuous. We denote a topological group.G; m; O/ usually just by the letter G. The neutral element will be denoted by e (also 1 is in use and 0 for abelian groups). The left translation lg W G! G, x 7! gx by g 2 G in a topological group is continuous, and the rules lg lh D lgh and le D id show it to be a homeomorphism. For subsets A and B of a group G we use notations like aB D fab j b 2 Bg, AB D fab j a 2 A; b 2 Bg, A2 D AA, A1 D fa1 j a 2 Ag, and similar ones. A group G together with the discrete topology on the set G is a topological group, called a discrete (topological) group. The additive groups of the real numbers R, complex numbers C, and quaternions H with their ordinary topology are topological groups, similarly the multiplicative 16 Chapter 1. Topological Spaces groups R, C and H of the non-zero elements. The multiplicative group R C of the positive real numbers is an open subgroup of R and a topological group. The complex numbers of norm 1 are a compact topological group S 1 with respect to multiplication. The exponential function exp W R! R C is a continuous homomorphism with the logarithm function as a continuous inverse. The complex exponential function exp W C! C is a surjective homomorphism with kernel f2 i n j n 2 Zg, a discrete subgroup of C. The main examples of topological groups are matrix groups. In the vector space Mn.
R/ of real.n; n/-matrices let GLn.R/ be the subspace of the invertible matrices. Since the determinant is a continuous map, this is an open subspace. Matrix multiplication and passage to the inverse are continuous, since they are given by rational functions in the matrix entries. This makes the general linear group GLn.R/ into a topological group. Similarly for GLn.C/. The determinant is a continuous homomorphism det W GLn.R/! R with kernel the special linear group SLn.R/; similarly in the complex case. Let O.n/ D fA 2 Mn.R/ j At A D Eg be the group of orthogonal.n; n/matrices (At transpose of A; E unit matrix). The set O.n/ is a compact subset in Mn.R/. Hence O.n/ is a compact topological group (the orthogonal group). The open and closed subspace SO.n/ D fA 2 O.n/ j det.A/ D 1g of O.n/ is the special orthogonal group. Similarly the subgroup U.n/ D fA 2 Mn.C/ j At xA D Eg of unitary.n; n/-matrices is a compact topological group (unitary group). The topological groups SO.2/, U.1/, and S 1 are isomorphic. The special unitary group SU.n/ is the compact subgroup of U.n/ of matrices with determinant 1. The multiplicative group of quaternions of norm 1 provides S 3 with the structure of a topological group. This group is isomorphic to SU.2/. From linear algebra one knows about a surjective homomorphism SU.2/! SO.3/ with kernel ˙E (a twofold covering); for this and other related facts see the nice discussion in [27, Kapitel IX]. For more information about matrix groups, also from the viewpoint of manifolds and Lie groups, see [29]; there you can ﬁnd, among others, the symplectic groups Sp.n/ and the Spinor groups Spin.n/. The isomorphisms SU.2/ Š Spin.3/ Š Sp.1/ hold, and these spaces are homeomorphic to S 3. If G and H are topological groups, then the direct
product G H with the product topology is a topological group. The n-fold product S 1 S 1 is called an n-dimensional torus. The trivial subgroup is often denoted by 1 (in a multiplicative notation) or by 0 (in an additive notation). The neutral element will also be denoted 1 or 0. The symbol H C G is used for a normal subgroup H of G. The notation H K or H G K means that H and K are conjugate subgroups of G. A homomorphism f W G! H between topological groups is continuous if it is continuous at the neutral element e. If G is a topological group and H G a subgroup, then H, with the subspace topology, is a topological group (called a topological subgroup). If H G is a subgroup, then the closure of H is also. If H is a normal subgroup, then xH is also. 1.8. Transformation Groups 17 1.8 Transformation Groups A left action of a