2009 Honors Examination: Topology
1. Define the terms "compact," "Hausdorff," and "closed" in the following theorem, and prove the theorem from these definitions. (You may want to set up a couple of lemmas along the way.)
Theorem. Any continuous map from a compact space to a Hausdorff space is closed.
2. Let X be the set of all sequences of real numbers, x0, x1, . . ., with the topology determined by the metric given by
(a) First, check that this is indeed a metric.
(b) Let x ∈ X, and derive a condition on x which is equivalent to the statement that the function R → X given by t |→ tx is continuous. (Here t(x0, x1, . . .) = (tx0, tx1, . . .).)
(c) Given x ∈ X, describe the connected component containing x.
3. Let S be a compact surface without boundary, let D2 be the closed 2-disk, and let D2 → S embed D2 into S. If you like, you can take this embedding as a homeomorphism of D2 with the standard 2-simplex, followed by the inclusion of the 2-simplex in a triangulation of S.
Remove the interior D° of the image of D2. The result is a surface with boundary, and the boundary is equipped with a homeomorphism to S1 regarded as the boundary of D2.
Describe the Mobius band, and specify a homeomorphism of its boundary with S1. Now attach a Mobius band to S - D° by identifying the boundaries according to how they have been identified with S1. The result is the "blowup" of S at the center point of D°.
Discuss the resulting space. Is it a surface? What is its Euler characteristic (in terms of the Euler characteristic of S)? Is it orientable? What do you get if you start with S = S2? How about if you start with S = RP2?
4. (a) What is the fundamental group of the circle? Sketch a proof, in a sentence or two.
(b) Let k be a positive integer. Please construct a simply connected space with a properly discontinuous action of the cyclic group Ck of order k.
(c) Please describe a space whose fundamental group is the free product of Ck with a free abelian group on two generators.
(d) Define the term "regular" as applied to a covering map, and give a specific example of a covering map which is not regular.
5. Suppose that
is a "ladder": a map of long exact sequences. So both rows are exact and each square commutes. Suppose also that every third vertical map is an isomorphism, as indicated. Prove that these data determine a long exact sequence
· · ·→ An → A'n ⊕ Bn → B'n -→ An-1 → · · ·
6. Let A be a 2 × 2 matrix with integral coefficients. Let T2 be the torus, regarded as the quotient space of R2 by the translation action of the subgroup Z2. Explain why multiplication by A on R2 descends to a self map of T2. Let f: T2 → T2 be a continuous map which is homotopic to the map induced by A. Investigate when you can guarantee that f fixes a point in T2, in terms of invariants of the matrix A.
Is every continuous self-map of T2 of this form?