Honors Examination in Topology 2007

1. Suppose f, g: X → Y are two continuous functions from X to Y. Define the coincidence set of f and g to be the subset of X

C(f, g) = {x ∈ X | f(x) = g(x)}.

a) Show that if Y is Hausdorff, then C(f, g) is a closed set in X.

b) Suppose A ⊂ X is a dense subset of X and A ⊂ C(f, g). Deduce that f = g.

c) Suppose X = Y and g = id_X, the identity mapping on X. Show that C(f, id_X) is the fixed set of f, Fix(f) = {x ∈ X | f(x) = x}, and deduce that if X is Hausdorff, then Fix(f) is closed.

2. Let S ⊂ R² denote the spiral given in polar coordinates by

S = {(r, θ) | 1 ≤ θ < ∞ and r = (θ - 1)/θ}.

Let A = cls S, the closure of S in R². Show in detail that A is connected, but not path-connected.

3. If X is a topological space, then π₀(X) denotes the set of equivalence classes of X under the relation x ∼ y if there is a continuous path λ: [0, 1] → X with λ(0) = x and λ(1) = y.

a) Suppose G is a topological group, that is, there is a binary operation on G, denoted µ: G × G → G, which is continuous, makes G a group, and for which the mapping g |→ g^-1 is also continuous. Show that π₀(G) is also a group.

b) Let Gl_n(R) denote the multiplicative group of invertible n × n matrices with real entries. Show that the group π₀(Gl_n(R)) is isomorphic to Z/2Z. (Hint: Think about elementary matrices that carry out row operations and paths in Gl_n(R) between them. Or maybe think about the Gram-Schmidt Process.)

4. The suspension ΣX of a topological space X is the quotient space of X × [0, 1] under the equivalence relation given by the equality on points (x, t) for 0 < t < 1 and for all x, x' ∈ X, we have (x, 0) ∼ (x', 0) and (x, 1) ∼ (x', 1).

a) Suppose that X is connected and path-connected. Show that the sets U = {[(x, t)] | t > 1/3} and V = {[(x, t)] | t < 2/3} are open in ΣX and that U and V are homotopy equivalent to a point.

b) Show that U ∩ V is path-connected.

c) Let x₀ ∈ X. Use a) and b) to show that π₁(ΣX, [(x₀, 1/2)]) is the trivial group.

5. If p: X˜ → X is a covering space, then the lifting of loops in X to paths in X˜ leads to an action of the fundamental group π₁(X, x₀) on the set p^-1({x₀}).

a) Describe the construction of this action in detail.

b) The action of any group G on a set F leads to a homomorphism φ: G → Sym(F), where Sym(F) denotes the group of permutations of the set F. This homomorphism is defined by g |→ (x |→ g · x). Show that φ is injective.

c) It follows that we have an injective homomorphism φ: π₁(X, x₀) → Sym(p^-1({x₀}). Suppose X˜ is simply-connected, and the cardinality of p^-1({x₀}) is 2. Deduce that π₁(X, x₀) ≅ Z/2Z. What can you say if the cardinality of p^-1({x₀}) is 3?

6. The classification theorem for surfaces shows that a (closed, compact, connected) surface may be represented as a quotient space of a polygon. This representation allows one to triangulate the surface, compute its fundamental group, etc. A surface is nonorientable if and only if it has a embedded M¨obius band. Use the classification theorem to show that the connected sum of a projective plane with an orientable surface has an embedded M¨obius band. And show that the canonical presentation of edge identifications for nonorientable surfaces contains an embedded Mobius band.

7. If you pluck a single point from S², then you get a space homeomorphic (via stereographic projection) to R². Since R² is convex, it is a contractible space and we can write S²-{x₀} =≈ ∗, the one-point space. In this problem, let's consider what happens when you remove a point from F, a compact, closed, connected surface. Show that, in general, F - {x₀} =≈ S¹ ∨ S¹ ∨ · · · ∨ S¹ for some number of circles (possibly none).

Relate the number of circles in the bouquet to the genus of the surface when it is orientable, and generally to the Euler characteristic of F.

8. Suppose that G is a topological group and e ∈ G is the identity element. Then, in problem 3, we asserted that π₀(G), the set of path components of G, is a group. We next show that π₁(G, e) is a (left) π₀(G)-set, that is, there is an action of π₀(G) on π₁(G, e). Consider the mapping

µ: π₀(G) × π₁(G, e) → π₁(G, e), µ([g], [λ]) = [g · λ · g^-1],

where g · λ · g^-1(t) = gλ(t)g^-1 ∈ G.

a) Show that µ is well-defined, and that it satisfies the properties of a group action, that is, for all [g], [h] ∈ π₀(G) and [λ] ∈ π₁(G, e),

(1) [g] · ([h] · [λ]) = ([g][h]) · [λ]

(2) [e] · [λ] = [λ].

b) The topological group O(2) consists 2×2-matrices with real entries and orthonormal columns. Show that π0(O(2)) is isomorphic to Z/2Z and that the path component of the identity matrix Id is homeomorphic to S¹. It follows that π₁(O(2),Id) ≅ Z. Thus the action in a) is algebraically an action of Z/2Z on Z.

9. Consider the following 2-dimensional complex, denoted by K, given by taking a pair of tetrahedra, ABCD and A'B'C'D' with A'B'C'D' smaller and inside ABCD together with the edges AA', BB', CC', and DD' , as well as the faces ABB'A', ACC'A', ADD'A', BCC'B', BDD'B', and CDD'C'. The complex is pictured below.

a) What is the Euler characteristic of this complex?

b) Give a plausibility argument that this complex is simply-connected.

c) Deduce from b) that H₁(K) = {0} and from a) deduce the rank of H₂(K).

10. Suppose K and L are finite simplicial complexes. Let |K| and |L| denote the realizations of K and L as topological spaces. Finally, let [|K|, |L|] denote the set of homotopy classes of continuous functions from |K| to |L|. Use the Simplicial Approximation Theorem to show that this set is countable. Show how this implies that π₁(|K|) is a countable group if K is a finite complex.