topological group G on a topological space X is a continuous map W G X! X,.g; x/ 7! gx such that g.hx/ D.gh/x and ex D x for g; h 2 G, e 2 G the unit, and x 2 X. A (left) G -space.X; / consists of a space X and a left action of G on X. The homeomorphism lg W X! X, x 7! gx is called left translation by g. We also use right actions X G! X,.x; g/ 7! xg; they satisfy.xh/g D x.hg/ and xe D x. For A X and K G we let KA D fka j k 2 K; a 2 Ag. An action is effective if gx D x for all x 2 X implies g D e. The trivial action has gx D x for g 2 G and x 2 X. The set R D f.x; gx/ j x 2 X; g 2 Gg is an equivalence relation on X. The set of equivalence classes X mod R is denoted by X=G. The quotient map q W X! X=G is used to provide X=G with the quotient topology. The resulting space X=G is called the orbit space of the G-
space X. A more systematic notation for the orbit space of a left action would be GnX. The equivalence class of x 2 X is the orbit Gx through x. An action is transitive if it consists of a single orbit. The set Gx D fg 2 G j gx D xg is a subgroup of G, the isotropy group or the stabilizer of the G-space X at x. An action is free if all isotropy groups are trivial. We have Ggx D gGxg1. Therefore the set Iso.X/ of isotropy groups of X consists of complete conjugacy classes of subgroups. If it contains a ﬁnite number of conjugacy classes, we say X has ﬁnite orbit type. A subset A of a G-space is called G -stable or G -invariant if g 2 G and a 2 A implies ga 2 A. A G-stable subset A is also called a G -subspace. For each subgroup H of G there is an H -ﬁxed point set of X, X H D fx 2 X j hx D x for all h 2 H g: Suppose X and Y are G-spaces. A map f W X! Y is called a G-map or a G -equivariant map if for g 2 G and x 2 X the relation f.gx/ D gf.x/ holds. In general, the term “equivariant” refers to something related to a group action. Left G-spaces and G-equivariant maps form the category G- TOP. This category has products: If.Xj j j 2 J / is a family of G-spaces, then the topological j Xj together with the diagonal action.g;.xj // 7!.gxj / is a product product in this category. A G-map f W X! Y induces by passage to the orbit spaces a map f =G W X=G! Y =G. We have the notion of an equivariant homotopy or G -homotopy Ht : this is a homotopy such that each Ht is a G-map. Q (1.8.1) Proposition. Let X be a G-space, A G and B X. If B is open then AB is open. The orbit map p W
X! X=G is open. 18 Chapter 1. Topological Spaces Proof. la.B/ is open, since la is a homeomorphism. Hence is a union of open sets. Let U be open. Then p1p.U / D hence p.U / is open, by deﬁnition of the quotient topology. S a2A la.B/ D AB g2G lg.U / is open, S (1.8.2) Proposition..1/ Let H be a subgroup of the topological group G. Let the set G=H of cosets gH carry the quotient topology with respect to p W G! G=H, g 7! gH. Then l W G G=H! G=H,.x; gH / 7! xgH is a continuous action..2/ G=H is separated if and only if H is closed in G. In particular, G is separated if feg is closed..3/ Let H be normal in G. Then the factor group G=H with quotient topology is a topological group. A space G=H with the G-action by left multiplication is called a homogeneous space. The space of left cosets Hg is H nG; it carries a right action. (1.8.3) Example. Homogeneous spaces are important spaces in geometry. The orthogonal group O.n C 1/ acts on the sphere S n by matrix multiplication.A; v/ 7! Av. The action is transitive. The isotropy group of e1 D.1; 0; : : : ; 0/ is O.n/, here considered as the block matrices with B 2 O.n/. We obtain a homeomorphism of O.n C 1/-spaces O.n C 1/=O.n/ Š S n. In the complex case we obtain a homeomorphism U.n C 1/=U.n/ Š S 2nC1, in the quaternionic case a homeomorphism Sp.n C 1/=Sp.n/ Š S 4nC3. Other important homogeneous spaces are the projective spaces, the Grassmann manifolds, and the Stiefel Þ manifolds to be discussed later. 1 0 0 B (1.8.4) Proposition..1/ If x 2
