Math 627Prof. Menasco
2nd Homework
Fall 2015
1.
2.
3.
4.
TOTAL
Hakan Doga
1
1. The set O(n) of orthogonal matrices square matrices of order n is a Lie group,
called the orthogonal group of order n. O(n) is dened by the equation XX t = E
(X t is the transp
Math 628Prof. Menasco
4th Homework
Spring 20156
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2.
3.
4.
TOTAL
Hakan Doga
1
1.
A compact oriented connected 7-manifold M (without boundary) has the following
homology groups
if i = 1
ZZ 3
ZZ
if i = 2
Hi (M ; ZZ)
=
ZZ ZZ 3 if i = 3
Compute all the rem
Math 628Prof. Menasco
3rd Homework
Spring 20156
1.
2.
3.
4.
TOTAL
Hakan Doga
1
1.
For X a finite CW complex and F a field, show that the Euler characteristic (X)
can also be computed by the formula
(X) = n (1)n dimHn (X; F ),
the alternating sum of the di
Math 628Prof. Menasco
2nd Homework
Spring 2016
1.
2.
3.
4.
5.
TOTAL
Hakan Doga
1
1. Let X1 and X2 be subspaces of a space X . Prove that the following are equivalent:
a. The excision map (X1 , X1 X2 ) (X1 X2 , X2 ) induces an isomorphism
of homology.
b. T
Math 628Prof. Menasco
1st Homework
Spring 2016
1.
2.
3.
4.
TOTAL
Hakan Doga
1
1. (a) Show that H0 (X, A) = 0 i A meets each path-components of X . (b) Show
that H1 (X, A) = 0 i H1 (A) H1 (X) is onto and each path-component of X
contains at most one path-c