MATH 7512 Topology II

Homework #1

1. In class, we have encountered the following descriptions of the torus (see also Massey, pp. 6-7):
   1. S¹ x S¹

   2. {(x,y,z) in R³ | [(x²+y²)^1/2-2]²+z²=1}

   3. X/~, where X={(x,y) in R² | 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.

2. (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.

3. (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.
   1. Prove that if X is locally path connected and U is open in X, then U is locally path connected.

   2. Prove that Rⁿ is locally path connected.

   3. Prove that if X is locally path connected and connected, then X is path connected.

Homework #2

1. 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.]
   1. Prove that f ~ g if and only if the loop f·g(1-t) is equivalent to e_x, the constant path at x.

   2. 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}.]

2. Let A be a retract of X.
   1. Prove that if X is Hausdorff, then A is closed.

   2. Prove that if X is simply connected, then A is simply connected.

Homework #3

1. 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.

2. Show that R² and Rⁿ are not homeomorphic if n is different from 2.
   Hint: Consider the complement of a point in R² or Rⁿ.

3. Determine the structure of the fundamental group of the n-dimensional torus S¹ x S¹ x ··· x S¹.

Homework #4

1. Let S¹={z in C : |z|=1} be the set of all complex numbers of length one, and consider the map f:S¹ -> S¹ defined by f(z)=z^k, where k is an integer.
   Determine the induced homomorphism f_* from the fundamental group of S¹ (based at 1) to the fundamental group of S¹ (based at f(1)=1).

2. Let G and H be groups. Prove that the free product G*H is unique up to isomorphism.

3. 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 x₀. Determine the structure of the fundamental group of X based at the point x₀.

4. 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.)

Homework #5

1. 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.

2. 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.

Homework #6

1. Prove that, if X is simply connected and p:X~ --> X be a covering space, then p is a homeomorphism of X~ onto X.

2. Determine all covering spaces (up to isomorphism) of
   1. the circle
   2. the projective plane
   Exhibit an explicit covering space from each isomorphism class.

3. 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.

4. Let p_i:X_i --> X be covering spaces, let x₀ be in X and suppose x_i in X_i satisfies p_i(x_i)=x₀ for i=1,2. Prove that the two covering spaces are isomorphic if and only if the images of the fundamental groups of X₁ and X₂ (based at x₁ and x₂) under the homomorphisms induced by the projection maps p₁ and p₂ belong to the same conjugacy class of subgroups of the fundamental group of X (based at x₀).