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CW Complexes and the Projective Space Omar Ortiz Department of Mathematics and Statistics University of Melbourne Parkville, VIC 3010 Australia [email protected] August 14, 2012 1 CW Complexes A CW complex is a topological space X constructed inductively with the following data: 1. The 0-skeleton X 0 is a discrete set. The points of this set are the 0-cells. 2. The n-skeleton X n is formed inductively from X n−1 by attaching open disks of euclidean dimension n, or n-cells, enα via maps ϕα : Sn−1 −→ X n−1 . The map ϕα identifies the boundary of the (closed) Dαn = enα with a subset of X n−1 , so that F disk n n−1 n X is the quotient space of X α Dα under the identifications x ∼ ϕα (x) for n n n x ∈ ∂Dα = eα . Thus as a set, X = X n−1 tα enα . 3. If the process at some n ∈ N then the CW complex is X = X n , if not, it is the S ends union X = n X n . In the latter case, X is given the weak topology: A set A ⊆ X is open (closed) if and only if A ∩ X n is open (closed) in X n for each n. 1.1 Examples 1. The Torus S1 × S1 is a CW complex with one 0-cell, two 1-cells and one 2-cell. 2. The orientable surface Mg of genus g is a CW complex with one 0-cell, 2g 2-cells and one 2-cell. 3. The complex projective space CP n is a CW complex as below. 1 2 Alternative Definition Let us first recall some basic definitions from topology. Let X be a topological space and S ⊆ X a subset of X. The interior S̊ of S is the union of all open sets contained in S. The closure S of S is the intersection of all closed sets containing S. The boundary of S is δS = S \ S̊. A CW-complex X is given by the following data: {(X n , {ϕnα }α∈Λn )}n∈N ⊆N where X 0 is a discrete set, Λ0 = ∅, Dαn ϕ : δDαn −→ X n−1 is a continuous map from the boundary δDαn ∼ = Sn−1 of the n-disk n n−1 ∼ , and = {x ∈ R | ||x|| ≤ 1} to X F X n−1 α Dαn n . X = x ∼ ϕnα (x) An n-cell enα is the interiorFD̊αn of an n-disk Dαn , i.e. enα = D̊αn = {x ∈ Rn | ||x|| < 1}. Note that as a set X n = X n−1 α enα . If N = {0, 1, . . . , n}, then X = X n is a finite dimensional CW-complex of dimension n. If in addition Λk is finite for all k ∈ N then X is a finite CW-complex. S If N = N, then X = n X n is an infinite dimensional CW-complex. In this case X is given the weak topology: A set A ⊆ X is open (closed) if and only if A ∩ X n is open (closed) in X n for each n. 3 Complex Projective Space The Complex Projective Space CP n is the space of 1-dimensional vector subspaces of Cn+1 , i.e. the space of complex lines through the origin in Cn+1 . That is, CP n = Cn+1 − {0}/ ∼ where z ∼ w ⇔ z = λw, for z = (z0 , . . . , zn ) , w = (w0 , . . . , wn ) ∈ Cn+1 , λ ∈ C − {0}. Or equivalently, CP n = S2n+1 / ∼ where z ∼ w ⇔ z = λw, 2 λ ∈ C such that |λ| = 1, and S2n+1 = {x ∈ R2n+2 ∼ = Cn+1 | ||x|| = 1} is the 2n + 1-sphere. Remark: these definitions are equivalent by the identification Cn+1 − {0}/ ∼ / S2n+1 / ∼ [z] / [z/kzk]. This is the same as regarding CP n as the orbit set of the (free) action of S1 on S2n+1 S1 × S2n+1 / S2n+1 (λ, z) / λz, i.e. CP n = S2n+1 /S1 . Remark: Since the S1 action on S2n+1 is free, then the orbit set is a manifold. 4 Infinite Complex Projective Space We have inclusions CP n−1 [z0 , . . . , zn ] / CP n . / [z0 , . . . , zn , 0] The infinite complex projective space is constructed as the union of all the finite complex projective spaces: [ CP ∞ = CP n n≥1 5 Topology CP n , with the quotient topology induced by the projection p : Cn+1 − {0} −→ Cn+1 − {0}/ ∼ = CP n , is second countable and locally compact since p is a surjective, continuous and open map, and thus preserves these properties from Cn+1 − {0}. 