Due Feb. 1 (exercises on quotient topology).
Due Feb. 8
1. Bredon gives the Inverse Function Theorem as a special case
of the Implicit Function Theorem. Show that the Implicit Function
Theorem follows from the Inverse Function Theorem.
2. Starting with phi_0 = 0, calculate the phi_1
and phi_2
produced by the contraction procedure of Lemma 1.4 as applied
to some simple examples with n=m=1. E.g. f(x,
y)=
x+y^2.
3. Apply the proof of the implicit function theorem to
g(x,y)=(x-1)^2 +(y-1)^2 -2.
Here the function
phi giving y
implicitly in terms of x, and satisfying phi(0)=0
is y = 1 - sqrt(1-x^2+2x)
as can easily be checked. Follow
the proof, starting with phi_0 = 0, to calculate phi_1
and phi_2. Note that these are the first and second Taylor
polynomial approximations of y = 1 - sqrt(1-x^2+2x).
4. (optional) Does this pattern continue?
Due Feb. 15 Bredon, p.75 #1,2; p.80 #1,2,3; p.88 #1,2.
Due Feb. 22 Bredon, p.82 #1,2,3,4,6; p.86 #1,2,3,4,5.
Due Feb. 29 The space of n x k matrices (real cofficients) is nk-dimensional. The Stiefel manifold V_k(R^n) is the set of such matrices whose k rows are pairwise orthogonal and of length 1; this can be defined, without speaking of matrices, as the manifold of k-frames in n-space. Special cases:
k=1 V_1(R^n) = S^(n-1); k=n V_n(R^n) = O(n) the orthogonal group.
Due Mar. 7 Bredon, p.264 #1,2,3; p.262: Prove
the ``It can be shown ... '' on line 7.
A Riemannian metric on a smooth manifold M is a smoothly varying
inner product on each tangent space TM_x, x in M.
Suppose M has a
Riemannian metric, and use it to construct an isomorphism between
the bundles TM and T*M.
Due Mar. 14 1. Prove the easy generalization
mentioned in Bredon p.266, line 5 from the bottom.
2. Calculate the integral of the 1-form x dy around
the perimeter of the ellipse x^2/a^2 + y^2/b^2 = 1.
Interpret
your result in the light of Stokes' Theorem.
3. Calculate the integral of the 2-form x dy ^ dz
over the surface formed by joining the disk x^2 + y^2 < =
R^2, z = 0
to the upper hemisphere x^2 + y^2 + z^2 = R^2,
z > = 0. Interpret
your result in the light of Stokes' Theorem.
Due Mar. 21 1. Elucidate (give careful proofs)
some loose ends from Bredon:
a. p.267 line 10 (the boundary DM is orientable if M is). Also,
is the hypothesis ``M orientable'' necessary?
b. p.269 ``one can see...'' on line 10
c. p.269 ``can be computed'' on line 18.
2. a. Show that if M is an oriented n-manifold
(without boundary) and
if omega is an (n-1)-form on M,
then the integral of d(omega) over M is 0.
b. On the 2-sphere with the usual coordinates theta = longitude east
and phi = co-latitude, can the 2-form
sin(phi)d(phi)^dtheta be the
d of a 1-form?
Due Mar. 28
1. Calculate H*S^n.
2. a. Prove that CP^n - {*} has CP^(n-1)
as deformation retract.
Prove that CP^n - CP^(n-1) is a
2n-dimensional disk.
Prove that CP^n - CP^(n-1) - {*} has
S^(2n-1) as deformation retract.
b. Use the homotopy theorem, the Mayer-Vietoris sequence and induction
(starting with CP^0 = {*} or CP^1 = S^2)
to calculate H*CP^n.
c. Why doesn't the same procedure work for RP^n? What additional
information is needed?
3. Fill in the gaps in the proof of Proposition B (a short exact
sequence of cochain complexes 0 --> A --> B --> C --> 0
gives a long exact sequence of cohomology groups), namely exactness
at H^p(A) and at H^p(C). You may use
Bredon's proof of Theorem 5.6 as a model, but remember that
there the differentials are D : A_k --> A_(k-1),
etc., and that the ``connecting homomorphisms'' are
D*: C_k --> A_(k-1).
Due Apr. 11
1. The object of this exercise is to watch how the contraction map
produces a sequence of functions converging to a solution.
a. Consider the autonomous O.D.E. on R given by f(x)=x.
Take
as initial condition alpha(0)=1. Set up the proof with a = 1/2,
so B_{2a}(x_0) = [0,1]. Check that L = 1 and
K = 1 are appropriate
bound and Lipschitz constant. Start the iteration with
alpha_0(t)=1,
the constant function. Work through the first few steps of the
iteration and prove that alpha_n is the sum of the first
n+1 terms of the Taylor series for e^t about 0. (Part of this
exercise was done in class).
b. Same exercise with f(x) = x+1,
again with alpha(0)=1. Prove
that alpha_n is the sum of the first n+1
terms of the Taylor series
about 0 of the solution (which you will identify).
2. Let G = Gl(n,R) be the Lie group of n x n
invertible matrices.
Topologically, G is an open set in R^{n^2}. So the tangent space
TG_g to G at any point g,
and in particular at the identity I
can be identified with R^{n^2} and
therefore with the space of all n x n real matrices.
a. For any A in TG_I, let V_A
be the vectorfield defined as
follows: V_A(g) = g_*A, where
g_* is short for the map induced
on tangent vectors by the diffeomorphism L_g: G
--> G (``left
multiplication by g'') which takes h to gh for every
h in G.
Check that
k_*(V_A(g))=V_A(kg),
for any group element k. (This
is why such vectorfields are called ``left-invariant.'')
b. Using the identification of TG_g with the space of
n x n
matrices, show that V_A(g) = gA.
c. Consider the series I + A + (1/2)A^2
+ (1/3!)A^3 + ... = exp(A).
Prove that this series converges for any A.
d. Show that the curve exp(At) is a solution of the autonomous
differential equation alpha'(t) =
V_A(alpha(t)), alpha(0) = I.
e. In Gl(2,R) find the solution corresponding to
0 -1
A = ( ).
1 0
0 -1 i 0 0 -i
A = ( ), A = ( ), A = ( ).
1 0 0 -i -i 0
Due Apr. 25
1. Continue with the notation of Assignment 10, Problem 2.
You showed that for the left-invariant vectorfield V_A, the
curve alpha(t) = exp(At) is the solution
to the autonomous differential
equation defined by V_A
with initial condition alpha(0) = I.
a. Show that g.exp(At) is the solution corresponding to initial
condition alpha(0) = g, for any group element g.
b. It follows that the flow defined by V_A is given by
phi_t(g) = g.exp(At).
This will enable you to calculate explicitly the Lie derivative of one left-invariant vectorfield with respect to another. Using part b of Assignment 10, Problem 2, and the matrix-power-series definition of exp(tA), show that the Lie derivative of the left-invariant vectorfield V_B ("corresponding to B") along the vectorfield V_A is the left-invariant vectorfield corresponding to the Lie bracket [A,B] = AB-BA.
2. Make up two problems suitable for a midterm or a final exam in this course. You may lift these from texts (with acknowledgement); the important thing is that they be doable in 15-20 minutes each by a student who knows the material. Give your solutions.
3. Bredon p.176 #1,2,3 (read Chapter IV section 3).