Download CW Complexes and the Projective Space

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