3 Furthermore, CP n is compact and connected because it is the image of S2n+1 through the composite map p i π : S2n+1 −→ Cn+1 − {0} −→ CP n , where i : S2n+1 ,→ R2n+2 ∼ = Cn+1 is the inclusion. It is readily verifiable that CP n is also a Hausdorff space. Using this topology and the inclusion maps CP n−1 ,→ CP n we can endow CP ∞ with the final topology. 6 CW structure The n-complex projective space has a cell structure consisting of one cell in each even dimension up to 2n CP n = e0 ∪ e2 ∪ · · · ∪ e2n where the k th cell is attached to the (k − 1)-skeleton via the quotient map S2k−1 −→ CP k−1 . To see this, note that it is also possible to obtain CP n as the quotient space of the (closed) disk D2n under the identifications v ∼ λv for v ∈ ∂D2n = S2n−1 . But S2n−1 modulo this relation is CP n−1 , so we are obtaining CP n by attaching a cell e2n to CP n−1 via the quotient map S2n−1 −→ CP n−1 . The cellular decomposition follows by induction. Similarly CP ∞ has a cell structure with one cell in each even dimension. 7 Homotopy groups Recall the homotopy groups for S1 , 1 πk (S ) = Z , if k = 1; 0 , if k 6= 1; and that πk (Sn ) = 0 if k < n. The long exact sequence of homotopy groups induced by the fiber bundle S1 ,→ S2n+1 −→ n CP is · · · −→ πk (S1 ) −→ πk (S2n+1 ) −→ πk (CP n ) −→ πk−1 (S1 ) −→ · · · When k 6= 1, 2, it follows that πk (CP n ) ∼ = πk (S2n+1 ). 4 For k = 1 we have 0 = π1 (S2n+1 ) −→ π1 (CP n ) −→ π0 (S1 ) −→ π0 (S2n+1 ) = 0, so, π1 (CP n ) ∼ = π0 (S1 ) = 0. Similarly, k = 2 part of the sequence: 0 = π2 (S2n+1 ) −→ π2 (CP n ) −→ π1 (S1 ) −→ π1 (S2n+1 ) = 0, implies π2 (CP n ) ∼ = π1 (S1 ) = Z. Therefore if k = 0, 1; 0, Z, if k = 2; πk (CP n ) = 2n−1 πk (S ) , if k > 2. 8 Cellular Homology The cellular chain complex for CP n is ··· dk+2 / Ck+1 dk+1 / Ck dk / Ck−1 dk−1 / ··· where Ck is the free abelian group with basis the k-cells of CP n . This is, Z{e2n } d2n / 0 d2n−1 / Z{e2n−2 } d2n−2 / 0 dn−1 / · · · 0 d1 / Z{e0 } . This shows that all the cellular boundary maps dk in this chain complex are zero. Then the cellular homology groups are free abelians with basis in one-to-one correspondence with the k-cells. This is Z , if 0 ≤ k ≤ 2n and k even; n Hk (CP ) = 0 , other case. 9 Smooth Structure Let Ui = {[z0 , . . . , zn ] ∈ CP n : zi 6= 0}, and 5 n φi : Ui −→ C : [z0 , . . . , zn ] 7−→ zi−1 zi+1 zn z0 ,..., , ,..., zi zi zi zi . Then A = {(Ui , φi )}ni=1 is an atlas for CP n , showing that CP n has complex dimension n (or real dimension 2n). Remark: Depending on the nature of the functions considered in the atlas (smooth, analytic, polinomial, etc.) we can regard CP n as a smooth manifold, complex manifold, algebraic variety, etc. 10 de Rham Cohomology Theorem 10.1 The de Rham Cohomology of CP n is: R , if 0 ≤ k ≤ 2n and k even; k n HDR (CP ) = 0 , other case. References [Hat] A. Hatcher. Algebraic Topology. Cambridge University Press. Cambridge, 2002. [Ma] Ib Madsen & J. Tornehave. From Calculus to Cohomology. De Rham cohomology and characteristic classes. Chapter 14. Cambridge University Press. Cambridge, 1997. [Sh] R.W. Sharpe. Differential Geometry. Cartan’s Generalization of Klein’s Erlangen Program. Section 1.1. Graduate Texts in Mathematics 166, Springer-Verlag. New York, 1997. 6