18.905 Problem Set 11
Due Wednesday, November 29 (post-break) in class
Five questions. Do all ve.
1. Hatcher, exercise 2 on page 257.
2. Hatcher, exercise 7 on page 258.
3. Show that the exterior cup product respects boundary homomorphisms.
In other words
18.905 Problem Set 7
Due Friday, October 27 in class
The method of acyclic models.
5 questions. Do any 4 for full credit; all 5 for bonus marks.
Suppose that C is a category, and F is a functor from C to abelian groups.
Suppose that for each i I , Mi is a
18.905 Problem Set 3
Due Wednesday, September 27 in class
1. Suppose that C and D are chain complexes, and f and g are maps of
chain complexes from C to D . Recall that a chain homotopy from f to
g is a collection of maps hn : Cn Dn+1 such that for all x
18.905 Problem Set 9
Due Wednesday, November 8 in class
1. Use naturality to show that if X is the disjoint union of subspaces X ,
then there is an isomorphism of rings
H (X )
H (X ),
where the latter ring has the standard componentwise ring structure
(r
18.905 Problem Set 2
Due Wednesday, September 20 in class
1. A simplicial complex consists of a pair (V, F ), where V is a set of vertices
and F (the set of faces) is a collection of nite subsets of V satisfying the
following properties.
We have cfw_v F
18.905 Problem Set 5
Due Wednesday, October 11 in class
1. Hatcher, Exercise 7 on page 155.
2. Hatcher, Exercise 9 on page 156.
3. Suppose X = A1 A2 An , each Ai is open, and that every nonempty
intersection Ai1 Ai2 Air is contractible. Show Hk (X ) = 0 f
18.905 Problem Set 10
Due Friday, November 17 in class
1. Hatcher, Problem 11 on page 205.
2. Hatcher, Exercise 1 on page 228.
3. Hatcher, Exercise 6 on page 229.
4. Hatcher, Exercise 13 on page 229.
18.905 Problem Set 6
Due Wednesday, October 18 in class
1. Use the universal coecient theorem to compute H (L(p, q ); Z/m) for all
m, where L(p, q ) are the lens spaces dened in class.
2. Hatcher, Exercise 1 on page 267.
3. Given an arbitrary nitely gener
18.905 Problem Set 12 (Final)
Due Friday, December 8 in class
1. Suppose f C p (X ), g C q (X ), and x Cp+q+r (X ). Show, using the
denition of the cap product given in class, that
g
(f
x) = (f
g)
x Cr (X ).
Show that this makes C (X ) = n Cn (X ) into a
18.905 Problem Set 1
Due Wednesday, September 13 in class
1. Prove that a CW complex X is a disjoint union of connected components,
and these connected components are also path components.
2. Suppose that f : K X is a map from a compact space K to a CW
co
18.905 Problem Set 4
Due Wednesday, October 4 in class
1. Let A be the 1-skeleton of 3 , i.e. the union of the vertices and lines. Let
CA, the cone on A, be the union of all the lines joining A to the center of
3 . Compute the homology groups H (CA, A).
2
18.905 Problem Set 8
Due Wednesday, November 1 in class
1. Hatcher, Exercise 2 on page 280.
2. For any k , compute the homology of the k -fold product
RP2 RP2
with Z and Z/2-coecients. (A recursive formula is ne.)
3. Hatcher, Exercise 4 on page 205.
4. Ha