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Projective Spaces Projective spaces Kim A. Frøyshov October 5, 2019 1 Real projective spaces × Let R be the multiplicative group of non-zero real numbers. For n ≥ 0 n+1 we introduce an equivalence relation in R − f0g by declaring that x ∼ y × n if and only if x = ty for some t 2 R . The quotient space RP is called real projective n{space. The equivalence class of a point (x0; : : : ; xn) in n+1 R − f0g will be denoted by [x0; : : : ; xn]. Note that there is a bijective n n+1 correspondence between RP and the set of lines in R through the origin. n We are going to show that RP is a compact connected n{manifold. Lemma 1.1 The projection n+1 n π : R − f0g ! RP is an open map. n+1 × Proof. Let U be an open subset of R − f0g. For every t 2 R the set tU := ftx j x 2 Ug n+1 is an open subset of R − f0g. Therefore, [ π−1π(U) = tU t2R× n is open. Hence, π(U) is open in RP by definition of the quotient topology. Lemma 1.2 Let f : X ! Y be a surjective, open, and continuous map between topological spaces. Let B be a basis for the topology on X. Then ff(U) j U 2 Bg is a basis for the topology on Y . 1 Proof. Let V ⊂ Y be open and y 2 V . Choose x 2 f −1(y). Since f −1(V ) is open, there is a basis element U 2 B such that x 2 U ⊂ f −1(V ): It follows that y 2 f(U) ⊂ V: n Lemma 1.3 RP is second countable. n+1 Proof. This follows from Lemmas 1.1 and 1.2 since R − f0g is second countable. n Lemma 1.4 RP is Hausdorff. n+1 Proof. Let x; y be points in R − f0g representing distinct points in n RP . Then x; y are linearly independent, so there are linear maps S; T : n+1 R ! R such that Sx = 1 = T y; Sy = 0 = T x: Let n+1 A := fz 2 R − f0g j jSzj > jT zjg; n+1 B := fz 2 R − f0g j jSzj < jT zjg: Then A; B are disjoint neighbourhoods of x; y. Since A and B are both saturated, their images π(A) and π(B) are disjoint neighbourhoods of π(x) and π(y). n Theorem 1.1 RP is an n{manifold. Proof. For j = 0; : : : ; n let n Uj := f[x0; : : : ; xn] 2 RP j xj 6= 0g: n n Then fU0;:::;Ung is an open covering of RP . Let φj : Uj ! R be defined by x0 cxj xn φj([x0; : : : ; xn]) := ;:::; ;:::; ; xj xj xj where the b indicates a term that is to be omitted. Using Lemma 1.1 it is easy to see that φj is continuous. In fact, φj is a homeomorphism, the n inverse map R ! Uj being given by −1 φj (y1; : : : ; yn) = [y1; : : : ; yj; 1; yj+1; : : : ; yn]: 2 n This shows that every point in RP has a neighbourhood homeomorphic n n to R . Since we have already verified that RP is second countable and n Hausdorff, it follows that RP is an n{manifold. n Theorem 1.2 RP is compact and connected. 0 Proof. For n = 0 the statement is trivial since RP consists of just one point. For n ≥ 1 observe that the projection map n n S ! RP is surjective and continuous. Since Sn is compact and connected, it follows n that RP has the same properties. n n Theorem 1.3 RP is homeomorphic to the quotient space S =±1 obtained by identifying antipodal points in Sn. n n n Proof. Let ρ : S ! RP be the projection. Two point x; y in S have the same image under ρ precisely when x = ±y. Therefore, ρ induces a bijective and continuous map n n f : S = ± 1 ! RP : n n Because S = ± 1 is compact and RP is Hausdorff, the map f is a homeo- morphism. 1 1 1 Example RP is homeomorphic to S . To see this, we regard S as the unit circle in the complex plane and consider the map f : S1 ! S1; z 7! z2: Arguing as in the proof of Theorem 1.3 we find that f descends to a home- omorphism S1= ± 1 ! S1. Theorem 1.4 For n ≥ 1 the map n−1 n RP ! RP ; [x0; : : : ; xn−1] 7! [x0; : : : ; xn−1; 0] is an embedding. n n n+1 Proof. It is convenient to identify R with R × f0g ⊂ R . The inclu- n n+1 sion map R − f0g ! R − f0g induces an injective continuous map n−1 n g : RP ! RP : n n+1 Because R −f0g is a closed subset of R −f0g, the map g is closed, hence an embedding. n−1 n Henceforth, we identify RP with its image in RP . 3 n n−1 n Theorem 1.5 The complement RP − RP is homeomorphic to R . Proof. The complement is just the domain of the homeomorphism φn : n Un ! R constructed in the proof of Theorem 1.1. 2 Complex projective spaces n n The complex projective space CP is defined in the same way as RP , except that the real numbers R are replaced with the complex numbers C. To be n n+1 explicit, CP is the quotient of C −f0g obtained by identifying two points w; z whenever w = tz for some non-zero complex number t. Arguing as n before we find that CP is a (2n){manifold. The reason why the dimension n is 2n is that the homeomorphisms φj now take values in C , which we 2n identify with R as a real vector space via the map (x1 + iy1; : : : ; xn + iyn) 7! (x1; y1; : : : ; xn; yn): The projection 2n+1 n S ! CP n is continuous and surjective, hence CP is compact and connected. Taking n = 1 we get the Hopf map 3 1 S ! CP : 1 The space CP is known as the Riemann sphere because of the following result. 1 2 Theorem 2.1 CP is homeomorphic to S . Proof. As in the proof of Theorem 1.1 let 1 U1 := f[z0; z1] 2 CP j z1 6= 0g: The complement of U1 consists only of the point [1; 0], and we have a home- omorphism φ1 : U1 ! C; [z0; z1] 7! z0=z1: On the other hand, stereographic projection provides a homeomorphism 2 2 τ : S − fNg ! R ; 1 2 where N is the \north pole". Since CP and S are compact Hausdorff spaces, the composite homeomorphism −1 2 τ ◦ φ1 : U1 ! S − fNg 1 2 extends to a homeomorphism CP ! S (see [1, Theorem 29.1]). 4 References [1] J. R. Munkres. Topology. Prentice Hall, 2nd edition, 2000. 5.
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