X is closed, then Gx is closed in G..2/ If X is a Hausdorff space, then X H is closed..3/ Let A be a G-stable subset of the G-space X. Then A=G carries the subspace topology of X=G. In particular X G! X! X=G is an embedding..4/ Let B X be closed and A X. Then fg 2 G j gA Bg is closed in G..5/ Let B X be closed. Then fg 2 G j gB D Bg is closed. T Proof. (1) The isotropy group Gx is the pre-image of x under the continuous map G! X, g 7! gx. (2) The set X g D fx 2 X j gx D xg is the pre-image of the diagonal under X! X X, x 7!.x; gx/, and X H D g2H X g. (3) Let C A=G be open with respect to the quotient map A! A=G. Then p1.C / A is open, and we can write p1.C / D A \ U with an open subset U X. We have A \ U D A \ GU, since A is G-stable. We conclude C D p.p1C / D A=G \p.GU /. Since GU is open, p.GU / is open, hence C is open in the subspace topology. By continuity of A=G! X=G, an open subset in the subspace topology is open in A=G. (4) ra W G! X, g 7! ga is continuous, hence r 1 a.B/ D fg 2 T a2A r 1 G j ga 2 Bg closed and therefore a.B/ D fg 2 G j gA Bg closed. (5) The set fg j gB D Bg D fg j gB Bg \ fg j g1B Bg is closed, by (4). (1.8.5) Proposition. Let r W G X! X be a G-action, A G and B X. 1.8. Transformation Groups 19 (1) If A and B are compact, then AB is compact. (2) If A is compact, then the restriction A X
! X of r is proper. If, moreover, B is closed, then AB is closed. (3) If G is compact, then the orbit map p is proper. Thus X is compact if and only if X=G is compact. (4) If G is compact and X separated, then X=G is separated. (5) Let G be compact, A a G-stable closed subset and U a neighbourhood of A in X. Then U contains a G-stable neighbourhood of A. Proof. (1) A B G X is compact as a product of compact spaces. Hence the continuous image AB of A B under r W G X! X is compact. (2) The homeomorphism AX! AX,.s; x/ 7!.s; sx/ transforms r into the projection pr W A X! X. The projection is proper, since A is compact (see (1.5.1)). Hence the image AB of the closed set A B is closed. (3) Let A X be closed. Then p1p.A/ D GA is closed, by (2). Hence p.A/ is closed, by deﬁnition of the quotient topology. The pre-images of points are orbits; they are compact as continuous images of G. (4) Since p is proper, so is p p. Hence the image of the diagonal under p p is closed. (5) Let U be open. Then p.X X U / is disjoint to p.A/. By (4), X X p1p.X X U / is open and a G-stable neighbourhood of A contained in U. The orbit category Or.G/ is the category of homogeneous G-spaces G=H, H closed in G, and G-maps. There exists a G-map G=H! G=K if and only if H is conjugate to a subgroup of K. If a1Ha K, then Ra W G=H! G=K, gH 7! gaK is a G-map and each G-map G=H! G=K has this form; moreover Ra D Rb if and only if a1b 2 K. An action G V! V on a real (or complex) vector space V is called a real (or complex) representation of G if the left translations are linear maps. After choice of a basis, a representation amounts
to a continuous homomorphism from G to GLn.R/ or GLn.C/. A homomorphism G! O.n/ or G! U.n/ is called an orthogonal or unitary representation. Geometrically, an orthogonal representation is given by an action G V! V with an invariant scalar product h; i. The latter means hgv; gwi D hv; wi for g 2 G and v; w 2 V. In an orthogonal representation, the unit sphere S.V / D fv 2 V j hv; vi D1g is G-stable. Let E be a right G-space and F a left G-space. We denote by E G F the orbit space of the G-action G.E F /! E F,.g;.x; y// 7!.xg1; gy/. A G-map f W F1! F2 induces a continuous map id Gf W E G F1! E G F2;.x; y/ 7!.x; f.x//: If E carries a left K-action which commutes with the right G-action (i.e., k.xg/ D.kx/g), then E G F carries an induced K-action.k;.x; y// 7!.kx; y/. This construction can in particular be applied in the case that E D K, G a subgroup 20 Chapter 1. Topological Spaces of K and the G- and K-actions on K are given by right and left multiplication. The assignments F 7! K G F and f 7! id Gf yield the induction functor indK G W G- TOP! K- TOP. This functor is left adjoint to the restriction functor resK G W K- TOP! G- TOP which is obtained by regarding a K-space as a G-space. The natural adjunction TOPK.indK G X; Y / Š TOPG.X; resK G Y / sends a G-map f W X! Y to the K-map.k; x/ 7! kf.x/; in the other direction one restricts a map to X Š G G X K G X. (Here TOPK denotes the set of K-equivariant maps.) (1.8.6) Theorem. Suppose the
