Math 411
HW 1, due Friday, February 8
1. Prove the rst clause of Prop. 1.7 in the topology handout.
2. Prove Prop. 1.9 in the topology handout.
3. Give the simplest example of a topological space that is not Hausdor.
4. Let X and Y be disjoint copies of R
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Homology of Simplicial Complexes
Math 411, David Perkinson
Introduction. This is an introduction to the homology of simplicial complexes suitable for a
rst course in linear algebra. It uses little more than the rank-nullity theorem. As you read,
note how
Math 411
HW 2, due Friday, February 15
1. Show that the composition
phys
geom
alg
phys
Tp M Tp M Tp M Tp M
is the identity.
2. Consider the projective plane P2 with homogeneous coordinates (x, y, z ), and let p =
(1, 1, 1) P2 . Dene
x
f (x, y, z ) = .
y
(
Math 411
HW 3, due Friday, February 22
1. (a) Let X and Y be sets. What are X Y and X Y ? Show they satisfy the
appropriate universal properties.
(b) Let X and Y be objects in any category. If A and B are both products of X and Y ,
use the universal prope
Math 411
HW 4, due Friday, March 1
1. Computations.
(a)
f : R2 R4
(x, y ) (x2 , 2x + y, y 4 , xy )
i. Let = y1 dy1 dy2 + (y1 y3 )dy3 dy4 2 R4 . Compute f and express
your answer in terms of the standard basis for 2 R2 .
ii. Consider the vector eld v = y x
Math 411
HW 5, due Friday, March 8
1. Let V and W be vector spaces over an arbitrary eld K , and suppose that V has nite
dimension n.
(a) Show that
hom(V, W ) W n = W W .
n factors
(b) There is a mapping of vector spaces
V W hom(V, W )
w [v (v )w]
Its in
Math 411
HW 6, due Friday, March 15
1. On P1 , we have the orienting atlas A = cfw_(Ux , h), (Uy , k ) where
h : Ux := cfw_(x, y ) P1 : x = 0 R
(x, y ) y/x
k : Uy := cfw_(x, y ) P1 : y = 0 R
(x, y ) x/y.
Dene 1 P1 whose local description with respect to h
HW 7, due Friday, March 29
Math 411
For this assignment, please refer to section 4 of our handout on topology, available from our
website.
1. Explain why the components and the path components of a manifold are the same.
(Quote the right result from the h
Math 411
HW 8, due Friday, April 5
1. Manifolds M and N are homotopy equivalent if there are maps f : M N and g : N
M such that g f idM and f g idN (where denote homotopy equivalence of
maps). Show that if M and N are homotopy equivalent then H k M H k N
Math 411
HW 9, due Friday, April 12
1. Let V be an oriented vector space with scalar product , of index s. Let e1 , . . . , en
be an orthonormal basis for V with ei , ei = i cfw_1, 1 for each i, and let V be the
volume form for V . For each k , let : k V
HW 10, due Friday, April 19
Math 411
1. Let P be the hexagon in the plane with vertices (0, 0), (1, 0), (0, 1), (2, 1), (1, 2), and
(2, 2).
(a) Draw the fan, , for the corresponding toric variety, X , labeling the rst lattice
points along its 1-dimensiona
Math 411
HW 11, due Friday, April 26
1. A linear subspace L of Pn of dimension r, also called an r-plane, is an (r + 1)
dimensional vector subspace L An+1 modulo the equivalence p p for every
p L and nonzero k . Given two linear subspaces L and M of Pn ,