Answers to some of homework is here; I willl
continue to add to it.
Homework #1 (due Friday Sep 11):

Homework #2 (due Friday Sep 18, answers here):
Homework #3 (due Friday Sep 25):
Homework #4 (due Friday Oct 2): book problems 2.1, 2.5, 2.10, 2.11.
Homework #5 (due Friday Oct 9): book problems 2.9, 3.6, 3.12, 3.16

Homework #6 (due Friday Oct 16):

Let M be a diagonalizable matrix, i.e. one with a basis v_1...v_n of
eigenvectors. Show that the dual basis elements are also eigenvectors,
of M^*.
book problems 3.15, 4.1
Homework #7 (due Friday Oct 23):

book problems 5.1, 5.2, 5.3
Let b: R^n x R^m -> R be a bilinear map.
Show that there exists a unique nxm matrix M such that
b(v,w) = trace(M w v^T) for all column vectors v,w.
Describe the two subspaces of R^n, R^m that we have to quotient by
in order to make this a perfect pairing, in terms of M.
Homework #8 (due Friday Nov 6):
Let M = R^2 \ 0, the punctured plane. We computed its
ordinary cohomology H^*(M) a few weeks ago.
Compute its compactly supported cohomology H^*_c(M), i.e. using the
d operator on compactly supported forms. (Careful: the set of vectors v in R^2
with |v| at most r is compact, but once you remove 0 it's not compact.)
Show that each pairing H^i_c(M) x H^{2-i}(M) -> R, by wedging then
integrating, is perfect.
book problem 8.6
Given orientations on a pair V < W of vector spaces, show how to
define an orientation on the quotient space W/V. Your rule should have
the property that if you flip V or W the rule flips W/V.
Homework #9 (due Friday Nov 13):
book problems 9.1, 9.2 (recall the book writes ab for a ^ b), 9.5
Homework #10 (due Friday Nov 20):
Let T: A->B, U: C->B. Break up A,B,C as direct sums in ways that T,U
respect, such that on each piece T,U are either 0 or isomorphisms.
Put in another space V: D->B, and assume B is 2-dimensional while A,C,D
are 1-dimensional. Describe all the possibilities, up to change of
basis in A,B,C,D.
Homework #11 (due Friday Dec 4):
Let M be a manifold. Define the **PoincarĂ© polynomial** of M as
p_M(t) := sum_{i=0}^{dim M} (dim H^i(M)) t^i,
and the **Euler characteristic** as Chi(M) := p_M(-1).
Use Mayer-Vietoris to show that if M = U union V (open submanifolds),
then Chi(M) = Chi(U) + Chi(V) - Chi(U intersect V).

Compute the Euler characteristic of a genus g surface using this.
What's the best statement you can make like that formula, relating
the PoincarĂ© polynomials?