Hausdorff group G is locally compact with countable basis. Let X be a locally compact Hausdorff space and G X! X a transitive action. Then for each x 2 X the map b W G! X, g 7! gx is open and the induced map xb W G=Gx! X a homeomorphism. Proof. If b is open, then xb is a homeomorphism. Let W be a neighbourhood of e,.Bi j i 2 N/ a countable basis, and g1 i 2 Bi. For each g 2 G there exists a j such that Bj W g1, g 2 gj W. Therefore the gj W cover the group. S Let V G be open and g 2 V. There exists a compact neighbourhood W of e such that W D W 1 and gW 2 V. Since G is the union of the gj W and the action is transitive, X D gj W x. Since W is compact and b continuous, gj W x is compact and hence closed in X. By (1.4.8), there exist j such that gj W x contains an interior point, and therefore W x contains an interior point wx. Then x is an interior point of w1W x W 2x and hence gx D p.g/ an interior point of gW 2x V x D p.V /. This shows that p is open. (1.8.7) Corollary. Let the locally compact Hausdorff group G with countable basis act on a locally compact Hausdorff space X. An orbit is locally compact if and only if it is locally closed. An orbit is a homogeneous space with respect to the isotropy group of each of its points if and only if it is locally closed. Problems 1. Let H be a normal subgroup of G and X a G-space. Restricting the group action to H, we obtain an H -space X. The orbit space H nX carries then an induced G=H -action. 2. Let a pushout in TOP be given with G-spaces A, B, X and G-maps. Let G be locally compact. Then there exists a unique G action on Y such that F; J become G-maps. The diagram is then a pushout in G-TOP. Hint: (2.4.6) 1.9. Projective Spaces
. Grassmann Manifolds 21 3. Let Y be a K-space and G a subgroupp of K. Then K G Y! K=G Y,.k; y/ 7!.kG; ky/ is a K-homeomorphism. If X is a G-space, then K G.X Y /!.K G X/ Y;.g;.x; y// 7!..g; x/; gy/ is a K-homeomorphism. 4. Let H be a closed subgroup of G. Then G=H is a Hausdorff space and therefore F D G=H H is closed. The relation gH 2 G=H H is equivalent to g1Hg H. Hence fg 2 G j g1Hg H g is closed in G. The normalizer NH D fg 2 G j g1Hg D H g of H in G is closed in G. The group H is a normal subgroup of NH and NH=H D WGH D W H is the Weyl group of H in G. The group NH always acts on the ﬁxed set X H, by restricting the given G-action to NH. The action G=H W H! G=H;.gH; nH / 7! gnH is a free right action by G-automorphisms of G=H. 1.9 Projective Spaces. Grassmann Manifolds Let P.RnC1/ D RP n be the set of one-dimensional subspaces of the vector space RnC1. A one-dimensional subspace of V is spanned by x 2 V X 0. The vectors x and y span the same subspace if and only if x D y for some 2 R D R X 0. We therefore consider P.RnC1/ as the orbit space of the action R.RnC1 X 0/! RnC1 X 0;.; x/ 7! x: The quotient map p W RnC1 X 0! P.RnC1/ provides P.RnC1/ with the quotient topology. The space RP n is the n-dimensional real projective space. We set p.x0; : : : ; xn/ D Œx0; : : : ; xn and call x0; : : : ; xn
the homogeneous coordinates of the point Œx0; : : : ; xn. In a similar manner we consider the set P.CnC1/ D CP n of one-dimensional subspaces of CnC1 as the orbit space of the action C.CnC1 X 0/! CnC1 X 0;.; z/ 7! z: We have again a quotient map p W CnC1 X 0! P.CnC1/. The space CP n is called the n-dimensional complex projective space. (It is 2n-dimensional as a manifold.) We describe the projective spaces in a different manner as orbit spaces. The subgroup G D f˙1g R acts on S n RnC1 by.; x/ 7! x, called the antipodal involution. The inclusion i W S n! RnC1 induces a continuous bijective map W S n=G!.RnC1 X 0/=R. The map j W RnC1 X 0! S n, x 7! kxk1x induces an inverse. The quotient S n=G is compact, since S n is compact. By (1.4.4), the quotient is a Hausdorff space. In a similar manner one treats CP n, but now with respect to the action S 1 S 2nC1! S 2nC1,.; z/ 7! z of S 1 on the unit sphere S 2nC1 CnC1 X 0. 