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(math323)[1993](s)final~PPSpider^_10506.pdf
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OLD MATH323 FINAL EXAMS
(Answers not absolutely guaranteed to be correct)
Math 323 Final, Spring 1993
(1)
(18 points) True or False
1.
If X1 . X2, and Y1 . Y2, then X1 Y1 . X2 Y2;
2.
If X = Y Z, such that Y , Z, and Y Z are contractible, then X is contractible;
3.
If X . Y , and X is not contractible, then Y is not contractible;
4.
If 1X
= 1Y , then X . Y ;
5.
Suppose f, g : X Y are homotopic. If we weaken the topology of X, then f and g are still homotopic;
6.
Suppose f, g : X Y are homotopic. If we weaken the topology of Y , then f and g are still homotopic.
(2)
(18 points) Prove that X is contractible if and only if for any Y , and continuous maps f, g : Y X, f and g are homotopic.
(3)
(18 points) Prove that if p : X. X and q : Y. Y are covering maps, then pq : X.Y. X Y is a covering map.
(4)
(18 points) The cone cX of a space X is obtained by collapsing one end X 0 of the cylinder X [0, 1] into a point. Find 1(cX).
(5)
(18 points)
1.
Find the covering transformation group of the following covering map;
2.
Find a covering of the .gure 8 with Z5 as the covering transformation group.
(6)
(10 points) Prove that if a covering map is a 1-1 correspondence, then it is a homeomorphism.
Math 323 Final, Spring 1995
(1)
(12 points) True or False (no reason needed)
1.
The image of a path connected space under a continuous map is path connected.
2.
The image of a compact space under a continuous map is compact.
3.
The image of a hausdor. space under a continuous map is hausdor..
4.
The image of a simply connected space under a continuous map is simply connected.
5.
The identity map id : X X is a covering map.
6.
The projection map p1 : X Y X is a covering map.
7.
For a subspace A of X, the inclusion i : A X is a covering map.
8.
If X is simply connected, then with a weaker topology, it is still simply connected.
9.
If two spaces have the same fundamental groups, then they are homeomorphic.
10.
If two spaces are not homeomorphic, then they have di.erent fundamental groups.
11.
If two spaces have di.erent fundamental groups, then they are not homeomorphic.
12.
If two spaces have di.erent fundamental groups, then they are not homotopic.
(2)
(16 points) Which of the following spaces are connected? compact? (no reason needed) a) Real projective space RP2 . b) B3 S1 {2 points}. c) B2 S3 (T 2 . 2 points). d) Spiral inside the unit disk: = { t eit :0 t< }.
1+t
e) Spiral and the unit circle: 1 = S1 .
f) Spiral and half of the unit circle 1/2 = {ei :0 }.
g) Rational vertical lines in the square X = {r [0, 1] : 0 r 1,r rational}.
h) Rational vertical lines with the boundary square Y = X ([0, 1] 0) ([0, 1] 1).
(3)
(12 points) Let p : Y X b