MTH3400 — GEOMETRY 1 — SPRING 1997

HOMEWORK #1

1. Let X be a locally Euclidean space.

(i) Show that each component of X is an open subset of X. (ii) Show that, if X is connected, then X is path-connected.

2. Let X and Y be connected, locally Euclidean spaces of the same dimension. If f : X → Y is bijective and continuous, show that f is a homeomorphism.

3. Let π_{±} : S^{n} \ p_{±} → ℝ^{n} be the stereographic projections. Write down explicit formulas for these maps, and prove that they are homeomorphisms.

4. Let D^{n} = {v ∈ ℝ^{n} | ||v|| ≤ 1} be the unit n-disk, with boundary ∂D^{n} = S^{n−1}. Prove that D^{n}/S^{n−1} is homeomorphic to S^{n}.

5. Define an equivalence relation ∼ on S^{n} = {v ∈ ℝ^{n+1} | ||v|| = 1} by writing v ∼ w if and only if v = ±w. The quotient space ℝℙ^{n} = S^{n}/ ∼ is called the (real) projective n-space. Let π : S^{n} → ℝℙ^{n}, π(v) = {±v} be the canonical projection. For each i, 1 ≤ i ≤ n + 1, define a subset U_{i} of ℝℙ^{n} by

U_{i} = {π(x^{1}, …, x^{n+1}) | x^{i} ≠ 0}.

Prove the following facts, which together show that ℝℙ^{n} is an n-manifold.

(i) U_{i} is open in ℝℙ^{n}. (ii) {U_{1}, …, U_{n+1}} covers ℝℙ^{n}. (iii) There is a homeomorphism ϕ_{i} : U_{i} → ℝ^{n}. (iv) ℝℙ^{n} is compact, connected, and Hausdorff.

6. Let X be an n-dimensional manifold with boundary. Prove:

(i) ∂X ∩ int(X) = ∅. (ii) ∂X is an (n − 1)-manifold. (iii) intX is an n-manifold.

7. Let M, N be manifolds with boundary. Prove that ∂(M × N) = ∂M × N ∪ M × ∂N.

8. Let V and W be real vector spaces. Show how a choice of bases e_{1}, …, e_{m} for V and f_{1}, …, f_{n} for W defines an isomorphism of Hom_{ℝ}(V, W) with the real vector space of n × m matrices with real coefficients, Mat_{n×m}(ℝ). Shows that this isomorphism transforms composition

Hom_{ℝ}(V, W) × Hom_{ℝ}(T, V) → Hom_{ℝ}(T, V)
(f, g) ↦ f ∘ g

into matrix multiplication

Mat_{n×m}(ℝ) × Mat_{m×k}(ℝ) → Mat_{n×k}(ℝ)
(A, B) ↦ A · B