22 Chapter 1. Topological Spaces Projective spaces are homogeneous spaces. Consider the action of O.n C 1/ on RnC1 by matrix multiplication. If V 2 P.RnC1/ is a one-dimensional space and A 2 O.n C 1/, then AV 2 P.RnC1/. We obtain an induced action O.n C 1/ P.RnC1/! P.RnC1/: This action is transitive. The isotropy group of Œ1; 0; : : : ; 0 consists of the matrices 0 0 B ; 2 O.1/; B 2 O.n/: We consider these matrices as the subgroup O.1/O.n/ of O.nC1/. The assignment A 7! Ae1 induces an O.n C 1/-
equivariant homeomorphism b W O.n C 1/=.O.1/ O.n// Š P.RnC1/: The action of O.n C 1/ on P.RnC1/ is continuous; this follows easily from the continuity of the action O.n C 1/.RnC1 X 0/! RnC1 X 0 and the deﬁnition of the quotient topology. Therefore b is a bijective continuous map of a compact space into a Hausdorff space. In a similar manner we obtain a U.n C 1/-equivariant homeomorphism U.n C 1/=.U.1/ U.n// Š P.CnC1/. Finally, one can deﬁne the quaternionic projective space HP n in a similar manner as a quotient of HnC1 X 0 or as a quotient of S 4nC3. We generalize projective spaces. Let W be an n-dimensional real vector space. We denote by Gk.W / the set of k-dimensional subspaces of W. We deﬁne a topology on Gk.W /. Suppose W carries an inner product. Let Vk.W / denote the set of orthonormal sequences.w1; : : : ; wk/ in W considered as a subspace of W k. We call Vk.W / the Stiefel manifold of orthonormal k-frames in W. We have a projection p W Vk.W /! Gk.W / which sends.w1; : : : ; wk/ to the subspace Œw1; : : : ; wk spanned by this sequence. We give Gk.W / the quotient topology determined by p. The space Gk.W / can be obtained as a homogeneous space. Let W D Rn with standard inner product and standard basis e1; : : : ; en. We have a continuous action of O.n/ on Vk.Rn/ and Gk.Rn/ deﬁned by.A;.v1; : : : ; vk// 7!.Av1; : : : ; Avk/ and such that p becomes O.n/-equivariant. The isotropy groups of.e1;
: : : ; ek/ and Œe1; : : : ; ek consist of the matrices Ek.k/; B 2 O.n k/ respectively. The map A 7!.Ae1; : : : ; Aek/ induces equivariant homeomorphisms in the diagram O.n/=O.n k/ Š Vk.Rn/ O.n/=.O.k/ O.n k// Š Gk.Rn/. 1.9. Projective Spaces. Grassmann Manifolds 23 This shows that Gk.Rn/ is a compact Hausdorff space. It is called the Grassmann manifold of k-dimensional subspaces of Rn. In a similar manner we can work with the k-dimensional complex subspaces of Cn and obtain an analogous diagram of U.n/-spaces (complex Stiefel and Grassmann manifolds): U.n/=U.n k/ Š Vk.Cn/ U.n/=.U.k/ U.n k// Š Gk.Cn/. Chapter 2 The Fundamental Group In this chapter we introduce the homotopy notion and the ﬁrst of a series of algebraic invariants associated to a topological space: the fundamental group. Almost every topic of algebraic topology uses the homotopy notion. Therefore it is necessary to begin with this notion. A homotopy is a continuous family ht W X! Y of continuous maps which depends on a real parameter t 2 Œ0; 1. (One may interpret this as a “time-dependent” process.) The maps f0 and f1 are then called homotopic, and being homotopic is an equivalence relation on the set of continuous maps X! Y. This equivalence relation leads to a quotient category of the category TOP of topological spaces and continuous maps, the homotopy category h-TOP. The importance of this notion is seen from several facts. (1) The classical tools of algebraic topology are functors from a category of spaces to an algebraic category, say of abelian groups. These functors turn out to be homotopy invariant, i.e., homotopic maps have the same value under the functor. (2) One can change maps by homotopies and spaces by homotopy equivalences.
