2010 Honors Exam in Topology

Point-set topology

(1) Let f: X → Y be a bijective continuous map of topological spaces.

(a) Find an example of such an f that is not a homeomorphism.

(b) Find conditions on X and/or Y that force f to be a homeomorphism, and prove your answer.

(2) Suppose f: X → Y is a map, not necessarily continuous, of topological spaces.

(a) If f is continuous, prove that f preserves convergence of sequences. That is, prove that if (x_n) is a sequence in X converging to x, and f is continous, then the sequence (f(x_n)) converges to f(x).

(b) Find the best hypothesis you can on X that makes the converse to part (a) true, and prove it. That is, prove that, under your hypothesis on X, if f preserves convergence of sequences, then f is continuous.

(c) Find an example of an f that preserves convergence of sequences, but which is not continuous.

(3) Prove that a connected metric space with more than one point and a countable dense subset must have exactly c points, where c is the cardinality of the real numbers.

Fundamental group and surfaces

(4) Suppose f, g: S → S¹ are continuous, f is homotopic to a constant map, and g is homotopic to the identity map. Show that there is some point x ∈ S¹ such that f(x) = g(x). (You might start with the case where g is the identity map).

(5) Suppose we take an 8-sided polygon and identify the edges in pairs according to the labelling scheme below.

(a) Prove that the resulting space K is a surface.

(b) Calculate π₁(K) and H₁(K).

(c) Identify this surface as one of the standard surfaces.

(6) Suppose we start with a path-connected space X and attach a 2-disk D² to X along a map f: S → X. That is, we form a new space Y as the quotient of the disjoint union of X and the unit disk D² in R², obtained by identifying x ∈ S¹ ⊆ D² with f(x) ∈ X for all x ∈ S¹.

(a) Determine the fundamental group of Y in terms of the fundamental group of X and the map f.

(b) Now suppose we attach a 3-disk D³ to X instead, along a map g: S² → X. Determine the fundamental group of the new space Z in terms of the fundamental group of X and g.

(7) Let X be a path connected, locally path connected, semilocally simply connected space. A path-connected covering space p: E → X of X is called abelian if it is regular and the group of covering transformations is an abelian group.

(a) Prove that there is a universal abelian covering space q: X~ → X, in the sense that if p: E → X is any abelian covering space of X, then there is a covering map r: X~ → E such that pr = q.

(b) Describe this universal abelian covering space for the figure eight space S¹ ∨ S¹.

Homology

(8) (a) Define the degree of a continuous map f: Sⁿ →Sⁿ.

(b) Calculate, with some proof, the degree of the map f: Sⁿ → Sⁿ defined by f(v) = -v.

(c) Show that a map f: Sⁿ → Sⁿ of nonzero degree must be onto all of Sⁿ.

(9) An n-dimensional pseudomanifold is a simplicial complex with the following three properties:

1. Every simplex is a face of some n-dimensional simplex;

2. Every (n - 1)-simplex is a face of exactly two n-simplices.

3. Given any two n simplices σ and τ, there is a finite sequence of n-simplices

σ = σ₀, σ₁, σ₂, . . . , σ_k = τ

where σ_i and σ_i+1 intersect in an (n - 1)-simplex for each i.

If X is an n-dimensional pseudomanifold, there are two possibilities G₁ and G₂ for H_n(X). Find them, with proof of course. What property distinguishes pseudomanifolds with H_n(X) = G₁ from those with H_n(X) = G₂?

(10) Prove the butterfly lemma. This says that if we have the commutative diagram of abelian groups and homomorphisms below,

(collapse the C's to see the "butterfly"), where both the diagonals

A →ⁱ C →^p E

and

D→^j C →^q B

are exact at C, there is an isomorphism

im q/im f ≅ im p/im g.