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Home , Complex projective space, Projection (mathematics)

Projective spaces

Kim A. Frøyshov

October 5, 2019

1 Real projective spaces

× Let R be the multiplicative of non-zero real numbers. For n ≥ 0 n+1 we introduce an in R − {0} by declaring that x ∼ y × n if and only if x = ty for some t ∈ R . The quotient space RP is called real projective n–space. The of a point (x0, . . . , xn) in n+1 R − {0} will be denoted by [x0, . . . , xn]. Note that there is a bijective n n+1 correspondence between RP and the of lines in R through the origin. n We are going to show that RP is a compact connected n–. Lemma 1.1 The projection

n+1 n π : R − {0} → RP is an open .

n+1 × Proof. Let U be an open subset of R − {0}. For every t ∈ R the set tU := {tx | x ∈ U}

n+1 is an open subset of R − {0}. Therefore, [ π−1π(U) = tU t∈R× n is open. Hence, π(U) is open in RP by definition of the quotient .

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 {f(U) | U ∈ B} is a basis for the topology on Y .

1 Proof. Let V ⊂ Y be open and y ∈ V . Choose x ∈ f −1(y). Since f −1(V ) is open, there is a basis element U ∈ B such that x ∈ U ⊂ f −1(V ). It follows that y ∈ 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 − {0} is second countable.

n Lemma 1.4 RP is Hausdorff. n+1 Proof. Let x, y be points in R − {0} 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 := {z ∈ R − {0} | |Sz| > |T z|}, n+1 B := {z ∈ R − {0} | |Sz| < |T z|}. 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 := {[x0, . . . , xn] ∈ RP | xj 6= 0}. n n Then {U0,...,Un} 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 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- .

1 1 1 Example RP is homeomorphic to S . To see this, we regard S as the unit in the 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 × {0} ⊂ R . The inclu- n n+1 sion map R − {0} → R − {0} induces an injective continuous map n−1 n g : RP → RP . n n+1 Because R −{0} is a closed subset of R −{0}, 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 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 −{0} obtained by identifying two points w, z whenever w = tz for some non-zero 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 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 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 := {[z0, z1] ∈ CP | z1 6= 0}.

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, provides a homeomorphism 2 2 τ : S − {N} → 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 − {N} 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.

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