This fact allows for a great ﬂexibility. But still global geometric information is retained. Basic principles of topology are deformation and approximation. One idea of deformation is made precise by the notion of homotopy. Continuity is an ungeometric notion. So often one has to deform a continuous map into a map with better properties. (3) The homotopy notion leads in an almost tautological way to algebraic structures and categorical structures. In this chapter we learn about the simplest example, the fundamental group and the fundamental groupoid. The passage to the homotopy category is not always a suitable view-point. In general it is better to stay in the category TOP of topological spaces and continuous maps (“space level” as opposed to “homotopy level”). We thus consider homotopy as an additional structure. Then classical concepts can be generalized by using the homotopy notion. For instance one considers diagrams which are only commutative up to homotopy and the homotopies involved will be treated as additional information. One can also deﬁne generalized group objects where multiplication is only associative up to homotopy. And so on. The passage from TOP to h-TOP may be interpreted as a passage from “contin- uous mathematics” to “discrete mathematics”. The homotopy notion allows us to apply algebraic concepts to continuous maps. It is not very sensible to talk about the kernel or cokernel of a continuous map. 2.1. The Notion of Homotopy 25 But we will see later that there exist notions of “homotopy-kernels” (then called homotopy ﬁbres) and “homotopy-cokernels” (then called homotopy coﬁbres). This is the more modern view-point of a large variety of homotopy constructions. In general terms: The idea is to replace the categorical notions limit and colimit by appropriate homotopy notions. The prototype of a functor from spaces to groups is the fundamental group functor. Historically it is the ﬁrst of such functors. It was introduced by Poincaré, In general it is difﬁcult to determine the in different context and terminology. fundamental group of a space. Usually one builds up a space from simpler pieces and then one studies
the interrelation between the groups of the pieces. This uses the functorial aspect and asks for formal properties of the functor. We prove the basic theorem of Seifert and van Kampen which roughly says that the functor transforms suitable pushouts of spaces into pushouts of groups. This may not be the type of algebra the reader is used to, and it can in fact be quite complicated. We describe some related algebra (presentation of groups by generators and relations) and discuss a number of geometric results which seem plausible from our intuition but which cannot be proved (in a systematic way) without algebraic topology. The results are of the type that two given spaces are not homeomorphic – and this follows, if their fundamental groups are different. Finally we show that each group can be realized as a fundamental group (this is the origin of the idea to apply topology to group theory). The study of the fundamental group can be continued with the covering space theory where the fundamental group is exhibited as a symmetry group. This symmetry inﬂuences almost every other tool of algebraic topology (although we do not always carry out this inﬂuence in this text). The chapter contains two sections on point-set topology. We discuss standard spaces like spheres, disks, cells, simplices; they will be used in many different contexts. We present the compact-open topology on spaces of continuous maps; they will be used for the dual deﬁnition of homotopy as a continuous family of paths, and this duality will henceforth be applied to many homotopy constructions and notions. 2.1 The Notion of Homotopy A path in a topological space X from x to y is a continuous map u W Œa; b! X such that u.a/ D x and u.b/ D y. We say that the path connects the points u.a/ and u.b/. We can reparametrize and use the unit interval as a source Œ0; 1! X, t 7! u..1 t/a C tb/. In the general theory we mostly use the unit interval. If u W Œ0; 1! X is a path from x to y, then the inverse path u W t 7! w.1 t/ is a path from y to x. If v W Œ0; 1! X is another path from y to z, then the
