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18 1.2 Differential Topology and Cobordism Topological Manifolds. Definition 1.2.1 A subset M ⊂ R1+ × Rk = {(x0 , x1 , . . . , xk )|x0 ≥ 0, k = 1, 2, . . . , ∞}, with induced topology is called a topological manifold of dimension n < ∞ if: 1 (i) for any Rk containing x T point x ∈ M with x0 > 0 there is an open set W ⊂ R+ × n such that W M is homeomorphic to an open neighborhood of zero in R ; 1 (ii) for any Rk containing x T point x ∈ M with x0 = 0 there is an open set W ⊂ R+ × 1 such that W M is homeomorphic to an open neighborhood of zero in R+ × Rn−1 ; Exercise 1.2.1 Which letters of English alphabet are homeomorphic to a topological manifold? Remark 1.2.1 A set of points x of an n-dimensional topological manifold M which satisfy x0 = 0, forms a topological manifold ∂M of dimension n − 1. ∂M is called boundary of M and ∂ 2 (M ) = ∂(∂(M )) = ∅. We are unable to prove this statement because it rests on the following fact: Rn+ × Rm is homeomorphic to Rn+m if and only if n = 0. Which, in its turn, is a consequence of the Brauwer’s fixed point theorem and will be proved later. We shall also deduce later that the dimension of a (topological) manifold is a correctly defined notion. Example 1.2.1 An open ball Int(D n ) and a sphere S n are topological manifolds of dimension n without boundary, non-compact and compact respectively (as sets, of course). It is obvious in the case of Int(D n ), while in the case of S n the map ¶ µ q 2 2 (x1 , . . . , xn ) → x1 , . . . , xn , x1 + · · · + xn is a homeomorphism between Int(D n ) ⊂ Rn and an upper semi sphere S+n ⊂ Rn+1 , xn+1 > 0 (suitable rotations of this construction give coordinate systems around every point of S n ). In the same vein D n is a topological manifold with boundary. The embedding into R1+ × Rn required is µ ¶ q 2 2 (x1 , . . . , xn ) → 1 − x1 + · · · + xn , x1 , . . . , xn . Obviously, in this case ∂D n = S n−1 . 19 K. E. Feldman In the literature you can find a different definition of topological manifolds. The advantage of such another definition is that it allows one to study some properties of a manifold internally, without referring to the Euclidean space it is supposed to be embedded into by Definition 1.2.1 Definition 1.2.2 A topological manifold M n of dimension n is a topological space which (i) has countable base (ii) is Hausdorff and such that every point of x ∈ M n has a neighborhood U homeomorphic to an open subset of R1+ × Rn−1 . If we denote this homeomorphism by h: h : U → V ⊂ R1+ × Rn−1 , and projections onto coordinate axes in R1+ ×Rn−1 by r1 , . . . , rn , then compositions xi = ri ◦h are coordinate functions on U and (U, x1 , . . . , xn ) is a coordinate system in U . Because every subset of Rn is second countable and Hausdorff, a topological manifold in the sense of Definition 1.2.1 is also a topological manifold in the sense of Definition 1.2.2. The converse is also true. Let us discuss two different approaches to the proof of this statement. Theorem 1.2.1 Every compact topological manifold in the sense of Definition 1.2.2 is homeomorphic to a subset of R1+ × Rk . The latter subset must be a topological manifold in the sense of Definition 1.2.1. Proof. Let n be the dimension of M . We can cover every point x of our manifold by an open set Ux homeomorphic to an open ball B (0, rx ) around zero in R1+ × Rn−1 . Denote the corresponding homeomorphism by hx : hx : Ux → B (0, rx ) . For each ball B (0, rx ) we define a continuous function 1, |y| < r3x rx 2 − 3|y|/rx , 3 ≤ |y| < 2r3x fx (y) = 2rx 0, ≤ |y| < rx 3 We can extend each fx as a continuous function on all M , by defining it to be equal to zero outside Ux . Because M is compact, there is a finite collection I of α ∈ M such that [ h−1 α (B (0, rα /3)) α∈I 20 Differential Topology and Cobordism ¡ ¡ ¢¢ covers every point of M . Let us denote these α by α1 , . . . , αm , the sets h−1 αj B 0, rαj /3 by Vj , j = 1, . . . , m, and the corresponding extended functions fα by f1 , . . . , fm . Consider a map f : M → R(m+1)n : f (x) = (f1 (h1 (x)) · h1 (x), f1 (h1 (x)), f2 (h2 (x)) · h2 (x), f2 (h2 (x)), . . . , . . . fm (hm (x)) · hm (x), fm (hm (x))) , where we assume that f1 (h1 (x)) · hj (x) = 0 ∈ Rmn if hj (x) is not defined. Let us check that the map f : M → R(m+1)n is an injection. Indeed for any x 6= y, either x, y ∈ Vj for some j = 1, . . . , m, or x ∈ Vj and y ∈ / Vj . In the first case fj (hj (x)) = fj (hj (y)) = 1 and hj (x) 6= hj (y) because hj is a homeomorphism on Vj . Therefore, fj (hj (x)) · hj (x) 6= fj (hj (y)) · hj (y) and f (x) 6= f (y). In the second case 1 = fj (hj (x)) > fj (hj (y)) and again f (x) 6= f (y). A continuous injection of a compact space into a Hausdorff space is a homeomorphism with the image. If there are boundary points, then the image of f belongs to a subset of type Rr+ × Rmn+n−r for some r > 0 with the boundary points having at least one coordinate from Rr+ equal to zero. In that case we identify Rr+ × Rmn+n−r with R1+ × Rmn+n−1 by means of a homeomorphism G and the image of M under G ◦ f is a topological manifold with boundary in the sense of Definition 1.2.1. Now we consider an alternative way. Definition 1.2.3 An order of an open cover U = {Uα | α ∈ I} of a topological space (X, T ) is the maximum (if it exists) over all points of X of the number of elements of U which contain the given point of X. Definition 1.2.4 A space has Lebesgue covering dimension m if for every open cover of that space, there is an open cover that refines it such that the refinement has order at most m+1. If a space does not have Lebesgue covering dimension m for any m, it is said to be infinite dimensional. Theorem 1.2.2 A compact metrizable space of Lebesgue dimension ≤ m is homeomorphic to a subset of a unit cube I2m+1 = I1 × · · · × I1 , {z } | 2m+1 where I1 = [0, 1]. Moreover, the set of all continuous bijections Homeo(X, I2n+1 ) is dense in the set of all continuous maps C 0 (X, I2n+1 ). Proof. Obviously, the first statement of the theorem is an easy consequence of the second. We, thus, will be working with the second part only and start the proof with one useful definition: 21 K. E. Feldman Definition 1.2.5 Let X be a compact metric space and ² > 0. We say that a continuous map g : X → Y is an ²-map if the pre image g −1 (y) of any point y ∈ g(X) ⊂ Y has diameter less than ². Lemma 1.2.1 If g : X → Y is an 1/i-map for any i = 1, 2, . . . , then g : X → g(X) is a homeomorphism. The converse is also true. Indeed, g must be a bijection and a continuous bijection of a compact space to a Hausdorff space is a homeomorphism with the image. The converse is obvious. Lemma 1.2.2 Let (X, ρ) be a compact metric space. Then for a metric space (Y, d) and any ² > 0 the set G² of all ²-maps is open in the set C 0 (X, Y ) of all continuous maps from X to Y . Let g be an ²-map. We define η = inf {d(g(x), g(x0 )) | (x, x0 ) ∈ X × X : ρ(x, x0 ) ≥ ²} where the infinum is taken over all pairs (x, x0 ) ∈ X × X with ρ(x, x0 ) ≥ ². Because X is compact there is a pair of points (x, x0 ) such that d(x, x0 ) ≥ ², and η = d(g(x), g(x0 )). Because g is an ²-map we must have η > 0. Let f : X → Y be a map on a distance less than η/2 from g: 1 d(f (x), g(x)) ≤ η, ∀x ∈ X. 2 If f (u) = f (v) for some u, v ∈ X, then d(g(u), g(v)) ≤ d(g(u), f (u)) + d(f (u), f (v)) + d(f (v), g(v)) ≤ 1 1 η + 0 + η = η. 2 2 Thus, f is an ²-map. Lemma 1.2.3 Let (X, ρ) be a compact metric set of Lebesgue dimension ≤ n. Then for any ² > 0 the set G² of all ²-maps is dense in the set C 0 (X, I2n+1 ) of all continuous maps X → I2n+1 . Proof. Consider f ∈ C 0 (X, I2n+1 ) and η > 0. Let us denote the metric on I2n+1 by d and