Math 6510 Homework, due Nov 27, 2012.
(1) In this problem we will use Poincaré-Hopf to prove that χ(CP n ) = n + 1.
Consider the action of the circle S 1 = {λ ∈ C | |λ| = 1} on CP n given in
homogeneous coordinates by
λ · [x0 : x1 : · · · : xn ] = [x0 : λx1 : λ2 x2 : · · · : λn xn ]
and let v be the associated vector field that to a point x assigns the velocity
vector of the curve λ 7→ λ · x at λ = 1 ∈ S 1 .
(a) By working in the standard charts Ui = {xi 6= 0} show that there are
exactly (n + 1) points where v vanishes, and identify these points.
(b) Find the explicit formula for v in each chart Ui .
(c) Show that the index of v at each zero is 1.
(d) Deduce that χ(CP n ) = n + 1. For n = 1 we have CP 1 = S 2 . Picture
the circle action and the vector field.
(2) Compute the Lefschetz number of the mapping
fk : CP n → CP n
given by
[x0 : x1 : · · · : xn ] 7→ [xk0 : xk1 : · · · : xkn ]
where k = 2, 3, · · ·. What happens when k = 1? Hint: Find the fixed points
and compute the derivative.
(3) In this problem we will compute de Rham cohomology of CP n . We will
i
show that HDR
(CP n ) = R if 0 ≤ i ≤ 2n with i even, and 0 otherwise.
When n = 1 we proved this in class since CP 1 = S 2 . Now use induction on
n.
(a) Show that CP n − {[0 : 0 : · · · : 0 : 1]} is homotopy equivalent to CP n−1
(in fact, show that this set deformation retracts to CP n−1 = {xn = 0},
i.e. that there is a 1-parameter family of maps ft : CP n − {[0 : 0 : · · · :
0 : 1]} → CP n − {[0 : 0 : · · · : 0 : 1]} for t ∈ [0, 1] so that f0 is identity,
f1 is a retraction to CP n−1 , and every ft fixes all points of CP n−1 ).
(b) Run the Mayer-Vietoris argument with U = CP n − {[0 : 0 : · · · : 0 : 1]}
and V = {xn 6= 0}. What is the homotopy type of V and U ∩ V ?
Conclude the inductive step.
(4) (a) Let ω be a smooth 1-form on a manifold M such that
Z
ω=0
C
for
smooth closed curve C : S 1 → M in M (by which I mean
R every
∗
C (ω) = 0). Show that ω is exact.
S1
1
(b) If M is simply-connected, prove that H 1 (M ) = 0. By simply-connected
I mean that every smooth map S 1 → M can be extended to a smooth
map of the 2-disk.
2