- In class, we have encountered the following descriptions of the
torus (see also Massey, pp. 6-7):
- S
^{1}x S^{1} - {(x,y,z) in
**R**^{3}| [(x^{2}+y^{2})^{1/2}-2]^{2}+z^{2}=1} - X/~, where X={(x,y) in
**R**^{2}| 0≤x,y≤1}, and ~ is the equivalence relation given by

(0,y)~(1,y) for 0≤y≤1, (x,0)~(x,1) for 0≤x≤1, and (x,y)~(x,y) for 0<x,y<1.

Show that the spaces given in these descriptions are all homeomorphic.

- S
- (Exercise 5.1 on Massey, p. 15) Let P be a polygon with an
even number of sides. Suppose that the sides are identified in pairs
in accordance with any symbol whatsoever. Prove that the quotient
space is a compact surface.
- (See Exercise 2.1 on Massey, p. 56) A topological space X
is said to be
*locally path connected*or*locally arcwise connected*if for every x in X and each open neighborhood U of x, there is a path connected open neighborhood V of x which is contained in U.- Prove that if X is locally path connected and U is open in
X, then U is locally path connected.
- Prove that
**R**^{n}is locally path connected. - Prove that if X is locally path connected and connected, then X is path connected.

- Prove that if X is locally path connected and U is open in
X, then U is locally path connected.

- Let f,g:I -> X be two paths with initial point x and terminal
point y.

[Then for instance g(1-t) is a path with initial point y and terminal point x.]- Prove that f ~ g if and only if the loop f·g(1-t) is
equivalent to e
_{x}, the constant path at x. - Prove that f and g give rise to the same isomorphism from
the
fundamental group of X based at x to the fundamental group of X based at y
if and only if the equivalence class [f·g(1-t)] belongs to the
center of the fundamental group of X based at x.

[The center Z(G) of a group G is the subgroup Z(G)={a in G | ab=ba for all b in G}.]

- Prove that f ~ g if and only if the loop f·g(1-t) is
equivalent to e
- Let A be a retract of X.
- Prove that if X is Hausdorff, then A is closed.
- Prove that if X is simply connected, then A is simply connected.

- Prove that if X is Hausdorff, then A is closed.

- In class, we stated the following:

**Theorem**. Let X be a space which can be written as the union of two simply connected open sets U and V in such a way that the intersection of U and V is path connected. Then X is simply connected.

Use the Lebesgue Lemma to prove this theorem, by showing that any loop in X (based at a point in the intersection of U and V) is equivalent to a product of loops, each of which is contained in either U or V. - Show that
**R**^{2}and**R**^{n}are not homeomorphic if n is different from 2.

Hint: Consider the complement of a point in**R**^{2}or**R**^{n}. - Determine the structure of the fundamental group of the n-dimensional torus S
^{1}x S^{1}x ··· x S^{1}.

- Let S
^{1}={z in**C**: |z|=1} be the set of all complex numbers of length one, and consider the map f:S^{1}-> S^{1}defined by f(z)=z^{k}, where k is an integer.

Determine the induced homomorphism f_{*}from the fundamental group of S^{1}(based at 1) to the fundamental group of S^{1}(based at f(1)=1). - Let G and H be groups. Prove that the free product G*H is unique up to isomorphism.
- Let X be a Hausdorff space such that X is the union of A and B, where A and B are each homeomorphic to a torus, and the intersection of A and B is a point, say x
_{0}. Determine the structure of the fundamental group of X based at the point x_{0}. - State and prove a generalization of problem 3.

(We will eventually do something quite similar in class, but I want you to think about this now.)

- Let p:X~ --> X be a covering space (with X~ path connected) and suppose f:X~ --> X~ is continuous and that p(f(x))=p(x) for all x in X~. If f(x)=x for some x in X~, prove that f is the identity map.
- Suppose that p:X --> Y is a covering space, and that X and Y are both Hausdorff spaces. Prove that X is an n-manifold if and only if Y is an n-manifold.

- Prove that, if X is simply connected and p:X~ --> X be a covering space, then p is a homeomorphism of X~ onto X.
- Determine all covering spaces (up to isomorphism) of
- the circle
- the projective plane

- Construct three distinct (that is, not isomorphic) path-connected four-fold covering spaces of a bouquet of two circles. Explain how you know they are distinct.
- Let p
_{i}:X_{i}--> X be covering spaces, let x_{0}be in X and suppose x_{i}in X_{i}satisfies p_{i}(x_{i})=x_{0}for i=1,2. Prove that the two covering spaces are isomorphic if and only if the images of the fundamental groups of X_{1}and X_{2}(based at x_{1}and x_{2}) under the homomorphisms induced by the projection maps p_{1}and p_{2}belong to the same conjugacy class of subgroups of the fundamental group of X (based at x_{0}).