construct g ∈ C 0 (X, I2n+1 ) such that supx∈X d(f (x), g(x)) < η, and g ∈ G² . (1.1) 22 Differential Topology and Cobordism On a compact set X a continuous function f is uniformly continuous. Thus, there is δ < ² such that for all x, y ∈ X with ρ(x, y) < δ: 1 d(f (x), f (y)) < η. 2 Consider a cover of X by open sets of diameter less than δ and pick a finite refinement U1 , . . . , Ur of order ≤ n. Let us pick points which are in general position in R p1 , . . . , pr ∈ I2n+1 2n+1 and 1 d(pi , f (Ui )) < η. 2 For any point x ∈ X we associate r weights: wi (x) = ρ(x, Xi \ Ui ), i = 1, . . . , r. Obviously, wi (x) > 0 for all x ∈ Ui and wi (x) = 0 for all x 6∈ Ui . Every point x ∈ X has at least one non zero weight because U1 , . . . , Ur cover the whole X. To each point x ∈ X we associate a point g(x) ∈ I2n+1 which represents the centrum of masses of points p1 , . . . , pr with weights w1 (x), . . . , wr (x). It is easy to see that g : X → I2n+1 is continuous. Let us check that it satisfies (1.1). For a given point x we can relabel sets U1 , . . . , Ur , so that for some s we have that wi (x) > 0 if i ≤ s and wi (x) = 0 for i > s. Because x ∈ Ui for i ≤ s we conclude that d(pi , f (x)) < η, ∀i ≤ s. From here it follows that the centrum of masses g(x) of points p1 , . . . , ps with weights wi (x), i = 1, . . . s, also satisfies d(g(x), f (x)) < η. For the second part we consider all sets Ui1 , . . . , Uis , which contain x ∈ X. Because order of the cover is ≤ n we must have s ≤ n + 1. Consider (s − 1)-dimensional subspace L(x) ∈ R2n+1 which contains p i1 , . . . , p is . 23 K. E. Feldman Obviously, g(x) ∈ L(x). Let x0 be some other point of X and assume that L(x0 ) passes through pj1 , . . . , pjt , t ≤ n + 1. T If L(x) L(x0 ) 6= ∅ then the minimal affine subspace V of R2n+1 which contains both L(x) and L(x0 ) has dimension dim V ≤ dim L(x) + dim L(x0 ) = s + t − 2 ≤ 2n. Because p1 , . . . , pr are in general position it follows that at least one point pi appears in both lists: pi1 , . . . , pis , and pj1 , . . . , pjt . T Thus, pi ∈ L(x) L(x0 ). Assume that g(x) = g(x0 ). Then L(x) and L(x0 ) have non empty intersection which also contains some point pi : \ i = {i1 , . . . , is } {j1 , . . . , jt }. But then x, x0 ∈ Ui and, therefore, ρ(x, x0 ) < δ < ². Thus, (1.1) is satisfied and Lemma 1.2.3 is verified. To finish the proof of the main theorem we observe that the set H= ∞ \ G1/i i=1 is dense in C 0 (X, I2n+1 ) due to the following exercises. Exercise 1.2.2 Check that for any topological space X the set C 0 (X, I2n+1 ) is a complete metric space with respect to the metric d(f, g) = supx∈X |f (x) − g(x)|. Exercise 1.2.3 Intersection of countably many open sets which are dense in a complete metric space Y , is dense in Y . Finally, according to Lemma 1.2.1 the set H consists of continuous bijections. The proof of Theorem 1.2.2 is now completed. 24 Differential Topology and Cobordism Definition 1.2.6 A simplicial complex is a subset K of RN which can be represented as a union of closed simplexes ∆(p1 , p2 , . . . , pm ) = {w1 p1 + · · · + wm pm | w1 + · · · + wm = 1, wj ≥ 0, j = 1, . . . , m}, where p1 , . . . , pm ∈ RN are in general position, and such that if ∆(p1 , p2 , . . . , pm ) ⊂ K, ∆(q1 , q2 , . . . , qn ) ⊂ K and \ ∆(p1 , p2 , . . . , pm ) ∆(q1 , q2 , . . . , qn ) 6= ∅ then ∆(p1 , p2 , . . . , pm ) for some \ ∆(q1 , q2 , . . . , qn ) = ∆(r1 , r2 , . . . , rk ) {r1 , r2 , . . . , rk } = {p1 , p2 , . . . , pm } \ {q1 , q2 , . . . , qn } Definition 1.2.7 Let X be a compact space and U its finite open cover. A nerve of U is a simplicial complex K(U) which has vertexes pU1 , pU2 , . . . labeled by elements of U and such that ¡ ¢ ∆ pUi1 , . . . , pUim ⊂ K(U) if and only if Ui1 \ ··· \ Uim 6= ∅. Corollary 1.2.1 For any η, a compact metric space (X, ρ) of Lebesgue dimension ≤ n and a