Math 730. Fundamental Concepts of Topology (Fall 1998)

Homework Assignments
 

Tentative Schedule

 
due date assignment (in Bredon, unless otherwise noted)
Wednesday, September 9 Ch. I, p. 3, #1, 3; p. 7, #3, 4, 5; p. 10, #3, 8.
Monday, September 14 Ch. I, p. 12, #1, 2, 3, 5; p. 14, #1, 2, 3, 4, 5.
Wednesday, September 23
  1. A topological space X is called sequentially compact if every sequence {xn} in X has a convergent subsequence {xnk}.
    1. Give your own proof that a metric space is compact if and only if it is sequentially compact (the definition you may already be familiar with). (If you get stuck, see Theorem 9.4.)
    2. Give (with proof) an example of a space which is compact but not sequentially compact.
  2. Bredon, Ch. I, pp. 24-25, #5, 6, 9.
Monday, September 28
  1. Suppose X is a locally compact Hausdorff space, and X+ is its one-point compactification. Show that X+ is the topological coproduct of X and a one-point space if and only if X is compact.
  2. Bredon, Ch. I, p. 28, #2 and p. 31, #2.
Wednesday, October 7 Ch. I, pp. 43-44, #2, 4, 5, 6, 8.
Wednesday, October 14 Ch. I, pp. 50-51, #4, 5, 6, 9; pp. 55-56, #3, 4, 5, 7.
Wednesday, October 21 See assignment here (postscript format) or here (dvi format).
Monday, October 26 Mid-term Exam on Ch. I, first part of Ch. II
Monday, November 2 Ch. II, p. 82, #3, 4, 6, 7; p. 89, #3, 5.
Solutions to p. 82, #7, and p.89, #5 below.
Wednesday, November 11 Assignment on Ch. II, section 10:
  1. Prove the following extension of the Whitney Embedding Theorem: Let Xm be a smooth (compact) manifold with boundary. Show that if N > 2m, then any smooth embedding of the boundary of Xm in RN extends to an embedding of Xm in RN.
  2. An isotopy means a homotopy through embeddings. Apply the result of (1) to show that if Mn is a smooth (compact) manifold without boundary, then any two embeddings of Mn into RN are isotopic, provided that N > 2m + 2. (Take X = M x [0,1], m = n + 1. An embedding of M x [0,1] restricts for each t in [0,1] to an embedding of M.)
Monday, November 16 Ch. III, p. 138, #1, 2, 3.
Monday, November 23 Ch. III, p. 143, #2, 4; pp. 145-156, #1, 2.
Monday, November 30 Ch. III, p. 147, #2; p. 154, #1, 2, 3.
Wednesday, December 9 Ch. III, pp. 162-164, #3, 5, 7, 8, 10, 12.
Solution to p. 163, #10 below.
Tuesday, December 15 Final Exam

Selected Solutions
Ch. II, p. 82, #7:
One has to be quite careful here because the multiplication is non-commutative. If one allows multiplication from both the left and the right, then the "Fundamental theorem of Algebra" fails completely. For example, the "linear" "polynomial"
z --> i z + z i - k
has no roots.
Ch. II, p. 89, #5:
This problem requires two facts:
  1. A sphere is stably parallelizable, that is, the Whitney sum (see Bredon, p.113) of TSn and a trivial one-dimensional bundle is a trivial bundle of rank n + 1. (You have to use this fact twice, once for Sn and once for Sk.) To prove this, use the map TSn x R --> Sn x Rn + 1 given by (v, w, t) --> (v, w + tv). (Here v lies on Sn and w in Rn + 1 is orthogonal to it.)
  2. An odd-dimensional sphere Sn has an everywhere non-vanishing vector field (obtained by differentiating the action of S1 by complex multiplication on the unit sphere of a complex vector space).
Fact 2 says that we can split TSn as a Whitney sum of a vector bundle E and a trivial one-dimensional bundle. Combining that one-dimensional bundle with TSk, we get (by Fact 1) a trivial bundle of rank k + 1. Split this as a Whitney sum of a trivial bundle of rank k - 1 and two trivial one-dimensional bundles. Add one of the trivial one-dimensional bundles back to E to get TSn, then add the other and use Fact 1 again (applied to Sn) to get a trivial bundle.
Ch. III, p. 163, #10:
For simplicity let's first assume that f is bijective, which will be the case exactly when the determinant ad - bc is 1 or -1. Let T1 and T2 be the two solid tori. Then X is the union of T1 and T2, and their intersection is a torus. Note also that even though T1, T2, and their intersection are closed rather than open, Van Kampen's Theorem still applies here, since each of these submanifolds is the deformation retract of a small open neighborhood in X. So the fundamental group of X is the free product of two copies of Z, the fundamental groups of the two solid tori, amalgamated along a copy of Z2 mapping to (in fact, onto) both. Since Z2 surjects onto both copies of Z, the fundamental group of X is a quotient of this group by the relations corresponding to the inclusions of T2 into the two solid tori. Viewing Z2 as the fundental group of the boundary of the second solid torus, the inclusion of S1 x S1 into S1 x D2 kills off the second generator (0,1), and the inclusion of S1 x S1 into D2 x S1 kills off the image under f of the first generator (1,0), in other words, (a, d). So if a = 0, the fundamental group of X is infinite cyclic, and if a is non-zero, it's finite cyclic of order a.