# 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

- (2 points) State the definition of a topological manifold.



- (2 points) Prove that real projective space $RP^2$ is 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 $X$ relative to $x_0 \in X$.
- (4 points) Compute the fundamental group of $R^n$ relative to the origin.

Problem 4
- (2 points) Give a pair of spaces that are homotopic equivalent, but not homeomorphic.
- (4 points) Let $A$ be a subspace of $R^n$, and $h : (A, a_0) \to (Y, y_0)$. Show that if $h$ is extendable to a continuous map of $R^n$ into $Y$, then
  $$h_* : \pi_1(A, a_0) \to \pi_1(Y, y_0)$$
  is the trivial homomorphism (the homomorphism that maps everything to the identity element).