Topics in Geometry - Homework 8 - Spring 2009 | MATH 6490, Assignments of Geometry

Download Topics in Geometry - Homework 8 - Spring 2009 | MATH 6490 and more Assignments Geometry in PDF only on Docsity! TOPICS IN GEOMETRY: SHEAF THEORY MATH 6490, SPRING 2009 HOMEWORK 8 Exercise 1. Let i : A ↪→ X be a closed embedding. Show that i∗ is an exact functor. Conclude that for any sheaf F on A we have Hq(A,F) ' Hq(X, i∗F). Exercise 2. Let f : X → Y be a continuous map. Show that f pulls back constant sheaves to constant sheaves, i.e., for any abelian group A, show that f∗AY = AX . Where is this exercise necessary in our discussions on functoriality? Exercise 3. Let I = [−1, 1] be the closed one-dimensional unit disc, and let A be an abelian group. Show that Hq(I,A) = 0 for all q ≥ 1. (Hint: See exercises 18–21 in page 65 of Harder’s book.) Some exercises in point set topology. Exercise 4. Let X be a locally compact Hausdorff space. For any open set U and for any x ∈ U show that there is an open set V which is relatively compact (that is V is compact) and x ∈ V ⊂ V ⊂ U . (This can be rephrased as: the set of relatively compact neighbourhoods of any point forms a neighbourhood basis at that point.) Exercise 5. Let X and Y be locally compact Hausdorff spaces. Let X+ and Y + be their one-point compactifications (look this up from some book on topology) with ∞X and ∞Y being the points at infinity in X+ and Y +. Let f : X → Y be a continuous map. Show that f is proper if and only if f extends to a continuous map f+ : X+ → Y + with f+(x) = f(x) for all x ∈ X and f(∞X) = ∞Y . Exercise 6. Let X be a Hausdorff space which is exhausted by compact subsets, i.e., there is a sequence of compact subsets K1 ⊂ K2 ⊂ · · · ⊂ Kn ⊂ · · · such that X = ∪Kn and Kn ⊂ Int(Kn+1). Show that X is paracompact. (Here Int(A) stands for the interior of the subset A.) ∗ ∗ ∗ ∗ ∗ ∗ ∗