product path u v, deﬁned by t 7! u.2t/ for t 1=2 and v.2t 1/ for t 1=2, is a path from x to z. We also have the constant path kx with value x. 26 Chapter 2. The Fundamental Group From these remarks we see that being connectible by paths is an equivalence relation on X. An equivalence class is called a path component of X. We denote by 0.X/ the set of path components and by Œx the path component of the point x. A space X is said to be path connected or 0-connected if it has one of the following equivalent properties: (1) 0.X/ consists of a single element. (2) Any two points can be joined by a path. (3) Any continuous map f W @I D f0; 1g!X has a continuous extension F W I! X. (Later we study the higher dimensional analogous problem of extending maps from the boundary @I n of the n-dimensional cube to I n.) A map f W X! Y induces 0.f / W 0.X/! 0.Y /, Œx 7! Œf.x/. In this way 0 becomes a functor from the category TOP of topological spaces to the category SET of sets1. We will see that this functor is the beginning of algebraic topology, although there is no algebra yet. Thinking in terms of categories and functors is a basic method in (algebraic) topology. The size of 0.X/ is a topological property of the space X. A functor transports isomorphisms to isomorphisms. Thus a homeomorphism f induces a bijection 0.f /. Suppose f W X! Y is a homeomorphism; then f induces a homeomorphism X XA! Y Xf.A/ for each subset A X. Suppose f W R! Rn is a homeomorphism; the space R X x has two path components (intermediate value theorem of calculus), and Rn X y is path connected for n > 1; we apply the functor 0 and conclude that R is not homeomorphic to Rn for n > 1. This example seems almost trivial, but the reasoning is typical. Here is another simple example of this type: The subspace X D R 0 [ 0 R of R2 is not homeomorphic to R
since X contains a point x D.0; 0/ such that 0.X X x/ has four elements whereas 0.R X y/ has always two elements. 2.1.1 Path categories. Forming the product path is not an associative composition. We can remedy this defect by using parameter intervals of different length. So let us consider paths of the form u W Œ0; a! X, v W Œ0; b! X with u.a/ D v.0/ and a; b 0. Their composition v ı u D w is the path Œ0; a C b! X with w.t/ D u.t/ for 0 t a and w.t/ D v.a t/ for a t a C b. In this manner we obtain a category W.X/: Objects are the points of X; a morphism from x to y is a path u W Œ0; a! X with u.0/ D x; u.a/ D y for some a 0; and composition of morphisms is as deﬁned before; the path Œ0; 0! X with value x is the identity of the object x. A continuous map f W X! Y induces a functor Þ W.f / W W.X/! W.Y /, x 7! f.x/, u 7! f u. 1Our general conventions: space = topological space, map = continuous map. A set map between spaces is a map which is not assumed to be continuous at the outset. 2.1. The Notion of Homotopy 27 A space is connected if it is not the topological sum of two non-empty subspaces. Thus X is disconnected if and only if X contains a subset X which is open, closed, and different from ; and X. A decomposition of X is a pair U; V of open, nonempty, disjoint subsets with union X. A space X is disconnected if and only if there exists a continuous surjective map f W X! f0; 1g; a decomposition is given by U D f 1.0/, V D f 1.1/. The continuous image of a connected space is connected. Recall from calculus: A R is connected if and only if A is an interval. (An interval is a subset which contains with x; y also Œx; y.) (2
