MTH3483 — TOPICS IN TOPOLOGY — FALL 1998

PROF. A. SUCIU

Homework 3

1. Let ξ be a k-plane bundle with transition functions {ϕ_{ij}}. Show:

(a) If k = 1, then ξ^{⊗m} := ξ ⊗ ⋯ ⊗ ξ(m factors) has transition functions {ϕ_{ij}^{m}}.

(b) The determinant bundle det ξ := ∧^{k}ξ has transition functions {det ϕ_{ij}}.

(c) det(T^{∗}(G_{k}(ℝ^{n}))) ≅ (det γ_{k}(ℝ^{n}))^{⊗n}.

2. Let γ be the tautological complex line bundle over ℂℙ^{n}.

(a) Find the transition functions of γ with respect to the usual cover of ℂℙ^{n}.

(b) Show: {sections of γ^{m}} ≅ {homogeneous polynomials in ℂ^{n+1} of degree m}.

3. Let G̃_{k}(ℝ^{n}) be the Grasmannian of oriented k-planes in ℝ^{n}.

(a) There is a double covering ℤ_{2} → G̃_{k}(ℝ^{n}) → G_{k}(ℝ^{n}).

(b) There is a principal bundle SO(k) × SO(n − k) → SO(n) → G̃_{k}(ℝ^{n}).

(c) Interpret the covering map G_{k}(ℝ^{n}) → G_{k}(ℝ^{n}) from part (a) as a map of homogeneous spaces, SO(n)/SO(k) × SO(n − k) → O(n)/O(k) × O(n − k).

(d) Show that G̃_{2}(ℝ^{4}) is diffeomorphic to S^{2} × S^{2}.

(e) What is the involution on S^{2} × S^{2} that corresponds to the non-trivial covering map G̃_{2}(ℝ^{4}) → G̃_{2}(ℝ^{4})?

4. Let X be an n-connected CW-complex and Y an m-connected CW-complex. Show that their join, X ∗ Y, is (n + m + 1)-connected.

5. Show that the inclusion-induced map π_{i}(O(k) → π_{i}(O(k + 1)) is an epimorphism for i = k − 1 and an isomorphism for i < k − 1. Deduce that π_{i}(V_{k}(ℝ^{n})) = 0.

6. Let G be a Lie group and H a closed subgroup. Show:

(a) If K is a closed subgroup of H, there is as an H-bundle H/K → G/K → G/H.

(b) If K is also a normal subgroup of H, this in fact a principal H/K-bundle.

(c) There is an H-bundle G/H → BH → BG.

7. Let ξ be a principal G-bundle over B with classifying map f : B → BG. Let H be a

closed subgroup of G, and i : H → G the inclusion. Show:

ξ reduces to H ⟺ ∃ g : B → BH such that Bi ∘ g ∼ f.