continuous bijection f : X → I2n+1 there is a simplicial complex K = K(η) ∈ I2n+1 such that ∀x ∈ X supy∈K d(f (x), y) < η. The corresponding K(η) can be chosen as a nerve of a cover of X with a diameter less than δ, where δ is chosen in such way that if ρ(x, y) ≤ δ then 1 d(f (x), f (y)) < η. 2 To apply Theorem 1.2.2 in order to prove Theorem 1.2.1 we need the following result. Lemma 1.2.4 A topological manifold in the sense of Definitions 1.2.1,1.2.2 and in particular Rn has Lebesgue dimension n. Proof. The proof rests on Brauwer’s fixed point theorem (Theorem ??) and, therefore, will be given later. 25 K. E. Feldman Remark 1.2.2 The second proof can be generalized to the case of non-compact manifolds. Roughly speaking this is done by replacement of the notion ²-map on the corresponding notion given in terms of covers. For a cover U of a topological space X a map f : X → Y is an U-map if for any point y ∈ Y there is an open neighborhood V ∈ Y , y ∈ V such that f −1 (V ) is contained in some element of U. Using this notion and the same scheme as in the proof of Theorem 1.2.2 we can deduce the same result for non compact spaces of Lebesgue dimension n. Example 1.2.2 Real projective space RP n is a topological manifold. Indeed, pick a point l ∈ RP n . It represents a line l ⊂ Rn+1 passing through zero. We introduce a positive definite inner product on Rn , for example, the standard one: hx, yi = n+1 X xi yi . i=1 Let Ul ⊂ RP n be a set of all lines which are not orthogonal to l. We also introduce an orthogonal complement αl to l in Rn+1 and two orthogonal projections: prl : Rn+1 → l, prα : Rn+1 → αl . Then for any l0 ∈ Ul the map prl : l0 → l is an isomorphism of one-dimensional real vector spaces. Therefore, we can define a map ¯ Ll (l0 ) = prα ◦ prl ¯−1 l 0 : l → αl . Obviously, Ll (l0 ) ∈ Hom(l, αl ) ∼ = Rn . Introducing any basis into l and α we obtain a bijection n between Ul and R . In fact this bijection is a homeomorphism. To see this we have to check that the map n S n ⊃ pr−1 S 0 (Ul ) → Ul → R , is continuous when prS 0 : S n → RP n is the antipode quotient. But this is obvious, as two sufficiently close points on S n define arbitrarily close linear functional Ll (l0 ) ∈ Hom(l, αl ). Thus, RP n is a topological manifold. Exercise 1.2.4 Prove similarly, that CP n , HP n are topological manifolds. Recall that a topological space (X, T ) is connected if every its closed–open subset, is either ∅ or X. A topological space (X, T ) is path–connected if for any two points x, y ∈ X there is a continuous map f of a unit interval I = [0, 1] ⊂ R1 into X such that f (0) = x and f (1) = y. 26 Differential Topology and Cobordism Exercise 1.2.5 Give an example of a connected but not a path–connected space. Exercise 1.2.6 Verify that a topological manifold is connected if and only if it is path– connected. Example 1.2.3 A Cartesian product of two topological manifolds M1m1 and M2m2 is a manifold of dimension m1 + m2 represented by a Cartesian product of sets M1m1 × M2m2 . If U is a neighborhood of x in M1m1 homeomorphic to an open subset of R1+ × Rm1 −1 and V is a neighborhood of y in M2m2 homeomorphic to an open subset of R1+ × Rm2 −1 , then U × V is a neighborhood of (x, y) in M1m1 × M2m2 homeomorphic to an open subset R2+ × Rm1 +m2 −2 . From a homeomorphism of spaces R2+ × Rm1 +m2 −2 ≈ R1+ × Rm1 +m2 −1 we deduce that the Cartesian product is in fact a topological manifold. Exercise 1.2.7 Let M ⊂ Rk be a topological manifold without boundary. Prove directly, that M × I, where I = [0, 1] is a closed interval, is a topological manifold realizable as a F subset of R1+ × Rk in such a way, that the boundary M × {0} M × {1} belongs to {0} × Rk ⊂ R1+ × Rk . Definition 1.2.8 A differential structure F of class C r , r = 1, . . . , ∞ on a topological manifold M is a set of coordinate systems F = {(Uα , φα ) : α ∈ A} which satisfies to the following three properties S (i) α∈A Uα = ¯ M; −1 ¯ (ii) φα ◦ φβ φβ (Uα T Uβ ) is of class C r for any α, β ∈ A; ¯ (iii) F is¯ maximal, i.e., if (U, φ) is a coordinate system on M such that φ α ◦ φ−1 ¯φ(Uα T U ) ¯φ (U T U ) are of class C r for any α, β ∈ A then (U, φ) ∈ F. and φ ◦ φ−1 β β β Exercise 1.2.8 Prove, that the collection of all sets Ul ⊂ RP n , where l is any line in Rn+1 , together with maps Ll : Ul → αl = l⊥ , satisfies to the conditions (i) and (ii) of Definition 1.2.8 for any r > 0. Check that the same happens to CP n and HP n . K. E. Feldman 27 Theorem 1.2.3 There is a unique differential structure on kP n (k = R, C, H) which contains coordinate systems (Ul , Ll ) for any line l ∈ k n+1 . Remark 1.2.3 We shall call it as the canonical differential structure on kP n (k = R, C, H). T Proof. Consider two coordinate systems (V, φ) and (W, ψ) on M such that V W 6= ∅. Assume that for any l ∈ kP n maps ¯ ¯ \ \ ¯ ¯ −1 ¯ −1 ¯ T ¯L (V T U ) , Ll ◦ φ−1 ¯¯φ (V (W Ul ) , U ) , ψ ◦ L , L ◦ ψ φ ◦ L−1 ¯ ψ l l L (W U ) l l l l l l T T are C r -smooth. Chose a line l ∈ V W ⊂ kP n . For a small neighborhood Z ⊂ V W is defined in Ll (Z) and the map Ll ◦ ψ −1 is defined in ψ(Z). The of l the map φ ◦ L−1 l composition of these maps is of class C r and, therefore, the composition φ ◦ ψ −1 is of class T r C in some neighborhood of φ(l). Because T the choice of l ∈ V W is arbitrary, the map φ ◦ ψ −1 is C r -smooth everywhere in V W . Varying (V, φ) we deduce that (W, ψ) belongs to the same differential structure on kP n as all (Ul , Ll ). Exercise 1.2.9 Check that there is a unique differential structure F1 × F2 on a Cartesian product of topological manifolds M1 and M2 which contains all products (Uα × Vα , φα × φβ ), where (Uα , φα ) ∈ F1 and (Vβ , φβ ) ∈ F2 with F1 and F2 to be some differential structures on M1 and M2 respectively. Example 1.2.4 Grassmann manifold (or Grassmannian) kGnm (k = R, C, H) is define as a set of all m-dimensional planes in n-dimensional vector space over k. It is a topological manifold with a differential structure introduced by means of a collection of such open neighborhoods Uα of points α ∈ kGnm which consist of all m-dimensional subspaces in k n with isomorphic orthogonal projection prα onto α. The coordinates are introduced by means of identification: ¯ ¢ ¡ ⊥ hα (β) = prα⊥ ◦ prα ¯−1 β ∈ Hom α, α ? ? Exercise 1.2.10 Check that this definition is correct, i.e., that hα0 ◦ h−1 α : R → R is a C ∞ -diffeomorphism where it is defined, and that there is a unique differential structure which contains all (Uα , hα ). What is the dimension of kGnm ? Exercise 1.2.11 Prove that Grassmann manifold kGm k is compact. Definition 1.2.9 A C r -manifold is a topological manifold with a C r -differential structure. 28 Differential Topology and Cobordism Problems Problem 1.2.1 Prove that homeomorphic spaces have the same Lebesgue dimension. Problem 1.2.2 Let ∆2n+2 be a simplex of dimension 2n + 2 and let Pn ⊂ ∆n be a subset of all faces of ∆2n+2 of dimension ≤ n. Prove, that it is not homeomorphic to a subset of R2n . (This is an example of a topological space of Lebesgue dimension ≤ n (will be proved later) which cannot be realized as a subset of 2n-dimensional Euclidean space. Thus, Theorem 1.2.2 is sharp.) Problem 1.2.3 Let F be the standard differential structure on R1 , i.e., the structure which contains the identity map id : R1 → R1 . Prove that there is another differential structure F 0 on R1 which contains the map t → t3 and F 6= F 0 . Problem 1.2.4 Let Z2 acts on S 2 × S 2 by simultaneous involutions (central symmetries) and consider another action of Z2 on S 2 × S 2 which interchanges the factors. Prove that the set of orbits of the Cartesian product of these two actions on S 2 × S 2 with factor topology is homeomorphic to S 4 .