.1.2) Proposition. Let.Aj j j 2 J / be a family of connected subsets of X such that Ai \ Aj 6D ; for all i; j. Then j Aj D Y is connected. Let A be connected and A B xA. Then B is connected. S The union of the connected sets in X which contain x is thus a closed connected subset. We call it the component X.x/ of x in X. If y 2 X.x/, then X.y/ D X.x/. A component of X is a maximal connected subset. Any space is the disjoint union of its components. A space is totally disconnected if its components consist of single points. Since intervals are connected a path connected space is connected. A product …j Xj is connected if each Xj is connected. The component of.xj / 2 …j Xj is the product of the components of the xj. (2.1.3) Example. The space X D Œ1; 0 0 [ 0 Œ1; 1 [ f.x; sin.x1/ j 0 < x 1g is connected but not path connected. The union S of X with f˙1g Œ2; 0 [ Œ2; 2 f2g is called the pseudo-circle. The complement R2 X S has two path components. T A pseudo-circle S has a sequence K1 K2 of compact neighbourhoods Þ i Ki D S and Ki homeomorphic to S 1 Œ0; 1. with Let X and Y be topological spaces and f; g W X! Y continuous maps. A homotopy from f to g is a continuous map H W X Œ0; 1! Y;.x; t/ 7! H.x; t/ D Ht.x/ such that f.x/ D H.x; 0/ and g.x/ D H.x; 1/ for x 2 X, i.e., f D H0 and g D H1. We denote this situation by H W f'g. One can consider a homotopy as a dynamical process, the parameter t is the time and Ht is a time-dependent family of maps. One also says that f is deformed continuously into g. Another (dual) view-point of a homotopy is: a parametrized family of paths. We use the letter
I for the unit interval Œ0; 1. If we write a homotopy in the form Ht, we understand that H W X I! Y,.x; t/ 7! Ht.x/ is continuous in both variables simultaneously. We call f and g homotopic if there exists a homotopy from f to g. (One can, of 28 Chapter 2. The Fundamental Group course, deﬁne homotopies with Œ0; 1 X. While this does not affect the theory, it does make a difference when orientations play a role.) The homotopy relation'is an equivalence relation on the set of continuous maps X! Y. Given H W f'g, the inverse homotopy H W.x; t/ 7! H.x; 1t/ shows g'f. Let K W f'g and L W g'h be given. The product homotopy K L is deﬁned by.K L/.x; t/ D ( K.x; 2t /; L.x; 2t 1/; 0 t 1 2 ; 1 2 t 1; and shows f'h. The constant homotopy H.x; t / D f.x/ shows f'f. The equivalence class of f is denoted Œf and called the homotopy class of f. We denote by ŒX; Y the set of homotopy classes Œf of maps f W X! Y. A homotopy Ht W X! Y is said to be relative to A X if the restriction Ht jA does not depend on t (is constant on A). We use the notation H W f'g (rel A) in this case. The homotopy relation is compatible with the composition of maps: Let H W f'g W X! Y and G W k'l W Y! Z be given; then.x; t/ 7! G.H.x; t/; t/ D Gt Ht.x/ is a homotopy from kf to lg. We see that topological spaces and homotopy classes of maps form a quotient category of TOP, the homotopy category h-TOP, when composition of homotopy classes is induced by composition of representing maps. If f W X! Y represents an isomorphism in h-TOP, then f is
called a homotopy In explicit terms this means: f W X! Y is a equivalence or h-equivalence. homotopy equivalence if there exists g W Y! X, a homotopy inverse of f, such that gf and fg are both homotopic to the identity. Spaces X and Y are homotopy equivalent or of the same homotopy type if there exists a homotopy equivalence X! Y. A space is contractible if it is homotopy equivalent to a point. A map f W X! Y is null homotopic if it is homotopic to a constant map; a null homotopy of f is a homotopy between f and a constant map. A null homotopy of the identity id.X/ is a contraction of the space X. 2.1.4 Categories of homotopies. We generalize (2.1.1) and deﬁne a category W.X; Y /. The objects are the continuous maps f W X! Y. A morphism from f to g is a homotopy H W X Œ0; a! Y with H0 D f and Ha D g. Composition Þ is deﬁned as in (2.1.1). As in any category we also have the Hom-functors in h-TOP. Given f W X! Y, we use the notation f W ŒZ; X! ŒZ; Y ; g 7! fg; f W ŒY; Z! ŒX; Z; h 7! hf 2.1. The Notion of Homotopy 29 for the induced maps2. The reader should recall a little reasoning with Homfunctors, as follows. The map f is an h-equivalence, i.e., an isomorphism in h-TOP if and only if f is always bijective; similarly for f. If f W X! Y has a right homotopy inverse h W Y! X, i.e., f h'id, and a left homotopy inverse g W Y! X, i.e., gf'id, then f is an h-equivalence. If two of the maps, and gf are h-equivalences, then so is the third. Q Homotopy is compatible with sums and products. Let pi W j 2J X

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