MTH3483 — TOPICS IN TOPOLOGY — FALL 1998

PROF. A. SUCIU

Homework 2

1. Let ξ be a principal G-bundle. Show:

(a) f^{∗}(ξ) is a principal G-bundle.

(b) If G acts effectively on F, then f^{∗}(ξ)[F] ≅ f^{∗}(ξ[F]).

2. Given a principal G-bundle P → B and a closed subgroup H of G, show:

(a) P/H → B is a G-bundle with fiber G/H associated to P → B.

(b) If H is also a normal subgroup of G, then P/H → B is a principal G/H-bundle.

3. Let G be a topological group acting on a space X. Show:

(a) If A ⊂ X is open, then G · A is open.

(b) If G is compact and A ⊂ X is closed, then G · A is closed.

(c) If G is compact and A ⊂ X is compact, then G · A is compact.

(d) If X is Hausdorff, then X^{G} is Hausdorff.

(e) If G is compact and X is Hausdorff, then X/G is Hausdorff.

(f) The compactness assumption in part (e) is necessary.

[Hint: Find an ℝ-action on ℝ^{2} such that ℝ^{2}/ℝ is not Hausdorff.]

4. Let ξ_{n} be the principal ℤ_{n}-bundle p_{n} : S^{1} → S^{1}, where p_{n}(z) = z^{n}. Consider the associated ℤ_{n}-bundle η_{n} = ξ_{n}[S^{1}], where ℤ_{n} ⊂ S^{1} acts on S^{1} by left-translation. Show that η_{n} is not trivial as a ℤ_{n}-bundle, but it is trivial as a (principal) S^{1}-bundle.

5. Consider the ℤ_{n}-action on S^{3} given by (z_{1}, z_{2}) ↦ (ξz_{1}, ξz_{2}), where ξ = e^{2πi/n}. Let L_{n} be the orbit space.

(a) Define a principal S^{1}-bundle L_{n} → S^{2}.

(b) What is the clutching function of this bundle?

(c) Show that L_{1} = S^{3} and L_{2} = SO(3).

6. Let p : E → B be a real vector bundle of rank k. Let

F (E) = {(b, f) | b ∈ B and f = (f_{1}, …, f_{n}) is a basis for p^{−1}(b)}

be the space of frames of E, and let q : F (E) → B be given by q(b, f) = b. Show that:

(a) q : F (E) → B is a principal GL(n, ℝ)-bundle.

(b) The given vector bundle is associated to its frame bundle via the natural action of GL(n, ℝ) on ℝ^{n}, i.e., E = F (E) ×_{GL(n,ℝ)} ℝ^{n}.