**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+2*x*)
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+2*x*).

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=1V_1(R^n) =S^(n-1);k=nV_n(R^n) = O(n) the orthogonal group.

Show that

You may consult references if necessary, but if you do you must say so on your homework sheets.

**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 T*M*_*x*, *x* in *M*.
Suppose *M* has a
Riemannian metric, and use it to construct an isomorphism between
the bundles T*M* 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* d*y* 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* d*y* ^ d*z*
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 D*M* 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*)^d*theta* be the
d of a 1-form?

**Due Mar. 28**

1. Calculate *H***S*^*n*.

2. a. Prove that C*P*^*n* - {*} has C*P*^(*n*-1)
as deformation retract.

Prove that C*P*^*n* - C*P*^(*n*-1) is a
2*n*-dimensional disk.

Prove that C*P*^*n* - C*P*^(*n*-1) - {*} has
*S*^(2*n*-1) as deformation retract.

b. Use the homotopy theorem, the Mayer-Vietoris sequence and induction
(starting with C*P*^0 = {*} or C*P*^1 = *S*^2)
to calculate *H**C*P*^*n*.

c. Why doesn't the same procedure work for R*P*^*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*_{2*a*}(*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
T*G*_*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 T*G*_*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 T*G*_*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

f. Check that a.b.c.d. work for Gl(n,C) and find the solutions corresponding to

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).