7810 Algebraic Topology II
Problem Set 3
Chun-Ju Lai
Due day: December 06, 2011
Exercise 3.3.6
Proof.
Let M1 = A B, M2 = C B where A, C :open and A B, C B deformation-retract to S n1 ,
then (M1 #M2 , S n1 ) is a good pair with LES
0 = Hn (S n1 ) Hn (M1 #M
7810 Algebraic Topology II
Problem Set 2
Chun-Ju Lai
Due day: November 20, 2011
Exercise 3.2.3
Proof.
a.
Case: RP
Consider f : RPn RPm (n > m), it induces
f :
H (RPm ; Z2 ) Z2 [x1 ]/xm+1
1
x1
Hence we have
n
H (RPn ; Z2 ) Z2 [y1 ]/y1 +1
cy1
0 = f (0)
7810 Algebraic Topology II
Problem Set 1
Chun-Ju Lai
Due day: September 20, 2011
Exercise 3.1.2
Proof.
n
1. Consider the map H H . Since that any abelian group has a free resolution 0 R
F H 0, so by Lemma 3.1 it is sucient to prove that the following cha