1.0 KiB
1.0 KiB
Math4202 Topology II Exam 1 Practice
In the following, please provide complete proof of the statements and the answers you give. The total score is 25 points.
Problem 1
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(2 points) State the definition of a topological manifold.
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(2 points) Prove that real projective space
RP^2is a manifold. -
(2 points) Find a 2-1 covering space of
RP^2.
Problem 2
- (2 points) State the definition of a CW complex.
- (4 points) Describe a CW complex homeomorphic to the 2-torus.
Problem 3
- (2 points) State the definition of the fundamental group of a topological space
Xrelative tox_0 \in X. - (4 points) Compute the fundamental group of
R^nrelative to the origin.
Problem 4
- (2 points) Give a pair of spaces that are homotopic equivalent, but not homeomorphic.
- (4 points) Let
Abe a subspace ofR^n, andh : (A, a_0) \to (Y, y_0). Show that ifhis extendable to a continuous map ofR^nintoY, thenh_* : \pi_1(A, a_0) \to \pi_1(Y, y_0)is the trivial homomorphism (the homomorphism that maps everything to the identity element).